# MATHEMATICAL MODELING OF THE IMMUNE SYSTEM IN HOMEOSTASIS, INFECTION AND DISEASE

EDITED BY : Gennady Bocharov, Burkhard Ludewig, Andreas Meyerhans and Vitaly Volpert PUBLISHED IN : Frontiers in Immunology

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ISSN 1664-8714 ISBN 978-2-88963-461-3 DOI 10.3389/978-2-88963-461-3

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# MATHEMATICAL MODELING OF THE IMMUNE SYSTEM IN HOMEOSTASIS, INFECTION AND DISEASE

Topic Editors:

Gennady Bocharov, Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia Burkhard Ludewig, Kantonsspital St. Gallen, Switzerland Andreas Meyerhans, Catalan Institute for Research and Advance Studies (ICREA), and Infection Biology Group, Department of Experimental and Health Sciences, University Pompeu Fabra, Barcelona, Spain Vitaly Volpert, UMR5208 Institut Camille Jordan (ICJ), France

The immune system provides the host organism with defense mechanisms against invading pathogens and tumor development and it plays an active role in tissue and organ regeneration. Deviations from the normal physiological functioning of the immune system can lead to the development of diseases with various pathologies including autoimmune diseases and cancer. Modern research in immunology is characterized by an unprecedented level of detail that has progressed towards viewing the immune system as numerous components that function together as a whole network. Currently, we are facing significant difficulties in analyzing the data being generated from high-throughput technologies for understanding immune system dynamics and functions, a problem known as the 'curse of dimensionality'.

As the mainstream research in mathematical immunology is based on low-resolution models, a fundamental question is how complex the mathematical models should be? To respond to this challenging issue, we advocate a hypothesis-driven approach to formulate and apply available mathematical modelling technologies for understanding the complexity of the immune system. Moreover, pure empirical analyses of immune system behavior and the system's response to external perturbations can only produce a static description of the individual components of the immune system and the interactions between them. Shifting our view of the immune system from a static schematic perception to a dynamic multi-level system is a daunting task. It requires the development of appropriate mathematical methodologies for the holistic and quantitative analysis of multi-level molecular and cellular networks. Their coordinated behavior is dynamically controlled via distributed feedback and feedforward mechanisms which altogether orchestrate immune system functions. The molecular regulatory loops inherent to the immune system that mediate cellular behaviors, e.g. exhaustion, suppression, activation and tuning, can be analyzed using mathematical categories such as multi-stability, switches, ultra-sensitivity, distributed system, graph dynamics, or hierarchical control.

GB is supported by the Russian Science Foundation (grant 18-11-00171). AM is also supported by grants from the Spanish Ministry of Economy, Industry and Competitiveness and FEDER grant no. SAF2016-75505-R, the "María de Maeztu" Programme for Units of Excellence in R&D (MDM-2014-0370) and the Russian Science Foundation (grant 18-11-00171).

Citation: Bocharov, G., Ludewig, B., Meyerhans, A., Volpert, V., eds. (2020). Mathematical Modeling of the Immune System in Homeostasis, Infection and Disease. Lausanne: Frontiers Media SA. doi: 10.3389/978-2-88963-461-3

# Table of Contents

*05 Editorial: Mathematical Modeling of the Immune System in Homeostasis, Infection and Disease*

Gennady Bocharov, Vitaly Volpert, Burkhard Ludewig and Andreas Meyerhans

*08 Supplemented Alkaline Phosphatase Supports the Immune Response in Patients Undergoing Cardiac Surgery: Clinical and Computational Evidence*

Alva Presbitero, Emiliano Mancini, Ruud Brands, Valeria V. Krzhizhanovskaya and Peter M. A. Sloot

*23 Integrative Computational Modeling of the Lymph Node Stromal Cell Landscape*

Mario Novkovic, Lucas Onder, Hung-Wei Cheng, Gennady Bocharov and Burkhard Ludewig


Marco Blickensdorf, Sandra Timme and Marc Thilo Figge

*58 Model-Based Assessment of the Role of Uneven Partitioning of Molecular Content on Heterogeneity and Regulation of Differentiation in CD8 T-Cell Immune Responses*

Simon Girel, Christophe Arpin, Jacqueline Marvel, Olivier Gandrillon and Fabien Crauste

*74 Quantitative Mechanistic Modeling in Support of Pharmacological Therapeutics Development in Immuno-Oncology*

Kirill Peskov, Ivan Azarov, Lulu Chu, Veronika Voronova, Yuri Kosinsky and Gabriel Helmlinger

*85 Linking Cell Dynamics With Gene Coexpression Networks to Characterize Key Events in Chronic Virus Infections*

Mireia Pedragosa, Graciela Riera, Valentina Casella, Anna Esteve-Codina, Yael Steuerman, Celina Seth, Gennady Bocharov, Simon Heath, Irit Gat-Viks, Jordi Argilaguet and Andreas Meyerhans

*98 Mathematical Modeling Reveals That the Administration of EGF Can Promote the Elimination of Lymph Node Metastases by PD-1/PD-L1 Blockade*

Mohamed Amine Benchaib, Anass Bouchnita, Vitaly Volpert and Abdelkader Makhoute


Filippo Castiglione, Dario Ghersi and Franco Celada


Martin Meier-Schellersheim, Rajat Varma and Bastian R. Angermann

*224 An Integrated Pipeline for Combining* in vitro *Data and Mathematical Models Using a Bayesian Parameter Inference Approach to Characterize Spatio-temporal Chemokine Gradient Formation* Dimitris I. Kalogiros, Matthew J. Russell, Willy V. Bonneuil, Jennifer Frattolin, Daniel Watson, James E. Moore Jr., Theodore Kypraios and Bindi S. Brook

*239 Interleukin-15 Signaling in HIF-1*α *Regulation in Natural Killer Cells, Insights Through Mathematical Models* Anna Coulibaly, Anja Bettendorf, Ekaterina Kostina, Ana Sofia Figueiredo, Sonia Y. Velásquez, Hans-Georg Bock, Manfred Thiel, Holger A. Lindner and Maria Vittoria Barbarossa

*259 Immunological Paradigms, Mechanisms, and Models: Conceptual Understanding is a Prerequisite to Effective Modeling* Zvi Grossman

# Editorial: Mathematical Modeling of the Immune System in Homeostasis, Infection and Disease

Gennady Bocharov 1,2 \*, Vitaly Volpert 3,4,5, Burkhard Ludewig<sup>6</sup> and Andreas Meyerhans 7,8

<sup>1</sup> Marchuk Institute of Numerical Mathematics of the Russian Academy of Sciences, Moscow, Russia, <sup>2</sup> Institute for Personalized Medicine, Sechenov First Moscow State Medical University, Moscow, Russia, <sup>3</sup> Institut Camille Jordan, Université Lyon 1, Villeurbanne, France, <sup>4</sup> INRIA Team Dracula, INRIA Lyon La Doua, Villeurbanne, France, <sup>5</sup> Peoples' Friendship University of Russia (RUDN University), Moscow, Russia, <sup>6</sup> Institute of Immunobiology, Kantonsspital St. Gallen, St. Gallen, Switzerland, <sup>7</sup> Infection Biology Laboratory, Department of Experimental and Health Sciences, Universitat Pompeu Fabra, Barcelona, Spain, <sup>8</sup> Institució Catalana de Recerca i Estudis Avançats (ICREA), Barcelona, Spain

Keywords: immune system, mathematical modeling, complexity, multiscale (MS) modeling, spatial organization, regulation, immune-related diseases

#### **Editorial on the Research Topic**

#### **Mathematical Modeling of the Immune System in Homeostasis, Infection and Disease**

The immune system is a dynamic and multi-level biological system that protects host organisms against invading pathogens and tumor development, and plays an active role in tissue homeostasis and organ regeneration. As such, it needs to respond to a vast diversity of threats while minimizing damage to its own cells and organs. To perform this task, it is organized as a tightly regulated, hierarchically controlled and spatially distributed network with soluble and cellular components. Recent technical advances including life-imaging, multi-color phenotyping, and "omics" technologies provide with an unprecedented level of detail of a functional immune system. Since mathematics is the universal language for expressing causal and functional relationships between observations, it is natural to use mathematical tools for mechanistically describing immune system dynamics and functioning. However, as both the immune system's complexity and experimental data sets are huge, it is a substantial challenge to connect these in a mechanistic way. The major problems in this respect are known as "curse of dimensionality" and "combinatorial explosion." The mainstream research in this field is still based on low-resolution models that often provide only limited descriptions of individual immune system components and their interactions after external stimulations. Shifting this simplistic perception of the immune system to a dynamic, multi-level, and spatially resolved system description with molecular and cellular networks is daunting and requires the combination of a solid understanding of the underlying systems biology with the application of appropriate mathematical methodologies. This may ultimately improve the biological relevance of the generated models and contribute to a better mechanistic understanding of immune system functioning as well as making biologically and clinically relevant predictions for diagnosis and treatment of human diseases.

The aim of this Research Topic was to present current state-of-the-art research on using mathematically driven exploration of the complexity of the immune system. A series of articles were collected, giving a comprehensive overview of conceptual frameworks and emerging topics including the "spatial organization of the immune system" and "multifactorial immunerelated diseases."

In his conceptual review, Grossman focused on the "smart surveillance" theory of how T cells individually and collectively respond to self- and foreign antigens depending on contextual parameters. He highlighted that the physiological messages to cells are encoded not only in the biochemical connections of sets of signaling molecules to the cellular machinery

Edited and reviewed by:

Vitaly V. Ganusov, The University of Tennessee, Knoxville, United States

> \*Correspondence: Gennady Bocharov bocharov@m.inm.ras.ru

#### Specialty section:

This article was submitted to Viral Immunology, a section of the journal Frontiers in Immunology

Received: 05 November 2019 Accepted: 02 December 2019 Published: 08 January 2020

#### Citation:

Bocharov G, Volpert V, Ludewig B and Meyerhans A (2020) Editorial: Mathematical Modeling of the Immune System in Homeostasis, Infection and Disease. Front. Immunol. 10:2944. doi: 10.3389/fimmu.2019.02944

**5**

but also in their magnitude, kinetics, and time and space contingencies. The "dynamic tuning hypothesis" is a central component of his theory and sets the ground for further theoretical and experimental exploration of immune tolerance, homeostasis, and diversity. Moreover, Grossman used his conceptual postulates to discuss conflicting models of HIV pathogenesis. Castiglione et al. addressed the underlying mechanism of cross-reactive immune responses and antigenic sin that may be beneficial, neutral, or detrimental for the host. They studied the relationship of clonal dominance with memory cell attrition with an agent-based model. They propose that attrition could serve as a curbing mechanism for the memoryanti-naive phenomenon.

Several studies describe novel modeling tools and their applications. Lanzarotti et al. developed a model for the prediction of cognate T cell receptor (TCR) targets. It is based on the similarity to a database of TCRs with known targets and may have important implications for the rational design of T cellbased therapies. With their time-resolved experimental data of splenic transcriptomes from mice infected with the lymphocytic choriomeningitis virus (LCMV), Pedragosa et al. addressed the problem of linking gene expression changes from whole tissue with immune cell dynamics. To this end, they combined weighted gene co-expression network analysis—with digital cell quantifier—providing a novel approach to bridge the genomic with the cellular level during antiviral immune responses. Meier-Schellersheim et al. discuss how mechanistic rule-based modeling can be used to test immunological hypotheses through quantitative simulations. They considered as an example Gprotein-coupled receptor signaling that is utilized by cells to respond to a wide range of extracellular stimuli and explore the cross-talk of multiple cytokine pathways, thereby providing basis for deriving cell population behavior from single-cell models and bridging a current scale gap. Finally, Enciso et al. demonstrated how discrete dynamic models can be transformed to continuous dynamic models using Fuzzy logic. This approach enables a better description of growth and differentiation of T lymphocytes in various microenvironments.

The consequences of an uneven partitioning of molecular contents on cell fate regulation were studied by Girel et al. They introduced a multi-scale mathematical model of CD8 T cell responses in lymph nodes and showed that the degree of unevenness of molecular partitioning affects the outcome of the immune response and memory cell generation. Huang et al. considered virus and interferon spread within an infected host as two competing processes and analyzed a well-mixed vs. a spatially segregated scenario. They defined the conditions under which the interferon response works most effectively and suppresses the infection.

A series of six publications considered the architecture and functioning of lymph nodes within an immune response. Novkovic et al. reviewed available computational lymph node models with the focus on the structure and organization of stromal cells. The authors pointed out that hybrid- and multiscale models in combination with high-resolution imaging will be important to unravel the complex immune mechanisms that are initiated in lymph nodes. The study by Moses et al. is focused on the definition of rules specifying the search strategies of T cells for antigen. They discovered striking similarities between the strategies ant colonies use to forage and the immune cells use to find pathogens. The strategies are based on a variety of search behaviors including directional movement using chemokine gradients, random motion using correlated random walk, and movement along physical networks. Kalogiros et al. developed a mathematical framework to characterize spatiotemporal chemokine gradient formation. With their Bayesian parameter inference approach, they provided a building block for subsequent multi-scale modeling. Azarov et al. developed an agent-based model to investigate the role of T cell-dendritic cell (DC) chemoattraction in T cell priming in the lymph node. They stressed that the balance of naive and activated antigen-specific T cells that are both chemotactically attracted to the neighborhood of DCs determine the overall amplitude of the specific T cell response. Grebennikov et al. developed a physics-based model of T cell motility in lymph nodes. The cell dynamics is determined by a superposition of autonomous locomotion, intercellular interactions, and viscous dumping. The model was then used to predict the required CD8 T cell frequencies necessary to detect HIV-infected cells before they start releasing virus particles. McDaniel and Ganusov studied lymphocyte recirculation in sheep. With a series of mathematical models, they estimated the distribution of residence times in ovine lymph nodes.

Finally, six publications addressed various aspects of multifactorial immune-related phenomena and diseases. Presbitero et al. described the role of alkaline phosphatase (AP) during cardiac surgery. They developed a mathematical model of systemic inflammation and suggested that supplemented AP provides a patient benefit by inducing liver-type tissue non-specific AP production. Coulibaly et al. formulated a mathematical model that describes the molecular mechanisms involved in the IL-15-induced signaling cascade of the hypoxiainducible factor 1α (HIF-1α) pathway in natural killer cells. In combination with experimental work, they identified mammalian target of rapamycin (mTOR), the nuclear factor-kB (NF-kB), and the signal transducer and activator of transcription 3 (STAT3) as central regulators of HIF-1α accumulation. Benchaib et al. studied the interaction between cancer and immune cells in the lymph node. They delineated with mathematical models the conditions for the three possible outcomes, namely, tumor elimination, equilibrium, and tumor evasion. The study of Blickensdorf et al. compared fungal infections with Aspergillus fumigatus in murine and human lungs. They analyzed the spatial infection dynamics with a hybrid agent-based model that accounts for the specific lung physiologies. Infections are more efficiently cleared in mice due to their smaller alveolar surface areas. Peskov et al. reviewed the state of the art of quantitative systems models describing tumor and immune system interactions and discussed approaches for biomarker identification. Finally, Nikolaev et al. studied fundamental interactions between a pathogen with a tumor. Their work is based on the recent finding that an acute influenza infection in the lung promotes melanoma growth in the dermis of mice. Using models of complex intracellular biochemical reaction networks, they analyzed virus-specific

and melanoma-specific CD8 T cells in the lung. They proposed that the observed melanoma growth results from sequestering of tumor-specific effector cells in the lung due to their loss of motility via PD-1 interactions. In contrast, virus-specific T cells remain functional and clear the influenza infection since they adapt to the strong stimulation by their cognate antigen locally.

Collectively, this Research Topic highlighted the ongoing attempts to quantitatively describe and mechanistically understand the complex interactions inherent in immune system functioning during normal conditions and in disease. While far from providing a complete view, important mathematical elements of systems immunology are emerging that are based on genuine collaborations between experimentalists and applied mathematicians. Only with such multi-disciplinary efforts will we be able to enrich immunological research with analytical and predictive modeling tools that complement the impressive advances in observational technologies. This area of research is and will continue to flourish.

# AUTHOR CONTRIBUTIONS

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

# FUNDING

GB and AM are supported by the Russian Science Foundation (grant 18-11-00171). VV is supported by the RUDN University program 5-100. AM is also supported by a grant from the Spanish Ministry of Economy, Industry and Competitiveness and FEDER grant no. SAF2016-75505-R (AEI/MINEICO/FEDER, UE) and the María de Maeztu Programme for Units of Excellence in R&D (MDM-2014-0370).

# ACKNOWLEDGMENTS

We wish to convey our appreciation to all the authors who have participated in this Research Topic and the reviewers for their insightful comments.

**Conflict of Interest:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2020 Bocharov, Volpert, Ludewig and Meyerhans. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# Supplemented Alkaline Phosphatase Supports the Immune Response in Patients Undergoing Cardiac Surgery: Clinical and Computational Evidence

Alva Presbitero1†, Emiliano Mancini 2†, Ruud Brands 3,4, Valeria V. Krzhizhanovskaya1,2 and Peter M. A. Sloot 1,2,3 \*

#### Edited by:

*Gennady Bocharov, Institute of Numerical Mathematics (RAS), Russia*

#### Reviewed by:

*Jimmy Thomas Efird, University of Newcastle, Australia Kirill Peskov, Modeling and Simulation Decisions, Russia*

> \*Correspondence: *Peter M. A. Sloot p.m.a.sloot@uva.nl*

*†These authors have contributed equally to this work*

#### Specialty section:

*This article was submitted to Molecular Innate Immunity, a section of the journal Frontiers in Immunology*

Received: *29 June 2018* Accepted: *20 September 2018* Published: *11 October 2018*

#### Citation:

*Presbitero A, Mancini E, Brands R, Krzhizhanovskaya VV and Sloot PMA (2018) Supplemented Alkaline Phosphatase Supports the Immune Response in Patients Undergoing Cardiac Surgery: Clinical and Computational Evidence. Front. Immunol. 9:2342. doi: 10.3389/fimmu.2018.02342* *<sup>1</sup> High Performance Computing Department, ITMO University, Saint Petersburg, Russia, <sup>2</sup> Institute for Advanced Studies and Computational Science Laboratory, University of Amsterdam, Amsterdam, Netherlands, <sup>3</sup> Complexity Institute, Nanyang Technological University, Singapore, Singapore, <sup>4</sup> Alloksys Life Sciences BV, Wageningen, Netherlands*

Alkaline phosphatase (AP) is an enzyme that exhibits anti-inflammatory effects by dephosphorylating inflammation triggering moieties (ITMs) like bacterial lipopolysaccharides and extracellular nucleotides. AP administration aims to prevent and treat peri- and post-surgical ischemia reperfusion injury in cardiothoracic surgery patients. Recent studies reported that intravenous bolus administration and continuous infusion of AP in patients undergoing coronary artery bypass grafting with cardiac valve surgery induce an increased release of liver-type "tissue non-specific alkaline phosphatase" (TNAP) into the bloodstream. The release of liver-type TNAP into circulation could be the body's way of strengthening its defense against a massive ischemic insult. However, the underlying mechanism behind the induction of TNAP is still unclear. To obtain a deeper insight into the role of AP during surgery, we developed a mathematical model of systemic inflammation that clarifies the relation between supplemented AP and TNAP and describes a plausible induction mechanism of TNAP in patients undergoing cardiothoracic surgery. The model was validated against clinical data from patients treated with bovine Intestinal AP (bIAP treatment) or without AP (placebo treatment), in addition to standard care procedures. We performed additional *in-silico* experiments adding a secondary source of ITMs after surgery, as observed in some patients with complications, and predicted the response to different AP treatment regimens. Our results show a strong protective effect of supplemented AP for patients with complications. The model provides evidence of the existence of an induction mechanism of liver-type tissue non-specific alkaline phosphatase, triggered by the supplementation of AP in patients undergoing cardiac surgery. To the best of our knowledge this is the first time that a quantitative and validated numerical model of systemic inflammation under clinical treatment conditions is presented.

Keywords: alkaline phosphatase, innate immune response, cardiac surgery, ODE model, in-silico, clinical trial

# INTRODUCTION

Alkaline Phosphatase (AP) is an enzyme originally known for its pivotal role in skeletal mineralization (1) but also for its capability to reduce inflammation. AP is in fact capable to reduce inflammation in animals by dephosphorylating inflammation triggering moieties like bacterial lipopolysaccharides (LPS) and extracellular nucleotides (2–9). In addition, several studies demonstrated that AP has a key function in maintenance and restoration of physiological barriers (10) in addition to this anti-inflammatory role of AP. In fact, many, if not all of these barriers may become hyper-permeable and or dysfunctional during such systemic ischemic and inflammation triggering insult. Extracellular nucleotides, like Adenosine Triphosphate (ATP) and Adenosine Diphosphate (ADP), having pivotal energy housekeeping functions, intracellularly act as ITM as soon they have leaked out of cells exposed to ischemic insults (11–13). LPS (14), a major component of the Gram negative bacterial outer membrane that is responsible for mediating septic shock (15). These inflammation triggering moieties (ITMs) are pro-inflammatory signals that may start local and systemic inflammatory responses in the innate immune system (16–18). Clinical trials involving the parenteral administration of AP to patients with severe sepsis, showed significant improvement in renal function (19, 20).

Humans have four distinct AP isozymes: tissue-nonspecific AP (liver/bone/kidney type AP), which is the most predominant circulating form of isozyme, intestinal-, placental-type, and germ cell AP. The anti-inflammatory effects of AP have been confirmed in settings with intestinal-, placental-, and liver-type AP.

Coronary artery bypass grafting (CABG) is one of the most common types of open-heart surgery which very often triggers a systemic inflammatory response, the clinical impact thereof is specific for the specific patient and depends on multiple factors like age, underlying diseases, and other confounding factors. The average annual number of CABG procedures in Western practice is about 62.2 per 100 000, ranging from 29.3 procedures in Spain to 135.4 procedures in Belgium (21). According to the Society of Thoracic Surgeons National Database, CABG-mediated complications contribute to 1.8% in-hospital and 2.2% operative mortalities, but caused 24% post-operative atrial fibrillation incidences in 151,474 patients in 2015 (22). We focus on the experiments by Kats et al. where we assume a systemic insult due to the amounts of ITMs introduced and generated peri- and post-cardiac surgery. Cardiac surgery invokes a vigorous systemic inflammatory response where massive amounts of ITMs are simultaneously generated from various sources in the body: (a) CABG and valve surgery under CPB (Cardio Pulmonary Bypass) induces sheer stress on blood cells damaging them and releasing a massive amount of ITMs in the process, (b) surgical area where tissue is damaged locally, and (c) reperfusion damage by accumulated ITMs that have crossed the gut barrier during hypoperfusion and become systemically available upon re-circulation (23). The body thereby deals with a massive amount of ITMs that enter the circulation and are transported into the tissue via blood flow, and circulate in the blood stream due to the effects of cardio pulmonary bypass grafting and reperfusion injury.

De novo synthesis and release in circulation of AP induced by AP prophylaxis could be the body's way to improve its defense mechanism. A study in 2012 by Kats et al. (24) demonstrated that intravenous bolus administration and continuous infusion of bovine intestinal Alkaline Phosphatase [bIAP, bRESCAP, and APPIRED studies by Alloksys Life Sciences (7, 9)], in patients undergoing CABG (with or without valve surgery) results in the release of endogenous tissue non-specific AP (TNAP), most likely liver-type AP. This release exhibits a unique feat that was not before observed in septic shock patients (19). Induction of liver-type non-specific AP supports the idea that AP contributes significantly to the immune response. Additional Phase III clinical trials are currently on the way to confirm the beneficial effects of AP previously reported in CABG and valve surgery.

If indeed excess AP or the release of additional liver-type TNAP is beneficial to the clinical outcome of patients undergoing major surgeries as well as for individuals suffering from acute and chronic inflammation, then there is an urgent need to develop computational models that can reproduce and predict the dynamics of induced TNAP in circulation. The developed model could then pave way to better understand when and more importantly how much of this liver type TNAP is expressed and released back into circulation through in-silico experiments. We develop a new model of systemic inflammation based on existing models of the innate immune response to acute inflammation (25–32), with the purpose of describing, and gaining further insight on the dynamics of the innate immune system response through in-silico experiments (33). We thereby report, to the best of our knowledge, the first calibrated and validated mathematical model for systemic inflammation.

The human innate immune system (HIIS) is the body's first line of defense to an infection or trauma. This is commonly manifested in the form of acute inflammatory response, resulting from these and other oxidative stress conditions (34). Numerous studies were reported aiming to understand the acute inflammatory response based on the response of a single population of white blood cells to invading pathogens (25– 29). Unlike in previous models that only deal with a general population of invading and invaded entities, Dunster et al. (30) distinguished between populations of white blood cells by incorporating activated macrophages, activated, apoptotic, and necrotic neutrophil populations. More specific mathematical models of HIIS that further distinguish white blood cells into distinct populations have also been developed. For instance, Su et al. (31) used a system of partial differential equations (PDE) that capture the spatial and temporal dynamics of the innate and adaptive immune response via the following stages: recognition, initiation, effector response, and resolution of infection. This model of the human innate immune response was adapted by Pigozzo et al. (32) who focused on the dynamics of LPS, neutrophils, pro-inflammatory, and anti-inflammatory cytokines.

This paper focuses on how the concentrations of the innate immune response components evolve over time. Partial differential equations (PDEs) provide ways to analyse both time and spatial dynamics of key aspects of HIIS. Since we are modeling a systemic insult, where a massive amount of ITMs are coming from various sources in the body and inflammation is not confined to a specific tissue or organ, we can assume that these moieties are present and distributed all throughout the organism. Thus, we regard the "tissue" as representative of the entire body. Given this assumption, the role of microscopic spatial effects for the dynamics of the system is negligible and we use ordinary differential equations (ODEs) to describe the dynamics of the immune response. However, we take into account spatial effects by modeling various compartments (liver, blood stream, and tissue) and the transport of cells and molecules between them due to the inflammation in blood and tissue. This compartmentalization of the organism allows us to account for chemotaxis at a macroscopic level using the change in permeability of the endothelium during the different stages of inflammation to affect the transport of immune cells and molecules between blood and tissue.

We therefore construct the HIIS model from Reynolds et al. (28), Su et al. (31), and Pigozzo et al. (32) by introducing the following key differences: compartmentalization of the organism into liver, blood and tissue; introduction of the dual pathway to neutrophils death, necrosis being pro-inflammatory and apoptosis being anti-inflammatory; introduction of the antiinflammatory action of AP and of the mechanism of AP induction; the dilution of cellular components in tissue typical of systemic inflammatory responses as opposed to the increased concentration of cellular components in a localized region typical of acute inflammatory responses.

# MATERIALS AND METHODS

#### Clinical Trial Data

Patients undergoing open-heart surgery were stratified according to a risk assessment score system called EuroSCORE (Type 1). EuroSCORE is a risk measure for severe complications (mortality) associated with this type of surgery. In addition to standard care treatment, patients undergoing cardiothoracic surgery were divided into two distinct categories based on the type of treatments they received: (a) placebo treatment (physiological buffer containing no AP) and (b) bovine intestinal AP (bIAP) treatment in the same buffer.

In the APPIRED I study patients with a 2 < EuroSCORE ≤ 6, were initially given either placebo or 1,000 IU of Bovine Intestinal AP (bIAP) followed by a continuous infusion during 36 h of either placebo (n = 31, with mean EuroSCORE = 3.7 ± 1.4) or 5.6 IU per kg body weight per hour (total 9,000 IU) (n = 32, with mean EuroSCORE = 3.6 ± 1.2).

In the APPIRED II study patients with a EuroSCORE of ≥ 5, were initially given either placebo (n = 25, with mean EuroSCORE 5.6 ± 2.6) or 1,000 IU of bIAP followed by a continuous infusion during 8 h (total 9,000 units) (n = 27 with mean EuroSCORE 5.8 ± 3.1) (total 9,000 units).

Where in APPIRED I, 63 patients underwent CABG only, in APPIRED II a total of 52 patients were included that underwent CABG combined with valvular surgery under CPB. This type of combined surgery is associated with an increased risk.

Further details of the APPIRED clinical trials have been described by Kats et al. (9). The induction of endogenous alkaline phosphatase in this trial was described in Kats et al. (24). Primary endpoints were cytokine levels peri- and post- surgery next to clinical outcome.

The APPIRED II data was used in sections Human Innate Immune System Model With the Induction Mechanism of TNAP 1 and HIIS Model Without the Induction Mechanism of TNAP 2 to validate the model. Data relative to APPIRED I was not used to validate the model due to the limited number of data points relative to AP. However, the secondary peak of ITMs observed in 16% of the patients in APPIRED I was used qualitatively to investigate the case of patients with complications in APPIRED II by adding a secondary source of ITMs in-silico in section Predicting The Innate Immune Response For Patients Having Excess ITMs Using Different AP Treatment Regimens. This secondary source of ITMs is relative to the documented median peak IL6 concentration in septic shock patients in Damas et al. (35).

Patients underwent CABG combined with valvular surgery. CABG pumps (heart-lung machine) were used during the surgery. The operations were primary, therefore operated specifically for open heart surgery, and were planned prior to the actual surgery, hence non-emergent. An intra-aortic balloon pump was not used during the surgery, except for one patient who exhibited cardiogenic shock with multi-organ failure. The surgery lasted for an average of 4.7 ± 1.4 h. The average perfusion time was 134 ± 40 min and average cross clamping time was 105 ± 40 min. Pre-medication such as relaxants, anesthetics, antibiotics, and blood products such as red blood cells or platelets were given prior and during the CABG surgery. We summarize patients' demographics, type and method of cardioplegia used, and patients' medical history in **Tables 1–3** of the **Supplementary Material** respectively.

The data used was approved by the Ethics committee with IRB approval number M09-1965. The set-up of the study as well as appropriate consent procedures, have been reviewed and approved by the Institutional Review Board (METC). The central Independent Ethics Committee of The Netherlands (CCMO) has been informed. Approval from the METC of ZOL Genk was obtained in 6. March, 2012. The Belgium Competent Authority office: FAGG (Federaal Agentschap voor Geneesmiddelen en Gezondheidsproducten) was informed about the METC approval to initiate the study.

# Biological Mechanism and Model Description

The HIIS model is constructed based on the biological mechanisms that occur in three separate compartments—blood, tissue, and liver. The blood and tissue are separated by the endothelial lining that acts as a modulated barrier toward accessing blood circulation derived components like immune cells. Models of acute inflammation commonly neglect the dynamics of immune cells in the blood compartment under the assumption that it plays a minor role on the dynamics of the innate immune response, acting as a reservoir of immune cells. Systemic inflammation is considerably different and it is characterized by a dilution of the immune cells in tissue caused by the delocalized inflammation. Additionally AP is known to act on ITM both in blood and in tissue. For this reason it is crucial to model both blood and tissue compartments to accurately capture the dynamic of resolution of a systemic inflammatory response.

#### Blood Compartment

Upon a systemic insult cytokines are released by both tissue cells and immune cells in tissue, with different rates. Cytokines then migrate first toward the endothelial barrier, with which they interact changing its permeability to recruit more immune cells into the inflamed tissue, and then in part migrate into the bloodstream. Since we only have data about the concentration of cytokines in blood, our model describes the dynamics of cytokines (IL-6 and IL-10) in the bloodstream. The rates of cytokines production used in our model represent the rate with which cytokines reach the bloodstream after being secreted by macrophages and necrotic neutrophils in tissue. These rates thus take into account not only the secretion rate by each cell type in response to the inflammatory state (presence of ITMs) but also of the mechanism of transport from tissue to the bloodstream.

In our model we assume that AP is the only component of the immune system that interacts with ITMs in the bloodstream, forming ITM-AP complexes that are later removed in the liver by Kupffer cells. Similarly to immune cells, the transport of AP and ITMs from the bloodstream into the tissue is controlled by a permeability factor.

#### Tissue Compartment

The innate immune response triggered in tissue by invasive cardiac surgery is shown in **Figure 1**. We assume that the presence of ITMs in tissue is mostly due to the migration into the tissue of ITMs released in bulk in the circulation through damaged blood cells and gut hypo-perfusion and later transported via the bloodstream. ITMs in tissue are also due to local tissue damage caused by the invasive surgery, but we consider this amount negligible compared to the other two sources of ITMs. An inflammatory response is triggered as soon as ITMs activate resting macrophages (MR) leading them to differentiate into "activated" macrophages residing in tissue (I). Activated macrophages (MA) secrete pro-inflammatory cytokines (CH), which result in increasing the permeability of the endothelial barrier (II) via a series of intermediate stages. Consequently, resting neutrophils (NR) in circulation are primed by circulating ITMs and then enter the tissue through the endothelial barrier via a process called "diapedesis" (III). In the context of the computational model, resting neutrophils are only rendered active when they enter the tissue through the endothelial barrier. Activated neutrophils (NA) phagocytose and/or release their granules to neutralize or antagonize inflammation (IV). If the inflammation is cleared, the neutrophils go into apoptosis or programmed death (V). Activated macrophages remove the apoptotic neutrophils (NDA) by phagocytosis and in the process induce an anti-inflammatory effect as shown in **Figure 1** by the green arrows (VI). If inflammation is too intense and not resolved rapidly, the neutrophils go into a necrotic (NDN) state (designated by the red arrows in **Figure 1**), which releases additional ITMs in the tissue (VII). The presence of yet another batch of ITMs in the tissue induces ongoing inflammatory responses that causes tissue damage, which in turn perpetuates overall inflammatory response by macrophage activation and neutrophil influx into the local inflamed tissue areas (VIII).

#### Liver Compartment

At the onset of surgery a very high concentration of ITMs is released in the blood stream as a consequence of the damage to blood cells caused by the cardiac surgery bypass. In response to this massive ITM insult the liver releases all its stored AP (∼5,300 IU) into the bloodstream. After 2–4 h the liver is able to supply newly synthesized AP again (36, 37). Endogenous AP is naturally produced by the body and is highly expressed at physiological barriers like the gut, placenta, lungs, kidney glomerulus, and the blood-brain barrier. Upon interaction with ITMs that are present in the bloodstream, endogenous AP is released from the apical membrane of specific physiological barriers expressing high levels of AP (like liver bile duct membrane, blood brain barrier, kidney, and gut) and brought into circulation or gut lumen as ITM-AP conjugates. ITM-AP complexes are eventually removed from circulation by the liver Kupffer cells (5). We assume that due to its size AP can, under normal non-inflammatory conditions, enter the tissue through the endothelial barrier fenestrae. Intravenously administered bovine AP (from here on noted as "bIAP treatment") follows the same mechanism as endogenous AP, where it enters the tissue and detoxifies local ITMs through dephosphorylation. Note that the supplemented AP also detoxifies circulating ITMs directly and is removed from circulation by the Kupffer cells. In the case of an oxidative stress insult, such as that induced by cardiac surgery, ITM-AP is removed from circulation by Kupffer cells (5). This removal is observed in pre-clinical and clinical studies as a decrease in AP concentration in plasma. This decrease serves as a "distress signal" for the liver to release its stored AP at the bile ductal membrane barrier indirectly into the bloodstream (38). The release of liver AP into circulation implies that AP residing at blood- brain-, kidney-, and gut-barriers may also be released, compromising the integrity of these barriers. This may result in clinical phenotypic conditions like kidney failure and cognitive impairment observed upon major surgery. The hypothesis central to the AP intervention is that by replenishing AP through either de novo synthesis or supplementation during surgery, the impairments can be circumvented by helping reduce the inflammation and preserving the integrity of such barriers. De novo synthesis, in the strictest sense, refers to the general production of an entity. In the context of our model, we use the term "de novo" synthesis as the continuous production of AP by the liver.

Our HIIS model takes into account the following key mechanisms: (a) activation and inhibition of MR, (b) changes in endothelial permeability, (c) phagocytosis of rest products of ITMs, (d) phagocytosis of ND<sup>A</sup> and NDN, (e) release of ITMs from necrotic cells, (f) natural death of immune cells and degradation of molecular entities, (g) production of CH and ACH

(h), induction of D, and finally (i) delay in necrosis and cytokine production. The following key mechanisms are used to model the dynamics of AP: (a) release of endogenous/stored AP from the liver bile canalicular membrane, (b) de novo synthesis of AP in the liver, and (c) administration of bIAP into the bloodstream. The model does not take into account the AP released from other physiological barriers, since the amount of AP present on these is negligible compared to the AP released from the liver.

# Code Implementation and Repository

We used Python 3.6.5 on a 3.30 GHz Intel <sup>R</sup> CoreTM i7-5820K CPU with 16.0 GB RAM in all our simulations. Python libraries used were: numpy, pandas, scipy, joblib, SALib, and scikit-learn. The python codes and sample data have been uploaded to https:// github.com/avpresbitero/HIIS.

# RESULTS

### Human Innate Immune System Model With the Induction Mechanism of TNAP

In the first sub-section we describe first the calibration process of the model with data from the bIAP branch of APPIRED II under the assumption that supplemented bIAP stimulates the liver cells to produce additional TNAP. The calibrated model is then used to predict the dynamics of the immune response for the placebo branch. These predictions are validated using data from the placebo branch of APPIRED II. In the second sub-section we show the dynamics of all cellular and molecular entities in the model and highlight the action of bIAP on the dynamics of systemic inflammation.

In this study we use the median for each branch of the APPIRED II clinical trial to designate the values that best represent the population of patients undergoing cardiac surgery. Since we assume that the induction mechanism of alkaline phosphatase is inherent to all patients injected with bolus alkaline phosphatase, we did not cluster patients into different sub-groups. Although clustering patients into sub-groups based on their response would lead to a deeper understanding of the induction mechanism, the current dataset is not large enough to look into the individual trends of the patients' blood parameters. Providing a personalized take on the modeling of the innate immune response and on the individual response to the supplemented AP, this endeavor is beyond the scope of the current research but will be investigated in future studies.

#### Calibration and Validation

Patients in the APPIRED studies were supplemented with a bolus of AP plus continuous infusion of AP and showed a surge of TNAP in the bloodstream (**Figure 2A**, bIAP Calibration). We calibrate the model parameters (summarized in 5 of the **Supplementary Material**) using three datasets: AP, pro-inflammatory, and anti-inflammatory cytokine profiles of patients in the bIAP treatment experiment. We summarize the results in **Figure 2**.

In response to a massive insult, the liver releases all its stored AP into the bloodstream. The liver then takes roughly 2 h to recover. We actually see this dynamics on the in-silico prediction of the model initially exhibited as a high concentration of AP at the onset of surgery. This is then followed by an immediate drop in AP concentration, corresponding to the time interval when the liver is still recuperating. Note that the effect of the AP bolus on the concentration disappears within 20 min after its supplementation as attributed to its short half-life and its interaction with ITMs. Then the liver begins supplying AP again at around 2 h after surgery.

A continuous supply of bIAP was administrated into the patients for 8 h, in addition to the initial concentration of 1,000 IU bovine AP. It was observed that liver-type TNAP is induced in these patients as is shown by the overall concentration of AP in circulation in **Figure 2A.** This supports the conjecture that, as a result of an ischemic condition, added AP serves as an indirect trigger for the liver to release more AP into the bloodstream. We therefore introduce an induction term <sup>r</sup>inducepeak 1+exp r induce(t−tAPdelay) (APstissue + APsblood) in Equation (15) that is dependent on the concentration of bolus AP supplied into the system. The induction mechanism of AP is modeled as a reverse sigmoid function that is centered at 1 h—corresponding to the lag of release of AP from the liver, having flushed all its contents, as the liver recuperates.

The rate at which AP is being used up by the system and the rate at which AP is replenished back into the bloodstream from the liver should be the same regardless of the treatment type. This is because the two groups of patients (bIAP and placebo branches) underwent the same type of cardiothoracic surgical procedure. Hence, we assume the same scale of insult, or the same amount of ITMs on both branches. We validate our model by using the parameter values that we have previously calibrated on the supplemented AP treatment branch to predict the AP profiles of patients in the placebo treatment branch. Our results are shown in **Figure 2B** (Placebo Validation).

#### Dynamics of the HIIS With the Induction Mechanism of TNAP

#### **Dynamics of macrophages.**

In the case of a massive insult, such as cardiac surgery, where ITMs are simultaneously originating from numerous sources in the body, the entire population of resting macrophages immediately becomes activated. This is evident in **Figure 3A** where we see, from simulated data, an immediate drop of resting macrophage population at the moment surgery is initiated (time = 0), which corresponds to an immediate rise of activated macrophage population as shown in **Figure 3B.**

#### **Dynamics of neutrophils.**

In the context of our model, resting neutrophils become "activated" when they enter the tissue from the bloodstream via the endothelial barrier. The recruitment of neutrophils is proportional to the concentration of pro-inflammatory cytokines that increase the permeability of the endothelial barrier. This means that the larger the insult, the more resting neutrophils are recruited from the blood stream into the tissue.

For placebo patients the model predicts (**Figure 4**) an increased level of neutrophils necrosis in tissue in response to slightly higher concentration of ITM in tissue. The increased number of necrotic neutrophils leads to the production of additional ITMs, which acts as a positive feedback for inflammation. In AP-treated patients, the presence of additional AP prevents or reduces the necrosis of neutrophils. This could explain the lower number of adverse events reported for the AP branch of the clinical trial compared to the placebo branch (9).

The model dynamics shown in **Figure 4** supports the idea that AP is an anti-inflammatory mediator that plays an important and active role in the human innate immune system, even though the impact of AP observed with the current levels of ITMs appears to be confined to a small shift in the ratio of apoptotic vs. necrotic neutrophils.

#### HIIS Model Without the Induction Mechanism of TNAP

In this section we present the model results under the assumption that supplemented bIAP does not stimulate the liver cells to produce additional TNAP. In this case we use the previous model without the induction term introduced in section Calibration and Validation. We first attempted to calibrate the parameters of this alternative model on the supplemented bIAP branch. However, we were not able to model the AP dynamics of the bIAP branch without the induction term. For this reason we calibrated the model with data from the placebo branch of APPIRED II. The calibrated alternative model is then used to predict the dynamics of the immune response for the bIAP branch. We compare these predictions with data from the bIAP branch of APPIRED II and observe that the calibrated model fails to predict the AP dynamics observed in the clinical trial. Since the model without the induction mechanism cannot be validated we do not show the detailed dynamics of cellular and molecular entities as we did in section Dynamics of the HIIS With the Induction Mechanism of TNAP.

#### Calibration and Validation

Instead of calibrating the parameters using the AP treatment and validating using the placebo treatment, we now reverse the process and calibrate instead the parameters in the placebo treatment first and validate them using the AP patient data. The aim of which is to find out whether we could model the induced amount of endogenous AP in the supplemented branch without using the induction term (see **Figures 5A–F**).

calibration column show the result of the calibration of the model parameters using data from the bIAP branch of APPIRED II. Data points are shown in red and correspond to the median value of the patients in this branch. The error bar shows the median absolute error. Blue lines correspond to the dynamics of the *in silico* model after calibration. (A) shows the dynamics of AP in blood (B) shows the dynamics of the pro-inflammatory cytokine represented in the model compared against IL6 data. (C) shows the dynamics of anti-inflammatory cytokines in the model against IL10 data. The three plots (B,D,F) on the placebo validation column show the validation of the model against data from the Placebo branch of APPIRED II. The model is able to predict the dynamics of placebo branch using the parameters calibrated with the data from the bIAP branch. The model predicts a protective effect of AP. As a consequence, the model predicts a greater concentration of pro-inflammatory cytokines in the placebo branch (D). See unit conversion of AP and cytokines from *molecules mm*3 in Equations (32) and (33) (section 10 of the Supplementary Material) respectively.

Using the calibrated parameters from the placebo study, we predict the dynamics of the bIAP study without the induction term. As shown in **Figure 5B**, the model without the induction term is not able to reproduce the AP profile of bIAP treatment patients. This clearly shows that an additional mechanism is missing and that the missing term has to account for additional production of TNAP to be released from the liver (the major source of stored AP) into the bloodstream. See **Supplementary Material** section 9 for details on the dynamics of the various compartments of the HIIS without the induction mechanism of TNAP.

# Predicting the Innate Immune Response for Patients Having Excess ITMs Using Different AP Treatment Regimens

The validation in section Human Innate Immune System Model With the Induction Mechanism of TNAP shows that the model with an induction mechanism for TNAP is able to predict the dynamics of the human innate immune system response in patients with systemic inflammation. Under the conditions of reported in APPIRED II patients in the placebo treatment branch have been able to resolve inflammation almost as effectively as patients in the bIAP branch, the only measurable difference being different levels of plasma AP and a reduced number of adverse events in the bIAP branch. Supplementation of AP under these conditions has an impact on the amount of necrotic neutrophils but did not drastically change the dynamics of immune cells from that of the placebo treatment group. However, since in the APPIRED I study 16% of patients show an excess amount of ITMs, the source of which is unknown, we explore the impact of supplemented AP in a system stressed by an additional source of ITMs after surgery. We model this scenario by adding a secondary source of insult in-silico and predicting how the human innate immune system would respond in different AP regimens using our model. The amount of this secondary source of ITMs is set to a reasonable value as indicated in Damas et al. (35). We perform two sets of in-silico experiments: in the first set we predict how the immune cells respond to an excess amount of ITMs; in the second set we predict how different AP regimens (i.e., different concentrations of bolus AP) affect the body's response to an additional source of ITMs as the one observed in APPIRED I.

#### In-silico Experiment #1: Innate Immune System Dynamics for Patients With Excess ITMs

The model predicts a protective effect due to supplemented AP when a patient is challenged by a second source of ITMs immediately after or during the surgery. **Figures 6**, **7** show the predicted dynamics for bIAP and placebo branches in case of a source of additional ITMs. Apoptotic neutrophils in supplemented branch have a concentration higher than in the placebo branch. On the other hand, necrotic neutrophils and ITMs in placebo branch are higher than in the supplemented treatment branch. The model predicts a more intense inflammation in the placebo branch as shown in the proinflammatory cytokines plot (**Figure 7C**). This is confirmed by (**Figure 7D**) which shows more anti-inflammatory cytokines in the supplemented than in the placebo branch.

#### In-silico Experiment #2: Treating Patients With Excess ITMs With Various Alkaline Phosphatase Regimen

Phase II and Phase IIIa clinical trials have observed an increased concentration of TNAP in circulation in the AP treatment group compared to the placebo group given an AP protocol. For instance in APPIRED I, a bolus of 1,000 IU of bovine AP was first injected to patients undergoing cardiac surgery followed by a continuous infusion of 5.6 IU/L per kg body weight for 8 h. In this section we predict the dynamics of the innate immune response under the conditions described in section in-silico Experiment #1: Innate Immune System Dynamics for Patients With Excess ITMs given two different AP supplementation regimens for which we increase the supplemented AP to twice and thrice the original protocol respectively. We then compare the results with the protocol tested in APPIRED II study.

The model predicts an increasing protective effect the higher the concentration of supplemented AP by showing an increasing

addition of bolus AP contributes to the human innate immune system as an anti-inflammatory mediator by reducing the amount of neutrophils that go into the necrosis pathway.

neutralizing effect on ITMs both in plasma and in tissue **(Figures 8A,B**). As the concentration of supplemented AP increases, more and more activated neutrophils are inclined to go into apoptosis rather than necrosis (**Figures 8C,D**). Pro-inflammatory profiles show that increasing supplemented AP decreases the amount of pro-inflammatory cytokines, while increasing the population of anti-inflammatory cytokines, indicating a better resolution of the systemic inflammation.

# DISCUSSION

The induction mechanism of liver-type Tissue Non-specific Alkaline Phosphatase (TNAP) into circulation observed in patients undergoing coronary artery bypass grafting with cardiac valve replacement could be the body's way of strengthening its defense mechanism against such a massive insult, making TNAP a de facto key player in the human innate immune

result of the calibration of the model parameters using data from the placebo branch of APPIRED II. Data points are shown in red and correspond to the median value of the patients in this branch. The error bar shows the median absolute error. Blue lines correspond to the dynamics of the *in silico* model after calibration. (A) shows the dynamics of AP in blood. (B) shows the dynamics of the pro-inflammatory cytokine represented in the model compared against IL6 data. (C) shows the dynamics of anti-inflammatory cytokines in the model against IL10 data. The three plots (B,D,F) on the bIAP validation column show the validation of the model against data from the bIAP branch of APPIRED II. Without the induction term, we are not able to reproduce the Alkaline Phosphatase profile in the AP-treated patients. See unit conversion of AP and cytokines from *molecules mm*3 in Equations (32) and (33) (section 10 of the Supplementary Material) respectively.

system. Hence, directing the attention to the role of Alkaline Phosphatase in systemic inflammation and understanding its role in supporting an appropriate innate immune response is of utmost immunological importance. Computational modeling makes it possible to mimic and understand such intricate details and mechanisms of the human innate immune system and offers a predictive power for experimental outcomes through in-silico experiments.

To our knowledge, we have developed the first mathematical model of systemic inflammation that is calibrated and validated on clinical data. We show that the amount of additional endogenous TNAP released in circulation in the supplemented branch is proportional to the total concentration of bolus AP being supplied. We also provide a plausible mathematical function that describes this mechanism. Our model predicts a protective effect of AP in the supplemented branch of APPIRED II, evidence of which can be concluded from the dynamics of the neutrophils. This effect, minimal under the conditions of APPIRED II, becomes obvious when patients exhibit an excess of ITMs after surgery (as observed in 16% of patients of APPIRED I study). We present a scenario where we mimic excess ITMs as seen in some patients by adding a secondary source of insult and see how the model reacts to different AP regimens by varying doses of AP supplied to patients. We show that additional AP has indeed a protective effect and this effect is more prominent in patients with excess ITMs. In this case in-silico experiments predict that the amount of apoptotic neutrophils in the supplemented AP branch is much higher than in the placebo branch. Additionally the amount of pro-inflammatory cytokines predicted for the supplemented AP branch is lower than in the placebo branch, giving further evidence that supplemented AP reduces the intensity of systemic inflammation. As expected the model predicts more anti-inflammatory cytokines in the supplemented branch than in the placebo branch. In other words, the dynamics predicted by the in-silico model show that resolution of inflammation is faster and more efficient with increasing concentration of AP. Hence, our findings suggest that AP indeed plays an important role in mitigating inflammation especially in systemic inflammation and that this protective effect can be modified through variation in AP protocol.

Our model for systemic inflammation is in fact similar to mechanistic models of physiological process, more specifically that of pharmacokinetic/pharmacodynamic (PKPD) models, that are used to develop insights on the dynamics and magnitude of the effect of a drug through quantitative analysis. These models are used to describe the dynamics of the physiological variables in different states. In our case, we model and validate the dynamics of bolus AP together with the human innate immune response for patients undergoing cardiac surgery in two treatment arms respectively: with or without AP supplementation. After validation, the model is used to understand and predict the effect of experimental perturbations, such as variations in AP regime, an approach that is referred to as "forward engineering." A synergy, henceforth, is created between systems biology and PKPD through iterations between computational/mathematical modeling and experimentation (39). Helmlinger et al. have

provided a comprehensive review of drug-disease modeling in the pharmaceutical industry (40). A robust application of systems pharmacology, or the application of systems biology in order to understand how drugs affect the human body, is detailed by Gadkar et al. (41).

The current model has three main limitations: the limited biological knowledge regarding the mechanism that regulates the induction of endogenous AP production triggered by supplemented AP, the potential bias introduced by the modeling approach, and the bias introduced by studying the median dynamics rather than attempting to cluster patients in groups with different dynamics.


neglected. It is possible that modeling the spatial properties of the system would result in a more accurate description of the dynamics, especially for the first 3 h during which there is a major displacement of cells between blood and tissue. For that, we would need a high resolution spatial-temporal clinical data.

3) Given the data at our disposal we decided to study the systemic inflammation via the median dynamics of the two branches of the APPIRED II clinical trial. The large variability in the dataset suggests that clustering patients in subgroups with different dynamics might provide a more accurate prediction of model parameters. However, we believe that the approach used in this paper is sufficient to prove the existence of the induction mechanism of AP and to provide a preliminary description of its dynamics.

The physiological relevance of this study is that to the best of our knowledge this is the first mathematical model describing systemic inflammation. Additionally this is the first model describing the role of Alkaline Phosphatase in the resolution of inflammation after invasive cardiac surgery, laying the foundations to understand systemic inflammatory response syndrome. The main clinically relevant result is the evidence of the existence of an induction mechanism triggered by

Cytokines for AP protocol APPIRED II (*red*) 2x the amount of AP, (*blue*), and 3x the amount of AP (*green*). Here we shoe that the model predicts an increasing protective effect the higher the concentration of supplemented AP by showing an increasing neutralizing effect on ITMs both in plasma and in tissue, increasing concentrations of apoptotic neutrophils and anti-inflammatory cytokines, and decreasing concentrations of necrotic neutrophils and pro-inflammatory cytokines. See unit conversion of cytokines from *molecules mm*3 in Equation (33) of section 10 of the Supplementary Material.

supplemented AP. This model provides a starting point to investigate the amount of endogenous AP induced in the body, and consequently, the optimal amount of supplemented AP to be administered during and after invasive cardiac surgery.

Using the proposed model, we have shown in-silico the dynamics of systemic inflammation within a period of 36 h using different AP regimens. This information is being taken into account in the planning of a clinical trial phase III b. Data from the new multi-center clinical trial will be used to further refine the model. This model and its future iterations will be useful to predict the dynamics of neutrophils during systemic inflammation (namely the balance between apoptotic and necrotic neutrophils and the subsequent resolution of inflammation) and act as a tool to optimize the administration of anti-inflammatory drugs (not necessarily AP) in clinical trials dealing with systemic inflammatory response syndromes.

We perform a global sensitivity test (see **Supplementary Material** section 8) where we vary our input parameters within intervals that correspond to the values found in literature.

The model provides evidence of the existence of an induction mechanism of liver-type tissue non-specific alkaline phosphatase, triggered by the supplementation of AP in patients undergoing cardiac surgery. We show that the AP branch of the clinical trial can only be explained using a mechanism that induces a release in circulation of liver TNAP that is proportional to the amount of supplemented AP. We provide a possible mathematical description of this induction mechanism. The model is validated using novel clinical AP, pro-inflammatory and anti-inflammatory cytokine profiles of placebo- and bIAPtreated patients. This is the first time that liver-type tissue non-specific alkaline phosphatase has been modeled together with the human innate immune system. To date, there are no other existing published clinical trials that tackle the exact mechanisms of liver-type TNAP induction, let alone, a model that describes systemic inflammation. To the best of our knowledge this is the first numerical model of a complex innate immune response that is quantitatively validated with clinical data. Our work paves the way to a deeper understanding of the immunological mechanisms underpinning this important innate immune response to oxidative stress mediated inflammation.

#### ETHICS STATEMENT

The clinical trial data used in this study was carried out in accordance with the recommendations of Institutional Review Board (METC) and the central Independent Ethics Committee of The Netherlands (CCMO) with written informed consent

#### REFERENCES


from all subjects. All subjects gave written informed consent in accordance with the Declaration of Helsinki. The protocol was approved by the institutional review board (METC) and the central Independent Ethics Committee of The Netherlands (CCMO) has been informed. Approval from Belgium was from the METC of ZOL Genk and the Belgium Competent Authority office: FAGG (Federaal Agentschap voor Geneesmiddelen en Gezondheidsproducten) was informed about the METC approval.

# AUTHOR CONTRIBUTIONS

All authors have contributed substantially to the conception and design of the work. All authors have drafted and revised the work for intellectual content. All authors have equally provided the approval for plausible publication of the content. All authors have agreed to be accountable for all aspects of the work, which includes ensuring the accuracy and integrity of all parts of the work.

#### ACKNOWLEDGMENTS

The clinical data from APPIRED I and II study were kindly provided by Alloksys Life Sciences BV, Wageningen, The Netherlands. Thanks to Macs Gallo for providing the image used in **Figure 1**. This research is financially supported by the Ministry of Education and Science of the Russian Federation, Agreement #14.575.21.0161 (26/09/2017), Unique Identification RFMEFI57517X0161.

## SUPPLEMENTARY MATERIAL

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fimmu. 2018.02342/full#supplementary-material


uridine diphosphate. AJP Gastrointest Liver Physiol. (2013) 304:G597–604. doi: 10.1152/ajpgi.00455.2012


**Conflict of Interest Statement:** RB was employed by company Alloksys Life Sciences BV.

The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2018 Presbitero, Mancini, Brands, Krzhizhanovskaya and Sloot. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# Integrative Computational Modeling of the Lymph Node Stromal Cell Landscape

Mario Novkovic<sup>1</sup> \*, Lucas Onder <sup>1</sup> , Hung-Wei Cheng<sup>1</sup> , Gennady Bocharov <sup>2</sup> and Burkhard Ludewig<sup>1</sup>

*1 Institute of Immunobiology, Kantonsspital St. Gallen, St. Gallen, Switzerland, <sup>2</sup> Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia*

Adaptive immune responses develop in secondary lymphoid organs such as lymph nodes (LNs) in a well-coordinated series of interactions between migrating immune cells and resident stromal cells. Although many processes that occur in LNs are well understood from an immunological point of view, our understanding of the fundamental organization and mechanisms that drive these processes is still incomplete. The aim of systems biology approaches is to unravel the complexity of biological systems and describe emergent properties that arise from interactions between individual constituents of the system. The immune system is greater than the sum of its parts, as is the case with any sufficiently complex system. Here, we review recent work and developments of computational LN models with focus on the structure and organization of the stromal cells. We explore various mathematical studies of intranodal T cell motility and migration, their interactions with the LN-resident stromal cells, and computational models of functional chemokine gradient fields and lymph flow dynamics. Lastly, we discuss briefly the importance of hybrid and multi-scale modeling approaches in immunology and the technical challenges involved.

Keywords: lymph node, stromal cells, systems biology, network topology, morphology, lymph flow, fibroblastic reticular cells, computational models

#### INTRODUCTION

The lymphatic vascular system extends throughout the body, collecting interstitial tissue fluid through a network of initial lymphatic vessels (1). The lymph is then carried to the collecting lymphatics and distributed through lymphoid organs before returning to the venous circulation. Secondary lymphoid organs such as lymph nodes (LNs) form at bifurcation points along the lymphatic vasculature and serve as checkpoints for immune cells (2, 3). Adaptive immune responses are initiated and maintained in LNs via coordinated interactions between T cells, B cells, dendritic cells (DCs) and the LN-resident stromal cells (4–6) (**Figure 1A**). Traditionally, stromal cells have been described as connective tissue cells which organize the underlying LN infrastructure and cellular compartmentalization, however in recent decades their critical roles in regulation and coordination of immune responses have been established (7–9).

#### Edited by:

*Matteo Iannacone, San Raffaele Hospital (IRCCS), Italy*

#### Reviewed by:

*Scott N. Mueller, The University of Melbourne, Australia Sophie Acton, University College London, United Kingdom*

> \*Correspondence: *Mario Novkovic mario.novkovic@kssg.ch*

#### Specialty section:

*This article was submitted to Viral Immunology, a section of the journal Frontiers in Immunology*

Received: *20 August 2018* Accepted: *02 October 2018* Published: *23 October 2018*

#### Citation:

*Novkovic M, Onder L, Cheng H-W, Bocharov G and Ludewig B (2018) Integrative Computational Modeling of the Lymph Node Stromal Cell Landscape. Front. Immunol. 9:2428. doi: 10.3389/fimmu.2018.02428*

# LYMPH NODE STROMAL CELL FRAMEWORK

CD45<sup>−</sup> non-hematopoietic stromal cells in LNs originate from mesenchymal and endothelial precursors and can be divided into four major subsets based on the expression of podoplanin (PDPN) and CD31; PDPN−CD31<sup>+</sup> blood endothelial cells (BECs), PDPN+CD31<sup>+</sup> lymphatic endothelial cells (LECs), PDPN+CD31<sup>−</sup> fibroblastic reticular cells (FRCs) and PDPN−CD31<sup>−</sup> double-negative cell fraction (10, 11). Stromal cell subsets form site-dependent niches customized for efficient interactions with immune cells, separating the LN into distinct regions (**Figure 1A**).

The lymph drains to the LN subcapsular sinus (SCS) through several afferent lymphatic vessels, carrying antigen, signaling molecules and immune cells. The SCS is lined with two types of LECs, the floor and ceiling LECs. It has been demonstrated that ceiling LECs express the atypical chemokine receptor CCRL1 (ACKR4) which binds CCR7 ligands CCL19 and CCL21, whilst floor LECs are devoid of its expression (12). Differential expression of CCRL1 creates chemokine gradients for DCs to migrate from the SCS to the LN parenchyma. The outer cortex of the LN under the SCS contains B cell follicles which are populated by several stromal cell subsets critical for B celldependent responses. B cells sense the CXCL13 gradient and migrate to the follicles in a CXCR5-dependent manner (13), where they interact with a dense network of CD21+CD35<sup>+</sup> follicular dendritic cells (FDCs) in order to sample antigens (14, 15). A monolayer of MadCAM1<sup>+</sup> marginal reticular cells (MRCs) also contributes to B cell homing by expression of CXCL13 (16) and they have also been shown to express RANKL (TNFSF11) in LNs (17). The specific expression of RANKL by MRCs was subsequently confirmed by single-cell RNA sequencing, although MadCAM1 expression could not be readily detected (18). It was previously shown that MRCs are able to proliferate and differentiate into FDCs during inflammation-induced remodeling of the B cell follicles (19), however the phenotype and function of MRCs still remain poorly understood.

Furthermore, the B cell zone-resident reticular cells alongside the expanding FDC network orchestrate germinal center formation during inflammation (20). Additional stromal cell subsets have been reported in the B cell follicle, such as the CXCL13-producing stromal cells surrounding inflamed B cell follicles (21) and a CXCL12<sup>+</sup> reticular stromal subset in the dark zone of the germinal center following infection (22). Clearly, the heterogeneity of B follicle stromal cells requires further dissection in order to identify the key players in the development of humoral immunity.

The LN is a highly vascularized organ as the blood vasculature needs to deliver oxygen and nutrients to cells in the LN parenchyma. Advances in microscopy technologies have enabled 3D imaging and quantification of the topology of the entire microvascular network in LNs (23). Importantly, during inflammation the vasculature must expand in order to accommodate the increasing metabolical demand of the LN, which is achieved through proliferation of BECs and subsequent return to homeostasis by stochastic deletion of both preexisting and newly generated blood vessels (24). The majority of lymphocytes enter the LN paracortex through specialized blood vessels called high endothelial venules (HEVs) which mediate transendothelial extravasation (25, 26). The specific roles of HEVs in lymphocyte motility and chemotaxis as opposed to capillary endothelial cells have been recently elucidated by transcriptional profiling (27).

Upon entering the LN parenchyma, T cells crawl along the FRC network searching for cognate antigen loaded on DCs (28–30). FRCs in the T cell zone (TRC) produce homeostatic chemokines CCL19 and CCL21, guiding T cells and DCs into the relevant compartments and facilitating T-DC interactions necessary for developing adaptive immunity and antiviral responses (31–33) (**Figure 1A**). The interaction between PDPN<sup>+</sup> perivascular FRCs and the platelet-derived C-type lectin-like receptor 2 (CLEC-2) has been shown to promote VE-cadherin expression by HEVs through local sphingosine-1-phosphate (S1P) release by platelets, effectively maintaining the vascular integrity of HEVs (34). While DCs crawl on the FRC network, the interaction between PDPN<sup>+</sup> FRCs and CLEC-2 on DCs induces actin cytoskeleton remodeling and promotes DC motility (35). Furthermore, the same axis permits stretching of the FRC network in order to accommodate rapid LN expansion during an immune response, whilst DC-derived lymphotoxin beta receptor (LTβR) ligands promote FRC survival by modulating PDPN expression (36–38). A recent study employing single-cell RNA sequencing has suggested the existence of nine non-endothelial stromal cell clusters within LNs, however the expression of the canonical stromal cell marker PDPN was not sufficient to distinguish the clusters on a single cell level (18).

In addition to the emerging roles of FRCs in regulation of immune responses (39), the FRC network serves a fundamental role in the formation of the LN conduit system (40, 41). The conduit system emerges as a complex branched mesh of microvessels from the floor of the SCS, comprising of a collagenrich core surrounded by a microfibrillar zone and a basement membrane. A sparse network of conduits enwrapped by FDCs pervades the B cell follicles and drains the lymph through the T cell zone where it forms a dense re-entrant loop network ensheathed by the TRCs (42). The conduits rapidly transport small signaling molecules, chemokines and soluble antigens with the lymph and deliver them to the relevant stromal cells and lymphocytes (43, 44). The conduits size exclusion criterion of <70 kDa for entry of lymph-borne antigens has been recently shown to be dependent on plasmalemma vesicle associated protein (PLVAP) expression by SCS and medullary LECs (45). Ultimately the lymph is carried through the conduit system to the medullary lymphatics where it drains out of the LN through an efferent lymphatic vessel. Egress of lymphocytes occurs at the cortical and medullary sinuses through sensing of sphingosine-1 phosphate (S1P) produced by LECs (46–48).

In conclusion, stromal cells exhibit niche-specific functions and heterogeneity, indicating the complexity of their specialized interactions with immune cells. Many questions still remain open regarding their development and plasticity in homeostasis and during ongoing immune responses.

cell; TRC, T cell zone fibroblastic reticular cell; MRC, marginal reticular cell; BEC, blood endothelial cell; LEC, lymphatic endothelial cell. (B) Network graphs of the TRC network and equivalent network models; Watts-Strogatz small-world network, Erdos-Renyi random network and 1D ring lattice network. Colors indicate nodes with low (blue) or high (red) betweenness centrality.

# STROMAL-IMMUNE CELL INTERACTION MODELS

T cell motility and migration patterns arise from cell-intrinsic cues such as actin polymerization and cell-extrinsic cues which include integrin-dependent adhesion, physical guidance of the microenvironment, and chemotactic gradients (49). Based on these observations it has been proposed that T cells switch between two modes of intranodal migration (50); anchorage-dependent motility mediated by engagement of LFA-1 with ICAM-1 on DCs and FRCs (31, 51), and anchorageindependent motility driven by FRC-derived chemokines and lysophosphatidic acid (LPA) (52). Moreover, a recent study demonstrated that LFA-1 and CCR7 contribute complementary and not sequentially to intranodal T cell migration. Interestingly, the authors also show that T cells migrate in a continuous sliding locomotion rather than in a caterpillar-like manner (53, 54).

Intranodal T cell motility is closely linked to search strategies employed in order to efficiently find cognate antigen loaded on DCs (55). Additionally, migration patterns are heterogeneous between T cell subsets such as CD4 and CD8, and whether they are naive, activated or memory T cells (56–59). Thus, T cells can exhibit a spectrum of search patterns, ranging from diffusive random walks analogous to Brownian motion, superdiffusive Lévy walks and subdiffusive random motion (49, 60). It has been shown in a recent report using Agent-Based Models (ABM) that naive T cells in LNs exhibit a type of superdiffusive walk which fits best as a lognormal modulated correlated random walk among the idealized computational models studied (61) (**Table 1**). Similarly, another ABM study demonstrated that T cell migration in inflamed LNs best fits an inverse heterogeneous correlated random walk (78).

Numerous computational T cell migration studies in LNs did not readily include the underlying reticular network structure. Furthermore, these analyses were performed using rule-derived modeling methodologies that are phenomenological in nature, rather than a biophysics-based approach (85). However, several modeling studies have simulated the TRC network with randomized connectivity and addressed its involvement in guiding T cell motion. In one study a 3D ABM approach was used to simulate infection responses in order to observe T-DC encounters and T cell differentiation in LNs under different antigen conditions (77). 3D Cellular Potts Models (CPM) offer a complementary modeling framework to simulate dynamics of T cell and DC migration alongside the TRC network. It was shown that the complex cell movement is determined by the densely packed LN environment, even though similar migratory behavior of T cells was observed whether they preferentially adhered to the TRC network or not (73). Interestingly, the study demonstrated the existence of small dynamic T cell streams within LNs, which the authors speculate occur alongside the TRC network fibers. Another study simulated migration of T cells and DCs on the TRC network and found that constraining cell movement on the TRC network does not increase the frequency of T-DC encounters compared to Brownian motion in free 3D space (75). A subsequent theoretical study confirmed in simulations that the TRC network has only a minor effect on the contact probability between T cells and DCs (76).

A question then naturally arises; do lymphocytes require the TRC network as a guiding structure for cellular movement? The answer seems evident from plethora of experimental work, corroborated by a recent reports demonstrating that deficiency in CCR7-mediated chemokine sensing and integrin LFA-1 dependent adhesion in T cells does not abrogate intranodal migration and firm attachment to the TRC network (53, 72). However, the existing theoretical models were characterized by poorly resolved sets of multi-scale control processes regulating various cell migration modes and antigen-driven functional states of immune cells. Ultimately, the theoretical framework of many modeling studies lacked the necessary quantitative data to faithfully recapitulate the stromal-immune cell interactions. In order to extend the analogy, the simulations would represent a "car with no fuel and no wheels, moving along a random road map."

An alternative approach to examine the TRC network at a fundamental level would be to employ the theory of complex networks, also called graph theory (86, 87). Within this mathematical framework the TRC network is denoted as a series of nodes (cells) connected with edges (cell protrusions). A recent study demonstrated that the TRCs organize as a nonstochastic small-world network with highly robust topological properties, ensuring that network failure does not occur even when up to half of the network is destroyed (66, 88). Specific genetic ablation of CCL19-producing TRCs led to highly reduced numbers of hematopoietic cells in LNs and impaired intranodal migration of T cells with marked reduction in average cell speed and motility. The few T cells that did enter the LNs exhibited undirected movement around the HEVs and were not able to migrate deeper into the paracortex, despite the conduit system still being present (20, 66). The loss of FRCs and HEVs is also associated with graft-vs.-host disease after allogeneic hematopoietic stem cell transplantation and it has been recently shown that FRCs can prime alloreactive T cells through Delta-like Notch ligands (89, 90). Moreover, the TRC network is capable of fully regenerating after complete ablation and this observation is indicative of a formation of a costeffective, optimally robust network structure that simply could not have a random configuration (91, 92) (**Figure 1B**). The heterogeneous topological properties of real world networks could not be explained by the random network model, thus it is likely that these networks evolved by optimizing two competitive selection criteria: high connectivity which confers efficiency of information transfer and low connection cost during formation of the network (93). Likewise, spatial embedding in many real world networks has significant confining effects on the overall topological structure by restricting the formation of long-distance connections (94, 95) (**Figure 1B**).

The intricate structure of LNs determines organ functionality, however the reverse also holds true; the diverse cellular interactions require a particular underlying structure to be present (96). Although it is widely accepted the TRC network serves as a "road system" for T cell and DC migration (29, 97), it remains unclear whether and to which extent dynamic cell movements are spatially constrained by the intricate network fibers (92, 98). Hence, incorporating quantitative data into integrative models may provide answers to these fundamental questions.

#### INTEGRATIVE LYMPH NODE MODELS

Maintenance of chemokine gradients by stromal cells is crucial for lymphoid organ development and spatiotemporal segregation of specialized immune cell compartments (99, 100). Chemotaxis in LNs has been modeled using ABMs in order to simulate large numbers of T cells in a computationally efficient manner. By modeling T cell motion as a persistent random walk and allowing for cell crowding on a 3D lattice, a basic T cell ingress-egress model in LNs could be constructed (74). Lymphocytes must navigate efficiently within spatially heterogeneous chemokine fields that also vary over the time course of an immune response. It was shown that temporal sensing of rising chemokine concentrations is required for directional persistence of DC and neutrophil migration (101). Moreover, chemotactic-driven directional movement of DCs is steered by soluble forms of CCL19 and CCL21, whilst immobilized form of CCL21 on FRCs induces both DC motility and integrin-dependent adhesion (102).


TABLE 1 | Integrative modeling frameworks for lymph node structures and processes.

*<sup>a</sup>ABM, Agent-based model; CPM, Cellular-Potts model; DDE, Delay differential equation; ODE, Ordinary differential equation; PDE, Partial differential equation.*

Functional chemokine gradients of CCL19 and CCL21 have been simulated in various LN regions using a fluid flow model where the intranodal chemokine dynamics are described by ordinary (ODE) and partial differential equations (PDE) (68). Similarly, using a reaction-diffusion PDE model, highly heterogeneous distribution of IFN-α has been found, where certain LN subdomains are highly protected, whilst others are characterized by much lower levels of the cytokine (62). In a recent theoretical study, it has been demonstrated by reaction-diffusion-advection modeling that hypersensitivity in antigen recognition by immune cells can occur when chemotactic strength is higher than a predicted threshold, leading to immune system instability (69). In the case of cytokine concentration fields, it has been demonstrated that the size of cytokine niches on a single-cell level are governed by a simple mechanism dependent on cytokine diffusion and the density of consumer cells present in the niche (103).

It is important to consider another relevant aspect of LN functionality, namely lymph flow dynamics which contribute greatly to antigen, cytokine and chemokine transport. In order to gain insights into the quantitative flow parameters regulating lymph transport, a computational lymph flow model of the LN was constructed (63, 65). Interestingly, the model predicted that 90% of lymph traveled the peripheral path through the SCS and medullary sinuses. In a subsequent study the authors expanded their computational model to include intranodal CCL19 and CCL21 chemokine gradients (68). An integrative LN model with realistic 3D geometry has been recently developed in order to study lymph transport phenomena (82). The relationship between the structural LN geometry and fluid pathways has been investigated using image-based modeling of fluid flow in order to study the permeability of the LN tissue (64, 71). Furthermore, fluid flow dynamics of the blood microvasculature and the conduit system have been successfully integrated in the existing LN model (67, 70). The model predicted high robustness of the conduit system, with 60–90% elimination of conduits required to halt the lymph flux. Moreover, computational simulations of lymph flow can be expanded on larger spatial scales by modeling the entire human lymphatic system interconnected between hundreds of LNs (84).

In order to model complex biological phenomena with continuous and discrete variables, and across several spatial scales, hybrid and multi-scale modeling approaches are necessary (104–108). In recent years these models have been used to describe spatial dynamics of immune responses in LNs (79– 81, 83). A summary of the integrative modeling frameworks described here and their implementation in elaborating LN processes and functions is available in **Table 1**. These multiscale modeling approaches will prove invaluable in unraveling complex mechanisms of immune system control in future studies.

#### CONCLUDING REMARKS

Systems biology approaches have made tremendous advances in the past decade due to a high demand for bioinformaticsbased computational methods necessary to describe biological systems on a global level (109). Likewise, quantitative and computational in silico models in immunology have become critical for understanding the emergent properties of both single cells and whole tissues (110, 111). However, the development of mathematical LN models is still confronted with technical challenges. Understanding the multi-layered compartmentalization of the LN is an important prerequisite so that the initial assumptions of the model reflect the functionality observed experimentally. To date, our knowledge of the heterogeneity of stromal cells that construct the underlying foundations of a LN is still incomplete. The directional cues und critical immunoregulatory functions of stromal cells enable the formation of specialized micro-environmental niches for immune cells within the LN, effectively facilitating immune responses (11). The described computational models largely do not take into account an additional layer of complexity, which is introduced by the fact that chemoattractant fields significantly change during inflammation and ongoing immune responses, influencing the migration and composition of immune cells. Moreover, the LN stromal compartment undergoes extensive remodeling in order to accommodate the increased LN size and proliferative demands of developing adaptive immune responses (9). Therefore, mathematical models must take into account how the spatial constraints of the LN and heterogeneous chemoattractant gradient fields affect the nonuniform distribution of immune cells, the spatiotemporal dynamics of cellular interactions and the anisotropy of non-Brownian immune cell movement patterns. To this end, quantitative data on immune cell motility metrics in homeostasis and disease/inflammatory states are critically needed for the development and calibration of biophysics-based models.

#### REFERENCES


One major difficulty lies in delineating the complexity of the fundamental LN architecture and simplifying the components to a degree necessary to obtain biologically meaningful conclusions. Morphometric studies have been instrumental in describing the structural framework of distinct LN regions. However, quantitative data is still lacking for the organization of lymphatic endothelium in the medullary region, a comprehensive description of the B follicular stromal cells has not been fully elaborated and the structure of the fine-grained conduit system has not been extensively studied. Absence of detailed structural parameters represents a major caveat in data-driven systems biology approaches (112). Nevertheless, novel high-resolution imaging technologies coupled with multi-scale computational models will give us valuable insights into the inner "clockwork" of the LN.

#### AUTHOR CONTRIBUTIONS

All authors contributed to writing the manuscript. Figure and data were generated by H-WC, LO, and MN, table was generated by GB.

#### FUNDING

This study received financial support from the Swiss National Science Foundation (grants 166500 and 159188 to BL) and the Russian Science Foundation (grant 18-11-00171 to GB).


orchestrate rapid memory CD8<sup>+</sup> T cell responses in the lymph node. Immunity (2013) 38:502–13. doi: 10.1016/j.immuni.2012.11.012


**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2018 Novkovic, Onder, Cheng, Bocharov and Ludewig. This is an openaccess article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# Immunobiochemical Reconstruction of Influenza Lung Infection—Melanoma Skin Cancer Interactions

#### Evgeni V. Nikolaev 1,2, Andrew Zloza3,4 and Eduardo D. Sontag5,6,7 \*

*<sup>1</sup> Center for Quantitative Biology, Rutgers University, Piscataway, NJ, United States, <sup>2</sup> Clinical Investigations and Precision Therapeutics Program, Rutgers Cancer Institute of New Jersey, New Brunswick, NJ, United States, <sup>3</sup> Section of Surgical Oncology Research, Division of Surgical Oncology, Rutgers Cancer Institute of New Jersey, New Brunswick, NJ, United States, <sup>4</sup> Department of Surgery, Rutgers Robert Wood Johnson Medical School, New Brunswick, NJ, United States, <sup>5</sup> Department of Electrical and Computer Engineering, Northeastern University, Boston, MA, United States, <sup>6</sup> Department of Bioengineering, Northeastern University, Boston, MA, United States, <sup>7</sup> Laboratory for Systems Pharmacology, Harvard Medical School, Boston, MA, United States*

#### Edited by:

*Gennady Bocharov, Institute of Numerical Mathematics (RAS), Russia*

#### Reviewed by:

*Zvi Gershon Grossman, National Institute of Allergy and Infectious Diseases (NIAID), United States Kirill Peskov, Modeling and Simulation Decisions, Russia*

\*Correspondence:

*Eduardo D. Sontag e.sontag@northeastern.edu*

#### Specialty section:

*This article was submitted to Viral Immunology, a section of the journal Frontiers in Immunology*

Received: *29 May 2018* Accepted: *02 January 2019* Published: *28 January 2019*

#### Citation:

*Nikolaev EV, Zloza A and Sontag ED (2019) Immunobiochemical Reconstruction of Influenza Lung Infection—Melanoma Skin Cancer Interactions. Front. Immunol. 10:4. doi: 10.3389/fimmu.2019.00004* It was recently reported that acute influenza infection of the lung promoted distal melanoma growth in the dermis of mice. Melanoma-specific CD8+ T cells were shunted to the lung in the presence of the infection, where they expressed high levels of inflammation-induced cell-activation blocker PD-1, and became incapable of migrating back to the tumor site. At the same time, co-infection virus-specific CD8+ T cells remained functional while the infection was cleared. It was also unexpectedly found that PD-1 blockade immunotherapy reversed this effect. Here, we proceed to ground the experimental observations in a mechanistic immunobiochemical model that incorporates T cell pathways that control PD-1 expression. A core component of our model is a kinetic motif, which we call a PD-1 Double Incoherent Feed-Forward Loop (DIFFL), and which reflects known interactions between IRF4, Blimp-1, and Bcl-6. The different activity levels of the PD-1 DIFFL components, as a function of the cognate antigen levels and the given inflammation context, manifest themselves in phenotypically distinct outcomes. Collectively, the model allowed us to put forward a few working hypotheses as follows: (i) the melanoma-specific CD8+ T cells re-circulating with the blood flow enter the lung where they express high levels of inflammation-induced cell-activation blocker PD-1 in the presence of infection; (ii) when PD-1 receptors interact with abundant PD-L1, constitutively expressed in the lung, T cells loose motility; (iii) at the same time, virus-specific cells adapt to strong stimulation by their cognate antigen by lowering the transiently-elevated expression of PD-1, remaining functional and mobile in the inflamed lung, while the infection is cleared. The role that T cell receptor (TCR) activation and feedback loops play in the underlying processes are also highlighted and discussed. We hope that the results reported in our study could potentially contribute to the advancement of immunological approaches to cancer treatment and, as well, to a better understanding of a broader complexity of fundamental interactions between pathogens and tumors.

Keywords: influenza, melanoma, PD-1/PD-L1, incoherent feedforward loop, mathematical modeling

#### 1. INTRODUCTION

It was recently reported that acute influenza A infection (A/H1N1/PR8) of the lung promoted distal B16-F10 skin melanoma growth in the dermis (1). It was also observed that melanoma-specific CD8+ T cells were shunted to the lung in the presence of the infection, where they expressed high levels of inflammation-induced cell-activation blocker PD-1, and became incapable of migrating back to the tumor site. At the same time, co-infection virus-specific CD8+ T cells remained functional while the infection was cleared. Finally, it was also unexpectedly found that blockade of PD-1 resulted in reversal of infection-mediated anti-tumor response disruption.

In this respect, it is very important to mention that the work by Kohlhapp et al. (1) was primarily motivated by two still unmet challenges: (i) emerging epidemiological studies reporting an increased cancer prevalence and cancer-specific deaths in patients with infections (1), and (ii) despite the fact that tremendous amount of work on immune response in the context of pathogenic co-infection has been done, findings in this field still remain discordant and a matter of debate, as also reviewed by Kohlhapp et al. (1) and Zloza (2).

Motivated by the need to provide a more conceptual and quantitative biology insight into "the previously unrecognized acute non-oncogenic infection factor" accelerating tumor growth (1) and more broadly into the interactions between pathogens and cancer, and specifically, in order "to harness these interactions to improve microbial-based cancer therapy" (2), we suggest a few immunobiochemical mechanisms and a simple mathematical model which may help to interpret the observed phenomena (1).

Our main results relate to two fundamental functional roles of immunity (3–5): (i) adaptation of immune function, and (ii) competition between excitation and de-excitation ("push-pull") factors possessing different response kinetics. In the context of this work, the loss of adaptation occurs in the expression of PD-1 receptors on anti-melanoma CD8+ T cells, a phenomenon that may constitute the essence of the previously unrecognized immunologic factor (1), while competing push-pull factors (3) correspond to opposite outcomes of the corresponding kinetic motifs identified as incoherent feedforward loops (IFFLs) in the classification of Alon (6). We briefly note that push-pull factors also play multiple fundamental roles in physiology (and biology) in general, e.g., Dampney et al. (7).

Our working hypothesis is that the melanoma-specific T cells shunted to the lung in the presence of the infection express high levels of inflammation-induced cell-activation blocker PD-1, which upon interacting with PD-L1 constitutively expressed in the lung, render T cell motility paralysis (8). At the same time, virus-specific cells adapt to strong stimulation by their cognate antigen by lowering the transiently-elevated expression of PD-1, remaining functional and mobile while the infection is cleared.

Although other important mechanisms may contribute to the previously unrecognized acute non-oncogenic infection factor (1), we focus our efforts on one concrete aspect of the problem, which is a gene regulatory network (GRN) that controls PD-1 expression. Indeed, the fact that many other factors may contribute to the enormously complex molecular makeup of the acute non-oncogenic infection effect, such factors, obviously, do not exclude the interaction PD-1:PD-L1 playing a role as clearly seen from the data collected in Kohlhapp et al. (1). Thus, the importance of the PD-1:PD-L1 signal sent by the data cannot be disputed. Moreover, it is the PD-1:PD-L1 signal "detected" experimentally in Kohlhapp et al. (1) that defines the scope of our work aimed in uncovering relevant molecular detail in an unbiased way. We then develop and use a simple mathematical model in order to further illuminate the PD-1:PD-L1 role.

Specifically, a core component of our PD-1 gene-regulatory network (GRN) is a kinetic motif, which we call a Double Incoherent Feed-Forward Loop (PD-1 DIFFL), and which reflects known interactions between IRF4, Blimp-1, and Bcl-6 transcription factors (TFs). The different activity levels of the PD-1 DIFFL components, as a function of (a) the cognate antigen levels, (b) the T cell receptor (TCR) activity, and (c) the given inflammation context, manifest themselves in the T cell phenotypically distinct outcomes discussed in our work.

The rest of our work is organized as follows. In section 2.1, the main results of Kohlhapp et al. (1) are briefly outlined. Alternative hypotheses potentially related to the unrecognized factor are discussed in section 2.2. Here, the motivation for the development of the PD-1 DIFFL is also given. The PD-1 DIFFL is reconstructed in section 2.3. We next attempt to falsify and validate the kinetic motif (PD-1 DIFFL) against key experiential observations in section 2.4. The results of our mathematical modeling are described in section 2.5. Finally, a literature review of the corresponding mechanistic detail, the model construction, and the model's parameter justification can be found in **Supplementary Material**.

# 2. RESULTS

We begin our analysis of the experimental data (1) by discussing a few alternative hypotheses, followed by the introduction of a number of mechanisms consistent with the discussed observations.

The selected mechanisms will then be formalized in terms of a relevant genetic molecular circuit (PD-1 DIFFL) that regulates PD-1 expression. Our proposed PD-1 DIFFL model is based upon molecular detail discovered previously, and is independent of the results obtained in Kohlhapp et al. (1).

We hope that the strong inference methodological approach (9) that guides our research will allow us to customize the PD-1 DIFFL to different inflammatory conditions (1) with the ultimate goal to capture both infection-tumor and infection-infection interactions at the mechanistic molecular level.

# 2.1. Linking Observations With Immunological Mechanisms

A key challenge in the study of T cells within different dual immunological self (tumor) and non-self (infectious) contexts, is the organization of large amounts of relevant molecular and biochemical information (section SI-1) compactly summarized in **Table 1**.

Specifically, **Table 1** highlights the following key observations (O1)–(O5) made in Kohlhapp et al. (1):


TABLE 1 | A summary of the immunological reconstruction of infection-tumor interactions<sup>a</sup>


.

*<sup>a</sup>Literature citations are directly inserted through the text (section SI-1).*

immune response and, additionally, (ii) reactivated antimelanoma CD8+ T cells sequestered in the infected lung may regain their motility and return back to the TME, where they also aid in the anti-tumor response.


# 2.2. From a Physiologic Systemic View of Lymphocyte Re-circulation to Systems Biology of PD1:PD-L1 Interactions

It is known (10, 11) that non-specific cardiovascular edema effects, caused by infection-induced inflammation in the infected site, redirect the blood-flow to the site of infection-induced inflammation. Therefore, it is highly appealing to explain the observed accumulation of anti-melanoma CD8+ T cells in the infected lungs, (O1), by non-specific inflammation effects only.

Note that the lymphocyte recirculation routes are phenotypedependent and significantly differ for naïve/memory/effector subsets (12). We leave the corresponding details specific to the different subsets out of the discussion that follows. What is relevant to our work is that all newly activated cytotoxic T lymphocytes (CTLs) exit the corresponding lymph nodes into lymph via lymphatic ducts before they enter circulation via the great veins, and then flow through the pulmonary circulation (**Figure 1**). Under resting non-inflamed conditions, re-circulation of lymphocytes between lungs and blood is very rapid, with the average residence time in the lungs less than one minute (16).

After leaving the heart and lungs, the traveling CTLs continue to flow into systemic circulation, followed by their ultimate but not instantaneous homing in the corresponding infectious or tumor sites. Indeed, lymphocytes on average must pass via vasculature of the lung or liver about 10 times or even more times (15) before they migrate to one of the secondary lymphoid tissues (12)[BOX 14.2]. For example, it was shown that if anti-tumor CTLs were activated in the breast, they would perform on average about eight circulatory transient cycles before extravasation into the tumor site (15).

Before reconciling the experimental observation (O1) with these studies, we have to briefly discuss a unique role that the lung plays in the physiology and immunology of trafficking lymphocytes under both resting and inflamed conditions.

Experimental studies have revealed that different subsets of lymphocytes, including naïve/memory/activated effector T cells, transiently accumulate in the lungs (17, 18) both by means of and, what is also extremely important, without specific antigendependent recruitment of CTLs to the lung (19). Anderson et al. (19) further discuss "numerous observations indicating that T cell trafficking withing the lung is starkly different from what is known about T cell trafficking in most nonlymphid tissues,"

FIGURE 1 | Schematic representation of lymphocyte re-circulation routes. There are two different routes for naïve and activated trafficking lymphocytes (12, 13). First, due to the data discussed in Owen et al. (12, Ch.14) and, independently estimated in Van den Berg (14, p. 23) after approximately 30 min. transit time in the blood, about 45% of all naïve lymphocytes (produced by the thymus and bone marrow) migrate to the spleen, where they reside for about 5 h. Another 45% of lymphocytes enter various peripheral nodes, where they remain for 12–18 h, scanning stromal cell surfaces. A smaller fraction of lymphocytes migrate to secondary lymphoid tissues (skin, gastrintestinal, etc.), to protect the organism against the external environment. Thus, about 5% of the lymphocytes are, at resting conditions, in the blood, and the majority resides in the lymph nodes. Second, as discussed in Poleszczuk et al. (15) activated CTLs enter the blood system via the great veins, flow through the pulmonary circulation, and, then, continue into systemic circulation. Venus blood from gastrointestinal tract and spleen goes to the liver through the hepatic portal vein. In all cases, lymphocytes migrate from the blood into lymph nodes through high-endothelial venues, specialized areas in postcappillary venues. (a) MALT is Mucosa Associated Lymphoid Tissue. (b) Lymph nodes have both afferent and efferent lymphatic vessels, while MALT, Spleen, and Thymus have efferent lymphatic only (12).

including the fact that lymphocyte extravasation into the lung is chemokine independent (20, 21). So, one must revisit the observation (O1) by taking into account the unique role that the lung may play in lymphocyte retention even in the absence of influenza A related antigen-induced chemokine gradients that would additionally force anti-melanoma CTLs to extravasate into the lung epithelium, should influenza A infection be present.

Unfortunately, the above results and the unique role of the lung to transiently retain lymphocytes still do not explain the difference in the observations (O1) and (O2), nor they explain the observation (O3), for the following reasons.

First, concerning the observations (O1) and (O2), both antimelanoma and anti-infection CTLs should follow the same pattern of multiple vascular re-circulation cycles as discussed above (**Figure 1**). However, under similar re-circulation patterns, the presence of IAV infection impedes tumor clearance, while, at the same time, both IAV and another concomitant infection (e.g., VACV) are cleared efficiently as one infection would be cleared in the absence of another. Specifically, the question "Why are antimelanoma CTLs retained in the infected lung, while anti-VACV infection CTLs are not?" remains unanswered.

Given the large literature body on the importance of PD-1 receptors in immune response and the observations (O3) and (O5), we decided to explore theoretically whether molecular signaling pathways initiated by PD-1 ligated with PD-L1 would provide at least one plausible mechanism to explain the results obtained in Kohlhapp et al. (1).

We have excluded PD-L2 from our model and only consider PD-L1 in the analysis that follows. Indeed, PD-L2 has restricted expression on macrophages, dendritic cells (DCs), and mast cells, while PD-L1 is expressed more broadly in order to mediate T cell tolerance in non-lymphoid tissues (12, 22). Besides, mathematical simulations based on the biophysical and expression data have revealed an unexpectedly limited contribution of PD-L2 to PD-1 ligation during interactions of activated T-cells with APCs (23).

To this end, the immune system has apparently evolved the inhibitory PD-1/PD-L1 pathway as a result of the need to control the degree of inflammation at locations expressing the antigen in order to secure normal tissue from damage and also to maintain peripheral tolerance (4, 22). This includes the constitutive expression of PD-L1 in large quantities in various tissues such as lungs, pancreatic islets, ovary, colon, etc. (24–29) by which cross-reactive effectors that survive positive selection are also muted to maintain the peripheral tolerance (2).

#### 2.3. Incoherent Feed-Forward Regulation of PD-1 Expression

PD-1 expression on CD8+ T cells is known to be regulated at the level of transcription of its gene pdcd1 (30). More precisely, two upstream conserved regulatory CR-B and CR-C regions (30) are key for PD-1 expression in response to CD8+ T cell activation. Specifically, TCR signaling induces PD-1 gene expression through the transcriptional activator, Nuclear Factor of Activated T cells, cytoplasmic 1 (NFATc1) (**Figure 2**), which binds to CR-C after translocation to the nucleus (30, 31).

Next, the down-regulation of PD-1 during acute infection (32) suggests that there exists a mechanism that directly represses its expression after initial activation events. Indeed, Blimp-1 (B Lymphocyte-Induced Maturation Protein 1) (33) has been found to be induced during the later stages of CD8+ T cell activation and was shown to be required for the efficient terminal differentiation of effector CD8+ T cells (30). When Blimp-1 is suppressed, the same data suggest that in the absence of Blimp-1, PD-1 expression is maintained by NFATc1 (**Figure 2**).

For the sake of completeness, we recall that the existing data also suggest that Blimp-1 represses PD-1 gene expression in CD8+ T cells using three distinct molecular mechanisms (30):


(3) eviction of the activator NFATc1 from its site that controls PD-1 expression.

In addition, Blimp-1 has been found to be a transcriptional antagonist of proto-oncogene Bcl-6 (B cell lymphoma 6 transcription factor), and vice versa (**Figure 2**) (i.e., Blimp-1 and Bcl-6 are known to mutually repress one another) (34–38). Specifically, Blimp-1 can bind to the bcl6 promoter (39).

Although it is not known exactly how Bcl-6 inhibits Blimp-1 in T cells, it is well known that in B cells Bcl-6, in association with a corepressor MTA3, represses prdm1 by binding to sites in prdm1 intron 5 and intron 3 (34, 40, 41). We take this fact into consideration because signaling pathways and their activation are similar in both B and T cells (12). Additionally, Bcl-6 binds its own promoter and inhibits its own transcription (**Figure 2**), thus implementing an autoregulatory loop (42, 43) (**Figure 2**).

Competing with Bcl-6 for intron 5 in prdm1, IRF4 (Interferon Regulatory Factor) (44–47) is shown to be a direct activator of prdm1 (**Figure 2**) by binding to a site in intron 5 (34). At the same time, IRF4 directly represses gene bcl-6 by binding to a site within its promoter (34, 45), which is rich in IRF4-binding sites (43).

Because IRF4 is known to be activated both directly via TCR and by NF-κB (48, 49), we have then sought to determine who activates NF-κB in this context and found that NF-κB is activated by TCR signaling (34, 37, 50, 51). Several potential NFκB binding sites in the prdm1 promoter have been suggested. It is also known that IRF4 can bind to its own promoter, supporting a positive feedback mechanism by which high IRF4 expression can be maintained (43, 52).

There are additional signaling routes leading to the activation of IRF4 (e.g., via Akt-mediated pathways) which are not discussed here (34).

After a careful analysis of the reconstructed molecular interactions, we have come to the conclusion that this intricate reaction network consists of two subnetworks (**Figure 2**). Both subnetworks have the same input from the activated TCR, while the outputs of the subnetworks are different. Namely, PD-1 is the output of the subnetwork color-coded in blue and red, while Bcl-6 is the output of the subnetwork color-coded in green and red. The two subnetworks share a number of common species and interact with one another via repressive interactions mediated by the three key species color-coded in red, (i) IRF4, (ii) Blimp-1, and (iii) Bcl-6.

Each of the two subnetworks corresponds to a gene-regulatory network (GRN) motif known as an incoherent feed-forward loop (IFFL) (6). Because the PD-1 circuit is formed of two such IFFLs, we call it a Double Incoherent Feed-Forward Loop (DIFFL).

Our IFFL network may be viewed as a mechanistic instantiation of a conceptual signal discrimination model based on a competition between "excitation" and "de-excitation" factors possessing different response kinetics, as initially introduced by Grossman and Paul (3). The latter concept has been gradually applied successfully in multiple studies since 1992 as reviewed in Grossman and Paul (5). In that sense, we address with our model the following goal formulated in Grossman and Paul (3): "More explicit rules of organization, or models, need to be explored. Such rules should suggest, in particular, how the functional segregation of immunological responses may reasonably come about."

#### 2.4. PD-1 Expression Within Different Inflammatory Contexts

We next attempt to validate the PD-1 DIFFL motif (**Figure 2**) against all observations reported in Kohlhapp et al. (1) by following the falsification and validation methodology (53), which is also fundamental to any modeling study. **Figure 3** will be instrumental in our analysis that follows.

**Figure 3A** shows a biochemical reaction network reconstruction customized for the case of an anti-influenza cytotoxic effector T cell, TEFF, in the presence of large amounts of cognate Ag in the infected lung. In this case, the immunological complexity of interactions involving cytokines is already overwhelming (5, 54–58). For example, IL-2 activates and is simultaneously repressed by active Blimp-1 both directly and indirectly (31, 59).

The abundance of the cognate viral Ag in the infected lung leads to a strong TCR activation which, in turn, results in the simultaneous activation of Blimp-1 and degradation of Bcl-6 (section 2.3) followed by suppression of PD-1 transcription with its subsequent degradation. The biochemical detail can explains transient and rapid PD-1 expression followed by downregulation of PD-1 expression in the presence of acute infection (32), see also section SI-1.2.

All this may also explain why anti-infection CD8+ T cells are not exhausted during the first phase of the biphasic response of the PD-1 DIFFL-circuit (section 2.3) despite the fact that bystander and tissue cells express large amounts of PD-L1 caused by large concentrations of pro-inflammatory cytokines such as INFγ (SI-1.1). Recall that large amounts of PD-L1 are already constitutively expressed in the lung under resting condition in the absence of any infection (section 2.3).

**Figure 3B** shows the response of the reconstructed circuit in the tumor microenvironment (TME). Specifically, antimelanoma CD8+ T cells overexpress PD-1 in the presence of large amounts of tumor-specific cytokines such as IL-6, a welldescribed regulator of Bcl-6 expression (38). Due to relatively low levels of tumor Ag and a weak self-Ag TCR signal (60) of antitumor CD8+ T cells, the TCR is not activated strongly enough to activate Blimp-1 and, at the same time, the weak activation of the TCR sets the first phase of the biphasic response of the dose-dependent PD-1 DIFFL motif (**Figure 2**).

Indeed, the PD-1 DIFFL strongly activates Bcl-6 for small and medium TCR strengths, and weakly activates Bcl-6 for high activity levels of TCR. As a result, Bcl-6 is overexpressed, while Blimp-1 is not expressed in the melanoma TME (38), which leads to the overexpression of PD-1 on the surface of anti-tumor CD8+ T cells.

**Figure 3C** shows the PD-1 DIFFL in an anti-melanoma TEFF relocated into the infected lung. In this case, the conditions discussed just above to introduce **Figure 3B** play the role of a spark plug that activates the transcription of Bcl-6, which represses prdm1 even after the relocation of the anti-tumor TEFF into the lung.

These relocated TEFF can now sense the elevated levels of INFγ and TNF-α, which are abundant in the infected site, and which are produced by professional antigen presenting cells (APCs) (section 2.2).

The cytokines strongly stimulate the expression of both PD-1 and PD-L1 (61), as well as maintain the expression of PD-1 on the surface of the anti-melanoma TEFF, initially sparked by the ligation of TCRs with cognate tumor Ags during the time when the TEFF cells were present in the TME before their relocation to the lung.

Because the tumor Ag is absent from the infected lung, the TCR is not ligated, and, hence, all routes leading to the activation of Blimp-1 and IRF4 are disabled. We can thus propose that the major route contributing to PD-1 overexpression here is mediated by INFγ and TNFα. The corresponding PD-1 expression activation route is marked by sign + inside a circle in **Figure 3C**.

Recall that large quantities of PD-L1 are constitutively expressed in the lung already under resting conditions (section 2.2). PD-1 mediated control of immune responses depends on interactions between PD-1 on CD8+ T cells and PD-L1 in tissues (62). Importantly, such PD-1:PD-L1 interactions can result in CD8+ T cell motility paralysis (8, 28, 63).

We introduce the paralysis mechanism (**Figure 4**) in detail in (O1-M6) (section SI-1.3) and believe that this mechanism can provide a valuable insight into the previously unrecognized factor contributing to the retention of anti-melanoma CD8+ T cells shunted to the influenza A infected lung (1). Of course, other yet unknown mechanisms may exist and need to be elucidated in order to provide a more complete explanation of the retention effect (1). Therefore, additional experimental observations should be obtained.

The study conducted by Cheng et al. (23) reports that "it now seems that very stable complexes are not prerequisite for potent inhibitory PD-1:PD-L1 signaling" because measurements of the human and mouse PD-1 binding to PD- L1 affinities suggest that potent inhibitory signaling can be mediated by weak interactions.

Zinselmeyer et al. (8) further stress: "Prolonged motility arrest is an excellent host strategy to decrease T cell efficiency and likely facilitates exposure to multiple regulatory pathways. PD-1:PD-L1 blockade is known to restore function to virus-specific and tumor-specific T cells, and has shown some promise in recent clinical trials."

Although dissociation and association of the complex PD-1:PD-L1 are assumed to be fast (64, 65), this however does not preclude the long-known loss of T cell motility due to multiple PD-1:PD-L1 interactions (66, 67).

**Figure 3D** shows the PD-1 DIFFL in an anti-melanoma TEFF cell in the infected lung after administration of PD-1 (αPD-1) blockade. Recall that the NF-κB pathway is downregulated in exhausted CD8+ T cells (38). To this end, the PD-1 blockade (marked by symbol αPD-1 color-coded in red) in **Figure 3D**, removes the brake (68) from the corresponding T cell signaling pathways (see section 2.1, observation O(3), and **Table SI-1.1**) leading to overexpression of NF-κB (66, 69). Additionally, NF-κB activation is positively regulated through TNFR (TNF Receptor) and TLR (Toll-like Receptor) sensing TNFα and viral materials in the infected lung, respectively (70–72).

As discussed earlier, NF-κB activates IRF4 (34), and the latter directly represses Bcl-6 (34). In turn, the repression of Bcl-6 removes the brake from the overexpression of Blimp-1, which then leads to reduced numbers of PD-1 receptors on the surface of reactivated anti-melanoma effector cells. This may allow the reactivated TEFF to become mobile (**Table SI-1.1**) with a potential to relocate back to the melanoma TME with the lymph flow and blood circulation as discussed in the mechanism (O1-M6). Indeed, it is well known that after the TEFF re-circulation in the blood (15), effector T cells are preferentially found in the lymph nodes in which their activation occurred, and in the area drained by those lymph nodes (73).

The above conclusions are also based on the experimental evidence that PD-1:PD-L1 interactions contribute to reduced T cell motility on day 7, and therapeutic blockade of PD-1:PD-L1 restore CD8+ T cell motility within 30 min (8). Although we use the references (8, 63) in order to support our hypotheses, additional experimental research is needed to understand deeper the paralysis phenomenon (28, 63).

We conclude our discussion of the PD1 DFFIL motif by noting that the core of the reconstruction (**Figure 2**) fits well to all discussed inflammatory contexts (**Figure 3**).

### 2.5. Probing Immunobiochemical Reconstruction Modeling

Our modeling goal here is quite simple. Given the discussed specificity of PD-1 expression (section 2.4) with respect to different amounts of antigen available in the medium and different values of TCR affinity in terms of the values of the offrate constant koff for the Ag:TCR bond (74, 75), we focus on the analysis of the dependence of the levels of key species, Bcl-6, IRF4, Blimp-1, and PD-1, on the two parameters, (i) the antigen concentration, Ag, and (ii) the values of koff defined in sections SI-2 and SI-3.

#### 2.5.1. Modeling PD-1 Expression in the Absence of PD-L1

We first consider the case when the PD-1:PD-L1 interaction is absent from the model by setting φL(P) ≡ 8(P) ≡ 1 corresponding to the condition L = [PD-1:PD-L1] = 0 in both Equations (SI-2.1c) and (SI-2.2a).

Typical plots for the (non-dimensional) steady-state (76) concentration levels of PD-1, Bcl-6, Blimp-1, and IRF4 in the absence of the PD-1:PD-L1 interaction and at the different values of koff are shown in **Figure 5**. The model's nondimensionalization is done in sections SI-2 and SI-3.

We next discuss the case of small values of koff from the set of the values given in the legend of **Figure 5**. We observe from **Figure 5** that the level of PD-1 (**Figure 5A**) becomes rapidly elevated already at very small values of the scaled Agconcentration (section SI-1). A further increase in the scaled Agconcentration results in the formation of the PD-1 level plateau, followed by a drop in PD-1 levels.

The increase in the level of PD-1 (**Figure 5A**) is fully aborted when the level of Blimp-1 (**Figure 5C**) reaches the threshold sufficient to suppress PD-1 expression initiated by TCR activation. We interpret the top (left) plateau in the level of PD-1 (**Figure 5A**) as corresponding to the homeostasis maintained by both the PD-1 DIFFL and the negative feedback activation of TCR which we discuss shortly below. At the same time the bottom (right) plateau in the level of PD-1 (**Figure 5A**) can be interpreted as an adaptation to high levels of Ag (3), a direct consequence of adaptive properties of IFFLs (6, 77– 82).

We further observe that in complete agreement with the theory of IFFLs demonstrating biphasic steady-state behavior (6, 77, 78), the levels of Blimp-1 and IRF4 first increase and then decrease, and, at the same time, the level of Bcl-6 first decreases and then increases, while the level of Ag is constantly increased. Remarkably, the levels of all the three species almost perfectly adapt to their respective original states formed initially at very low levels of Ag, when the level of Ag becomes high enough to establish adaptation. A similar adaptive phenotype is discussed using an example of a generalized enzyme network in Chiang et al. (79).

Consider now the case of large values of koff from the set of the values given in the legend of **Figure 5**. In this case, the response of the PD-1 DIFFL becomes abnormal, when all remarkable adaptive properties are completely lost. Even in the case of a very large value of koff, the model predicts a tonic expression of PD-1 corresponding to very small nonzero values coded in black color in **Figure 5A**. We believe that this tonic expression of small PD-1 levels can be attributed to the immune tolerance discussed in section SI-1.

FIGURE 3 | The PD-1 DIFFL motif in the context of complex influenza-tumor interactions. (A) Shows the PD-1 DIFFL response in an anti-influenza CD8+ T cell in the infected lung. (B) Shows the response of the PD-1 DIFFL circuit in an anti-tumor CD8+ T cell in the TME. (C) Shows the PD-1 DIFFL response in an anti-tumor CD8+ T cell in the influenza-infected lung. (D) Shows the PD-1 DIFFL response in an anti-tumor CD8+ T cell in the influenza-infected lung after PD-1 blockade. Gray color corresponds to weak or disabled reactions shaped by the given inflammation context. Symbol + inside a circle in (C) shows the additional PD-1 activation route initiated by external cytokines in the case when the Blimp-1 mediated repression of PD-1 expression is absent. This route does not play any significant role in the case when the expression of PD-1 is suppressed by active Blimp-1 as in (A). Arrows denote activation, and barred lines denote repression. The abbreviation APCs stands for (influenza) Antigen Presenting Cells.

correspond to *<sup>k</sup>*off <sup>=</sup> <sup>10</sup>−<sup>4</sup> , 2.03 × 10−<sup>4</sup> , 4.13 × 10−<sup>4</sup> , and 5.88 × 10−<sup>4</sup> , respectively. Two shades of blue color correspond to *<sup>k</sup>*off <sup>=</sup> 2.43 <sup>×</sup> <sup>10</sup>−<sup>3</sup> and 7.01 × 10−<sup>3</sup> , respectively. Four (top-down) shades of purple color correspond to *<sup>k</sup>*off <sup>=</sup> 2.03 <sup>×</sup> <sup>10</sup>−<sup>2</sup> , 4.13 × 10−<sup>2</sup> , 5.88 × 10−<sup>2</sup> , and 8.38 × 10−<sup>2</sup> . Magenta color corresponds to *k*off = 1.0. Black color corresponds to *k*off = 49.24. (A–D) Correspond to the levels of four species, PD-1, Blimp-1, Bcl-6, and IRF4, computed from the model developed in SI2, respectively.

To better see the role of IFR-4 and its impact on the level of PD-1, we then completely disabled IRF4 by setting the value of the parameter k<sup>b</sup> to zero, k<sup>b</sup> = 0 in the Equation (SI-3.1d). This computational experiment can be thought of as an "in silico IRF4-knockout." The corresponding plot of PD-1 levels against the Ag-concentration is shown in **Figure 6A**.

Surprisingly, the shapes of all PD-1 level plots obtained for the same set of koff values as in **Figure 5** are preserved, and only the magnitudes of the corresponding levels are changed by a factor of 40 or more.

Motivated by these computational predictions, we checked if IRF4 knockout results were previously reported in the literature and found that irf4-deficient CD4+ T cells display increased expression of PD-1 associated with T cell dysfunction (83, 84). However, the role of IRF4 is still poorly understood as it can be completely opposite in the cases of acute and chronic infections (83, 85).

The second interesting observation (**Figure 6B**) is that while the PD-1 DIFFL regulatory function is lost due to in silico knockout of IRF4, the adaptation of PD-1 expression with respect to Ag levels (**Figure 6A**) is still preserved by the negative feedback regulation of TCR activity (**Figure 6B**) (5, 86–88). Both the TCR activation and the negative feedback are interpreted as another IFFL in Lever et al. (74). Collectively, we can thus conclude that the PD-1 transcription and its adaptation to high levels of antigen is regulated by multiple incoherent feed-forward loops.

#### 2.5.2. Modeling PD-1 Expression in the Presence of PD-L1

We observe that in the presence of PD-1:PD-L1 interactions, the maximum levels of PD-1 and Bcl-6 increase (by a factor of 6.75 and 7.86, respectively, but, of course, these numbers are only meaningful in our model and with the parameters used, and they do not have biological significance) (**Figure 7**). At the same time, the levels for Blimp-1 and IRF4 are negligibly small, which allows us to interpret that the transcription of these two species is almost fully suppressed (**Figure 7**).

From our comparison of the PD-1 level plots in **Figures 5**, **7**, we can conclude that the PD-1:PD-L1 interaction plays the role of an amplifier of transient activation of PD-1 transcription, initiated by the ligation of TCR with Ag presented with an MHC (section SI-1).

PD-1:PD-L1 interactions may terminate signal transduction pathways, including those pathways that lead to the activation of IRF4 and Blimp-1, by recruiting phosphatases (68, 89, 90).

Our last computational experiment compares quantitatively the PD-1 level on the surface of an anti-melanoma CD8+ T cell shunted to the lung with the PD-1 level on the surface of an anti-influenza CD8+ T cell in the lung under the same conditions.

To conduct the computational experiment, the following conditions were taken into consideration: (i) the absence of distant tumor Ag in the lung, leading to the shutting down of the TCR signal (U = 0 in the Equations (SI-2.1a–d), (ii)

the abundance of inflammatory cytokines, including TNFα and IFNγ , known to induce the expression of both PD-1 and PD-L1 (SI-1), and (iii) the abundance of IL-2, which induces Blimp-1 (SI-1).

To account for the abundance of the lumped TNFα and IFNγ species, we have replaced the rate constant σ<sup>p</sup> in the Equation (SI-2.1b) by the rate expression (SI-2.6). To account for the abundance of IL-2 in the lung compartment, we have increased the value of the parameter a<sup>b</sup> by a factor γ in Equation (SI-2.1d). In this case, we assumed that IL-2 was secreted by activated T cells (50) and, hence, IL-2 affected Blimp-1 expression through autocrine and paracrine signaling, depending on the TCR activation strength.

In the case when the value of the parameter γ was set to one, the level of PD-1 was increased by a factor of 6 compared with the maximum level of PD-1 shown in **Figure 7** for both antiinfluenza and anti-melanoma cases. So, we can conclude that just the PD-1 DIFFL alone is not enough to counteract the effect of the pro-inflammatory cytokines. Only when a "strong action of IL-2" was taken into consideration by setting γ > 5,000, the level of PD-1 was suppressed for anti-influenza T cells.

### 3. DISCUSSION

Below we discuss our modeling studies conducted in order to complement our immunobiochemical reconstruction toward a better understanding of the previously unrecognized acute non-oncogenic infection factor (1). We then discuss potential implications of our research to further stimulate ongoing efforts toward developing and improving physiological and functional cure approaches based on the host's ability to eliminate non-self foreign invaders and, at the same time, the host's inability to install strong altered-self (cancer) responses (2).

#### 3.1. What We Learn From the Model

Our PD-1 DIFFL reconstruction (**Figure 2**), when combined with the mathematical modeling (**Figures 5**, **7**), suggests that it is the loss of Ag dose-dependent adaptation of the expression of PD-1 receptors in the anti-tumor CD8+ T cells that could be one of major factors resulting in the multiple effects in the presence of acute non-oncogenic infection (1). Specifically, in the case of acute infection, the level of PD-1 receptors on the surface of Agexperienced anti-infection CD8+ T cells first increases and then decreases to lower levels in the course of the virus replication (**Figure 8B**), the hallmark of a fundamental biological adaptation (3). Therefore, based on the discussion around **Figure 3**, we can conclude that chances that the cells with the phenotype shown in **Figure 8B** will loose their motility due to PD-1:PD-L1 interactions in the infected lung are low (**Figure 4**).

In contrast, in the case of Ag-experienced anti-tumor CD8+ T cells, due to the much smaller levels of tumor antigens presented with MHCs in the TME, the strength of the TCR signal in anti-tumor CD8+ T cells may not be enough to activate Blimp-1 and IRF4 species to suppress PD-1 expression (**Figures 2**, **3**). The lack of the expression of Blimp-1 in melanoma is known experimentally (38). As a result, chances that T cells bearing large numbers of PD-1 receptors (**Figure 8A**) will be paralyzed in the infected lung due to PD-1:PD-L1 interactions are high.

Importantly, the higher levels of PD-1 receptors on antimelanoma CD8+ T cells compared with much lesser levels of PD-1 receptors on anti-influenza CD8+ T cells co-localized in the same infected lung were observed in Kohlhapp et al. (1). This supports the two different phenotypes shown in **Figures 8A,B**, respectively.

Our quantitative estimates obtained from the model (**Figures 5**, **7**) show that the Ag level should be increased by several orders of magnitude required to move the Agexperienced T cell from phenotype (A) to phenotype (B) (**Figure 5**). This means that at least a 1000-fold increase in cognate Ag levels (**Figures 5**, **7**) may be required for the adaptation of PD-1 expression to strong antigen-mediated stimulation.

Although more research into the novel adaptation effect illuminated by our model as well as into the lymph motion (93, 94) and molecular mechanisms by which cells are rapidly moved with the blood (95) is undoubtedly needed, we believe that it is worth providing some "biological" numbers that support

our findings. For example, for the LCMV system, a gold standard for infectious biology, the virus titer was increased by factor about 10<sup>3</sup> from day 2 to day 5 (92, Figure 4.4). We digitized the corresponding data points and plotted them in **Figure 8** next to **Figure 8B**. Similar data are reported for influenza A infection (96, 97). The examples of the population measurements are well translated to our modeling studies because in all cases we use dimensionless ratios of the corresponding concentrations.

Of course, one also needs to make sure whether a T cell would be capable to provide a large number of TCRs sufficient to accommodate the above huge increase in Ag-levels. Indeed, the typical number of TCR molecules is estimated in the range of 3 × 10<sup>4</sup> (98), which is a reasonable number to match up with the model-suggested transition from phenotype (A) to phenotype (B) shown in **Figure 8**. At the same time it is highly unlikely for tumor cells to divide as fast as the viruses do to build enough antigen that would be sufficient to change phenotype (A) to

digitized from the corresponding plots in Kuznetsov et al. (91) and Bocharov et al. (92), respectively. Comparing (C,D), we observe that the changes in the tumor Ag levels within the first 7 days are small, corresponding to the fold change less than 10 as seen from (C). At the same time, the viral Ag levels change significantly, corresponding to the 10<sup>4</sup> -fold increase during the first seven days as seen from (D). The small 7-day tumor Ag-level increase shown in (C) corresponds to the red solid "snapshot" circle in (A), while the large 7-day increase in the viral Ag level shown in (D) corresponds to the green solid "snapshot" circle in (B). Additional detailed explanations of (A–D) are provided in the main text.

phenotype (B) within a few days. Indeed, the doubling time for virus particles can be 43–65 min (99), while the doubling time for malignant mouse melanoma B16 cells may take up to 2.8 days or longer (100, 101).

To support the above argument, we note that (91) uses experimental data where the number of tumor cells is increased by factor about 10<sup>2</sup> in the time span of just 40 days. We digitized the corresponding data points and plotted them in **Figure 8** next to **Figure 8A**. We can thus conclude that due to our modeling estimations (**Figures 5**, **7**), such a slow increase in Ag levels may not be enough to change between the prototypes shown in **Figure 8** for short periods of time (days), when acute infection develops and is cleared (1). Similar data can be learned from other independent studies (102).

Note that the discussed transient elevation of PD-1 receptor levels as function of antigen, **Figures 8A,B**, was experimentally observed and was also used as a "window of opportunity" in the context of the combined radiotherapy (RT) and anti-PD-1:PD-L1 treatments (103). Our theoretical work provides additional valuable insight into, and add in the development of combined RT/anit-PD-1:PD-L1 therapy.

# 3.2. Harnessing Anti-infection and Anti-bacterial Responses Against Cancer

By addressing the "the previously unrecognized acute nononcogenic infection factor" revealed through systematically collected heterogeneous experimental data encompassing different pathogens and tumor types (1), we have suggested and discussed concrete molecular mechanisms which allowed us to delineate inherently weak anti-cancer (i.e., altered-self) immune responses from inherently strong anti-infection (i.e., non-self, foreign) responses, including co-infections.

Our findings may thus have potential clinical relevance particularly in the context of ever-expanding immunotherapy efforts and FDA approvals involving PD-1/PD-L1 axis immune checkpoint blockade. Two relevant scenarios to consider, include (1) that patients with cancer treated with such blockade may also be experiencing a concomitant diagnosed or sub-clinical undiagnosed infection in a tissue distant to their tumor, and (2) that selective patients with cancer are being treated with oncolytic viruses (OVs), which preferentially infect tumor cells, but can also infect cells in tissues distant to their tumor (104, 105). In both scenarios, checkpoint blockade may have less-recognized effects discussed here (e.g., releasing the T cell motility paralysis caused by an infection in a tissue distant to the tumor) and thus such blockade may improve patient outcomes, including in the context of combination with OVs (106). As additional clinical information is collected from patients receiving checkpoint blockade (including about infection status and OV viral loads in non-injected sites), future efforts may provide the data necessary to reveal and model this blockade effect further.

We conclude this work with a hope that our theoretic analysis of the newly discovered infection-tumor interaction (1), made by combining solid immunobiochemical reconstruction with appropriate mathematical modeling may also be useful in current developments of both "physiological" and "functional cures" (2). Specifically, our mechanistic molecular-based analysis of the novel immunologic phenomenon uncovers important competing push-pull processes fundamentally inherent in immunity (3–5). We believe that the results reported may have broader implication toward developing (i) physiological

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cure approaches in order to completely eliminate tumors as it happens in the case of rapid (one week long) clearance of acute infection, and, alternatively, toward undertaking (ii) functional cure treatments to maintain long-term immunologic control as in the cases of controlled chronic infection and other disorders as, for example, hypertension (7). However, research (1) clearly suggests that all such cures must be developed with care.

### AUTHOR CONTRIBUTIONS

AZ and ES conceived the work. AZ contributed data. EN and ES designed research and wrote the manuscript. EN analyzed the data and performed the research.

# FUNDING

Partially supported by grants AFOSR FA9550-14-1-0060 and NSF 1817936.

#### SUPPLEMENTARY MATERIAL

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fimmu. 2019.00004/full#supplementary-material


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**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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# Comparative Assessment of Aspergillosis by Virtual Infection Modeling in Murine and Human Lung

Marco Blickensdorf 1,2, Sandra Timme1,2 and Marc Thilo Figge1,2 \*

<sup>1</sup> Research Group Applied Systems Biology, Leibniz Institute for Natural Product Research and Infection Biology-Hans Knöll Institute, Jena, Germany, <sup>2</sup> Faculty of Biological Sciences, Friedrich Schiller University of Jena, Jena, Germany

Aspergillus fumigatus is a ubiquitous opportunistic fungal pathogen that can cause severe infections in immunocompromised patients. Conidia that reach the lower respiratory tract are confronted with alveolar macrophages, which are the resident phagocytic cells, constituting the first line of defense. If not efficiently removed in time, A. fumigatus conidia can germinate causing severe infections associated with high mortality rates. Mice are the most extensively used model organism in research on A. fumigatus infections. However, in addition to structural differences in the lung physiology of mice and the human host, applied infection doses in animal experiments are typically orders of magnitude larger compared to the daily inhalation doses of humans. The influence of these factors, which must be taken into account in a quantitative comparison and knowledge transfer from mice to humans, is difficult to measure since in vivo live cell imaging of the infection dynamics under physiological conditions is currently not possible. In the present study, we compare A. fumigatus infection in mice and humans by virtual infection modeling using a hybrid agent-based model that accounts for the respective lung physiology and the impact of a wide range of infection doses on the spatial infection dynamics. Our computer simulations enable comparative quantification of A. fumigatus infection clearance in the two hosts to elucidate (i) the complex interplay between alveolar morphometry and the fungal burden and (ii) the dynamics of infection clearance, which for realistic fungal burdens is found to be more efficiently realized in mice compared to humans.

#### *Edited by:*

Burkhard Ludewig, Kantonsspital St. Gallen, Switzerland

#### *Reviewed by:*

Lalit Kumar Dubey, Université de Lausanne, Switzerland Joana Vitte, Aix-Marseille Université, France

> *\*Correspondence:* Marc Thilo Figge thilo.figge@leibniz-hki.de

#### *Specialty section:*

This article was submitted to Molecular Innate Immunity, a section of the journal Frontiers in Immunology

*Received:* 25 October 2018 *Accepted:* 17 January 2019 *Published:* 05 February 2019

#### *Citation:*

Blickensdorf M, Timme S and Figge MT (2019) Comparative Assessment of Aspergillosis by Virtual Infection Modeling in Murine and Human Lung. Front. Immunol. 10:142. doi: 10.3389/fimmu.2019.00142 Keywords: virtual infection modeling, *Aspergillus fumigatus* lung infection, mouse model, human model, hybrid agent-based computer simulations

#### INTRODUCTION

The concept of systems biology constitutes a powerful approach to investigate biological phenomena by combining wet-lab and dry-lab investigations that mutually support and complement each other (1–3). However, systems biology of infection faces problems that can interrupt the experiment-theory-cycle of systems biology (4–6). First, since in vivo experiments are predominantly conducted in animals, the general transferability of findings in the context of immunology to the human system is a matter of ongoing dispute (7, 8). Secondly, even in animal experiments it may be impossible to capture the spatio-temporal dynamics of infection processes. For example, in the case for lung infection in vivo time-lapse imaging is challenging due to animal breathing. In these cases, virtual infection modeling is of particular importance, since it has the potential to advance our knowledge despite the aforementioned limitations and to generate hypotheses that direct future experiments in a targeted manner (9, 10). In particular, building in silico models of infection on the available experimental data basis, gives rise to realistic to-scale models that can be used to compare the outcome of computer simulations for animal and human systems.

In this study, we use virtual infection modeling to investigate Aspergillus fumigatus lung infections. A. fumigatus is an environmentally wide-spread fungus that is an opportunistic pathogen causing severe infections in immunocompromised patients (11–14). The fungal conidia are small in size of 2–3µm (12, 13) and can reach the alveoli in the lower respiratory tract of the lung. Because alveoli make up about 50% of the lung volume and also make the largest contribution to lung surface area, they are by far the most likely niche for infection (15). If not efficiently removed by the innate immune system, A. fumigatus can cause invasive pulmonary aspergillosis (IPA) with high mortality rates of 30–90% (11). The resident immune cells in the lung are alveolar macrophages (AM) that constitute the first line of immune defense by phagocytosing the inhaled conidia (11, 14, 16). Without efficient clearing by innate immunity, A. fumigatus conidia can undergo morphological changes: Upon contact to the surfactant layer, which covers the alveolar epithelial cells (AEC) (15), resting conidia can swell and after ∼6 h start forming hyphae. These hyphae are able to penetrate the epithelial tissue of the alveolus and can thereby reach the bloodstream, from where they may disseminate and cause severe systemic infections (12, 13, 17). The first six hours after entrance of the conidia in the lung are therefore considered as a critical time frame, during which conidia need to be found in order to prevent damage of host tissue. This implies that the role of adaptive immunity can be neglected compared to a required rapid response by innate immunity, e.g., involving the complement system as well as phagocytic activity by AM and neutrophils. The condition of neutropenia, i.e., the considerable reduction in the absolute neutrophil count, poses a major risk factor for IPA (14, 18). Therefore, the nowadays increasing number of immunocompromised patients leads to a rising clinical prevalence, making A. fumigatus a relevant target for fungal infection research. Due to its complex interactions with the host immune system and its ability to adopt different morphologies, various levels of pathogenicity have to be considered in the development of effective therapy (13, 19).

Various mammalian species have been used for experimental research on A. fumigatus infection. Besides rats, rabbits, and guinea pigs, mice models have been used most extensively (20). It is important to note that—in order to provoke measurable numbers of interactions between pathogens and host cells—the experimentally applied infection doses typically are orders of magnitude higher compared to the natural inhalation dose for humans, which ranges between a few hundred and thousands of conidia per day (21–25). Thus, in addition to studying animal systems with host environments that are quite different from the human system, the significant differences in the applied infection doses need as well to be taken into consideration in the knowledge transfer from animals to humans. However, little is known about the comparability and transferability of mouse infection models in wet-lab and natural A. fumigatus infections in human. Therefore, in this study we compare A. fumigatus infection in mice and humans using virtual infection modeling to account for the respective lung morphologies and study the impact of the infection doses. In passing we note that, even though daily inhalation doses will be associated with homeostatic clearance and will typically pass unnoticed, we here use throughout the more general term infection clearance involving inflammation, tissue damage and a multifactorial host response in the case of high fungal doses.

In previous studies, we already implemented an infection model for the simulation of A. fumigatus infection in humans. The agent-based model (ABM) was built on an extensive experimental data basis available from literature and represents a typical human alveolus in three-dimensional continuous space (26, 27). The human alveolus was composed of AEC of type I and II, as well as of Pores of Kohn (PoK) representing connections between neighboring alveoli (28, 29). Our computer simulations revealed that AM performing random walk migration are not able to reliably detect a conidium in the alveolus before the onset of germination, i.e., before 6 h post infection (17, 26). This led to the hypothesis that a not yet experimentally identified chemotactic signal must exist that guides AM to the position of the conidium in the alveolus (26). The virtual infection model was then extended to explicitly incorporate chemokine secretion and diffusion by solving partial differential equations in a hybrid ABM (27). Scanning all unknown parameters within reasonable ranges, we determined those relevant for efficient pathogen clearance. For example, we found that a preferably high ratio of chemokine secretion by AEC with rate sAEC over chemokine diffusion with diffusion coefficient D is required to establish a chemokine gradient that facilitates AM to detect a conidium before the onset of germination.

While these studies considered the immune response in human alveoli for daily inhalation doses of A. fumigatus conidia, the focus of the present study is on comparing A. fumigatus infections in mice and humans taking into account natural as well as experimental infection doses. Thus, we significantly adapted the agent-based virtual infection model to the toscale morphometry of mouse alveoli. This enables generating comparative and quantitative predictions on the influence of morphological factors as well as dose-dependent effects during A. fumigatus infection in mice and humans.

# RESULTS

Aspergillus fumigatus lung infection is commonly investigated using mouse models (20), where the pathogens can be administered in different ways (30): Intranasal deposition and intra-tracheal/intra-bronchial instillation bring the conidia directly in the nose or trachea/bronchia and are based on liquid solutions, while a more natural administration is realized in inhalation chambers with air-soluted conidia. All methods have in common that relatively high doses of 10<sup>6</sup> − 10<sup>8</sup> conidia are applied; however, the amount of conidia which is actually reaching the lower respiratory tract, i.e., the fungal burden in the alveoli of the mouse lung is found to be in the range of 10<sup>3</sup> − 10<sup>5</sup> conidia (31, 32). On the other hand, it is reported that the distribution of conidia is fairly uniform only for administration by inhalation, whereas intranasal administration is accompanied with the accumulation of conidia in specific lung sections, i.e., inducing distributions with local variations in the fungal burden (33). This implies that our in silico experiments need to incorporate three major issues that differ from simulations of the human infection scenario: (i) implementing the differences in the morphometry of the lung for human and mouse, (ii) scanning for a larger range of infection doses, and (iii) studying the limit of high local fungal burdens due to the non-uniform distribution of conidia for administration based on liquid solutions.

As a measure of fungal clearance, we introduced an infection score ISs=H,M, where the superscript refers to the human (s = H) or mouse (s = M) system and ISs=H,<sup>M</sup> = 0 (ISs=H,<sup>M</sup> = 1) implies that infections were cleared in each (none) simulations (for details see Materials and Methods section, Readout of Simulations).

#### Putative Morphology-Related Impact on Infection Clearance in Humans and Mice

As can be seen in **Figure 1**, the alveoli for human and mouse have been implemented as to-scale models that are composed of AEC of type I and II, as well as PoK. Given the differences in the size and composition of alveoli for the two organisms (see **Table 1** and **Supplementary Table 1**), it can be expected that infections may be cleared with different efficiency. For example, the surface area of the human alveolus is about 20 times larger compared to that of the murine alveolus and the number of AM per alveolus is about 6 times higher in the human alveoli. This gives rise to a scanning area per AM, which is about three times higher in humans suggesting that mice could cope much better with the detection of alveolar pathogens. However, the situation is complicated by the fact the number of PoK per alveolar area is higher by a factor 5.7 in the mouse alveolus, which together with the alveolar entrance ring gives rise to an increase of the relative alveolus' open boundary length by a factor 3.4 compared to the human alveolus. On the one hand, since AM can enter and leave the alveolus only across these boundaries (28, 29), this may result in a faster infection dynamics of the murine system. On the other hand, chemotactic signaling molecules can as well flow out of the alveolus via these boundaries implying that their increased length in the murine alveolus may be of disadvantage with regard to establishing an efficient chemokine gradient. Again, this argument may only be valid for a low pathogen density in the alveolus, because for high pathogen densities the induced chemokine profile may

TABLE 1 | Comparison of morphometric parameters and innate immune cells.


The parameters of the human alveolus were taken from the literature search by Pollmächer et al. (26) and those of the mouse alveolus have been retrieved from the indicated references.

FIGURE 1 | Visualization of a to-scale alveolus in the hybrid agent-based model for mouse (A) and human (B). The alveolar entrance ring (left) and Pores of Kohn (black) represent entry/exit points for AM (green) and chemokine flow (white isolines) induced by conidium (red). Alveolar surface is covered with epithelial cells of type 1 (yellow) and 2 (blue).

provide an ambiguous signal for AM guidance. For the same fungal burden in mice and humans, the pathogen density is much higher in the murine alveolus, due to their much lower number and smaller size. Therefore, A. fumigatus may be much more efficiently cleared from the human lung. Taken together, these considerations imply that the efficiency of the infection dynamics will depend on the combination of the alveolar morphometry and the fungal burden that together impact on the chemokine profile for AM migration in a way, which is impossible to quantitatively predict without performing comparative computer simulations of to-scale models.

## Case of Low Fungal Burden: *A. fumigatus* Infection More Efficiently Cleared in Mice

We first consider the case of low fungal burden, which we define as the case where one A. fumigatus conidium per alveolus is the highest alveolar occupation number (AON) that is statistically expected to occur in the whole lung. The corresponding fungal burden can be derived from the binomial distribution (see Methods section for details) and is 2.5 × 10<sup>3</sup> in mice and 3 × 10<sup>4</sup> in humans (see **Figure 2**). This implies that the limit of low fungal burden covers the dose of daily inhalation for humans, but is relatively low for experimental conditions in typical mice experiments. Examples of the infection dynamics can be seen for humans and mice in **Supplementary Videos 1, 2**, respectively.

Our previous work on A. fumigatus infection in human alveoli for low fungal burden revealed that a high secretion rate sAEC of chemotactic molecules combined with a low diffusion coefficient D of the chemokine is beneficial for a small infection

burden, which is reached in typical mice model experiment.

score IS<sup>H</sup> in humans (27). In the present study, we screened the diffusion coefficient and the secretion rate in the regimes, respectively, D = - 20, 6 × 10<sup>3</sup> µm<sup>2</sup> / min and sAEC = - 1.5 × 10<sup>3</sup> , 5 × 10<sup>5</sup> min−<sup>1</sup> for alveoli of mice and humans. The numerical results for the quantitative comparison between human and mouse is shown by the infection scores ISH,<sup>M</sup> in **Figure 3A**. It can be observed that, for all combinations of D and sAEC, the infection score in mice is significantly smaller: IS<sup>M</sup> < ISH. Furthermore, it can be seen that the relation of a high secretion rate and a low diffusion coefficient also leads to a more efficient infection clearance in mice. The relative difference in the infection scores of the two organisms, 1IS = 1 − ISM/ ISH, is in the range 50 − 90 %, indicating that the murine system performs always better than the human system in the limit of a low fungal burden.

#### Case of Low Fungal Burden: Size of Alveolus Governs Infection Dynamics

To dissect whether the infection dynamics is governed by the chemotaxis or the alveolar size, we compared the probability of directed AM migration resulting from one conidium in the alveolus of mice and humans. The chemokine concentration itself falls off with the distance from the source AEC, i.e., the AEC in contact with the conidium. In order to avoid that AM perform mostly random walk migration, the chemokine gradient (i) must not exceed a certain value to avoid saturation of AM chemokine receptors and (ii) must not fall below a certain value to provide a detectable signal. As a qualitative measure of gradient efficiency we calculated the probability that AM follow the gradient depending on the distance to the source AEC. This probability reflects the impact of the chemokine gradient on AM migration and was computed as explained in **Supplemantary Methods** (see section on AM Migration) for optimal chemokine parameters (D s opt, s s AECopt ) in the human (s=H) and mouse (s=M) system. The optimal parameters were computed from the 36 scanned parameter combinations, {D<sup>1</sup> . . . D6} × {sAEC<sup>1</sup> . . .sAEC<sup>6</sup> }, for the diffusion coefficient and the secretion rate as follows: Based on the simulation results in terms of the infection score ISD<sup>i</sup> ,sAECi and the limits of its respective 95%-confidence interval, we computed the optimal diffusion coefficient as Dopt = <sup>P</sup> 1 <sup>i</sup> w<sup>i</sup> · P <sup>i</sup> w<sup>i</sup> ·D<sup>i</sup> with weights w<sup>i</sup> = 1 − ISD<sup>i</sup> ,sAECi for all those parameter combinations that had infection scores not exceeding the minimal upper limit of all confidence intervals (see **Supplementary Video 3**). The optimal secretion rate sAECopt was determined in the same way yielding for the human host Dopt <sup>H</sup> = 34µm<sup>2</sup> min−<sup>1</sup> and s H AECopt = 1.5 × 10<sup>4</sup> min−<sup>1</sup> and for the murine host Dopt <sup>M</sup> = 61µm<sup>2</sup> min−<sup>1</sup> and s M AECopt = 4.9 × 10<sup>4</sup> min−<sup>1</sup> as the optimal parameters in the limit of low fungal burden.

The probability of directed AM migration for both host systems and for their respective optimal chemokine parameters is plotted in **Figure 3B**. The two curves exhibit quantitative similarity suggesting that the infection dynamics in the case of a low fungal burden is mainly governed by the size of the alveolus rather than the chemokine profile itself. Thus, in contrast to the significantly larger human alveolus, AM in the murine

counterpart will typically perform directed migration across the entire alveolus.

## *A. fumigatus* More Efficiently Cleared in Mice for Any Alveolar Occupation Number

Increasing the AON from one to higher conidia numbers, we again performed computer simulations for various infection scenarios that differ in the parameters for chemokine secretion sAEC and diffusion coefficient D. However, multiple conidia within the alveolus can lead to more complex chemokine profiles derived from the various conidia-associated AEC that are simultaneously serving as sources of chemokine secretion. In **Figure 4A** the infection scores IS obtained from 10<sup>3</sup> simulations are summarized for AON between one and six and for selected secretion rates sAEC, while the numerical results for the full range of studied parameter values is shown for human and mouse in **Supplementary Figure 1**. Parameter regimes of efficient infection clearance in these plots resemble those previously found for one conidium in the human alveolus (27), indicating that low ratios D/sAEC are as well preferred in the mouse system.

Extending the computation of optimal chemokine parameters for one conidium to larger AON enables computing for both systems the average optimal parameter set (see **Supplementary Figure 2**). We obtain for one to six conidia per alveolus the averaged optimal values Dopt <sup>H</sup> = 26±6.6 µm2min−<sup>1</sup> and s H AECopt = 1.1 × 10<sup>4</sup> ± 6 × 10<sup>3</sup> min−<sup>1</sup> for the human host and Dopt <sup>M</sup> = 74 ± 22.4 µm2min−<sup>1</sup> and s M AECopt = 8.0 × 10<sup>4</sup> ± 4, 1 × 10<sup>4</sup> min−<sup>1</sup> for the murine host. In **Figure 4B**, we show that the resulting infection score IS as a function of the AON is always significantly lower in mice compared to humans.

# Case of High Fungal Burden: Chemokine Profile Can Deteriorate Clearance Efficiency

Due to morphometric differences between the lungs of mice and humans, the AON is not directly related to the fungal burden. This follows from our earlier statistical considerations on the highest AON that is expected to occur in the whole lung for a given fungal burden (see **Figure 2**) exhibiting a significant quantitative difference between mice and humans. Since the number of more than 10<sup>8</sup> alveoli in the human lung exceeds that of mice by more than two orders of magnitude, even in the case of an extremely high fungal burden with 10<sup>6</sup> conidia in the lung, the maximal AON for humans does not exceed two. In contrast, the same fungal burden in the lung of mice yields a maximal AON between five and six conidia in one alveolus. It thus follows that a comparison between mice and humans for the same fungal burden requires contrasting infection scenarios with different AON. Of note, our analysis focuses on the maximal AON for a given fungal burden, because it is argued that this configuration will be directly correlated with the estimated time needed to clear all occupied alveoli from the pathogen. In **Figures 4C,D** the numerical results for the infection score IS are shown for mice and humans as a function of the fungal burden, respectively, for identical chemokine parameters and for the respective optimal chemokine parameters. **Supplementary Figure 3** shows the infection score IS as a function of the fungal burden for all the scanned parameter combinations. It can be seen by the smaller infection scores in the murine host that infections are still more efficiently cleared for the entire experimentally relevant range of 103−10<sup>5</sup> conidia in the lung. In **Supplementary Video 3** we indicated all combinations of chemokine parameters for which the infection score reaches values below the threshold of IS<sup>t</sup> = 5%.

D = 200 µm<sup>2</sup> min−<sup>1</sup> (A,C) and for optimal chemokine parameters Dopt and sAECopt (B,D) in mice and men. Dashed-dotted black line indicates the threshold infection score at IS<sup>t</sup> = 5%. Error bars represent 95% confidence intervals. Black line represents the experimental range of fungal burden, which is reached in typical mice model experiment.

However, as we have mentioned before, administration of conidia based on liquid solutions is reported to be associated with higher local fungal burdens due to a more non-uniform distribution of conidia (33). It can be seen in **Figure 2** for a uniform distribution of conidia that a high fungal burden in the range 10<sup>5</sup> − 10<sup>6</sup> conidia per lung is associated with an AON of two in the human system, whereas this value ranges between three and six for the murine system. Consequently, for a nonuniform distribution of conidia, such high AON can be reached in the murine lung and these can result in infection scores that are much higher than for the human system with AON of two, even if the respective optimal chemokine parameters are applied (see **Figure 4B**). Our spatio-temporal computer simulations of the infection scenarios reveal that higher AON are associated with chemokine profiles that deteriorate clearance efficiency. Since the mouse alveolus contains more than 10 times fewer AEC compared to the human alveolus (see **Table 1**), multiple randomly positioned conidia will occupy most of the alveolus' AEC associated with chemokine secretion from various source AEC. First of all, this can lead to chemokine saturation that will turn directed AM migration into random walk migration. Secondly, if the number of conidia is increased further, this will not alter the chemokine gradient anymore. Consequently, AM will perform the inefficient random walk migration until a sufficient number of conidia is detected, such that AM migration becomes again dominated by the chemokine gradient. Obviously, this complex interplay between the morphometry of the alveolus and the chemokine profile will be much less pronounced for the larger human alveolus that consists of many more AEC. To validate this hypothesis, we computed the mean values of the chemokine concentration across all alveolar surface grid points in the simulations and found that significant deviations arise between the human and mouse alveolus starting at AON of four. As can be seen in **Figure 5**, for AON above four the mean concentration value in the murine alveolus does change only slightly providing no additional chemotactic guidance to AM, whereas it is still increasing in the human alveolus and can provide chemotactic guidance associated with lower infection scores IS in the human alveolus and in the limit of fungal burdens well above the typical experimental range.

# Simulation Results Are Qualitatively Robust Against Variations of Model Parameters

In the quantitative comparison of infection scenarios in mice and humans, we so far assumed the same values for the model parameters. For example, we assumed that the chemokine secretion rates from human and murine alveolar epithelial cells are similar. However, it may be argued that this does not reflect the physiological reality correctly, since murine AEC are effectively about 33% smaller in area and may thus exhibit a reduced potential of chemokine secretion. Similarly, it is an open question today whether the postulated chemotactic signals in human and mice are transmitted by chemokines that are structural homologs and can therefore be expected to have similar diffusion coefficients in the surfactants of mice and humans. While these uncertainties cannot be avoided, we estimated the impact of variations in these parameters on the infection score in humans and mice. To this end, we calculated the relative infection score between the human and murine model 1IS = 1−ISM/ ISH, over all simulated parameter combinations in the experimental range of fungal burdens. Setting both diffusion parameters in humans and mice to identical values, the mouse shows lower relative infection scores with a median value of 1IS = 0.49.

Next, we analyzed the robustness of the infection outcome with regard to the diffusion coefficient and the secretion rate. To this end, we compared the infection scores for humans and mice for those simulated parameter combinations that obey the scaling factors f<sup>D</sup> = D <sup>H</sup>/D <sup>M</sup> for the diffusion coefficient and fsAEC = s H AEC/s M AEC for the chemokine secretion rate. For example, comparing diffusion coefficients with scaling factor f<sup>D</sup> = 3 −1 (i.e., D <sup>H</sup> = (20, 200, 2000) µm<sup>2</sup> / min, D <sup>M</sup> = (60, 600, 6000) µm<sup>2</sup> / min) revealed a reduction in the median

value of the relative infection score to 1IS = 0.27 over the scanned fungal burdens. This indicates that the infection score in mice is higher in > 50% of all selected parameter combinations, even if the diffusion coefficient is three times higher in the murine alveolus (see **Figure 6**). The scaling factor of the secretion rate fsAEC has a reversed impact on the relative infection score reflecting that a high ratio sAEC/D induces low infection scores (see **Figure 6**).

Taken together, our simulation results imply that our main conclusions are qualitative robust against variations in the chemotaxis parameters. As long as the associated scaling factors have values f<sup>D</sup> > 10−<sup>1</sup> or fsAEC < 10, the murine system still shows better infection scores in more than half of all screened fungal burdens, even if chemotactic signaling becomes deteriorated. We therefore conclude that within these limits our simulation results are qualitatively robust against variations in the chemotaxis parameters.

### DISCUSSION

In this study, we investigated clearance of Aspergillus fumigatus infection from the lung of mice and humans by computer simulation of the complex interplay between alveolar morphometry and fungal burden in the dynamics of infection clearance. Since in vivo live cell imaging of these processes in the whole lung is still not possible today, we here extended a previously developed model of IPA in humans (25, 26) to the murine alveolus. The virtual infection model represents a realistic to-scale representation that was built on detailed experimental data available on the morphometry of the alveolus in the two hosts. Furthermore, alveolar macrophages as well as chemokine secretion and diffusion were incorporated into the model and we screened the physiologically relevant parameter ranges for as small as possible infection scores IS, which represent the

percentage of simulations for which clearance of all A. fumigatus conidia from the lung took longer than 6 h.

One important finding of this study is that, for realistic fungal burdens comprising daily inhalation doses in humans as well as typical doses in mice experiments, infection clearance is more efficiently realized in mice compared to humans. This result holds true in the limit of low fungal burden, where at most one conidium is present in the alveolus, as well as for larger fungal burdens with a maximal number of two and three conidia, respectively, in the alveolus of humans and mice. As we observed before for the human system (27), a low ratio of chemokine diffusion over secretion, D/sAEC, leads to more efficient infection clearance in the murine system. However, our simulations revealed that in the limit of low fungal burden the dominating factor of efficient infection clearance in mice is the relatively short distances between AM and conidia in the relatively small murine alveolus. On the other hand, the chemokine profile played a dominant role in the limit of high fungal burden, because for four and more conidia in the relatively small murine alveolus this is associated with a featureless chemokine profile that cannot provide sufficient guidance to AM.

A quantitative comparison revealed that distinct optimal chemokine parameters exist that ensure minimal infection scores IS in the different alveoli of the two hosts. We therefore performed simulations comparing the infection results for both identical and optimal chemokine parameters. It should be noted that, even for the same chemotactic molecule in mice and humans, differences between optimal chemokine parameters can be induced by various factors that are different in the two hosts, such as the secretion competence of AEC and the viscosity of the alveolar surfactant. In any case, the importance of a well-established chemokine gradient as well as the functional sensing by AM is reflected by the fact that conidia, which are not detected within 6 h post infection, pose the risk of germination, invasion, and systemic infection. We also studied the case of non-uniform conidia distribution in the lung leading to locally high AON in alveoli. In this limit, which is more likely realized by the administration of conidia based on liquid solutions, our calculations predict that four and more conidia per alveolus can occur, leading to infection scores that are clearly higher in mice than in humans. However, in general, clearance of uniformly distributed conidia in the lung seems to be more efficiently realized in mice than in humans and we have demonstrated that this results are qualitatively robust over a broad range of variations in the chemokine parameters. These considerations are important with regard to the comparability and transferability of mouse infection models to the human system, e.g., with regard to estimating the efficiency of new therapeutics. Virtual infection modeling in the scope of systems biology has been applied to a broad range of biological systems and pathogens, such as bacteria (48) and fungi (9, 10, 49–53), since it provides a valuable tool to investigate infection processes that are not directly accessible in experiment. Moreover, this approach can direct future experiments by identifying key factors that govern the counterplay of infection and inflammation and require most attention. It should be mentioned that our results, indicating that AM are not able to clear the infection in the limit of a high fungal burden, are in line with previous findings based on a more phenomenological modeling approach. We applied evolutionary game theory on graphs to simulate several aspects of the immune response against A. fumigatus lung infection, including the complement system, phagocytosis by AM as well as recruitment and phagocytosis by neutrophils in one comprehensive model framework (54). This enabled us to reconcile the contradictory view on AM in the literature (55, 56) and predicted an infection dose-dependent switch in their function: While under low infection doses AM manage infection clearance, their role switches to a regulatory function under high infection doses by recruiting neutrophils (54).

In the future, validation of theoretical predictions needs to be addressed in experimental investigations. To date, one of the main limiting factors in understanding host response during A. fumigatus infections is the poor experimental accessibility and stable cultivation of alveolar tissue. However, new research approaches including organ-on-a-chip systems, which reduce the physiological complexity and bring nature closer to the simplifying virtual infection models, are promising for a better validation of e.g., alveolar epithelium properties or chemokine parameters (57–59). A lung-on-a-chip model will enable testing chemokine candidates for AM guidance, such as IL-8 that binds to the AM surface receptor CXCR2 (60). Similarly, the chemoattractant C5a is known to be activated by A. fumigatus conidia along the alternative pathway of the complement system (61, 62) and is able to trigger the secretion of macrophage inflammatory protein-2 and neutrophil chemoattractant-1 by AEC (63). Once chemokine parameters will have been identified and inflammatory conditions in terms of cytokine profiles will be accessible, the next step will be to extend the hybrid ABM toward neutrophil recruitment and an explicit phagocytosis model along the lines of our previous investigations based on evolutionary game theory (54). This will allow for the investigation of migration and phagocytic dynamics of AM, neutrophils and AEC in the alveolar environment during the interaction with pathogens. Furthermore, morphological changes of A. fumigatus including swelling and hyphae formation have a strong impact on phagocytosis of the fungus (17, 64) and can be included in such a virtual infection model. A further advancement will be in the scale-up of the alveolus to the higher organizational units of alveolar sacs for a more comprehensive simulation of infection scenarios.

#### MATERIALS AND METHODS

In this study, we extended our previously developed ABM of in silico infections by Aspergillus fumigatus in the human alveolus (26, 27) to the mouse alveolus in order to perform comparative analyses. The ABM is a spatio-temporal multi-scale model that simulates host-pathogen interactions on the cellular and molecular level. Thus, cells like the fungal conidia and AM are simulated as individual agents that migrate and interact in a rule-based fashion, while the chemokine secretion by AEC and the molecular diffusion of chemokines is simulated using partial differential equations. Chemokines are uniformly secreted with rate sAEC at the surface of each AEC, which is associated with at least one conidium. The implementation of the ABM is described in more detail in the **Supplementary Material**, while here the

focus is on the main aspects associated with the extension to the mouse alveolus.

# Morphometry of the Mouse Alveolus and Implementation

A comprehensive literature research was performed to design the virtual infection model of the mouse alveolus as realistic as possible. The most important morphology parameters are summarized and compared with the human alveolus in **Table 1**, from which other characteristics can be derived (see for examples **Supplementary Table 1**). For example, it can be seen that the radius (surface area) of a typical human alveolus is about 4.5 (20) fold larger compared to a murine alveolus. The numbers of AEC of type 1 and 2 differ significantly in both organisms, i.e., a human alveolus contains about 12.0-fold more type 1 and 21-fold more type 2 AEC. Furthermore, the number of PoK is about 3.4 times higher in the human alveolus. A video of both model alveoli is provided in the (**Supplementary Videos 1, 2**).

# Implementation of Mouse Alveolus in Virtual Infection Model

The ABM was adjusted for the implementation of the mouse alveolus with parameters as summarized in **Table 1** and **Supplementary Table 1**. This also required changes in the algorithm for cell positioning on the alveolar surface. Type 1 AEC were placed as described before around the three-quarter sphere (see **Supplementary Material** for details). Previously, type 2 AEC and PoK were placed at the borders between type 1 AEC. However, due to the larger ratio of type 2 AEC and PoK with respect to type 1 AEC in mice, the positioning of PoK and type 2 AEC had to be changed. We adjusted the position of type 2 AEC and PoK uniformly across the whole border of the type 1 AEC. While these changes in the cell positioning were required for realistic configurations of mouse alveolus morphometries, quantitative results of the ABM for the human alveolus remained within the 95%-confidence interval. Moreover, the smaller size of the mouse compared to the human alveolus required adjustment of the Delaunay-triangulated grid, on which the diffusion equation is solved (27). The number of grid points could be reduced from 10<sup>4</sup> in the human alveolus to only 5.1 × 10<sup>2</sup> , keeping the spatial resolution in the mouse alveolus the same as in the human system (see **Supplementary Table 1**).

## Readout of the Simulations

As a measure of fungal clearance we compute for various infection scenarios the first-passage-time (FPT) of AM, i.e., the time required for migrating AM to find all conidia in a particular alveolus (26, 27). The relation between the FPT and the time point of conidia germination, which corresponds to about 6 h post conidia arrival, is obtained from repeating the simulation of each infection scenario 10<sup>3</sup> times. From the corresponding FPT distribution, we then compute an infection score, IS, as the percentage p of simulations with FPT above 6 h: ISs=H,<sup>M</sup> = p(FPT > 6 h), where the superscript refers to the human (s = H) or mouse (s = M) system and ISs=H,<sup>M</sup> = 0 (ISs=H,<sup>M</sup> = 1) implies that conidia were cleared in each (none) of the 10<sup>3</sup> simulations. The various infection scenarios correspond to scanning the parameter space in terms of AM migration, chemokine secretion, and diffusion, as well as conidia infection doses in alveoli of mice and humans.

# Comparison of Fungal Burden

For a given fungal burden δ, the conidia are distributed across all alveoli nalv of the host's lung. Assuming an independent and uniform distribution of these conidia, we can describe the probability of having ncon conidia present in one alveolus by the Binomial distribution Bcon δ, p, ncon with probability of p = 1 nalv for δ repeats. To estimate the maximal AON that is associated with a specific fungal burden, we computed ncon from the 1 − 1 nalv -quantile of the distribution Bcon δ, p, ncon . The resulting number corresponds to the maximal AON that can be expected to occur in the whole lung for a specific fungal burden (see **Figure 2**). The corresponding IS was determined by linear interpolation of the results from our simulations for various AON.

# DATA AVAILABILITY STATEMENT

The raw data supporting the conclusions of this manuscript will be made available by the authors, without undue reservation, to any qualified researcher.

# AUTHOR CONTRIBUTIONS

MTF conceived and designed this study. MTF provided computational resources. Data processing, implementation and application of the computational algorithm were done by MB and ST. MB, ST, and MTF evaluated and analyzed the results of this study. MB, ST, and MTF drafted the manuscript and revised it critically for important intellectual content and final approval of the version to be published. MB, ST, and MTF agree to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved.

# FUNDING

This work was financially supported by the Deutsche Forschungsgemeinschaft (DFG) through the excellence graduate school Jena School for Microbial Communication (JSMC) the International Leibniz Research School for Microbial and Biomolecular Interactions (ILRS) and the CRC/TR124 FungiNet (project B4 to MTF).

# ACKNOWLEDGMENTS

We acknowledge numerous helpful discussions with Johannes Pollmächer.

# SUPPLEMENTARY MATERIAL

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fimmu. 2019.00142/full#supplementary-material

#### REFERENCES


**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2019 Blickensdorf, Timme and Figge. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# Model-Based Assessment of the Role of Uneven Partitioning of Molecular Content on Heterogeneity and Regulation of Differentiation in CD8 T-Cell Immune Responses

#### Edited by:

Simon Girel 1,2

Fabien Crauste1,2

\*, Christophe Arpin<sup>3</sup>

\*

Gennady Bocharov, Institute of Numerical Mathematics (RAS), Russia

#### Reviewed by:

Filippo Castiglione, Italian National Research Council (CNR), Italy Yinghong Hu, Emory University, United States

#### \*Correspondence:

Simon Girel girel@math.univ-lyon1.fr Fabien Crauste crauste@math.univ-lyon1.fr

#### Specialty section:

This article was submitted to Molecular Innate Immunity, a section of the journal Frontiers in Immunology

Received: 05 December 2018 Accepted: 28 January 2019 Published: 19 February 2019

#### Citation:

Girel S, Arpin C, Marvel J, Gandrillon O and Crauste F (2019) Model-Based Assessment of the Role of Uneven Partitioning of Molecular Content on Heterogeneity and Regulation of Differentiation in CD8 T-Cell Immune Responses. Front. Immunol. 10:230. doi: 10.3389/fimmu.2019.00230 <sup>1</sup> Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, Villeurbanne, France, <sup>2</sup> Inria, Villeurbanne, France, <sup>3</sup> CIRI, Centre International de Recherche en Infectiologie, Univ Lyon, Inserm, U111, Université Claude Bernard, Lyon 1, CNRS, UMR5308, ENS de Lyon, Lyon, France, <sup>4</sup> Univ Lyon, ENS de Lyon, Univ Claude Bernard, CNRS UMR 5239, INSERM U1210, Laboratory of Biology and Modelling of the Cell, Lyon, France

, Jacqueline Marvel <sup>3</sup>

, Olivier Gandrillon2,4 and

Activation of naive CD8 T-cells can lead to the generation of multiple effector and memory subsets. Multiple parameters associated with activation conditions are involved in generating this diversity that is associated with heterogeneous molecular contents of activated cells. Although naive cell polarisation upon antigenic stimulation and the resulting asymmetric division are known to be a major source of heterogeneity and cell fate regulation, the consequences of stochastic uneven partitioning of molecular content upon subsequent divisions remain unclear yet. Here we aim at studying the impact of uneven partitioning on molecular-content heterogeneity and then on the immune response dynamics at the cellular level. To do so, we introduce a multiscale mathematical model of the CD8 T-cell immune response in the lymph node. In the model, cells are described as agents evolving and interacting in a 2D environment while a set of differential equations, embedded in each cell, models the regulation of intra and extracellular proteins involved in cell differentiation. Based on the analysis of in silico data at the single cell level, we show that immune response dynamics can be explained by the molecular-content heterogeneity generated by uneven partitioning at cell division. In particular, uneven partitioning acts as a regulator of cell differentiation and induces the emergence of two coexisting sub-populations of cells exhibiting antagonistic fates. We show that the degree of unevenness of molecular partitioning, along all cell divisions, affects the outcome of the immune response and can promote the generation of memory cells.

Keywords: multiscale modeling, immune response, asymmetric division, agent-based models, immune memory

# 1. INTRODUCTION

Following acute infection, the activation of naive CD8 T-cells by antigen presenting cells (APCs) triggers the synthesis of proteins controlling cell proliferation and differentiation up to the memory state. While CD8 T-cell population dynamics have been widely described, it is of great interest to better understand the molecular mechanisms driving the CD8 T-cell response. In particular, determining the effects of molecular events on the generation of memory cells is necessary for vaccine design improvement. In vivo and in vitro studies have demonstrated that a single presentation of the antigen to naive CD8 T-cells is sufficient to trigger a complete CD8 T-cell immune response (1–5). Then, once initiated, antigen-independent molecular pathways drive a program of CD8 T-cell proliferation and differentiation (6, 7).

The CD8 T-cell immune response occurs through four main phases. First the activation of naive CD8 T-cells in secondary lymphoid organs such as lymph nodes (LN) or spleen by APCs through MHC class I antigenic peptide/T-cell receptor (TCR) binding, surface co-receptor/ligands interactions and soluble cytokines secretion. Once activated, CD8 T-cells proliferate quickly during the expansion phase, which expands the initial population by a factor of 10<sup>3</sup> to 10<sup>5</sup> (6, 8). Concomitantly, activated cells differentiate into effector cells, able to kill infected cells through cytotoxicity. At the end of the expansion phase, known as the peak of the response, the CD8 T-cell population begins a contraction phase, where most of the responding cells die yet leaving a quiescent population of cells with strong re-activation potential: the memory cells. The memory cell population survives the contraction phase and may remain for years in the organism (memory phase) to ensure faster and stronger host-protection against subsequent infection by the same pathogen.

The responding effector population is composite and two subsets with antagonistic fates have been described (9): memory precursor effector cells (MPEC) and short-lived effector cells (SLEC), characterised by the expression of two proteins KLRG1 and CD127 (IL-7 receptor). Both MPEC (KLRG1loCD127hi) and SLEC (KLRG1hiCD127lo) express effector features (cytotoxicity, proliferation) but MPEC are capable of differentiation into memory cells while SLEC are destined to die during the contraction phase (9). Thus, CD8 T-cell population dynamics arise from cell phenotypic heterogeneity, itself resulting from molecular-content heterogeneity.

Among the genes, transcription factors and proteins involved in the CD8 T-cell response, some seem to play key roles in the differentiation processes. Transcription factors Tbet and Eomesodermin (Eomes) appear to play critical roles in the acquisition of effector and memory phenotypes. It has been shown that the expression of Tbet induces the development of SLEC and represses the development of MPEC profiles (9– 11). Eomes is not involved in the SLEC vs. MPEC fate choice (12, 13). However, Eomes is necessary for the development of several properties of memory cells [survival, lymph node homing capacities, responsiveness to second infection (11, 12, 14)]. Along the differentiation from effector to memory, the concentration of Tbet in a CD8 T-cell decreases, while the concentration of Eomes increases (11, 15).

Since a unique initial antigenic signal can trigger a complete response, additional mechanisms are necessary to generate the observed molecular-content heterogeneity. Arsenio et al. (16), Chang et al. (17, 18), and Ciocca et al. (19) showed that TCR binding to MHC-class-I peptide-complex results in polarised segregation of proteins in activated CD8 T-cell: some proteins migrate on the TCR side of the T-cell, other migrate on the opposite side. The subsequent division of the activated CD8 Tcell splits the mother cell perpendicularly to the polarisation axis, such that the daughter cell coming from the TCR side (proximal cell) receives more proteins associated to effector lineage, including Tbet, while the other one (distal cell) receives more proteins associated to memory lineage. Asymmetric division of polarised naive CD8 T-cells appears to be one of the major mechanisms regulating CD8 T-cell fate decision.

Nevertheless, the role of asymmetric division of polarised naive cells in the T-cell differentiation process appears to be controversial (20). While there are several evidences for asymmetric division of polarised naive CD8 T-cells (21), it remains uncertain how this polarisation quantitatively depends on the affinity of the TCR for the MHC-class-I peptidecomplex, the duration of the binding, external chemokines and interactions with homotypic CD8 T-cells (21). Since the asymmetric partitioning of Tbet has been evidenced in mice CD8 T-cells, it will be considered hereafter.

Less is known about the partitioning of molecular content in the course of subsequent cell divisions. However, several studies support the hypothesis that when a cell divides, a random, uneven partitioning of the molecular content occurs (22–29). Partitioning of CFSE dye, a cell staining dye used to track cell proliferation through dye dilution, during lymphocyte proliferation has been mathematically studied by Bocharov et al. (23) and Luzyanina et al. (26). Based on comparison with in vitro experimental data, these studies suggest that uneven partitioning, which does not result from cell polarisation, occurs at T-cell division.

We emphasize that the asymmetric first division of naive cells, which goes through an active polarisation of the cell, has to be distinguished from the random partitioning of the molecular content during the subsequent divisions of non-polarised cells, hereafter referred to as uneven partitioning (29).

In a recent work (30), we studied how stochastic uneven molecular partitioning, repeated at each cell division, could regulate the effector vs. memory cell-fate decision in a CD8 T-cell lineage. To do so, we analysed an impulsive differential equation describing the concentration of the protein Tbet in a CD8 T-cell subject to divisions, where impulses were associated with uneven partitioning of Tbet. In this work, high and low Tbet concentrations were associated with effector and memory phenotypes, respectively. We concluded that, for a low degree of unevenness of molecular partitioning, a CD8 T-cell expressing a moderate concentration of Tbet can still generate both memory and effector cells. If the concentration of Tbet in this cell is high or low enough, the phenotype of the cell and its progeny becomes irreversible, with low Tbet-expresser and high Tbet-expresser differentiating in memory or effector cells, respectively. Moreover, our study indicates that the increase in cell cycle length throughout the immune response (31, 32) favours irreversible cell differentiation.

Several works [see (33) and the references therein], focused on modeling molecular mechanisms of the immune response coupled to cell population dynamics. Most of these works involve agent-based models.

Gong et al. (34, 35) developed a two-compartment model to study how the number of dentritic cells and the level of MHCpeptides on their membrane influence the size and composition of T-cell populations. Since they did not model any dynamics at the molecular level, they were limited in studying the molecular origins of cell differentiation and heterogeneity.

Prokopiou et al. (36) and Gao et al. (37) designed a multi-scale agent-based model of the early CD8 T-cell immune response (Day 3–5.5 post-infection). At the population scale, a discrete population of CD8 T-cells and APCs in a LN is modeled by a cellular Potts model (CPM) (38). At the molecular scale, the dynamics of a simplified molecular regulatory network (MRN) containing some key molecular factors is modeled by a system of differential equations, embedded in each cell of the population, whose state determines cell phenotype and fate. Cells communicate with each other through cell-cell contact and secretion of the cytokine IL2 such that the environment of a cell affects its molecular profile. Parameter calibration resulted in good agreement with in vivo data of an immune response in murine LN after influenza infection, at both cellular and molecular levels.

The model presented in this article has been developed from the multi-scale agent-based model previously introduced in Prokopiou et al. (36) and Gao et al.(37). Since the authors in Prokopiou et al. (36) and Gao et al. (37) focused on early events following CD8 T-cell activation, they did not consider processes leading to the generation of memory cells. We enriched their model in order to study a complete response, from the activation of naive cells to the generation of memory cells. In particular, Eomes has been added to the MRN.

In this paper, we are interested in understanding how, from the activation of naive CD8 T-cells, an antigen-independent regulation of intra-cellular molecular content can drive a complete CD8 T-cell response. We particularly focus on the role of molecular-content heterogeneity among a CD8 T-cell population in the generation of memory cells. We first verify our model's ability to reproduce in vivo data at both cellular and molecular scales. Then we study, in an in silico CD8 Tcell population, the impact of molecular-content heterogeneity on the emergence of sub-populations, characterised by their expression of proteins Tbet and Eomes. We discuss how uneven distribution of molecular content at cell division affects the cellular dynamics (population size, cell differentiation, and death) and suggest that memory cell generation efficiency is maximal for a moderate degree of unevenness. Finally, we show that memory cells generated by our model are able to reproduce some features of a secondary CD8 T-cell immune response. Indeed, when restimulated by antigen in silico they generate more cells at the peak of the response and in the memory phase.

## 2. MATERIALS, METHODS, AND MODELS

#### 2.1. Data

4 × 10<sup>5</sup> naive CD8 T-cells from CD45.1+ F5 TCR transgenic mice (B6.SJL-PtprcaPepc<sup>b</sup> /BoyCrl-Tg(CD2-TcraF5, CD2- TcrbF5)1Kio/Jmar) recognizing the NP68 epitope were transferred intravenously in congenic CD45.2+ C57BL/6 mice (C57BL6/J). The day after recipient mice were inoculated intranasally with 2×10<sup>5</sup> PFU (plaque forming units) of a vaccinia virus expressing the NP68 epitope (39). From day 4 to day 22 post-infection, the spleens of infected animals where harvested and the number of F5 transgenic CD8 responder T-cells was assessed by flow cytometry, based on CD8/CD45.1/CD45.2 expression, to distinguish F5 TCR-transgenic responder (CD45.1+CD45.2−) from host (CD45.1−CD45.2+) CD8 Tcells. Naive (CD44<sup>−</sup> Mki67<sup>−</sup> Bcl2+), effector (CD44<sup>+</sup> Bcl2-) and memory (CD44<sup>+</sup> Mki67<sup>−</sup> Bcl2+) CD8 T-cells have been identified (40). All experimental procedures were approved by an animal experimentation ethics committee (CECCAPP; Lyon, France), and accreditations have been obtained from the French government.

OT1 CD8 T cells mRNA expression data time courses come from the ImmGen project (http://www.immgen.org). According to the information provided on ImmGen.org, the in vivo mRNA data (**Figure 4**) were generated for OT1 T-cells stimulated in similar experimental settings i.e., the response of transferred OT1 TCR-transgenic CD8 T-cells following infection by vesicular stomatitis virus expressing their cognate antigen.

#### 2.2. Molecular Regulation and IL2 Diffusion

We aim at describing the molecular regulation within each CD8 T-cell during a response to an acute infection, and how the dynamical molecular state of a cell characterises its differentiation stage. We present on **Figure 1** the MRN that

will be used throughout this manuscript and give a detailed description in **Table 1**. It contains several key molecular factors involved in CD8 T cell proliferation, differentiation, apoptosis, and cell communication. This is an updated version of the MRN developed in Prokopiou et al. (36) and Gao et al.(37) that was limited to the description of differentiation up to the effector stage. To account for differentiation into memory cells, we introduced the protein Eomes and its interactions with the rest of the network as documented in the literature. Indeed, Eomes is involved in the development of essential properties of memory cells such as survival, lymph node homing capacities or faster response to antigenic stimulation (11, 12, 14).

#### 2.2.1. Molecular Regulatory Network

This MRN is initiated upon antigen presentation to a naive CD8 T-cell, through the engagement of the TCR. Antigenic stimulation triggers the synthesis of interleukine-2 (IL2) by the CD8 T-cell and the production of IL2 receptors (IL2R) on the cell membrane (44). The synthesised IL2 is then released in the environment and can bind its receptor (41) to form IL2-IL2R complex, hereafter referred to as activated IL2R. Activated IL2 receptors enhance the expression of IL2 receptors (44) as well as IL2 synthesis (44). In the meantime, activated IL2 receptors, jointly with protein Tbet (see below), inhibit the activation of the IL2 gene through the action of the mediator protein Blimp1 (45, 46).

Antigenic stimulation independently stimulates Tbet synthesis (43), a protein involved in the acquisition of cell cytotoxicity. Indeed, Tbet is known to induce the expression of Fas ligand (FasL) (52), a transmembrane protein that can bind to the transmembrane protein Fas to induce cell apoptosis via the activation of Caspases in the Fas-expressing cell (53). Caspases are a family of proteins playing essential role in cell apoptosis (60). There exist several types of Caspases involved in CD8 T-cell apoptosis yet, for the sake of simplicity, we aggregated them in a unique variable [Cas]. Moreover, Tbet induces its own synthesis (via the gene Tbx21) (54, 55).

Eomes expression, involved in the acquisition of memory phenotype (12), is first inhibited during the activation phase due to engagement of the TCR (via activation of the Akt/mTOR pathway and inhibition of FOXO1 and TCF7) (7, 13, 57). Eomes is induced later (11, 61) and its expression is enhanced by the activation of IL2 receptors (7, 13, 56). Eomes promotes the development of new IL2 receptors on cell membrane (14).

The activation of IL2 receptors, of the TCR and the protein Eomes prevents apoptosis by inhibiting the activation of Caspases, in particular through the mediator protein Bcl2 (12, 50, 51)

#### 2.2.2. Intracellular Molecular Dynamics

Based on the above-described reactions, and from the equations used in Prokopiou et al. (36) and Gao et al. (37), we describe the dynamics of the concentrations of non-activated IL2 receptors ([R]), activated IL2 receptors ([L • R]), Tbet ([Tb]), activated Fas ([Fs<sup>∗</sup> ]), Caspases ([Cas]) and Eomes ([E]) in a CD8 T-cell with the following system of equations

$$\frac{\mathbf{d}}{\mathbf{d}t}[R] = \lambda\_{R1} f\_{A^{\rm PC}} + (\mu\_{L2}^{-} + \lambda\_{R2})[L \bullet R] + \lambda\_{E1}[E]$$

$$-\left(\mu\_{L2}^{+}[IL2^{cm}] + k\_{R}\right)[R],\tag{1}$$

IL2

$$\frac{\mathbf{d}}{\mathbf{d}t}[L\bullet R] = \mu\_{\mathrm{IL}2}^{+}[\mathrm{IL}2^{cm}][R] - \mu\_{\mathrm{IL}2}^{-}[L\bullet R] - k\_{\mathrm{e}}[L\bullet R],\tag{2}$$

$$\frac{\mathbf{d}}{\mathbf{d}t}[Tb] = \lambda\_{T1} f\_{\text{APC}} + \lambda\_{T2} \frac{[Tb]^n}{\lambda\_{T3}^n + [Tb]^n} - k\_T [Tb], \tag{3}$$

$$\frac{\mathbf{d}}{\mathbf{d}t}[\![F\mathbf{s}^\*]\!] = H\mu\_F^+[\![Tb^{cm}]\!] \left(\frac{\lambda\_F}{k\_F} - [\![F\mathbf{s}^\*]\!]\right)$$

$$-\mu\_F^-[\![F\mathbf{s}^\*]\!] - k\_F[\![F\mathbf{s}^\*]\!],\tag{4}$$

$$\frac{\mathbf{d}}{\mathbf{d}t}[\text{Cas}] = G\lambda\_{c1} \frac{1}{1 + \lambda\_{c2}[L \bullet R]} \cdot \frac{1}{1 + \lambda\_{c3}f\_{APC}} \cdot \frac{1}{1 + \lambda\_{E2}[E]}$$

$$+\lambda\_{c4}[F\mathbf{s}^\*] - k\_c[\text{Cas}], \tag{5}$$

$$\frac{\mathbf{d}}{\mathbf{d}t}[E] = \frac{1}{1 + \lambda\_{E5} f\_{APC}}.$$

$$\left(\frac{\lambda\_{E3} [L \bullet R]}{\lambda\_{E6} + [L \bullet R]} + \frac{G \lambda\_{E4}}{1 + \lambda\_{E7} [Tb]}\right) - k\_E[E]. \tag{6}$$

All parameters are positive. Parameters λ are associated to induction and inhibition effects, µ are associated to activation and deactivation of transmembrane proteins and k are degradation and dilution rates. The concentrations of System (1– 6) are assumed to be null in naive CD8 T-cells, and remain null until TCR engagement.

The effects of the external environment on the intracellular system (1–6) are taken into account through five variables. The variable fAPC (Equations 1, 3, 5, 6) is equal to the number of APCs bound to the considered CD8 T-cell and accounts for TCR engagement. The variable G (Equations 5, 6) is equal to 0 in naive CD8 T-cells and to 1 otherwise, i.e., in cells that have already met with an APC. It accounts for the fact that up-regulation of Caspases and Eomes described by parameters λc<sup>1</sup> and λE<sup>4</sup> is not active in naive cells. The variable H (Equation 4) accounts for the expression of FasL by effector and memory T-cells and for the activation of Fas through cell contact. Hence, H is equal to 1 in a non-naive considered CD8 T-cell in contact with an effector or a memory CD8 T-cell, and equal to 0 otherwise. The variable [IL2 cm] is equal to the concentration of IL2 at the cell membrane, in the extracellular environment. Finally, [Tbcm] is defined as the sum of Tbet concentrations in effector and memory CD8 T-cells in contact with the considered CD8 T-cell and acts as a proxy for the expression of Fas in those cells.

We introduced the variable [E] and the associated Equation (6) to the system used in Gao et al. (37) in order to account for the synthesis of protein Eomes and its interactions with other molecular factors. The term λE1[E] in (1) accounts for the up-regulation of IL2 receptors by Eomes. Eomes also limits cell apoptosis by activating Bcl-2 gene, as do IL2 and activated TCR. This communal target explains the multiplicative form of the inhibition of Caspases by Eomes, IL2 and TCR in Equation (5). We also introduced the function G in (5) to update the dynamics of Caspases concentration from Prokopiou et al. (36) and Gao et al. (37).



The positive feedback loop on Tbet is modeled with an order n Hill function in order to allow bistable behaviour of Tbet. As discussed in the introduction, the concentration of protein Tbet can be associated to the level of differentiation of an effector CD8 T-cell, with high level of Tbet correlating with fully differentiated effector cell, while low Tbet levels are associated to memory precursor effector cells. Proposition 1 below, reproduced from Girel and Crauste (30), gives necessary and sufficient conditions to allow bistable behaviour of Tbet concentration.

Proposition 1 (30). Assume fAPC <sup>=</sup> <sup>0</sup>, <sup>n</sup> <sup>&</sup>gt; <sup>1</sup> and <sup>λ</sup>T2(n−1) <sup>n</sup>−<sup>1</sup> <sup>n</sup> > nkTλT3, then Equation (3) has exactly three non-negative steady states: 0 < [Tb]<sup>u</sup> < [Tb]<sup>s</sup> , such that 0 and [Tb]<sup>s</sup> are locally asymptotically stable and [Tb]<sup>u</sup> is unstable.

In the following, we will assume that the conditions n > 1 and <sup>λ</sup>T2(<sup>n</sup> <sup>−</sup> 1) <sup>n</sup>−<sup>1</sup> <sup>n</sup> > nkTλT<sup>3</sup> are fulfilled (see section 3.2).

System (1–6) is embedded in every CD8 T-cell. Nevertheless, cell-cell contacts, stochastic events (cell cycle length, protein distribution at division) and external concentrations of IL2 affect the evolution of the system such that each CD8 T-cell develops a unique molecular profile based on its own history.

#### 2.2.3. Extracellular IL2 Diffusion

The secretion of IL2 by CD8 T-cells and its isotropic diffusion in the extracellular domain (with periodic boundary conditions) are modeled by the following PDE, introduced by Prokopiou et al. (36),

$$\begin{split} \frac{\partial \left[ IL2 \right]}{\partial t} &= D \nabla^2 \left[ IL2 \right] + \\ \left( \lambda\_{R3} \frac{\left[ L \bullet R \right]}{\lambda\_{R4} + \left[ L \bullet R \right]} + \lambda\_1 f\_{APC} \right) \frac{1}{1 + \lambda\_{T4} \left[ Tb \right]} - \delta \left[ IL2 \right], \text{ (7)} \end{split}$$

where [IL2] is the IL2 concentration. CD8 T-cells react to extracellular IL2 through their IL2 receptors by means of the [IL2 cm] term, in (1–2), defined as the sum of [IL2] at the considered cell membrane.

#### 2.3. Cell Differentiation and Division

Rules controlling cell division (including protein distribution at the division), apoptosis and differentiation are summarised in **Table 2** and detailed hereafter. It must be noted that cells properties result from their molecular profile. For example, the properties observed in vivo in memory cells (survival, low IL2 secretion, low cytotoxicity) are not imposed by model rules but acquired as a consequence of their molecular profile. One exception is cell cycle duration (see 2.3.2).

#### 2.3.1. Differentiation

We designed a set of rules based on the linear, irreversible differentiation scheme from Prokopiou et al. (36) and Gao et al. (37), allowing the description of a full CD8 T cell response, from the activation of naive cells up to the generation of memory cells. The differentiation pathway is illustrated in **Figure 2**.


TABLE 2 | Main rules applying to APCs and CD8 T-cells in the model.

✔: able, ❍: unable.

A naive CD8 T-cell binding an APC becomes pre-activated and maintains the contact with the APC thanks to good adhesion properties (cf. Section 2.4 and **Table S1**). If the concentration [L • R] of activated IL2 receptors in a preactivated CD8 T-cell reaches a given threshold IL2Rth, the preactivated CD8 T-cell becomes activated, leaves the APC, and starts to proliferate. When an activated CD8 T-cell divides, it gives birth to two CD8 T-cells whose states are determined by their respective concentrations of protein Tbet by comparison with a given threshold Tbetth: activated if [Tb] < Tbetth, effector otherwise. Finally, if the concentration of protein Eomes is greater than the threshold Eomesth, a dividing activated or effector CD8 T-cell will differentiate into memory cell and stop proliferating.

#### 2.3.2. Cell Cycle Length

Division is considered only for activated and effector CD8 Tcells. The cell cycle length (hours) of a cell preparing its k-th division (k ≥ 0) is chosen, at cell birth, from uniform law U[ck−4,ck+4] where c<sup>k</sup> = 6 + 28k 2 /(k <sup>2</sup> + 100) such that the mean duration of the cycle length increases with the number of divisions and can range from 2 to 32 h (31, 32). At the outcome of a division, activated and effector CD8 T-cells immediately enter a new cycle.

#### 2.3.3. Protein Distribution Between Daughter Cells

When a CD8 T-cell divides, the molecular content of the mother cell is randomly divided between the two daughter cells. To account for protein distribution between daughter cells at each division and for each protein, let us introduce the parameter m, defined as the degree of unevenness. We say that divisions are m% uneven if at division one daughter cell inherits up to (50 + m/2)% of the mother cell's content, while the second daughter cell receives the rest, that is at least (50 − m/2)% of the mother cell's content. Then, the molecular content of each daughter cell evolves according to System (1–6) until the next division.

For the sake of clarity, we emphasise that the degree of unevenness m is not the percentage of proteins received by daughters cells at each division but indicates to what extent stochastic molecular partitioning can be uneven. Based on estimation from Luzyanina et al. (26), we consider that divisions are 10% uneven, so that the most uneven partitioning in this case would split 45 and 55% of the mother cell's proteins in the two daughter cells respectively.

The exact value of each daughter cell molecular content at birth is randomly chosen according to a probabilistic law, as detailed hereafter. Each protein concentration [i] of the six proteins in System (1)-(6) is unevenly distributed among daughter cells: one cell inherits ki[i] and the other (2 − ki)[i]. Coefficients k<sup>i</sup> , i = 1, . . . , 6, are different for each protein, each cell, and each division, and are chosen from the probability law U[1−m/100,1]. Unless otherwise indicated, we consider 10% uneven divisions (26), i.e., k<sup>i</sup> ∈ [0.9, 1] for i = 1, . . . , 6. One may note that k<sup>i</sup> ∈ [0, 1] so the quantity of molecular material is preserved at each division, given that the volume of each daughter cell is half the volume of the mother. Different degrees of unevenness will be considered in section 3.3.

One special case of division is the asymmetric division, and its associated unequal repartition of Tbet between daughter cells. To account for polarisation of naive cells by antigenic signalling and the consecutive asymmetric divisions, the first division of a CD8 T-cell following its activation by an APC is characterised by a very specific uneven distribution of protein Tbet only between the two daughter cells: a coefficient K is randomly chosen from the uniform law U[0.5,1], one of the daughter cells is arbitrarily designated as the proximal daughter and receives a concentration (2 − K)[Tb] for protein Tbet while the other one is designated as the distal one and receives a concentration K[Tb] where [Tb] is the Tbet concentration in the mother cell, so that Tbet accumulates in proximal cells (17, 18). Other proteins concentrations are partitioned according to the previously mentioned rule, see paragraphs above.

CD8 T-cell apoptosis occurs as soon as Caspases concentration [Cas] reaches the threshold Caspasesth. APCs are present from the beginning of the simulation and their lifetime is randomly chosen from the uniform law U[48,96] (hours). APCs' only role is to activate naive CD8 T-cells, so we do not model any molecular activity within APCs. Dead cells are removed from the domain.

# 2.4. Spatial Modeling and Cellular Interactions

At the cell population scale, we use a cellular Potts model (CPM), also known as Glazier-Graner-Hegeweg model (38), to describe a population of CD8 T-cells and APCs evolving in a two-dimensional domain. Basically, a CPM is a time-discrete algorithm where cells, or agents, are defined as sets of nodes and move on a lattice, one node at a time, according to probabilistic rules based on the minimisation of the energy of the system, known as the Hamiltonian.

In our model, based on that from Prokopiou et al. (36) and Gao et al. (37), the domain is a square lattice of S = 150 × 150 nodes with periodic boundary conditions. Each node xE bears an index σ(xE). A set of nodes bearing the same index σ defines a cell, also denoted by σ. Finally, each cell σ has a type τ (σ) defining its properties. In our case, the different types are: extracellular medium, APC, naive, pre-activated, activated, effector and memory CD8 T-cell. Note that, technically, the extracellular medium is considered as a cell, denoted by σe.

Cell (including extracellular medium) size variation and displacement result from the succession of copies of index from nodes to neighbour nodes, based on the minimisation of the Hamiltonian [see Equation (8)], thanks to a simulated annealing algorithm. More precisely, at each iteration, known as Monte Carlo Step (MCS), of the CPM, the following algorithm is executed N = 3 × S times:


Note that it is conventional to consider N = S pixel copy attempts per MCS. However, in that case the maximum speed cells can reach is limited to approximatively 0.1 pixel per MCS (62), which eventually defines a finer time resolution than expected for the integration of differential equations. We emphasise that this limitation can be removed by increasing this number (here N = 3 × S).

The Hamiltonian is computed using the following formula:

$$\Omega = \underbrace{\lambda\_{\text{per}} \underbrace{\Sigma\_{\text{\tiny(\sigma)\#\sigma\_{\text{\text{\tiny(\pi)}}}} \left(p\_{\sigma} - P\_{\text{\tiny(\sigma)}}\right)^{2}}\_{\text{perimeter}} + \underbrace{\lambda\_{\text{actual}} \sum\_{\sigma \neq \sigma\_{\text{\text{\tiny(\pi)}}}} \left(a\_{\sigma} - A\_{\text{\tiny(\sigma)}}\right)^{2}}\_{\text{area}}$$

$$+ \underbrace{\sum\_{\text{neighbours (\vec{x}, \vec{x}^{\ast})}} I\_{\text{\tiny(\sigma(\vec{x})), \text{\tiny(\sigma(\vec{x}^{\ast}))})} \left(1 - \delta\_{\sigma(\vec{x}), \sigma(\vec{x}^{\ast})}\right)}\_{\text{contact}},\tag{8}$$

where Jτ1,τ<sup>2</sup> accounts for the contact energy between two cells of types τ<sup>1</sup> and τ2. Thanks to the term 1 − δ σ(xE),σ(xE∗) , two neighbour nodes belonging to the same cell do not generate contact energy. pσ and aσ are the actual perimeter and area of cell σ, respectively, whereas P<sup>τ</sup> (σ) and A<sup>τ</sup> (σ) are the target perimeter and area, respectively, for a cell of type τ (σ) ; perimeter and area constraints then penalize the configurations where the effective perimeter and area are distant from the target ones. Parameters λarea and λpm define the weights of those two constraints. The perimeter constraint has been added to the definition used in Prokopiou et al. (36) and Gao et al. (37) in order to avoid potential cell fragmentation.

The energy 1motility is defined by

$$\Delta\_{\text{moility}} = \nu(\sigma(\mathbf{x}\_{\text{s}})) \left( \cos(\theta(\sigma(\mathbf{x}\_{\text{s}}), t)), \sin(\theta(\sigma(\mathbf{x}\_{\text{s}}), t)) \right) \cdot (\mathbf{x}\_{\text{g}} - \mathbf{x}\_{\text{s}}), \tag{9}$$

where v(σ(xs)) is the weight associated to the motility energy for the cell σ(xs) and θ(σ(xs), t) is the privileged angle of direction for the cell σ(xs) at time t, randomly updated along the simulation. The operator "·" stands for the dot product. Thus, 1motility is all the more high (and then the copy is all the more probably accepted) that the copy direction (x<sup>g</sup> − xs) aligns with (cos(θ(t)),sin(θ(t))).

#### 2.5. Numerical Resolution

The initial cell population is composed of 30 naive CD8 Tcells and 3 APCs. A simulation requires 30,000 iterations (MCS) corresponding to 20 days and 20 h in the real time, that is, 1 MCS represents 1 min. When a simulation starts, APCs are already present in the LN, ready to activate naive CD8 T-cells. We consider the initial time to be day 4 post-infection (D4 p.i.) since our in vivo data set starts D4 p.i..

We assume that a node of the lattice corresponds to 4 × 4µm<sup>2</sup> for biological interpretation. The target cell area is chosen to be 9 nodes (144µm<sup>2</sup> ) for CD8 T-cells and 140 nodes (2, 240µm<sup>2</sup> ) for APCs. The target perimeter for CD8 T-cells is 48µm in order to favour compact shapes ; there is no constraint on APC perimeter. The simulations have been performed using CC-IN2P3 servers on Compucell3D software (62) with, unless otherwise stated, the parameter values from **Tables S1**–**S4**. Simulation files are provided in **Supplementary File 2**.

In section 3.4, we study the ability of our model to simulate a secondary response, also called memory response. Our model has first been calibrated in order to reproduce an in vivo primary response against Listeria monocytogenes (Lm) infection from Badovinac et al. (63) (see **Figure 8**). Then, the same parameter values have been used to simulated both a primary and secondary responses. However, secondary response simulations are performed with initially 3 APCs and 30 memory CD8 Tcells (instead of 30 naive CD8 T-cells for the primary response) that are able to bind an APC to become pre-activated, then the differentiation scheme presented in section 2.3.1 applies. The molecular profile of the initial memory cells is set as the asymptotic molecular profile developed by memory cells at the end of a primary response, as discussed in section 3.2.

#### 2.6. Model Calibration

Parameters of Equations (1–9) have been calibrated on in vivo data using parameter values from Prokopiou et al. (36) and Gao et al. (37). Since handling big cell populations with an agentbased model implies expensive computation time, we focused on fitting the proportion, rather than the number, of CD8 T-cells in each state of differentiation among the whole cell population. In order to compare in silico and in vivo data at both cellular and molecular scales we minimised the metric D = Dcell+Dprot where

$$D\_{cell} = \frac{1}{\text{(\#S)}\text{(\#V)}} \sum\_{\text{simulation } \mathcal{S}} \cdot \sum\_{\text{mouse } \mathcal{V}} \cdot \sum\_{\text{cellular type } \mathcal{C}} \cdot$$

$$\sum\_{\text{time step } t} |\mathcal{S}\mathcal{C}\mathcal{C}, t\rangle - \mathcal{V}\mathcal{C}, t\rangle|\tag{10}$$

and

$$D\_{\text{prot}} = \frac{1}{\langle \#\mathcal{S} \rangle \langle \#V \rangle} \sum\_{\text{simulation } \mathcal{S}} \cdot \sum\_{\text{mouse } \mathcal{V}} \cdot$$

$$\sum\_{\text{protein } \mathcal{P}} \cdot \sum\_{\text{time step } t} |\mathcal{S} \langle \mathcal{P}, t \rangle - \mathcal{V} \langle \mathcal{P}, t \rangle|,\tag{11}$$

with #S the number of simulations performed with a given set of parameters and #V the number of mice from which in vivo data have been collected. S(C, t) (resp. V(C, t)) is the ratio between the number of cells of type C and the size of the CD8 T-cell population at time t in the simulation S (resp. the mouse V). S(P, t) (resp. V(P, t)) is the ratio between the mean concentration of protein (resp. expression of mRNA) P among the CD8 T-cell population at time t in the simulation S (resp. the mouse V) and the maximal concentration (resp. expression) observed among all the time steps.

Since pre-activated and activated cellular types are not identified in in vivo data, we gathered pre-activated with naive T-cells and activated with effector T-cells. Then cellular types C in Equation (10) are: naive/pre-activated, activated/effector and memory. In Equation (11), quantities P are the ones for which we have relevant in vivo mRNA expression data at our disposal: IL2 receptors, Tbet and Eomes.

Note that we did not perform a parameter estimation procedure, but a calibration of our model based on experimental data. Evaluation of accuracy and sensitivity of parameter values have been investigated in previous studies (36, 37). Since we modified the model to account for differentiation in memory cells, a sensitivity analysis of our model to parameter Eomesth is presented in section 2 (**Figures S1**, **S2**) of **Supplementary File 1**.

#### 3. RESULTS

# 3.1. Modeling the CD8 T-Cell Immune Response at Both Cellular and Molecular Scales

We first briefly illustrate our model's ability to reproduce in vivo dynamics at both cellular and molecular scales. The evolution of the composition of a CD8 T-cell population from D4 to D22 p.i. is presented on **Figure 3A**. In both in vivo and in silico data, naive CD8 T-cells are negligible after D6 p.i.. At the peak of the response, occurring D8 p.i. both in vivo and in silico, more than 94% of the CD8 T-cells are in the activated or effector state, while the memory population emerges during the subsequent contraction phase. As a result of effector cell death and differentiation, memory cells represent the major part of the population on D22 p.i.. **Figure 3B** shows the size, in number of cells, of the CD8 T-cell population. The qualitative in vivo dynamics is quite well-reproduced: antigen presentation to naive CD8 T-cells triggers clonal expansion, population size reaches a peak D8 p.i. followed by a contraction phase where most cells (64 and 67% in vivo and in silico respectively) die.

On **Figure 4**, in silico predictions are compared to the mean IL2 receptors, Tbet and Eomes mRNA expression levels of CD8 T cells activated in vivo. The kinetics of IL2R and Tbet are well-reproduced. Indeed, as a result of TCR engagement, IL2R concentration sharply increases and reaches a peak D5 p.i., allowing cells to capture IL2 and get activated. Then IL2R concentration decreases until D8 p.i. and slowly increases from D8 to D15 p.i. Tbet concentration increases from D4 to D6 p.i. and remains stable until D8 p.i., then decreases until D15 p.i. Mean Tbet concentration consistently correlates with the size of effector CD8 T-cell population (**Figures 3A,B**) and is in agreement with its role in the control of cytotoxicity and cell apoptosis. Regarding Eomes concentration, the in vivo increase between D4 and D8 p.i. is well-reproduced by our model, however the increase observed between D8 and D15 p.i. does not match the in vivo data. As cells evolve toward a memory phenotype, in silico Eomes concentration increases and upregulates the expression of IL2R (**Figure 1**) to exacerbate the sensitivity of memory cells to IL2. It should be noted that various works support that Eomes expression increases in effector cells progressing toward a memory phenotype (11, 15, 43), contrary to what is observed in the mRNA dataset from Immgen.

#### 3.2. Cellular Dynamics Arise From Cellular Heterogeneity

In our model, each cell develops its own molecular profile, resulting in a heterogeneous cell population. Consequently, studying the mean concentration of a given protein among the population, as shown on **Figure 4** for example, is not sufficient to understand the molecular dynamics among the CD8 Tcell population.

To study the molecular-content heterogeneity and its role in cellular dynamics, we show in **Figure 5** the in silico concentrations of Tbet, Eomes, and Caspases in each CD8 Tcell of the population at different times of the response. Cells

were ranked according to their Tbet content. D5 to D8 p.i., corresponding to the clonal expansion phase (see **Figure 3**), concentrations are heterogeneous but uniformly distributed around the mean value. Most of that heterogeneity comes from the conditions of activation and from molecular partitioning at cell division. Yet from D8 to D24 p.i., corresponding to the contraction phase, two sub-populations of cells clearly emerge: one with high concentration of Tbet (centred around [Tb]<sup>s</sup> ≈ 118 mol/L) and one with low concentration of Tbet (≈ 0 mol/L). The unstable steady state of (3), defined in Proposition 1 and separating the stable equilibria 0 and [Tb]<sup>s</sup> , is given by [Tb]<sup>u</sup> ≈ 21 mol/L. Moreover, cells expressing high levels of Tbet express high levels of Caspases and low levels of Eomes, a molecular profile associated with cell death and poor memory potential. On the contrary, cells expressing low levels of Tbet have good survival and memory differentiation properties since they express low levels of Caspases and high levels of Eomes. Progressively, cells with high concentrations of Tbet die (when their concentrations of Caspases reach the threshold Caspasesth ≈ 19 mol/L) and cells with low concentrations of Tbet differentiate into memory

cells and stop proliferating (when the concentration of Eomes reaches Eomesth = 16 mol/L). On D24 p.i. there is no cell with intermediary profile, most of the cells have differentiated into memory cells while a few effector cells with high Tbet concentrations still survive. One can observe that the molecular profiles of memory cells converge to the same state where [Tb] = 0 mol/L, [E] ≈ 26 mol/L and [Cas] ≈ 9 mol/L.

The coexistence of two sub-populations characterised by their concentrations of Tbet explains the population dynamics observed on **Figure 3**. That is, the contraction of the cytotoxic effector cell population simultaneously with the emergence of a memory cell population with survival properties.

As discussed in the introduction, responding CD8 T-cells can be distinguished between short-lived (SLEC) and memory precursor (MPEC) effector cells based on the expression of two proteins: KLRG1 and CD127 (7, 15). In this section, we investigated how, in our model, the heterogeneity of Tbet concentrations among a CD8 T-cell population explains the emergence of two sub-populations of CD8 T-cells. The first one, expressing high concentrations of Tbet, could be comparable to SLEC that exhibit properties such as apoptosis and cytotoxicity, a process regulated by Tbet. The second one (memory potential, survival) would be similar to the MPEC population. This is consistent with the litterature, since Tbet is known to favour the development of SLEC, to the detriment of MPEC (9–11).

# 3.3. Moderate Uneven Molecular Partitioning Favours Efficient Generation of Memory Cells

A major source of heterogeneity in our model is the uneven molecular partitioning at cell division determined by the degree of unevenness m (see section 2.3.3). We compare on **Figure 6** the sizes of the CD8 T-cell population at the peak of the response as well as the sizes of the memory population on D25 p.i. for different degrees of unevenness, that is the extent of unevenness of the stochastic molecular partitioning. We do not however modify the degree of unevenness of the asymmetric first division, consecutive to the polarisation of the cell due to APC binding (17, 18), see section 2.3.3.

First, **Figure 6** shows that the size of the CD8 T-cell population at the peak of the response decreases as the degree of unevenness increases. Indeed, the more uneven the molecular partitioning, the sooner CD8 T-cells expressing high levels of Caspases or Eomes appear and then the sooner cells die by apoptosis or differentiate in non-proliferating memory cells.

Second, the relation between the degree of unevenness and the size of the memory population generated at the end of the response is not monotonous: the biggest memory populations are observed when considering a moderate unevenness (10–50%).

In section 3.2, the role of Tbet concentration in determining the fate (death or memory differentiation) of an effector CD8 T-cell has been discussed. Additionally, we showed in Girel and Crauste (30) that the progression of a cell lineage toward death or memory differentiation can be slowed down or reversed by molecular partitioning depending on cell cycle length, initial Tbet concentration and the degree of unevenness. This stressed, on a simplified model, the influence of the degree of unevenness on cell fate choice regulation.

On the opposite, when molecular partitioning is symmetrical (m = 0) and no further T-cell-APC interactions are assumed, there is no more source of stochasticity and consequently all the CD8 T-cells of the same lineage express the same concentration of Tbet. As a consequence of Proposition 1, this concentration irreversibly converges either to [Tb]<sup>s</sup> (high Tbet concentration) or to 0 mol/L (low Tbet concentration). This irreversibly leads to apoptosis (high Tbet concentration) or memory differentiation (low Tbet concentration) of the whole cell lineage.

Thus, our result clearly stresses that uneven partitioning allows the maintenance of a CD8 T-cell compartment with undetermined fate for some time, through cell fate reversibility. As long as it is maintained, this compartment is able to produce both effector cells destined to die and memory cells.

We also showed in Girel and Crauste (30) that the higher the degree of unevenness, the more reversible the cellular fate. Surprisingly, strong unevenness (65 − 80%) results in smaller memory cell populations (**Figure 6**). In fact, strong unevenness favours the fast emergence of daughter cells with very high or low concentrations of Tbet such that those cell lineages are likely to die or to generate memory cells. In particular, effector cells with high memory potential poorly expand before they differentiate hence this leads to the generation of fewer memory cells.

To discuss the efficiency of memory cell generation, we compare on **Figure 6** the number of memory cells generated at the end of the response to the number of cells at the peak of the response, viewed as an indicator of the energetic cost of the response for the organism (red crosses). **Figure 6** suggests that the degree of unevenness in molecular partitioning impacts memory generation, with the better ratio (more than 30%) obtained when considering 50% uneven molecular partitioning.

#### 3.4. Memory Response

One of the characteristics of memory cells is their capacity to mount more rapid effector response than naive cells and to generate an increased fraction of memory cells (64). To test whether the memory cells generated by our model exhibit some of these features we compared the in silico primary response with a secondary response of in silico generated memory cells.

**Figure 7** shows the in silico memory response (or secondary response), obtained with an initial population of 30 memory T-cells, as described in section 2.5. This secondary response is compared to the primary immune response starting with 30 naive CD8 T-cells (section 3.1). The in silico secondary response is characterised by a bigger CD8 T-cell population, at any time of the response. From the primary to the secondary response, there is a small increase in the size of the subpopulation of activated and effector cells but the major change is in the size of the memory population. Indeed, the number of memory CD8 T-cells increases much faster during the secondary response such that D29 p.i. the memory population is two times bigger than during the primary response. This can be explained by the fact that memory cells are activated faster than naive cells, thanks to their molecular profile. Indeed, memory cells express higher concentrations of IL2 receptors than naive cells, since it is sustained by the expression of Eomes. Consequently, the threshold IL2Rth (see section 2.3.1)

is reached sooner when starting with memory cells than with naive cells. As a result, the concentration of Tbet, upregulated during APC binding, is lower after the activation of a memory cell than after the activation of a naive cell, and low Tbet level is associated to memory precursor fate and low cytotoxicity.

On **Figure 8**, we compare in silico primary and secondary responses from our model with in vivo primary and secondary responses against Lm infection from Badovinac et al. (63). Since our model has been calibrated to fit the primary response data, we do not aspire to reproduce the quantitative dynamics of the secondary response, but rather to study its qualitative properties. Namely, the secondary response is characterised by a slower and weaker contraction phase, from the peak of the response D7 p.i. to the last time point D29 p.i.. This weaker contraction could be explained by an early production of memory cells that leads to a large population of memory cells, as it is the case in our model (**Figure 7**).

#### 4. DISCUSSION

Activation of naive CD8 T-cells triggers a primary immune response, characterised by a well-orchestrated program of cell proliferation, differentiation, death and migration. It is now well-known that the responding CD8 T-cell population is heterogeneous and that a single naive T-cell can generate differently fated cells (65). However, evaluating how cellular and molecular events contribute to that heterogeneity and identifying its consequences on the outcomes of the immune response remain fundamental questions.

With this in mind, we expanded a hybrid multi-scale model of the CD8 T-cell immune response, where cell behaviour is determined by intracellular molecular dynamics. Model parameters have been calibrated using in vivo data at both cellular and molecular scales. Because of expensive running time, we were led to simulate small cell populations so that we focused on semi-quantitative fitting criteria. After calibration, our model succeeded in reproducing the temporal dynamics of the response regarding the size of the CD8 T-cell population and the proportion of cells in each differentiation stage. Apart from a discordance between in silico and in vivo mean concentration of Eomes on day 15 p.i., our model captured the dynamics of the mean concentration of IL2 receptors, Tbet and Eomes, which play key roles in the differentiation processes.

In addition to reproduce primary responses, our model easily produces secondary responses. Memory cells generated during the in silico primary response succeeded in mounting a stronger secondary response upon antigenic stimulation (**Figures 7**, **8**). It should be noted that the differences between outcomes of the primary and secondary in silico immune responses only depend, in this work, on the difference between the molecular profile of memory and naive CD8 T-cells and do not take into account a lot of characteristics of the secondary response described in the literature such as: biggest initial CD8 T-cell population (8), shorter cell cycle (66) or sensitivity to inflammatory cytokines, such as IL12 (15).

FIGURE 6 | Size of the CD8 T-cell population at the peak of the response (black squares, left axis) and size of the memory CD8 T-cell population at the end of the response D25 p.i. (blue diamonds, left axis) as functions of the degree unevenness of molecular partitioning (mean ± standard deviation over 5 simulations). Red crosses (right axis) show memory cell generation efficiency, measured as the ratio between the size of the memory CD8 T-cell population D25 p.i. and the size of the CD8 T-cell population at the peak of the response (mean over 5 simulations).

We discussed how a deterministic description of molecular concentration dynamics combined with stochastic events, such as uneven partitioning of molecular content at division, can regulate the emergence and the maintenance of two sub-populations of CD8 T-cells. Those sub-populations, characterised by their molecular profiles, coexist but express different properties and antagonistic fates, comparable to those of SLEC and MPEC described in the literature (9). From that observation, we showed that the dynamics observed at the cellular scale (cell differentiation, population size) could be explained by molecular-content heterogeneity among the cell population, which mostly originates from uneven partitioning of molecular content. We did not however consider the effect of stochastic fluctuations of gene expression, known to be an important source of heterogeneity (67). Interestingly, Huh and Paulsson (25) showed that both stochastic gene expression and stochastic partitioning of molecular content are equally good to explain the heterogeneity observed at cell division and suggested that much of the heterogeneity usually attributed to the former actually results from the latter.

FIGURE 8 | Number of CD8 T-cells, normalised by CD8 T-cell population size D7 p.i., during in silico primary (black full line) and secondary (blue dashed line) responses (mean over 10 simulations) compared with in vivo primary (black squares) and secondary (blue triangles) responses against Listeria monocytogenes from Badovinac et al. (63).

FIGURE 7 | Number of (A) activated/effector and (B) memory CD8 T-cells during in silico primary (dashed line) and secondary (full line) responses. Mean ± standard deviation over 10 simulations.

In our model, cell phenotypic heterogeneity, associated with molecular-content heterogeneity, first arises upon asymmetric division of polarised naive cells. This heterogeneity is thereafter continuously regulated throughout the whole response by means of uneven partitioning of molecular-content at each division. This is in agreement with the observations of Lemaître et al. (68) who state that T-cell diversification is a continuous process, spread over the whole response, including the asymmetric first division and late events occurring throughout subsequent divisions. Besides, Lemaître et al. (68) pointed out that cellular heterogeneity, that could result from variations in naive T-cell responsiveness to cytokines or TCR signalling, pre-exists prior to the first division. In this article we did not consider preexisting heterogeneity among the naive T-cell pool, that could be achieved by varying the parameter values of System (1–6) associated to each naive T-cell. We can expect that it would confer to each naive T-cell a predisposition to engender a cell lineage oriented toward either apoptosis or memory differentiation. Moreover, the initial heterogeneity among naive T-cells could be conserved through the response, then leading to a heterogeneous pool of memory cells, a feature that is not reproduced by our model (69). Note that the in vivo responses presented in this article also result from transgenic CD8 T-cells bearing the same TCR.

Polarisation of naive cells upon antigenic stimulation has been observed in CD4 T-cells (21, 70) and B-cells (21, 71, 72). This polarisation results in asymmetric division of naive cells and may induce heterogeneous cell fates (70–72). Regarding the subsequent divisions, it can be thought that they are subject to uneven and random partitioning of molecular content since this phenomenon has been reported in many types of cells, including yeast, bacteria and T-cells (22–27). However, the contribution of uneven and random partitioning of non-polarised cells in the development of heterogeneous cell fate has not been studied yet. To that end, it would be interesting to extend the approach developed in our study to the differentiation of other lymphocytes, such as B-cells or CD4 T-cells.

In our study, increasing the degree of unevenness of molecular partitioning reduces the expansion size of the whole CD8 T-cell population whereas the size of the sub-population of memory cells is maximal for intermediate degrees of unevenness. As a consequence, the ratio between the number of memory cells generated and the magnitude of the response at its peak, viewed as a measure of memory generation efficiency, is maximised when considering a 50% degree of unevenness. As discussed above, molecular partitioning is not the only regulator of heterogeneity. In this regard, we can believe that our evaluation overestimates the value of this optimal degree of unevenness and rather indicates that generating a moderate heterogeneity all along the immune response leads to efficient memory generation.

In our manuscript, when the degree of unevenness is m = 10%, each daughter cell inherit from 45 to 55% of the mother cell's molecular content, with uniform probability distribution. The unevenness of molecular partitioning remains difficult to measure experimentally. Based on in vitro experimental data of CFSE dye expression, Luzyanina et al. (26) estimated that the two daughter cells inherit of 42,3% and 57,7% of the mother cell's molecular content, respectively. Rather than considering a uniform probability distribution and a degree of unevenness, we could consider that the molecular partitioning is a binomial phenomenon (24), i.e., each protein has the same probability to be attributed to each daughter cell. Such a discrete distribution can be approximate by a continuous and truncated (to avoid negative values) normal distribution whose variance would characterise the level of unevenness.

Note that, in works dealing with the CD8 T-cell immune response, it is usual to consider that 5 to 10% of the cells present at the peak of the response survive the contraction phase and differentiate into memory cells (8). This is consistent with our results only for symmetric divisions or for divisions with high (65–80%) degrees of unevenness. However, this hypothesis can be challenged, as pointed out in (40), as for the actual in vivo data presented in **Figure 3**, D22 p.i. the memory population size is 19.5% of the whole population at the peak of the response, D8 p.i. This suggests that the amplitude, and possibly the kinetics, of the cellular contraction is not only an inherent feature of the CD8 immune response but also depends on external factors such as inflammatory factors.

In many mathematical models of the CD8 T-cell immune response, as those referenced in (6), cell proliferation and differentiation depend on the amount of pathogen, in the manner of prey-predator models used in ecology. In our model a brief initial antigenic stimulation of naive CD8 T-cells is sufficient to trigger an autonomous program of proliferation and differentiation, as stated in the literature (1–3). However, while dispensable, in vivo inflammatory signals can affect the immune response outcome (73). A motivating perspective is to evaluate the respective contributions of both the autonomous program and extrinsic inflammatory factors to the immune response, so that the latter could be tuned by mastering the inflammatory environment. For example, extending our model by incorporating the inflammatory cytokine IL12, secreted by APCs, could markedly affect the effector/memory cell balance since IL12 is known to respectively promote and repress Tbet and Eomes synthesis (9, 47, 74).

Cell cycle length depends in our model on the number of divisions the cell has undergone. It would be instructive to introduce a molecular control of cell proliferation, since the putative existence of coexisting sub-populations with disparate cycle lengths could considerably impact the cellular dynamics. One could for instance consider the transcription factor Foxo1, known to induce Eomes expression while repressing that of Tbet and inhibiting cell cycle progression (75), suggesting that the TbetloEomeshi memory precursor cells discussed in section 3.2 might adopt a longer cycle than the Tbethi Eomeslo cells.

In conclusion, our agent-based multiscale model successfully reproduced several aspects of the CD8 T-cell immune response at both molecular and cellular scales. Even though we cannot infer quantitative conclusions from this study, it suggests that uneven partitioning of molecular content at cell division, as a source of heterogeneity, can modulate cell fate decision and act as a regulator of the magnitude of the response and of the size of the memory cell pool. Actually, we did not consider intermediaries, namely DNA transcription and mRNA translation, between gene activation and protein synthesis. Consequently, our molecular model is an amalgam between gene activity and protein synthesis. Therefore, while our argumentation is based on uneven partitioning of the molecular content, it could also stand for the situation where, when a cell divides, the two daughter cells inherit different gene activity levels for each gene. All in all, our study focuses on molecular heterogeneity generation upon cell division in general, rather than the specific case of molecular partitioning. It stresses that dynamics observed at the cellular scale—including the initiation of the contraction phase and the origin of memory cells—can be explained by structural molecular-content heterogeneity, that is continuously regulated along the response, as CD8 T-cells divide.

#### DATA AVAILABILITY

The datasets generated for this study can be found in the Open Science Framework repository https://osf.io/xbq9r.

mRNA expression data analysed in this study come from the ImmGen project (http://www.immgen.org).

#### AUTHOR CONTRIBUTIONS

All co-authors discussed the problem, approach and results. SG, OG, and FC designed the model. SG ran simulations and performed analysis. CA and JM conducted the experimental studies. SG wrote the paper. All the authors approved the final version.

#### REFERENCES


#### FUNDING

This work was performed within the framework of the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program Investissements d'Avenir (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR) and has been supported by ANR grant PrediVac ANR-12-RPIB-0011.

#### ACKNOWLEDGMENTS

We thank the Centre de Calcul de l'Institut National de Physique Nucléaire et de Physique des Particules de Lyon (CC-IN2P3) for providing computing resources, particularly Pascal Calvat and Yonny Cardenas for their valuable help. We also thank the BioSyL Federation and the LabEx Ecofect (ANR-11- LABX-0048) of the University of Lyon for inspiring scientific events. We acknowledge the contribution of SFR Biosciences (UMS3444/CNRS, US8/Inserm, ENS de Lyon, UCBL) facilities. We acknowledge the contributions of the CELPHEDIA Infrastructure (http://www.celphedia.eu/), especially the center AniRA in Lyon.

#### SUPPLEMENTARY MATERIAL

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fimmu. 2019.00230/full#supplementary-material

Supplementary File 1 | Parameter value tables and sensitivity analysis to parameter Eomesth.

Supplementary File 2 | CompuCell3D simulation files used to generate the results presented in this manuscript.


of the transcription factor T-bet during T lymphocyte division. Immunity (2011) 34:492–504. doi: 10.1016/j.immuni.2011.03.017


**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2019 Girel, Arpin, Marvel, Gandrillon and Crauste. This is an openaccess article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# Quantitative Mechanistic Modeling in Support of Pharmacological Therapeutics Development in Immuno-Oncology

Kirill Peskov 1,2 \*, Ivan Azarov <sup>1</sup> , Lulu Chu<sup>3</sup> , Veronika Voronova<sup>1</sup> , Yuri Kosinsky <sup>1</sup> and Gabriel Helmlinger <sup>3</sup>

*<sup>1</sup> M&S Decisions, Moscow, Russia, <sup>2</sup> Computational Oncology Group, I.M. Sechenov First Moscow State Medical University of the Russian Ministry of Health, Moscow, Russia, <sup>3</sup> Quantitative Clinical Pharmacology, Early Clinical Development, IMED Biotech Unit, AstraZeneca Pharmaceuticals, Boston, MA, United States*

#### Edited by:

*Gennady Bocharov, Institute of Numerical Mathematics (RAS), Russia*

#### Reviewed by:

*Luis De La Cruz-Merino, Hospital Universitario Virgen Macarena, Spain Eric Vivier, INSERM U1104 Centre d'immunologie de Marseille-Luminy, France*

\*Correspondence: *Kirill Peskov kirill.peskov@msdecisions.ru*

#### Specialty section:

*This article was submitted to Cancer Immunity and Immunotherapy, a section of the journal Frontiers in Immunology*

> Received: *28 January 2019* Accepted: *10 April 2019* Published: *30 April 2019*

#### Citation:

*Peskov K, Azarov I, Chu L, Voronova V, Kosinsky Y and Helmlinger G (2019) Quantitative Mechanistic Modeling in Support of Pharmacological Therapeutics Development in Immuno-Oncology. Front. Immunol. 10:924. doi: 10.3389/fimmu.2019.00924* Following the approval, in recent years, of the first immune checkpoint inhibitor, there has been an explosion in the development of immuno-modulating pharmacological modalities for the treatment of various cancers. From the discovery phase to late-stage clinical testing and regulatory approval, challenges in the development of immuno-oncology (IO) drugs are multi-fold and complex. In the preclinical setting, the multiplicity of potential drug targets around immune checkpoints, the growing list of immuno-modulatory molecular and cellular forces in the tumor microenvironment—with additional opportunities for IO drug targets, the emergence of exploratory biomarkers, and the unleashed potential of modality combinations all have necessitated the development of quantitative, mechanistically-oriented systems models which incorporate key biology and patho-physiology aspects of immuno-oncology and the pharmacokinetics of IO-modulating agents. In the clinical setting, the qualification of surrogate biomarkers predictive of IO treatment efficacy or outcome, and the corresponding optimization of IO trial design have become major challenges. This mini-review focuses on the evolution and state-of-the-art of quantitative systems models describing the tumor vs. immune system interplay, and their merging with quantitative pharmacology models of IO-modulating agents, as companion tools to support the addressing of these challenges.

Keywords: immuno-oncology, mechanistic models, tumor vs. immune system, systems pharmacology, pharmacokinetics, pharmacodynamics, molecular and cellular biomarkers

# INTRODUCTION

Immunotherapy of cancer has had a long history of development, starting from pioneering efforts in using coley toxins to treat patients—a therapeutic approach named after Dr. William Coley (1). Even though these earlier efforts never turned into a standard treatment, further investigations on the relationships between tumor cells and the immune system led to discoveries which unveiled fundamental principles underlying cancer progression, such as immune surveillance (2, 3), cancer dormancy (4), cancer immuno-editing (5), and the cancer immunity cycle (6). These discoveries were foundational for clinical successes and corresponding regulatory approvals in recent years, of therapies targeting the CTLA-4, PD-1, and PD-L1 immune checkpoints. In the wake of these successes, there has been an explosion in the development of immuno-modulating, anti-cancer pharmacological modalities, leading to the initiation of, literally, thousands of clinical trials (7, 8). However, from the discovery phase to late-stage clinical testing and regulatory approval, challenges in the development of immuno-oncology (IO) drugs are multi-fold and complex (9), with related complexities in the design of clinical trials; if unaddressed, these may lead to a decreased probability of success (10). Some of these challenges can be mapped to an incomplete mechanistic understanding of immune response dynamics and the interplay of such immune responses with tumor infiltration processes and tumor cell growth (11). These quantitative knowledge gaps hinder: (i) effective translation of novel promising therapeutic approaches into the clinic, (ii) identification of predictive response biomarkers, and (iii) search of therapeutic drug combinations which may overcome intrinsic or acquired resistance to existing standards of care (12). This mini-review focuses on quantitative, mechanisticallyoriented modeling approaches which have been sought in IO, to address, at least partially, the abovementioned challenges and knowledge gaps.

# EVOLUTION OF QUANTITATIVE, MECHANISTICALLY-ORIENTED IO SYSTEMS MODELING

Application of mathematical modeling in support of preclinical and clinical research, as well as decision-making in Oncology, has a long-standing history covering multiple problems and addressing a variety of research questions—today often referred to as computational oncology (13–15). Historical milestones include adaptations of the Gompertz model for treatment outcomes in breast cancer (16). These earlier efforts started from models with a simplistic empirical structure, based on an ordinary differential equation (ODE) describing tumor size growth using an exponential or sigmoidal function (17). Such a model, however, would not adequately describe the interplay between tumor cells and tissue vs. the immune system, since it entirely ignores the immune component (18). It is nevertheless valuable to mathematically describe treatment response effects following various chemotherapies, which are adequately captured by generalized Gompertzian kinetics (19). In fact, such modeling results provided a basis for the use of specific "dose-dense" chemotherapeutic regimens, which subsequently showed favorable outcomes in the treatment of breast cancer (20). Additionally, such empirical considerations allowed for a gradual evolution of modeling concepts, which today can be grounded in mechanistically-oriented principles, including for tumor vs. immune system interactions (**Figure 1**).

Earlier efforts to describe tumor vs. immune system relationships via a general mathematical description appeared in the 1980's, following the pioneering IO work that introduced the concept of immune surveillance (2, 3). These mathematical models considered the addition of a second variable describing the dynamics of cytotoxic immune cells, which are able to attack tumor cells (22–24). The resultant "two-ODE" model actually follows a typical "predator-prey" model introduced by Alfred Lotka and Vito Volterra, in much earlier days, at the turn of the 20th century. In such a model, tumor cells may be interpreted as the "prey," whereas cytotoxic immune cells may be viewed as the "predator": their dynamic interplay may result in one possible system behavior reflective of cancer dormancy (4). Given the relative simplicity of such a "two-ODE" model and since the behavior of such a model could be assessed analytically, it gained immense popularity within the oncology modeling community and led to several theoretical hypotheses underlying fundamental principles of cancer progression. For example, it was shown, through modeling, that key parameters controlling tumor regrowth under steady-state conditions of cancer dormancy were those relating to activities of the immune system (25). A corollary result was that it is a reduction in the probability of achieving tumor cell kill, rather than a reduction in the probability of tumor cells being recognized by cytotoxic cells, which best explained immune evasion by tumor cells (26). Interestingly, this key result, derived theoretically at the time, has recently been supported by elegant modeling work linking high-level immunological and epidemiological data, which suggests that age-related decline in T cell output correlates better with risk of cancer diagnosis vs. agerelated accumulation of somatic mutations in tumor cells (27).

With the explosive growth of experimental data surrounding the complexity of tumor vs. immune system interplay, "two-ODE" models experienced a further evolution with additional biological entities and mechanisms being taken into mathematical consideration. At this point and looking forward, many biological candidates were tested as the "third modeling variable," representing either specific immune cells or cytokines that modulate cytotoxic T lymphocyte (CTL) function (28). Such models were initially focused on including IL-2 function and effects, reflective of the potential importance of this cytokine and its associated dynamics in long-term tumor relapse (29). In further work, de Pillis et al. used a "three-ODE" model to reveal a difference between the dynamics of CD8<sup>+</sup> CTLs vs. natural killer cells, which supported the importance of considering multiple cell types in the overall anti-tumor immune activity (30). More recently, CD4<sup>+</sup> T helper cells were considered as the third component, in a quantitative, model-based investigation of adoptive cellular immunotherapy (31).

"Three-ODE" models, however, exhibit one significant structural limitation, namely they completely lack (an) immunosuppressive component(s), which would be crucial when considering immune evasion mechanisms (32). Therefore, embedding a fourth variable into such models, to describe immuno-suppression, would seem rather natural; however, choices for the most appropriate candidate in this role are multifold. Several types of immuno-suppressive cells or molecules could be suitable candidates, including regulatory T cells (Tregs), myeloid-derived suppressor cells (MDSCs), or Type 2 tumorassociated macrophages, as well as cytokines such as TGFβ or IL-10. Thus, Arciero et al. chose TGFβ as the fourth model variable (33), while de Pillis et al. used Tregs as the principal immuno-suppressive component in their model (34). While these two modeling examples focused on immuno-suppressive effectors, other "four-ODE" models abound, declining a vast

variety of immune players or tumor cell clones (35–41). Such a variety in potential key immuno-modulating factors made the generalization of any "three-ODE" or "four-ODE" model an overly difficult process, since any one of the models cited above can be challenged with newly generated experimental data featuring the importance of one vs. another immune factor. This may also explain, at least partially, the relatively minimal recognition, to date, of quantitative modeling approaches by immuno-oncologists (28, 42, 43).

On one hand, some of the biological complexities which compose the IO cycle, as summarized in recent reviews (6, 44, 45) clearly indicate the limitations of oversimplified models such as "prey-predator" models, which appear to be too remote from experimental reality and would not be applicable or of use for the majority of research relevant questions. On the other hand, increasing model complexity with additional mechanistic insights always comes with challenges of model calibration, as depicted in this famous quote by John von Neumann, "with four parameters, I can fit an elephant and with five, I can make him wiggle his trunk" [see in Dyson (46)]—pointing to the necessity of avoiding overparameterized "metastatic" models with unreliable extensions and loss of predictive power. Achieving such a balance in capturing necessary (not oversimplified) yet sufficient (not over-developed) features, and as constrained by the available data, is arguably one of the most difficult challenges in fit-for-purpose, parsimonious mechanistic model building and calibration. Overparameterization can easily negate all benefits brought forward by the incorporation of exquisite biological details of the system under consideration (47); models which attempt to explain everything may in fact not be useful, their predictive power remaining a question mark (48).

To address this challenge, part of the solution may reside in the combining of modeling methodologies developed previously and in other disciplines (49). This would result in a repository of prior information and knowledge validated elsewhere, to build mechanistic models in immuno-oncology which, on one hand, incorporate increasing system complexity and, on the other hand, avoid overparameterization based on newly generated data—thereby resulting, using terminology of a Bayesian mindset, in a posterior model based on existing, established quantitative priors (50).

If so, the question then becomes, "where to find such established prior models?" One obvious domain is quantitative immunology (51, 52), where the use of various modeling techniques by experimentalists has already gotten significantly more traction, arguably, than in other fundamental biological disciplines (53). For example, modeling has provided quantitative "inference frameworks" for immunology basics and fundamentals such as T cell activation, homeostasis or self / non-self recognition (54–57), immune receptor signaling (58), and understanding of T cell immunological memory (59). Prior models from quantitative immunology may then be combined with prior models from quantitative pharmacology (60–62), another field where modeling has provided quantitative "inference frameworks" (63) In the next section, we will discuss selected works which considered the combining of modeling methodologies, in attempts to develop pharmacologically-modulated posterior models, which were then used to prospectively address questions in the development of IO therapies (**Table 1**).

# MECHANISTIC MODELING IN SUPPORT OF IO THERAPY DEVELOPMENT

Applications of mechanistic modeling in support of preclinical and clinical research, commonly referred to as pharmacokinetic (PK)/pharmacodynamic (PD) modeling, are traditionally centered around the optimization of treatment dosing and scheduling—the "dose" representing a critical component of any drug development program (82). Such modeling approaches have thus been used in the development of IO agents such as PD-1 and PD-L1 inhibitors (83–85). In particular, mechanistic PKPD modeling has been applied in support of first-in-human dose selection of pembrolizumab, an anti PD-1 agent (65); this resulted in a seamless clinical trial design with a model-informed dose justification, which the US FDA accepted in the process of an accelerated regulatory review (86). Label updates with flat dosing schedules were subsequently granted, for both nivolumab and pembrolizumab, strongly supported by model-based simulations (87, 88). PKPD modeling has also been used in the translation of preclinical data for a conjugated IL-2 therapy, in particular to gain a better understanding of such a therapy's downstream effects (89). PKPD modeling has been further used in the development of bispecific biologics. Chen et al. used it for the estimation of the minimally anticipated biological effect level (MABEL) of a bispecific antibody targeting CD3 and p-cadherin (66), while Ribba et al. used it for guided dose escalation study design of cergutuzumab amunaleukin, a fusion protein consisting of IL-2 and a carcinoembryonic antigen (CEA) human monoclonal antibody (64). Such models are great examples of a "fit-for-purpose" quantitative approach, focused on addressing a specific pharmacological question. However, they do not take into account details of the tumor vs. immune system interactions, which would be critical to gain a better understanding of mechanisms of action (MoA) of immunotherapies.

Progressively adding components of tumor vs. immune system interactions into such PKPD models may well support the addressing of questions around pharmacologically-modulated IO biology, a topic of paramount importance in, for example, the search for therapeutic IO drug combinations (90). Such a systems approach may become an indispensable quantitative tool supporting "go/no-go" decisions in development programs, especially if sufficient biological knowledge for viable generalization is considered in the model (91). This prior knowledge is generally derived from two sources: (i) connectivity information to determine the system structure, e.g., molecular & cellular interactions, and their integration into patho-physiological processes; and (ii) quantitative data, for the calibration of model parameters. As discussed in the previous section, an imbalance in structural vs. quantitative information will in one way or another complicate integration into, and practical use of a mathematical model. For example, Lai and Friedman developed an elegant, yet complex model which includes a high number of biological elements, and considered their dynamics in space and time using partial differential equations (PDEs), to better understand the potential synergy between PD-(L)1 antagonists and either a GVAX vaccination or BRAFi/MEKi targeted therapies (72, 73). However, assessing the predictive power of such a model is impractical, given insufficient experimental data for model validation. Serre et al. provided another example of an elegant, yet insufficiently validated mathematical model describing the potential synergy between radiotherapy (RT) and immune checkpoint blockade (70).

One obvious way to improve model validation and hence model predictive power is to use rich experimental data, to rigorously constrain model parameters. This, however, requires the use of adequate statistical methods to properly quantify uncertainty and variability, which are inherent to any experimental biomedical and life sciences dataset (49, 92). In oncology drug development, quantitative data supporting MoA elucidation are typically generated at the preclinical stage. Parra-Guillen et al. for example, used a nonlinear mixed-effects (NLME) model and experimental data from syngeneic tumor models, to reveal the most influential immuno-adjuvant capable of boosting anti-tumor vaccination effects (21, 67). Such a modeling approach, which combines mechanistic features and mixed effects, allows one to incorporate individual-level data into the model, which may then describe not only mean trends, but also the full range of individual biomarker dynamics (93). A similar, combined mechanistic and mixed-effects approach was used to develop a model describing synergistic effects between RT and PD-(L)1 blockade in mice (68). This model, in fact, synthesizes a fit-for-purpose, yet sufficiently detailed mathematical description of the IO cycle, together with adequate model validation based on data from multiple experiments. As a result, this model can be used as a simulation tool for experimental study design, and is also adequate for determining optimal schedule and sequencing of RT + IO, and IO + IO treatment combinations (68, 69). Interestingly, despite the wellknown challenges in translating oncology preclinical results into

#### TABLE 1 | Mechanistic models in support of IO therapy development.


#### TABLE 1 | Continued


*<sup>a</sup>MoA, Mechanism of action; CEA, carcinoembryonic antigen; mAb, monoclonal antibody; NLME, nonlinear mixed effects; IO, immuno-oncology; PK, pharmacokinetics; PD, pharmacodynamics; MABEL, minimally anticipated biological effect level; FIH, first-in-human; RT, radiotherapy; ICD, immunologic cell death; ODE, ordinary differential equations; PDE, partial differential equation; ABM, agent-based modeling.*

the clinic, simulation results from this preclinical modeling exercise were recently supported, in a qualitative sense, with clinical data and a corresponding meta-analysis (94, 95). For a quantitative translation, the Kosinsky et al. model would require adjustments for multiple quantitative differences that exist between mouse vs. human immune systems, e.g., appropriate expressions of immune checkpoints and turnover of specific T cells (96). Another modeling approach aimed at supporting the development of such an RT + IO combination therapy was proposed by Poleszczuk et al. who developed a physiologicallybased model which considered a detailed incorporation of T cell trafficking and was used for the identification of an optimal site for RT administration, to maximally increase the probability of incremental anti-tumor immune effects (71). Predictions from such a comprehensive modeling effort were also recently supported by clinical results, which showed that RT administered to liver metastases triggered a higher immunological response (97). A mechanistic model has also been proposed by Peng et al. in the search of an optimal combination strategy against castration-resistant prostate cancer (74).

The modeling applications discussed to this point emphasize the importance of addressing multi-pronged questions, e.g., not only around dose finding, but also on the identification of an adequate time window for maximizing therapeutic benefits (98). This problem is particularly challenging in the development of combination therapies, where multiple options around which cancer indication, which combination agents, which scheduling per agent, and which sequencing of the agents make trial design enormously complex (99, 100). In recent years, platform design of clinical studies, driven by one master protocol, has gained momentum (101, 102)—a format which, in fact, benefits even further from a supportive quantitative mechanistic modeling approach (103).

#### MECHANISTIC MODELING IN SUPPORT OF IO BIOMARKER IDENTIFICATION

A third problem which is highly relevant in the development of IO therapies is the identification of predictive biomarkers. Indeed, there still is a lot of room for improving numbers of responder patients in pivotal IO trials, even in immunologicallyactive indications (104). Several computational models focusing on the identification of predictive biomarkers, with applications to personalized treatment against glioblastoma and prostate cancer have been developed (75, 77). These approaches have yet to find a general use in clinical practice. Part of the challenge arises from the biological complexity in the IO field, although there also are significant limitations from an experimental standpoint, such as differences in fresh vs. archived samples, difficulties in obtaining multiple biopsies per patient, with related risk and cost issues (105). One approach to alleviate some of these problems is the development of novel combinatorial biomarkers ("signatures") which may relate multiple, routinely measured markers with clinically meaningful biological phenotypes (106). In fact, such a consensus approach, "Immunoscore," has recently been validated in a large international study of colon cancer (107).

Another complicating factor in the development and interpretation of mechanistic modeling of IO data is the tremendous heterogeneity in tumor cell clones and elements of the surrounding immune microenvironment (108). A rapid development of novel experimental techniques may overcome this challenge, at least partially. Thus, the identification of specific gene expression signatures may help in further validating existing immunoscores and related biomarkers, even increasing their discriminatory ability (109), as recently shown with a PD-L1 expression signature which outperformed a standard PD-L1 immunohistochemistry (IHC) assay (110). Multiple immune signatures have now been identified, which allow for a better characterization of various aspects of anti-tumor immunity (111– 116). Recent technological breakthroughs such as cytometry by time-of-flight (CyTOF) and single-cell mRNA sequencing (scRNA-seq) may further advance the utility and robustness of these immune signatures (117, 118); these techniques may allow for a deeper, more granular profiling of tumor and immune cell phenotypes involved in response or resistance to immunotherapies, in multiple indications (119–123). The importance in using quantitative models toward the selection and qualification (within the chain of events, from dosing to patient response) of IO biomarker signatures cannot be overemphasized (108): immune biomarkers involve a high number of molecular and cellular species, and often exhibit complex temporal and spatial dynamics; these need to be properly framed in the context of a quantitative model, especially if the purpose is to relate multi-variate biomarker signatures to IO treatment effects and clinical endpoints (124). Quantitative modeling may also support the development of biomarkers in context, by integrating different data types, and following a model-based qualification of biomarkers as surrogate measures of efficacy and response. Such an approach has been proposed, recently, in the evaluation of neoantigen fitness as a surrogate measure of immunogenic quality of the existing neoantigen pool (125, 126). The progressive integration of such consensus, multivariate combinatorial biomarkers into a unified, quantitative and mechanistic modeling framework will help overcome some of the limitations in the clinical use of IO biomarkers (127, 128).

# OTHER MECHANISTIC MODELING APPROACHES WITH RELEVANCE TO IO

The above sections focused on traditional deterministic models, which make use of ODEs and PDEs for the description of IO systems dynamics. Other modeling techniques can be used to describe tumor vs. immune interactions. For example, cellular automata and agent-based models (ABMs) (129), as well as various hybrid models which link continuous and discrete modeling elements have been developed (130). Such models may be useful in raising new hypotheses, which may arise from emergent properties of the system based on existing data, rather than generating bona fide forward predictions. For example, a lattice gas automata technique has been used to gain a better understanding of a vaccination treatment mechanism and its corresponding anti-tumor immune response dynamics (131, 132). ABMs also represent a popular modeling technique, since they are well-suited to describe stochastic processes which do occur at various stages of the IO cycle. For example, Gong et al. developed an ABM to reveal spatio-temporal characteristics of PD-L1 blockade (79). In another publication, Kather et al. presented an elegant 2D ABM framework for an improved understanding of the role of stromal cells in colorectal cancer (CRC) (80). These authors determined that malignant cells hiding in the stroma cannot be eradicated completely, while stromal cells, at the same time, would not allow for rapid tumor progression. Consequently, simulations of an immunotherapy illustrated how stroma permeabilization, concomitantly with immune activation, were able to markedly increase response to therapy in silico. Additionally, it was shown that a stromatargeted therapy with insufficient activation of tumor-specific CTLs can lead to rapid tumor escape and hyper-progression (80). More recently, this model has been extended and generalized to a 3D spatial description, incorporating macrophage effects; it accurately reproduced the tissue architecture typically observed in CRC and can be used, similarly to ODE systems models, for the identification of effective IO therapeutic combinations (81).

# CONCLUDING REMARKS

Following the approval, in recent years, of the first immune checkpoint inhibitors, the landscape of cancer treatment has changed dramatically and has shifted to a deep reconsideration of the role of the immune system in cancer progression and treatment. This led to an unprecedented number of clinical trials and generation of clinical data in the IO field. Clinical success rates, however, while improving significantly, are still relatively low. The observed imbalance, between the amount of biological and clinical data being generated vs. probability of trial success is not uncommon in biomedical disciplines, and calls for the development and updating of a companion, integrative, quantitative modeling framework with predictive value for MoAs and simulation value for study design purposes. As described by Sidney Brenner in his "Sequences and Consequences" landmark paper: "We should welcome with open arms everything that modern technology has to offer us but we must learn to use it in new ways. Biology urgently needs a theoretical basis to unify it and it is only theory that will allow us to convert data to knowledge" (133). We propose that quantitative, mechanisticallyoriented modeling represents a means toward the establishment of such a "theoretical basis," pending proper integration of prior knowledge gained from biology and clinical research. One of the main factors limiting a wider application of quantitative systems modeling is its demand for rich experimental data necessary for precise parameter estimation. Historically, generation of such datasets in oncology research has been challenging, due to translational limitations of experimental preclinical models and sparse collection of tissue samples in clinical settings. Also, in the IO field, another challenge is the lack of predictive power for univariate biomarkers (e.g., PD-L1 IHC status or tumor mutational burden taken in isolation), which may unequivocally link immunologically-driven therapeutic effects to clinical response; a multi-variate approach is clearly needed (128). Recent developments in multi-modality biomarkers and associated molecular signatures, together with innovative pharmacologies and clinical design under platform trials (134) will help in the progressive build-out and qualification of such a unified quantitative modeling framework, which in turn may help in predicting patient responses based on a given pharmacological intervention choice and multi-variate biomarker signatures.

#### AUTHOR CONTRIBUTIONS

KP, VV, and YK generated **Figure 1**. The Table was generated by IA, VV, and LC. All authors contributed to the writing of the manuscript.

#### REFERENCES


#### FUNDING

This work was funded by AstraZeneca Pharmaceuticals and the Russian Academic Excellence Project 5-100 program.

#### ACKNOWLEDGMENTS

The authors would like to acknowledge that the modeling work presented here benefited from numerous discussions with associates from AstraZeneca and MedImmune, including members from Translational Sciences, Preclinical Biology & Pharmacology, Oncology Clinicians, and Quantitative Clinical Pharmacology.


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**Conflict of Interest Statement:** The authors declare that this study received funding from AstraZeneca. VV, IA, KP, and YK are employed and KP, YK are owners of M&S Decisions, a modeling consultancy received research funding from AstraZeneca. LC and GH are employed by, and GH is a shareholder of AstraZeneca.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2019 Peskov, Azarov, Chu, Voronova, Kosinsky and Helmlinger. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# Linking Cell Dynamics With Gene Coexpression Networks to Characterize Key Events in Chronic Virus Infections

Mireia Pedragosa1†, Graciela Riera1†, Valentina Casella<sup>1</sup> , Anna Esteve-Codina2,3 , Yael Steuerman<sup>4</sup> , Celina Seth<sup>1</sup> , Gennady Bocharov 5,6, Simon Heath2,3, Irit Gat-Viks <sup>4</sup> , Jordi Argilaguet <sup>1</sup> \* and Andreas Meyerhans 1,7 \*

#### Edited by:

Shokrollah Elahi, University of Alberta, Canada

#### Reviewed by:

Marina Cella, Washington University School of Medicine in St. Louis, United States Aikaterini Alexaki, United States Food and Drug Administration, United States

#### \*Correspondence:

Jordi Argilaguet jordi.argilaguet@upf.edu Andreas Meyerhans andreas.meyerhans@upf.edu

†These authors have contributed equally to this work

#### Specialty section:

This article was submitted to Viral Immunology, a section of the journal Frontiers in Immunology

Received: 30 January 2019 Accepted: 18 April 2019 Published: 03 May 2019

#### Citation:

Pedragosa M, Riera G, Casella V, Esteve-Codina A, Steuerman Y, Seth C, Bocharov G, Heath S, Gat-Viks I, Argilaguet J and Meyerhans A (2019) Linking Cell Dynamics With Gene Coexpression Networks to Characterize Key Events in Chronic Virus Infections. Front. Immunol. 10:1002. doi: 10.3389/fimmu.2019.01002 1 Infection Biology Laboratory, Department of Experimental and Health Sciences (DCEXS), Universitat Pompeu Fabra, Barcelona, Spain, <sup>2</sup> CNAG-CRG, Center for Genomic Regulation (CRG), Barcelona Institute of Science and Technology, Barcelona, Spain, <sup>3</sup> Universitat Pompeu Fabra, Barcelona, Spain, <sup>4</sup> Cell Research and Immunology Department, Tel Aviv University, Tel Aviv, Israel, <sup>5</sup> Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia, 6 Institute for Personalized Medicine, Sechenov First Moscow State Medical University, Moscow, Russia, <sup>7</sup> Institució Catalana de Recerca i Estudis Avançats (ICREA), Barcelona, Spain

The host immune response against infection requires the coordinated action of many diverse cell subsets that dynamically adapt to a pathogen threat. Due to the complexity of such a response, most immunological studies have focused on a few genes, proteins, or cell types. With the development of "omic"-technologies and computational analysis methods, attempts to analyze and understand complex system dynamics are now feasible. However, the decomposition of transcriptomic data sets generated from complete organs remains a major challenge. Here, we combined Weighted Gene Coexpression Network Analysis (WGCNA) and Digital Cell Quantifier (DCQ) to analyze time-resolved mouse splenic transcriptomes in acute and chronic Lymphocytic Choriomeningitis Virus (LCMV) infections. This enabled us to generate hypotheses about complex immune functioning after a virus-induced perturbation. This strategy was validated by successfully predicting several known immune phenomena, such as effector cytotoxic T lymphocyte (CTL) expansion and exhaustion. Furthermore, we predicted and subsequently verified experimentally macrophage-CD8 T cell cooperativity and the participation of virus-specific CD8<sup>+</sup> T cells with an early effector transcriptome profile in the host adaptation to chronic infection. Thus, the linking of gene expression changes with immune cell kinetics provides novel insights into the complex immune processes within infected tissues.

Keywords: systems biology, cell dynamics, coexpression networks, WGCNA, DCQ, LCMV, chronic infection

# INTRODUCTION

A virus infection of a host organism represents a major perturbation from homeostasis. It is temporary limited in case of an acute infection or maintained in a chronic infection. Nonetheless, in both types of virus infections, a large number of the host genes of a lymphatic tissue in which the immune response is initiated may be differentially expressed compared to the healthy steady state

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indicating the enormous complexity of the overall host protection response (1, 2). To define the key processes that determine virus infection fates, and to understand the underlying mechanisms, most analyses have concentrated on few immune cell subtypes or regulatory factors, without addressing the interactions between them [reviewed in (3)]. Higher resolution techniques like mass cytometry, single-cell technologies and mass spectrometry were then used to further characterize cell subtype populations (4–6) or fine-map intracellular processes of selected cell types (7, 8). The respective data demonstrated a large degree of functional diversity even within virus-specific immune cell subtypes and characterized specific functional cell states (9, 10).

An alternative, holistic strategy to analyse virus-induced host perturbations is to apply a high resolution technique like RNA-Seq for capturing all processes within a complete organ. This strategy however has the disadvantage that the use of total organ RNA eliminates all information about organ cell type composition at the time of analysis and cell origin of the RNAs. Nonetheless, recent work using infection of mice with influenza A virus and lymphocytic choriomeningitis virus (LCMV), and analyzing lung or splenic transcriptomes, respectively, gave important insights into the systems regulation. Altboum et al. developed a computational method named digital cell quantifier (DCQ) that infers quantitative changes of over 200 defined immune cell subpopulations from timeresolved lung transcriptome data (11). They then predicted and subsequently verified experimentally that different dendritic cell populations have specific roles at early and late time points of acute Flu infections. More recently, we used weighted gene coexpression network analysis (WGCNA) to characterize systems perturbations during acute and chronic LCMV infections (1). From spleen transcriptome-derived coexpression modules and subsequent immunological analyses we demonstrated a delicate adaptation process toward a chronic virus infection with both immunosuppressive and immunostimulatory processes involved. However, only a tiny fraction of the global information content has been utilized and the bulk awaits exploitation.

To gain better insights into the mechanisms of chronicity development and virus control, we here combined DCQ with WGCNA, and explored further our time-resolved splenic transcriptome data sets. We show that: (i) DCQ predictions fit well with the current knowledge of the immune cell dynamics during acute and chronic LCMV-infections, (ii) the combination of WGCNA and DCQ allows to better characterize the dynamic cell events occurring in complex tissues, and (iii) during the evolution toward the chronic infection state, the chemokine XCL1 is produced by CD8<sup>+</sup> T cells that express markers of early effector cells. Together this demonstrates the utility of combining DCQ and WGCNA for analyzing complex RNA-Seq data sets.

# MATERIALS AND METHODS

### Animals, Infections, and Depletion of Macrophages

Male C57BL/6J mice aged 4–8 weeks were purchased from Charles River Laboratories and maintained under specific pathogen-free conditions at the animal facility of the Parc de Recerca Biomèdica de Barcelona (PRBB). Animals were treated according to the Guidelines of the Basel Declaration and from the Generalitat de Catalunya (project number 9422), and approved by the ethical committee for animal experimentation (CEEA-PRBB, Spain; permit license number JMR-16-0046). Animals were infected intraperitoneally (i.p.) with either 2 × 10<sup>2</sup> (low dose, LD) or 2 × 10<sup>6</sup> (high dose, HD) plaque forming units (PFU) of the strain Docile of LCMV (4–7 animals per group) to induce an acute or chronic infection, respectively. Viral titers from spleens of infected mice were determined on MC57 cells using focus-forming assay (12). For in vivo depletion of macrophages, mice were injected i.v. with 300 µl of clodronateloaded liposomes (Liposoma BV; 5 mg/ml) (13), or PBS-loaded liposomes as a control.

# Cell Surface and Intracellular Cytokine Staining by Flow Cytometry

For Flow Cytometry analysis and cell sorting, spleens were harvested and single-cell suspensions were generated. Cells were then stained with the following antibodies to analyze B cells, and effector and regulatory T cells: CD4-PE (Clone H129.19), CD8-PECy5 (Clone 53-6.7), CD8a-PercpCy5.5 (Clone 53-6.7), CD25-APCCy7 (Clone PC61), CXCR5-PECy7 (Clone SPRCL5), CD83-Alexa Fluor 488 (Clone Michel-19), CD199- BV421 (Clone CW-1.2), CD153-BV421 (Clone RM153), CD19- FITC (Clone 1D3), CD43-PE (Clone eBioR2/60), CD5-APC (Clone 53-7.3), IgM-PECy7 (Clone II/41), CD23-eFluor450 (Clone B3B4), XCL1-Unconjugated (Clone 80222), mouse antirat IgG2a-Alexa Fluor 647 (Clone 2A8F4), IFNÈ-FITC (Clone XMG1.2), FOXP3-Alexa Fluor 647 (Clone MF23), and the polyclonal TLR7-FITC. To analyze monocyte/macrophage and neutrophil populations, cells were stained with CD3e-PECy7 (Clone 145-2C11), NK1.1-PECF594 (Clone PK136), CD11b-APC (Clone M1/70), and CD27-FITC (Clone LG.7F9) for natural killer T cells, and with CD45R-PECF594 (Clone RA3-6B2), NK1.1-PECF594 (Clone PK136), CD11c-PercpCy5.5 (Clone HL3), CD11b-PECy7 (Clone M1/70), Ly-6G-PE (Clone 1A8), and Ly-6C-FITC (Clone AL-21). For determination of XCL1- and TLR7-producing T cells, splenocytes were directly put into media containing Brefeldin A (Sigma Aldrich) without stimulation before intracellular cytokine staining (ICS). Staining of FOXP3 expressing cells was performed following the manufacturer instructions (eBiosciences). To visualize IFN production, cells were first stimulated with LCMV gp33 peptide for 3 h followed with the addition of brefeldin A for 2 h. All antibodies were purchased from either BD Biosciences, eBioscience, Biolegend or R&D Systems. A LSR Fortessa (BD Biosciences) was used for flow cytometry and data were analyzed using FlowJo 10.1 software. A FACSAria II SORP (BD Biosciences) sorter was used for cell sorting. All samples were kept at 4◦C during cell sorting. Sort purity was >95% for all cell populations.

#### Digital Cell Quantifier (DCQ)

DCQ was performed as previously described (11). Briefly, the DCQ took as an input: (i) an immune cell compendium of transcriptional profiles, consisting of 213 different immune cell subsets and their corresponding cell surface markers; and (ii) differentially expressed genes from spleens from acute and chronic LCMV-infected mice (1). We used the glmnet R package (14) with the parameters α = 0.05, lambda.min.ratio = 0.2. To evaluate the robustness of the predicted results, DCQ was run 100 times using only a random collection of 50% of the cell types in the compendium on each run, resulting in 100 different solutions. Standard deviations were calculated across these 100 solutions. The robustness score (significance of a predicted change in quantity) was assessed by evaluating whether the sample of relative quantities is significantly different from zero (p-value score). Significantly changing cell types were defined as those whose –Log<sup>10</sup> p-value score was lower than −20 (cell decrease) or higher than 20 (cell increase) in at least one of the infections (**Supplementary Table 1**).

## ImmGen Data

To compare gene expression levels between early and late effector CD8<sup>+</sup> T cells, the tool "Population Comparison" from the ImmGen data browser (http://www.immgen.org/) was used. This tool provided a ranked table of genes that are always expressed in OVA-specific effector CD8<sup>+</sup> T cells analyzed 12 and 24 h post-infection with Listeria (LisOva) and never expressed in the same cells analyzed at days 5, 6, and 8 post-infection with Listeria (LisOva) or Vesicular stomatitis virus (VSVOva). Default thresholds from the ImmGen tool were used. To analyze the pattern of expression of the genes Xcl1, Tnfsf8, Tlr7, Ccr9, and Cd83 across OVA-specific CD8<sup>+</sup> T cells in the ImmGen compendium, we used the tool "My GeneSet" (http://www. immgen.org/) using Microarray V1 data set. Expression values were obtained as the log<sup>2</sup> of each gene expression value/average expression value of all genes.

## RNA-Sequencing and Bioinformatic Analysis

Total RNA from sorted cells from uninfected (2 pools of 2 mice, day 0) or acute (2 pools of 2 mice, day 0, day 7) infected mice (5 × 10<sup>4</sup> cells per sample) was isolated according to the manufacturer's instructions using Qiagen RNeasy Micro kit (Qiagen). The quality and concentration of RNA were determined by an Agilent Bioanalyzer. RNA was submitted for sequencing to Macrogen Inc. (Seoul, Korea). Sequencing libraries were obtained after removing ribosomal RNA by a Ribo-Zero kit (Illumina). cDNA was synthesized and tagged by addition of barcoded Truseq adapters. Libraries were quantified using the KAPA Library Quantification Kit (KapaBiosystems) prior to amplification with Illumina's cBot. Four libraries were pooled and sequenced (single strand, 50 nts) on an Illumina HiSeq2000 sequencer to obtain 50– 60 million reads per sample. RNA-Seq reads were mapped to the reference mouse genome (GRCm38, gencode M18) with STAR (15) and genes were quantified with RSEM (16). Differential expression analysis was performed with DESeq2 (17). Genes with a false discovery rate (FDR)<5% were considered significant. Gene ontology (GO) enrichment analysis was performed with DAVID (http://david.ncifcrf.gov/) (18).

# Statistical Analysis

Statistical analyses were performed using GraphPad Prism software version 6.0 (GraphPad Software Inc., CA, USA). Data were analyzed using non-parametric one-way ANOVA or twotailed t-test. For correlation between modules and DCQ-inferred cell kinetics, Pearson's correlation was used. Fisher's exact test was used to quantify the significance of gene overlap between acute-brown module hub genes and genes from CD8<sup>+</sup> T cells and monocytes/macrophage cell subsets. Non-significant differences were indicated as "ns." P-values (p) below 0.05 were considered significant and were indicated by asterisks: <sup>∗</sup>p ≤ 0.05; ∗∗p ≤ 0.01; ∗∗∗p ≤ 0.001; ∗∗∗∗p ≤ 0.0001.

# Data Access

The complete RNA-Seq datasets are available from the Gene Expression Omnibus (accession number GSE123134).

# RESULTS

# Immune Cell Dynamics During Acute and Chronic LCMV Infection

To obtain a global view of the biological processes that participate in and control acute and chronic LCMV infection fates, we created a new computational approach that combines WGCNAderived gene coexpression networks with DCQ-inferred immune cell kinetics. As input, we used our previously generated RNA-Seq data set (1) that consists of time-resolved splenic transcriptomes from C57BL/6J mice infected with a low-dose (2 × 10<sup>2</sup> PFU; acute infection) or a high-dose (2 × 10<sup>6</sup> PFU; chronic infection) of LCMV strain Docile (LCMVDoc) (**Figure 1**). The time points [days 0, 3, 5, 6, 7, 9, and 31 post-infection (p.i.)] were selected according to the main viral and immunological features, and therefore represent the main states of an acute and a chronic LCMV infection. Thirteen thousand nine hundred seventy-one genes were identified as differentially expressed (DE) when compared to uninfected animals, and were analyzed by WGCNA to obtain modules of highly coexpressed genes (1) (**Figure 1**).

To predict the immune cell dynamics during acute and chronic LCMV infection, we used the expression kinetics of the DE genes as an input for DCQ (11). The DCQ output consisted of the suggested kinetics of 207 different immune cell subsets. Of these, 125 cell subsets had a significant change in their quantity between at least two consecutive time points (robustness score higher/lower than ± 20, see methods). A comprehensive map of the dynamic changes of these 125 cell subsets during the infection courses is shown in **Figure 2**. Sixty-eight cell subsets were predicted to increase and 57 were predicted to decrease in both, acute and chronic infection (**Supplementary Table 1**). Note that the different cell subsets are named according to the nomenclature of the immune cell compendium which was used to establish DCQ (11). The respective names are also used below and given in brackets when referring to the different subsets as of **Figure 2**.

Effector CD8<sup>+</sup> T cells play a critical role during LCMV infections. They control virus expansion in acute infection while CD8<sup>+</sup> T cell exhaustion is a hallmark of chronic

infection (19). After an acute LCMVDoc infection, virusspecific CD8<sup>+</sup> T cells expand at d6-d7 and their percentages remained high at d31. In contrast, during a chronic infection, IFN È-producing CD8<sup>+</sup> T cells drop in their numbers at d7-d9 (**Supplementary Figure 1A**), and maintain an elevated expression of inhibitory receptors such as PD-1 and TIM-3

module, we performed a Pearson's correlation analysis of the module eigengene with DCQ-inferred cell kinetics, and novel hypotheses are generated.

(1). In order to verify that DCQ can correctly predict changes in the dynamics of immune cell subsets with major roles during LCMV infection, we first focused on the DCQinferred kinetics of effector CD8<sup>+</sup> T cells. The original compendium of immune cells used as input for the DCQ contains effector CD8<sup>+</sup> T cells obtained at 5, 6, and 8 days post-infection with Listeria (T.CD8+EFF, OT-I & LIS) or Vesicular stomatitis virus (T.CD8+EFF, OT-I & VSV). DCQ correctly predicted an increase of these effector CD8+ T cells in both acute and chronic LCMV infections (**Figure 2** and **Supplementary Figures 1A,B**). Importantly, DCQ also predicted exhaustion in chronic infection, showing a drastic

decrease of these effector cells between days 7 and 9 postinfection with LCMV (**Supplementary Figure 1B**). Moreover, the kinetics of memory CD8<sup>+</sup> T cells (T.CD8+EFF, OT-I & VSV-15 days, and T.CD8+MEM, OT-I & LIS-100 days) also showed the failure of chronically infected mice to generate a memory T cell response, in contrast to acute infected mice (**Figure 2** and **Supplementary Figure 1B**).

DCQ also correctly predicted the changes in immune cell quantities of several other cell subsets with a specific role in chronic LCMV infection. For example, CD4<sup>+</sup> regulatory T cells (T.CD4+FP3+CD25+) only increased late in chronic infection (day 31 p.i.), as previously described (20) and further validated by flow cytometry (**Supplementary Figure 2A**). Two subsets of conventional dendritic cell (cDC) expressing the marker CD103 showed an increase from day 9 in chronic infection (cDC.CD103+CD11b–) (**Supplementary Figure 2B**). These CD103<sup>+</sup> CD11b<sup>−</sup> DCs that were sampled for the immune cell compendium from intestine, are also present in other tissues such as the spleen. They express CD8 and the chemokine receptor XCR1, and are specialized in antigen cross-presentation (21). Thus, the predicted DC kinetics likely represents the appearance of XCR1<sup>+</sup> DCs that we have recently described to contribute to the maintenance of an antiviral cytotoxic T cell response and viral control during the chronic infection phase (1). Finally, DCQ also predicted a transient increase of two neutrophil subsets (GN.ARTH) in acute infection that only in chronically infected mice remained elevated at day 31 p.i. (**Supplementary Figure 2C**). These neutrophils, which were monitored from arthritic mice in the immune cell compendium, likely represent the previously reported appearance of neutrophilic suppressor cells which have an immunomodulatory role during chronic infections (22).

Other predicted immune cell subset kinetics showed a similar overall behavior in acute and chronic infected mice. For example, despite previous reports that attributed different roles to NK cells in the two infection outcomes (23, 24), activated NK cells (NK and NK.H+, MCMV) showed a similar kinetic in acute and chronic infection (**Figure 2** and **Supplementary Figure 3A**), with an early peak at days 5 and 3 p.i., respectively. The predicted increase of activated NK cells was validated by analyzing the kinetics of NK cells at different maturation states by staining cells with anti-CD11b and anti-CD27 antibodies (25). Interestingly, only activated effector NK cells coexpressing these two surface marker showed the kinetics as predicted by DCQ while immature NK cells differed (**Supplementary Figure 3B**). This nicely demonstrates the ability of DCQ to predict the quantity of immune cell subsets in a particular functional state. DCQ-predictions of monocyte kinetics were also validated by flow cytometry, showing a rapid increase of inflammatory Ly6c<sup>+</sup> monocytes followed by an increase of resident Ly6c<sup>−</sup> monocytes at later time points in both acute and chronic infection (**Supplementary Figures 3C,D**). Finally, B cells showed a decrease in numbers in both acute and chronic infections (**Figure 2** and **Supplementary Figure 4**), in agreement with previous publications that reported a type I IFN- or NK-mediated depetion of B cells in LCMV infection (26–28).

# WGCNA-Derived Modules Representing T Cell Responses Correlate With Effector CD8<sup>+</sup> T Cells and Macrophages

Using the same RNA-Seq data set from acute and chronic LCMV infections, our group previously generated spleen transcriptomederived coexpression modules by WGCNA (1). This analysis provided relevant information about biological processes playing a major role in response to virus-induced host perturbations.

However, the gene coexpression analysis of total organ RNA did not provide information about the cell subsets that participate in the expression of the genes within the coexpression modules. In order to decipher which immune cell subsets are involved in spleen-derived gene coexpression modules, we hypothesized that, in some circumstances, the kinetics of a set of coexpressed genes will correlate with the kinetics of the cell subsets expressing them. On this basis, we performed a Pearson's correlation analysis between WGCNA-derived module eigengenes (1) and DCQ-inferred cell kinetics (**Figure 1**). We first analyzed which DCQ-inferred immune cell subset kinetics from acute infection correlated with the acute-brown module eigengene. This module was previously identified as the representative of the LCMVspecific CD8<sup>+</sup> T cell response induced in acute infection. Its eigengene expression kinetics highly correlated with the LCMV-specific CD8<sup>+</sup> T cell response and the 315 hub genes within the module revealed an enrichment for T cell activation genes (1). Twenty-three out of the one hundred twenty-five immune cell subsets inferred by DCQ showed a significant positive correlation (p < 0.05) with the acute-brown module eigengene (**Supplementary Table 2**). As expected, effector CD8<sup>+</sup> T cells (T.CD8+EFF, OT-I; monitored at days 5, 6, and 8 postinfection with LIS or VSV) showed correlation scores above 0.9 (**Figure 3A** and **Supplementary Table 2**), thus indicating that our approach correctly predicts the immune cell subsets responsible for the expression of genes within the module. Interestingly, several monocyte and macrophage cell subsets also showed high correlation scores (**Figures 3A,B**). To test whether these cells subtypes also express genes contained in the acute-brown module, we performed RNA-Seq analyses of sorted monocytes/macrophages and CD8<sup>+</sup> T cells from naive mice, and

animals infected with a low dose of LCMVDoc (acute infection). A total of 5291 genes were significantly upregulated at day 7 p.i. in activated CD44<sup>+</sup> CD8<sup>+</sup> T cells compared to CD44<sup>−</sup> CD8<sup>+</sup> T cells from uninfected naive mice. Monocytes/macrophages showed 3,520 genes significantly upregulated at day 7 p.i. compared to the cells from uninfected mice. To analyze whether the genes within the acute-brown module were significantly enriched for genes from these two cell subsets, we determined the gene overlap between the module hub genes and the genes upregulated in CD8<sup>+</sup> T cells and monocytes/macrophages by a Fisher's Exact Test (**Figure 4A**). The acute-brown module was highly enriched for genes upregulated in both cell subsets. From the 315 hub genes within the module, 113 overlapped with genes upregulated in activated CD8<sup>+</sup> T cells (p < 3.6 × 10−<sup>7</sup> ) and were enriched for genes involved in the processes of TCR signaling pathway, T cell activation and IL4 production (**Figure 4B** and **Supplementary Table 3A**). Importantly, 182 hub genes overlapped with genes upregulated in monocytes/macrophages (p < 1.6 × 10−16) and were enriched for genes involved in T cell response, TGF-β signaling and leukocyte migration, among others (**Figure 4C** and **Supplementary Table 3B**), thus indicating that the acute-brown module represents the complex process of induction of the adaptive T cell response that requires the coordination of monocytes/macrophages and CD8<sup>+</sup> T cells. To validate this hypothesis further, we analyzed CD8<sup>+</sup> T cell response and virus loads in spleens from acutely infected mice after depletion of macrophages (**Figure 5A**). Mice treated with clodronate liposomes showed a significant decrease in percentages of IFNÈ-producing cells and an increase of virus loads at day 8 p.i. (**Figure 5B**), thus demonstrating that macrophages contribute to the induction of the T cell response.

All together, these results demonstrate that the combination of DCQ and WGCNA is a very valuable tool to better characterize immune cell subsets that participate in a complex biological pathway represented by a gene coexpression module.

## XCL1-Producing Cells During the Adaptation Phase to a Chronic Infection Present an Immature Early Effector Phenotype

Analysis of infection-fate-specific modules in Argilaguet et al. (1) allowed to identify the biological pathways specific of chronic infection. In particular, the chronic-darkturquoise module contains the chemokine XCL1, showing a "two-peak" behavior with an expression peak at day 5 and a second upregulation from day 7 to day 9 p.i., at the time when exhaustion of CD8<sup>+</sup> T cells appears (**Figure 6B** and **Supplementary Figure 1A**). We showed that XCL1 expression resulted in the recruitment of crosspresenting dendritic cells that express the XCL1 receptor XCR1, and that these dendritic cells contributed to the maintenance of the antiviral cytotoxic T cell response and viral control in the chronic infection phase. XCL1 was mainly produced by LCMVspecific CD8<sup>+</sup> T cells expressing CXCR5, a marker of exhausted CD8<sup>+</sup> T cells that retain effector functions (29, 30). However, due to the complexity of effector and exhausted CD8<sup>+</sup> T cell subpopulations present during a chronic infection (4), a detailed phenotypic characterization of XCL1-producing CD8<sup>+</sup> T cells was lacking.

In order to better characterize the phenotype and activation state of XCL1-producing CD8<sup>+</sup> T cells in chronic LCMV infection, we used our approach to analyze which DCQ-inferred immune cell subset kinetics from chronic infection correlated with the chronic-darkturquoise module eigengene. Interestingly, only CD8<sup>+</sup> T cell subsets isolated at 12 and 24 h post-Listeria infection (T.CD8+EFF, OT-I & LIS-12 and 24 h) showed a

ImmGen (http://www.immgen.org/). (B) Mean fluorescence intensity (MFI) of Tnfsf8, Tlr7, Ccr9, and Cd83 in XCL1<sup>−</sup> and XCL1<sup>+</sup> CD8<sup>+</sup> T cells in naive mice and in chronically-infected animals at day 9 p.i. Significant differences were determined by one-way ANOVA. ns, non-significant; \*p ≤ 0.05; \*\*p ≤ 0.01; \*\*\*\*p ≤ 0.0001.

significant correlation (**Figure 6** and **Supplementary Table 2**), suggesting that XCL1-producing CD8<sup>+</sup> T cells have an immature early effector phenotype. Using the "Population Comparison" tool from the Immunological Genome (ImmGen) Project, we next analyzed which genes are upregulated in these two cell subsets compared to late effector CD8<sup>+</sup> T cells (T.CD8+EFF, OT-I; monitored at days 5, 6, and 8 post-infection with LIS and VSV; see methods). Three hundred eighty-five genes showed expression values significantly higher in early vs. late effector CD8<sup>+</sup> T cells. To note, within them we found XCL1 and CXCR5, with a mean fold-change of 25.14 and 4.94, respectively, further indicating that XCL1 is produced by CD8<sup>+</sup> T cells with a phenotype characteristic of immature early effector CD8<sup>+</sup> T cells. To validate this hypothesis further, we analyzed protein expression characteristic for early effector CD8<sup>+</sup> T cells in chronically infected mice at day 9 by flow cytometry. Using the "My GeneSet" tool from ImmGen, we selected four proteins with high gene expression values in early effector T cells: TNFSF8, a cytokine that induces proliferation of T cells and that has been shown to be upregulated in CD8<sup>+</sup> T cells primed during chronic infection (31); TLR7, a receptor selectively upregulated by exhausted CXCR5<sup>+</sup> CD8<sup>+</sup> T cells (29); CCR9, a chemokine receptor that regulates early phases of inflammation (32); and CD83, which is upregulated upon T cell stimulation during virus infection (33) (**Figure 7A**). Expression of TNFSF8, TLR7 and CD83 by XCL1-negative CD8<sup>+</sup> T cells was similar between naive mice and infected animals at day 9 p.i. (**Figure 7B**). By contrast, CCR9 showed higher expression levels in naive mice, in concordance with data from the ImmGen dataset, in which CCR9 showed a high gene expression value in naive OT1 cells (**Figure 7A**). Importantly, TNFSF8, TLR7 and CCR9 were highly expressed by XCL1<sup>+</sup> CD8<sup>+</sup> T cells at day 9 p.i., showing mean fluorescence intensities (MFI) significantly higher than that in XCL1<sup>−</sup> cells (**Figure 7B**). These results demonstrate that during the adaptation phase to a chronic infection, XCL1 is produced by CD8<sup>+</sup> T cells with characteristic early effector cell marker. All together, our results demonstrate that by combining WGCNA with DCQ, it is possible to define cell—cell cooperativity as well as the activation and differentiation status of cells participating in the immune response to virus infections.

#### DISCUSSION

Here we describe a versatile computational approach that combines WGCNA-derived gene coexpression networks with DCQ-inferred immune cell kinetics. It enables to generate hypotheses about complex immune system functioning after a virus-induced perturbation. We show, first, the ability of DCQ to predict changes of immune cell subsets that play major roles during an LCMV infection. Second, we characterize immune cell subsets involved in spleen-derived gene coexpression modules that cooperate in complex biological pathways. Finally, we predict and subsequently verify experimentally that virus-specific CD8<sup>+</sup> T cells in the chronic infection phase resemble early effector cells.

To derive information about the immune system functioning after a perturbation or under pathological conditions solely from transcriptome data of whole tissue specimen is challenging. Algorithms like DCQ (11), seq-ImmuCC (34), or CoD (35) can predict dynamic quantities of cell subtypes, however they fail to distinguish between immunological mechanisms like cell migration or cell differentiation. Gene coexpression analysis like WGCNA, on the other hand, can order the several thousand genes from RNA-Seq runs according to common dynamic features however cannot predict cell types. With the combination of both, hypothesis generation becomes easier and goes beyond what both methods can provide by their own. For example, we could identify macrophages as an important immune cell type that cooperates with CD8<sup>+</sup> T cells in the induction of an adaptive immune response (**Figures 4**, **5**). It is known that dendritic cells are not critical for CD8<sup>+</sup> T cell priming in LCMV (36). Moreover, they decline in numbers shortly after infection (**Figure 2**) (22). Since other antigen presenting cells like macrophages and B cells are also infected by LCMV (37–39), and can efficiently present virus-derived peptide in conjunction with MHC-I proteins on their surface, they seem to contribute to the activation of the CTL response and thereby participate in the control of virus expansion. Furthermore, we were able to characterize an important immunological event during chronic infection that appear concomitant with exhaustion. Indeed, we demonstrate that XCL1-producing CD8<sup>+</sup> T cells have an early effector phenotype and differ fundamentally from the effector cells during acute infections. This observation is a step forward in our understanding of the immune adaptation process to a chronic infection. However, further work is necessary to decipher whether these cells emerge from "de novo" primed naive CD8<sup>+</sup> T cells or from exhausted cells that "recover" an effector functional state. Thus, by combining WCGNA with DCQ one can identify cell cooperativity and specify cellular phenotypes participating in critical biological events during perturbations.

The novel combination of WGCNA and DCQ for interpreting time-resolved transcriptome data from acute and chronic virus infections is a step forward for our understanding of the complex immune responses during pathogen invasion. However, it is just a tiny part of the overall process that needs to be complemented with other measures like in situ imaging techniques, single cell analyses and whole organism studies (2, 6, 40). Furthermore, computational approaches will be necessary to integrate all available data and generate hypotheses about the underlying regulatory principles that make the highly complex, diverse and dynamic immune system so functionally robust against pathogens. While many of the required technologies are in place, the data integration will require tight collaborations across disciplines engaging biologists, clinicians, physicists, and mathematical modelers. With this, one can easily envision predictive frameworks that will help in the rational design of therapies in infectious diseases and cancers. An exciting time lies ahead.

# ETHICS STATEMENT

This study was carried out in accordance with the recommendations of the Guidelines of the Basel Declaration and from the Generalitat de Catalunya (project number 9422). The protocol was approved by the ethical committee for animal experimentation (CEEA-PRBB, Spain; permit license number JMR-16-0046).

# AUTHOR CONTRIBUTIONS

JA and AM designed the study. MP, GR, VC, and CS performed experiments. MP, AE-C, YS, SH, and IG-V performed the bioinformatic analyses of the RNA-Seq data sets including WGCNA and DCQ. MP and JA accessed the bioinformatic data output. JA, MP, GB, and AM interpreted the data and wrote the manuscript. All authors read and approved the final manuscript.

#### FUNDING

This work is supported by a grant from the Spanish Ministry of Economy, Industry and Competitiveness and FEDER grant no. SAF2016-75505-R (AEI/MINEICO/FEDER, UE) and the María de Maeztu Programme for Units of Excellence in R&D (MDM-2014-0370). GB and AM are also supported by the Russian Science Foundation (grant 18-11-00171). AE-C and SH

#### REFERENCES


are also supported by Instituto de Salud Carlos III (ISCIII) grant from the Spanish Ministry of Economy, Industry and Competitiveness and FEDER grant no. PT17/0009/0019. YS is supported by the Edmond J. Safra Center for Bioinformatics, Tel Aviv University and Shulamit Aloni Scholarship, Ministry of Science and Technology. IG-V is supported by European Research Council (637885).

#### SUPPLEMENTARY MATERIAL

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fimmu. 2019.01002/full#supplementary-material


**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2019 Pedragosa, Riera, Casella, Esteve-Codina, Steuerman, Seth, Bocharov, Heath, Gat-Viks, Argilaguet and Meyerhans. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# Mathematical Modeling Reveals That the Administration of EGF Can Promote the Elimination of Lymph Node Metastases by PD-1/PD-L1 Blockade

Mohamed Amine Benchaib<sup>1</sup> \* † , Anass Bouchnita2†, Vitaly Volpert 3,4,5,6 and Abdelkader Makhoute1,7

<sup>1</sup> Faculté des Sciences, Université Moulay Ismail, Meknes, Morocco, <sup>2</sup> Division of Scientific Computing, Department of Information Technology, Uppsala University, Uppsala, Sweden, <sup>3</sup> Institut Camille Jordan, Université Lyon 1, Villeurbanne, France, <sup>4</sup> INRIA Team Dracula, INRIA Lyon La Doua, Villeurbanne, France, <sup>5</sup> Peoples Friendship University of Russia (RUDN University), Moscow, Russia, <sup>6</sup> Marchuk Institute of Numerical Mathematics of the RAS, Moscow, Russia, <sup>7</sup> Faculty of Sciences, Université Libre de Bruxelles (ULB), Brussels, Belgium

#### Edited by:

Marcelo R. S. Briones, Federal University of São Paulo, Brazil

#### Reviewed by:

Fernando Martins Antoneli Jr., Federal University of São Paulo, Brazil Nona Janikashvili, Tbilisi State Medical University, Georgia

#### \*Correspondence:

Mohamed Amine Benchaib benchaibma@gmail.com

†These authors have contributed equally to this work

#### Specialty section:

This article was submitted to Bioinformatics and Computational Biology, a section of the journal Frontiers in Bioengineering and Biotechnology

> Received: 08 February 2019 Accepted: 24 April 2019 Published: 14 May 2019

#### Citation:

Benchaib MA, Bouchnita A, Volpert V and Makhoute A (2019) Mathematical Modeling Reveals That the Administration of EGF Can Promote the Elimination of Lymph Node Metastases by PD-1/PD-L1 Blockade. Front. Bioeng. Biotechnol. 7:104. doi: 10.3389/fbioe.2019.00104 In the advanced stages of cancers like melanoma, some of the malignant cells leave the primary tumor and infiltrate the neighboring lymph nodes (LNs). The interaction between secondary cancer and the immune response in the lymph node represents a complex process that needs to be fully understood in order to develop more effective immunotherapeutic strategies. In this process, antigen-presenting cells (APCs) approach the tumor and initiate the adaptive immune response for the corresponding antigen. They stimulate the naive CD4<sup>+</sup> and CD8<sup>+</sup> T lymphocytes which subsequently generate a population of helper and effector cells. On one hand, immune cells can eliminate tumor cells using cell-cell contact and by secreting apoptosis inducing cytokines. They are also able to induce their dormancy. On the other hand, the tumor cells are able to escape the immune surveillance using their immunosuppressive abilities. To study the interplay between tumor progression and the immune response, we develop two new models describing the interaction between cancer and immune cells in the lymph node. The first model consists of partial differential equations (PDEs) describing the populations of the different types of cells. The second one is a hybrid discrete-continuous model integrating the mechanical and biochemical mechanisms that define the tumor-immune interplay in the lymph node. We use the continuous model to determine the conditions of the regimes of tumor-immune interaction in the lymph node. While we use the hybrid model to elucidate the mechanisms that contribute to the development of each regime at the cellular and tissue levels. We study the dynamics of tumor growth in the absence of immune cells. Then, we consider the immune response and we quantify the effects of immunosuppression and local EGF concentration on the fate of the tumor. Numerical simulations of the two models show the existence of three possible outcomes of the tumor-immune interactions in the lymph node that coincide with the main phases of the immunoediting process: tumor elimination, equilibrium, and tumor evasion. Both models predict that the administration of EGF can promote the elimination of the secondary tumor by PD-1/PD-L1 blockade.

Keywords: PD-1/PD-L1, immune response, cancer immune interaction, hybrid modeling, multiscale modeling

# 1. INTRODUCTION

Malignant cells commonly infiltrate the local, regional, and distant lymph nodes during the advanced stages of primary cancers (Dowlatshahi et al., 1997). The lymph node represents a major component of the lymphatic system and the organ where the cytotoxic T-cells (CTLs) are produced. While these cells can leave the lymph node and eliminate the tumor cells at the site of the primary tumor, they can directly eradicate the tumor cells in the lymph node upon their infiltration. The immune response begins when circulating antigen-presenting cells (APCs) capture tumor antigens and present them to the naive CD4<sup>+</sup> and CD8<sup>+</sup> T-cells (cross-presentation). These two cells undergo a series of asymmetric divisions culminating in the generation of mature helper and cytotoxic cells (CTL) (Chang and Reiner, 2008). The fate decision of immune cells strongly depends on the intracellular concentrations of interleukin-2 (IL-2) (Khan et al., 2015) and type I interferon (IFN) (Welsh et al., 2012). Mature CD4<sup>+</sup> T-cells produce IL-2 which upregulates the maturation of T-cells while antigen-presenting cells (APCs) secrete type I IFN which increases their division. CD8<sup>+</sup> eliminate the tumor cells by inducing their apoptosis through the secretion of cytokines such as Fas-Ligand (FasL). They can also inject these cytokines directly into the tumor cells during cell-cell contact.

In cancer, cells acquire mutations that affect their genetic landscape and make them proliferate excessively. One of the main pathways that are commonly altered in cancer is the EGFR/ERK pathway (Sebolt-Leopold, 2008). The final product of this pathway, the ERK protein, becomes necessary for the proliferation of the cell upon its translocation to the nucleus. The most commonly observed mutations in the MAPK/ERK pathway concern the K-RAS, N-RAS, and B-RAF genes. Such alterations can be observed especially in secondary tumors like melanoma and lung cancer (Burotto et al., 2014).

Malignant cells can resist the immune response using different strategies such as dormancy and immune suppression. Tumor cells can survive longer in the LN as they become resistant when they are in the quiescent state. There are different mechanisms governing the dormancy of the proliferating cells. First, tumor cells may enter the quiescent state when faced by a lack of available growth factors or extracellular matrix (ECM) proteins. This stress-induced dormancy is typically observed when the ERK/p38 ratio of the cell becomes low. The cell can become once again proliferating when the same ratio becomes sufficiently high. The ECM proteins, such as fibronectin and collagens, promote the activation of dormant cells due to the cross-talk between integrins, urokinase receptor (uPAR), and EGFR (Bragado et al., 2012). The complex formed by α1β5 integrins and uPAR recruits the EGFR and FAK proteins which regulates the EGFR/p38 ratio in a fibronectindependent manner (Barkan and Chambers, 2011). The effect of the ECM proteins on tumor dormancy is especially interesting in the case of secondary tumor development in the lymph nodes. These organs consist of distinct regions with different densities of the ECM proteins. The outer region of the lymph node contains follicles and the interfollicular zone. The ECM proteins (fibronectin, collagen, laminins) are abundant in the interfollicular area and less expressed in the follicles (Castaños-Velez et al., 1995).

Another mechanism that can cause the quiescence of the tumor cells is the immune-induced dormancy (Romero et al., 2014). In this process, effector CD8<sup>+</sup> T-cells secrete type II IFN which induces and maintains the dormancy of tumor cells (Farrar et al., 1999). To escape immuno-surveillance, the malignant cells may resort to the inactivation of neighboring Tcells using immunosuppressive mechanisms. One of these most effective techniques used by tumor cells is the activation of the programmed-death 1 (PD-1) receptor present on the surface of T-cells (Zitvogel and Kroemer, 2012). After the interaction of PD-1 with its ligand PD-L1 present on the surface of tumor cells, the T-cells reduce its production of cytokines that induce apoptosis and becomes incapable of division. Therefore, the inhibition of the PD-1/PD-L1 pathway represents one of the most effective immunotherapies (Alsaab et al., 2017). Ultimately, the balance between these different mechanisms defines the three stages of immunoediting: tumor elimination, equilibrium, and tumor escape (Dunn et al., 2002).

To our knowledge, there is no mathematical model describing the interaction between secondary tumor progression and the adaptive immune response in the lymph node. Most of the existing mathematical models concern the cancer-immune dynamics at the site of the primary tumor. These models adopted different techniques and methods depending on the question of the study. The first type of developed models uses ordinary differential equations (ODEs) to simulate the population of cells over time. The simplest form of these models consists of two equations describing the competition between tumor and immune cells in a similar way to prey and predator models. In these models, tumor cells represent prey while immune cells represent predator (dOnofrio, 2005; Fory´s et al., 2006). These models can be used to describe dynamics of this interaction under normal conditions (Michelson et al., 1987) and also during chemotherapy (Fory´s et al., 2006; dOnofrio, 2008). Other models include more details, and therefore more equations, such as various subpopulations of immune cells (De Pillis and Radunskaya, 2003), diffusing cytokines (Dranoff, 2004; de Pillis et al., 2005), or tumor dormancy (Page and Uhr, 2005; Wilkie and Hahnfeldt, 2013). Overall, the strength of these dynamical systems is that they can be both analyzed mathematically and simulated numerically. Among the other non-spatial models, stochastic ODEs are often used to study the effect of fluctuations and noise on the interaction between tumor and immune cells (Lefever and Horsthemke, 1979). Another class of immunecancer interaction models considers partial differential equations (PDEs) to describe the spatiotemporal aspect of this interplay. In this context, diffusion terms can be added to the previous ODEs models to capture the mobility of cells. In these models, the spatial densities of cells and concentrations of cytokines can be both described by the same type of PDEs (Bellomo et al., 2004; Matzavinos et al., 2004). To describe the interaction between immune and tumor cells in more detail, agent-based models are considered. The most commonly used agent-based modeling framework to describe this specific problem is cellular automata (CA) models (Qi et al., 1993; Mallet and De Pillis, 2006). Finally, it is possible to build more sophisticated models of tumorimmune interaction by coupling agent-based and continuous models. The resulting models allow the description of the different mechanisms affecting the behavior of the system at different scales (Gong et al., 2017). One of these models combined CA with PDEs to simulate different growth regimes that can result from the cancer-immune interplay (Mallet and De Pillis, 2006). The fate of each cell is given by a probability which depends on the local concentration of cytokines, and the number of neighboring cells. However, this study is restricted to the interaction between cancer and innate immunity. Furthermore, it does not include the dormancy of tumor cells which plays an important role in the survival and evasion of tumors. These models are stochastic by design and can be used to study the effects of immunotherapeutic treatments such as PD-1 and PD-L1 inhibitors.

This study is devoted to the mathematical modeling of the cancer-immune system interactions in the lymph node. To perform quantitative numerical simulations and ensure an accurate description of the system, we develop two models complementing each other. The first model is deterministic and uses four PDEs to describe the densities of proliferating tumor cells, dormant tumor cells, immune cells, and the concentration of a growth factor. The second model reported in this study belongs to the hybrid discrete-continuous class. In this model, cells are represented as individual objects (soft spheres) that can move, divide, differentiate, and die by apoptosis. The fate decision of each cell depends on the concentrations of intracellular proteins and extracellular cytokines described, respectively, by ODEs and PDEs. We begin with the description of tumor growth dynamics in the absence of immune surveillance. Then, we introduce the immune response and quantify the combined effect of PD-1/PD-L1 inhibition and EGF concentration on the outcome of the tumor-immune interaction in the LN. Both models confirm the existence of the three regimes characterizing the immunoediting process: tumor elimination, equilibrium, and tumor evasion. Furthermore, they reveal that combined anti-PD-1/PD-L1 therapy with growth factors can be administered in order to eradicate the tumor.

#### 2. MATHEMATICAL MODELING OF THE TUMOR-IMMUNE INTERACTION IN THE LYMPH NODE

To capture the dynamics of tumor-immune interaction, we develop two models describing this complex process. The first model adopts a population dynamics approach and uses PDEs to describe the densities of different types of cells. The model is deterministic and computationally cheap which makes it appropriate for quantitative studies. The second model is a hybrid-discrete continuous model where cells are represented as individual objects. These cells can move, autorenew, differentiate, or die by apoptosis. Their fate is regulated by the concentration of intracellular and extracellular proteins described by ODEs and PDEs. This multiscale model is complex and computationally expensive. However, it is also more realistic as it integrates the most important mechanisms regulating the dynamics of cells. Several assumptions were considered during the development of the two models. First, we consider that the dormancy of tumor cells is mediated exclusively by the lack of EGF and the exposition to type II IFN, which is secreted by CD8<sup>+</sup> T-cells. Others mechanisms that induce the dormancy of tumor cells are not included in the present model. Second, the model is restricted to the interplay between the tumor and the adaptive immune response in the lymph node. Therefore, the interaction with the innate immune response is not captured in the two models. We have represented the main interactions characterizing the models and provided a screenshot of the hybrid discrete-continuous one in **Figure 1**.

# 2.1. Continuous Model of the Spatiotemporal Dynamics of Tumor-Immune Interaction

After their infiltration to the lymph node, tumor cells find themselves in the direct contact with the immune cells. We propose the following population dynamics model describing the interaction between malignant cells and immune cells in the lymph node. Let us consider the spatial variables x and y as well as the temporal variable t, we simulate the evolution of the epidermal growth factor (EGF, e<sup>g</sup> (x, y, t)) and three populations of cells: proliferationg tumor cells (cp(x, y, t)), quiescent tumor cells (cq(x, y, t)), and immune effector cells (i(x, y, t)). The model is solved in a 2D computational domain representing the lymph node. It studies the interactions that exist between secondary cancer and immune cells in the LN without considering explicitly the underlying biological mechanisms. We begin with the equation describing the concentration of the epidermal growth factor in the lymph node:

$$\frac{\partial e\_{\mathcal{g}}}{\partial t} = D\_1 \Delta e\_{\mathcal{g}} - k\_1 (c\_{\mathcal{p}} + c\_{\mathcal{q}}) e\_{\mathcal{g}} - k\_2 e\_{\mathcal{g}},\tag{1}$$

where the first term in the right-hand side of this equation describes EGF diffusion, the second term represents EGF consumption by tumor cells, and the third term describes its degradation. Next, we describe the density of proliferating tumor cells:

$$\begin{split} \frac{\partial c\_{\mathcal{P}}}{\partial t} &= D\_{\mathcal{D}} \Delta c\_{\mathcal{P}} + k\_{\mathcal{S}} (e\_{\mathcal{S}}) c\_{\mathcal{P}} (1 - (c\_{\mathcal{P}} + c\_{\mathcal{q}})) \\ &+ \frac{k\_{\mathcal{A}} (e\_{\mathcal{S}}) c\_{\mathcal{q}}}{1 + K\_{\mathcal{A}} i} - k\_{\mathcal{S}} c\_{\mathcal{P}} i - k\_{\mathcal{C}} c\_{\mathcal{P}} i - k\_{\mathcal{T}} (e\_{\mathcal{S}}) c\_{\mathcal{P}} - k\_{\mathcal{S}} c\_{\mathcal{P}}. \end{split} \tag{2}$$

Here, the first term in the right-hand side represents the motility of cancer cells. The second term characterizes the logistic growth of the tumor cell population which depends on the local concentration of EGF. We consider that this rate correlates linearly with the density of the growth factor and we set k3(e<sup>g</sup> ) = k ∗ 3 e<sup>g</sup> , where k ∗ 3 is a positive constant. The third term represents the activation of dormant tumor cells which is also promoted by e<sup>g</sup> . Similarly to the k3(e<sup>g</sup> ), we set k4(e<sup>g</sup> ) = k ∗ 4 e<sup>g</sup> . This activation can also be inhibited by effector immune cells whose density is

the same ratio becomes high. Cytotoxic immune cells can induce the dormancy of tumor cells. They can also eliminate these cells by secreting apoptosis-inducing cytokines. On the other hand, tumor cells can disable the CTLs using their immunosuppressive abilities. (B) A screenshot of a simulation using the hybrid model showing the proliferating and dormant tumor cells (magenta and brown, respectively), the APCs (green), the naive T-cells (white and black), the helper CD4<sup>+</sup> T-cells (yellow), and the effector CD8<sup>+</sup> T-cells (blue). The concentrations of extracellular cytokines are not shown.

denoted by i. The fourth and fifth terms describe the elimination and the induced dormancy by the immune cells, respectively. Here we consider that the cytotoxic cells directly induce the dormancy of cells by secreting type II IFN (Katsoulidis et al., 2005). The sixth term represent the natural dormancy while the last term represents cell apoptosis. Prolfierating cells enter the quiescent phase when there is a lack of EGF. Thus, we consider a negative correlation between the rate of dormancy and the local concentration of EGF k7(e<sup>g</sup> ) = k ∗ 7 (1 − e<sup>g</sup> ). Next, we describe the population of quisecent tumor cells (cq) as follows:

$$\frac{\partial c\_q}{\partial t} = -\frac{k\_4(e\_\emptyset)c\_q}{1 + K\_4 i} + k\_6 c\_p i + k\_7(e\_\emptyset)c\_p - k\_9 c\_q,\tag{3}$$

where the term − k4(eg )cq 1+K4i describes the activation of dormant cells. The terms k6c<sup>p</sup> and k7(e<sup>g</sup> )c<sup>p</sup> describe induced and normal dormancies, respectively. The last term represents cell apoptosis. We suppose that dormant cells are more resistant to elimination by immune cells and live much longer than proliferating cells. Therefore, the rate of apoptosis for dormant cells (k9) is taken much lower than the one for proliferating cells k8. Finally, we describe the population of cytotoxic T-cells in the lymph node as follows:

$$\frac{\partial i}{\partial t} = D\_3 \Delta i + k\_{10} (i^0 - i)(c\_p + c\_q) - k\_{11} i (c\_p + c\_q) - k\_{12} i. \tag{4}$$

As before, we describe cell motion with a diffusion term. The second term in the right-hand side of this equation represents the activation of naive T-cell lymphocytes by tumor antigens. The third term describes the elimination of immune cells by immunosuppression. The rate of immunosuppression depends on the PD-L1 expression of tumor cells. We set k<sup>11</sup> = k ∗ <sup>11</sup>Ksupp where Ksupp is the level of PD-L1 expression on the surface of tumor cells. The last term corresponds to cell apoptosis.

We consider a square computational domain of 25 mm × 25 mm which corresponds approximately to the maximum size reached by enlarged lymph nodes. Proliferating cells are initially located inside a circular domain (cp0(x, y) = 1 for p x <sup>2</sup> + y 2 < 75 µm) for all the simulations. We set the value of e<sup>g</sup> to be constant as an initial condition for the concentration of EGF and we prescribe the same value as the Dirichlet boundary condition at all boundaries (e<sup>g</sup> = eg0). We use the zero-flux condition at all boundaries for the populations of tumor cells and immune cells ( ∂cp <sup>∂</sup><sup>n</sup> <sup>=</sup> 0, <sup>∂</sup>c<sup>q</sup> <sup>∂</sup><sup>n</sup> <sup>=</sup> 0, and <sup>∂</sup><sup>i</sup> <sup>∂</sup><sup>n</sup> = 0). The finite difference method was used for the numerical implementation of the system. The values of parameters are provided in **Table A1**.

#### 2.2. A Hybrid Discrete-Continuous Model for Multiscale Modeling of Tumor Growth in the Lymph Node

Cancer-immune interaction is based on several mechanisms affecting cells at different scales. Here, we formulate a discretecontinuous multiscale model to describe the interaction between cancer cells and immune cells in the lymph node. We have previously used hybrid models to study various physiological systems such as erythropoiesis (Eymard et al., 2015; Bouchnita et al., 2016b), multiple myeloma (Bouchnita et al., 2016a, 2017a), the immune response (Bouchnita et al., 2017b), and HIV infection (Bouchnita et al., 2017c). The hybrid model is based on some hypotheses. The regulation of tumor cells is assumed to depend solely on the EGFR/ERK, p38, and Fas signaling pathways. Other pathways such as TGF-β and PI3K-Akt are not considered in the present model. To properly present this new model, we divide it into two submodels, one for the immune response and the other one for the tumor development. Let us begin with the description of the displacement of individual cells because all cells are subject to the same mechanical laws of motion.

#### 2.2.1. Cell Motion

Each cell is characterized by the coordinates of its center x<sup>i</sup> as well as by its radius. While immune cells are supposed to move randomly in the computational domain, tumor cells do not move unless they are pushed by the surrounding cells. In the process of cell division, cells increase their radius and push the surrounding cells. Each cell consists of a compressible part corresponding to the cytoplasm and an incopressible part corresponding to the nucleus. We consider a repulsive force between each two cells when the distance between their centers hij is lower than the sum of their radii r<sup>1</sup> + r2. The motion of each cell is described by Newton's second law:

$$m\ddot{\mathbf{x}}\_i + \mu \dot{\mathbf{x}}\_i - \sum\_{j \neq i} f\_{ij} - F\_i^r = \mathbf{0},\tag{5}$$

where m is the mass of the particle, µ is the friction factor due to contact with the surrounding medium. F r i denotes a random force applied only to the immune cells. The repulsive force between two cells is given by the formula:

$$f\_{ij} = \begin{cases} K \frac{h\_0 - h\_{ij}}{h\_{ij} - (h\_0 - h\_1)} & , \ h\_0 - h\_i < h\_{ij} < h\_0 \\ 0 & , \quad h\_{ij} \ge h\_0 \end{cases},$$

where hij is the distance between the centers of the two cells i and j, h<sup>0</sup> is the sum of their radii, K is a positive parameter and h<sup>1</sup> is the sum of the incompressible parts of the two cell. The force between the cells tends to infinity if hij decreases to h<sup>0</sup> − h1.

#### 2.2.2. The Immune Response

We adapt the previously developed model of adaptive immune response (Bouchnita et al., 2017b,c) to the specific case of tumor growth in the lymph node. The model includes different types of immune cells such as APCs, naive and mature T lymphocytes.

#### **Cell division and differentiation**

Every 20 h of physical time, APCs and T-cells enter the computational domain around the tumor with given proportions when there is an available space. APCs capture tumor antigens as they become sufficiently close to a tumor cells. Then, they begin secreting type I IFN which promotes the differentiation of Tcells. They also present tumor antigens to naive T-cell receptors (TCRs) and induce the asymmetric divisions of T-cells. In this process, the distant daughter cell remains undifferentiated, the proximal daughter cell becomes differentiated. We consider two levels of maturation of CD4<sup>+</sup> T-cells and three levels for CD8<sup>+</sup> T-cells. As they reach the last maturation stage, the CD4<sup>+</sup> Tcells become helper cells and start secreting IL-2. The CD8<sup>+</sup> Tcells develop into cytotoxic T-cells that can kill the tumor cells either by cell-cell contact or by secreting FasL. They can also induce the dormancy of tumor cells by secreting type II IFN. Differentiated CD8<sup>+</sup> and CD4<sup>+</sup> T-cells can die by apoptosis or by immunosuppression.

We suppose that cells start increasing their radii as they reach half of their life cycle. If the cell divides, then two daughter cells will appear. The direction of the axis connecting their two centers is chosen randomly between 0 to 2π. The cell cycle duration for each cell is considered to be 18 h with stochastic perturbation uniformly distributed between −3 and 3 h.

#### **Intracellular regulation**

Activated CD4<sup>+</sup> and CD8<sup>+</sup> T-cell lymphocytes can only survive when there is a sufficient amount of signaling via their IL-2 and type IFN receptors. This signaling depends on the concentration of these two cytokines in the proximity of the corresponding receptors. To describe the intracellular dynamics of these two molecules, we use ODEs that depend on the value of the extracellular cytokines at the vicinity of the cell.

Let us begin with the IL-2 dependent regulatory signal dynamics in individual cells. We can describe it by the following equation:

$$\frac{dI\_i}{dt} = \frac{\alpha\_1}{n\_T} I\_e(\chi\_i, t) - d\_1 I\_i. \tag{6}$$

Here I<sup>i</sup> is the intracellular concentration of signaling molecules accumulated as a consequence of IL-2 signals transmitted through transmembrane receptor IL2R downstream the signaling pathway to control the gene expression in the ith cell. The first term in the right-hand side of this equation shows the cumulative effect of IL-2 signaling. The extracellular concentration I<sup>e</sup> is taken at the center of the cell (xi). The second term describes the degradation of IL-2-induced signaling molecules inside the cell, n<sup>T</sup> is the number of molecules internalized by T cell receptors.

Similarly, we describe the IFN-α dependent regulatory signal dynamics in individual cells as follows:

$$\frac{dC\_i}{dt} = \frac{\alpha\_2}{n\_T} C\_e(\chi\_i, t) - d\_2 C\_i. \tag{7}$$

Here C<sup>i</sup> is the intracellular concentration of signaling molecules accumulated as a consequence of IFN-α signals transmitted through transmembrane receptor IFN-αR downstream the signaling pathway to control the gene expression in the i-th cell. The first term in the right-hand side of this equation shows the cumulative effect of IFN-α signaling. The extracellular concentration C<sup>e</sup> is taken at the center of the cell x<sup>i</sup> . The second term describes the degradation of IFN-induced signaling molecules inside the cell.

Finally, we describe the activation of the cell surface receptor PD-1 as a function of the local concentration of its ligand PD-L1:

$$\frac{dPD\_i}{dt} = \alpha\_3 PD\_\epsilon(\chi\_i, t)(1 - PD\_i) - d\_3 PD\_i. \tag{8}$$

As before, the first term in the right-hand side of this equation represents the cumulative effects of PD-1 activation by PD-L1, PDe(x<sup>i</sup> , t) is the sum of PD-L1 expression by surrounding tumor cells. The second term describes the inactivation of the PD-1 receptors at the cell surface.

Overall, the fate of each T-cell depends on the gene activation threshold for different signaling such as TCR, IL-2, IFNa, and PD-1 as shown in **Figure 2**. We consider the following decision mechanism to describe the fate regulation of activated T-cells as a function of the IL-2, type I IFN, and PD-1 signaling at different stages of the cell cycle.

C1 At the beginning of cell cycle: if the concentration of activation signals induced by type I IFN, C<sup>i</sup> , is greater than

some critical level C ∗ i and that of I<sup>i</sup> , is smaller than the critical level I ∗ i , then the cell will differentiate in a mature cell (Bouchnita et al., 2017b).


#### **Extracellular dynamics of cytokines**

The concentrations of IL-2 and type I IFN determines the differentiation and maturation of T-cells as described before. These two cytokines are produced by mature CD4<sup>+</sup> T cells and active antigen-presenting cells, respectively. We use reactiondiffusion equations to describe spatial distributions of their concentrations:

$$\frac{\partial I\_{\text{e}}}{\partial t} = D\_{IL}\Delta I\_{\text{e}} + W\_{IL} - b\_{1}I\_{\text{e}}.\tag{9}$$

Here I<sup>e</sup> denotes the extracellular concentration of IL-2 and D is the diffusion coefficient, WIL is the rate of its production by mature CD4<sup>+</sup> T cells, and the last term in the right-hand side of this equation describes its consumption and degradation.

Each mature CD4<sup>+</sup> T-cell secretes IL-2 in the lymph node. The production rate WIL only applies at the areas of the computational domain occupied by these cells. The consumption of IL-2 is considered implicitly in the degradation term.

We suppose that antigen-presenting cells secrete type I IFN upon their activation (through direct contact with tumor

antigens). The concentration of extracellular type I IFN is described by the same type of equation as IL-2:

$$\frac{\partial \mathcal{C}\_{\text{e}}}{\partial t} = D\_{\text{IFN}} \Delta \mathcal{C}\_{\text{e}} + W\_{\text{IFN}} - b\_2 \mathcal{C}\_{\text{e}}.\tag{10}$$

As before, the production rate WIFN equals ρIFN at the area filled by APC cells and zero otherwise. The consumption of type IFN is also considered implicitly in the degradation term.

We also consider that the cytotoxic cells (mature CD8<sup>+</sup> T cells) secrete Fas-Ligand (Fe). It is an apoptosis inducing cytokines that participate in the elimination of tumor cells. Fas-Ligand activates Fas receptors in tumor cells which induces their apoptosis. Its concentration in the extracellular matrix is described by the following equation:

$$\frac{\partial F\_{\text{e}}}{\partial t} = D\_{\text{F}L} \Delta F\_{\text{e}} + W\_{\text{F}L} - b\_3 F\_L. \tag{11}$$

We impose initial and boundary conditions for Equations (9–11).

#### 2.2.3. Tumor Growth

We present here the model describing secondary tumor development in the lymph node. The model includes two subtypes of malignant cells: proliferating and quiescent cells. The former can proliferate while the latter are more resistant to tumor elimination. A simplified representation of the considered intracellular regulation of tumor cells is provided in **Figure 3**. Each proliferating tumor cell can have three possible fates: proliferation, quiescence, and apoptosis. We consider that the tumor progression is mainly driven by the EGFR/ERK pathway because it is one of the most altered in secondary tumors (Burotto et al., 2014). These alterations are caused by gene mutations that upregulate the expression of ERK and cause the proliferation of the tumor cell. The most common mutations that affect this pathway are acquired by the K-RAS, N-RAS, and B-RAF genes.

As before for the immune cells, we describe the intracellular and extracellular mechanisms regulating the fate of cancer cells.

#### **Intracellular regulation**

We describe the intracellular concentration of three intracellular proteins: ERK (Ei), p38 (Pi), and Fas (Fi). ERK is the final product of the EGFR/ERK pathway. This signaling pathway is stimulated when the epidermal growth factor receptors present at the cell surface are activated. Subsequently, these receptors activate the Ras/Raf/MEK/ERK cascade which promotes cell proliferation (Li et al., 2016). To simplify the model, we only describe the intracellular concentration of the final product of this pathway ERK:

$$\frac{dE\_i}{dt} = \beta\_1 G\_F(\mathbf{x}\_i, t)(E^0 - E\_i) - \gamma\_1 P\_i E\_i - d\_4 E\_i. \tag{12}$$

Here we suppose that ERK activation depends on the extracellular concentration of the epidermal growth factor (EGF) denoted by GF(x<sup>i</sup> , t). The second term in the right-hand side of this equation represents the inhibition of active ERK by the protein p38. The last term describes the degradation of ERK. This protein is activated by the FAK protein which is recruited during a cross-talk between EGFR signaling, uPAR, and integrins α1β5 (Barkan and Chambers, 2011). Next, we describe the concentration of another important protein called p38. The ERK/p38 ratio plays an important role in regulating the dormancy of tumor cells. We describe the intracellular concentration of p38 as follows:

$$\frac{dP\_i}{dt} = \beta\_2 (G\_F^0 - G\_F(\chi\_i, t))(P^0 - P\_i) - d\_5 P\_i. \tag{13}$$

Here we consider that the lack of the cross-talk between EGFR, uPAR and integrins provokes the upregulation of p38 (Gao et al., 2012). The last term represents the degradation of p38. Another important protein whose signaling induce the dormancy of tumor cells is IFN-γ R. We describe its intracellular concentration as follows:

$$\frac{dB\_i}{dt} = \beta\_3 B\_\varepsilon(\chi\_i, t) - d\_6 B\_i,\tag{14}$$

where Be(x<sup>i</sup> , t) denotes the concentration of extracellular type II IFN at the center of the cell and d<sup>6</sup> is the degradation rate.

A similar equation is used for the concentration of Fas:

$$\frac{dF\_i}{dt} = \beta\_4 F\_e(\mathbf{x}\_i, t) - d\_7 F\_{i\bullet} \tag{15}$$

where the Fe(x<sup>i</sup> , t) represents the concentration of the extracellular FasL at the center of the cell and d<sup>7</sup> is the degradation rate.

We consider the following decision mechanism for the regulation of each tumor cell:


#### **Extracellular regulation**

ERK and p38 signaling depend on the local concentration of epidermal growth factor (EGF). We describe the normalized concentration of this growth factor as follows:

$$\frac{\partial G\_{\rm F}}{\partial t} = D\_{\rm G\rm F} \Delta G\_{\rm F} - W\_{\rm G\rm F} - b\_3 G\_{\rm F},\tag{16}$$

where DG<sup>F</sup> is the diffusion coefficient, WG<sup>F</sup> is the consumption rate by tumor cells, and b<sup>3</sup> is the degradation rate. We prescribe the Dirichlet boundary condition:

$$G\_F = G\_{F0\dots}$$

where GF<sup>0</sup> is a positive constant.

In addition, the fate of tumor cells depends on the activation rate of type II IFN (IFN-γ ) and Fas receptors on their surface. These receptors are activated upon their binding with the respective ligands. We describe the extracellular concentration of type II IFN (Be) as follows:

$$\frac{\partial B\_{\varepsilon}}{\partial t} = D\_{\text{IFN2}} \Delta B\_{\varepsilon} - W\_{\text{IFN2}} - b\_{4} B\_{\varepsilon},\tag{17}$$

where DIFN<sup>2</sup> is the diffusion coefficient, WIFN<sup>2</sup> is its production rate, and b<sup>4</sup> is the degradation rate. This protein is produced by effector CD8<sup>+</sup> T-cells and contributes to the dormancy of tumor cells. Initial and boundary conditions are set to zero for equations this equation.

#### 2.2.4. Computer Implementation

The code was written under C++ in the Object Oriented Programming style. The WxWidget library was used to visualize the simulations in real-time. The average CPU time of a numerical simulation of 3 weeks of tumor-immune interaction in the lymph node is 3 h on a computer with four cores and 6 GB of RAM. The source code is available upon request to one of the two first authors. To the opposite of the previously described continuous model, the hybrid model contains some random variables including the stochastic motion of immune cells and their introduction into the computational domain, the random direction of division, and fluctuations in the cell cycle duration. An example of numerical simulations with the hybrid model is shown in **Figure 4**. It shows a growing tumor surrounding by immune cells in the lymph node. The distributions of three cytokines are shown. The values of parameters are given in **Table A2**.

#### 3. RESULTS

#### 3.1. Model Validation

We begin by comparing the output of the continuous model with the available data. Experimental results describing the effects of PD-L1 on tumor evasion (Juneja et al., 2017) were used to calibrate the model. Although these experiments describe the development of primary colorectal adenocarcinoma outside of the lymph node, they are still useful for our study because mutated MC38 cells commonly migrate to neighboring lymph

nodes in the advanced stages of the disease. The aim of these experiments was to elucidate the effect of PD-1 blockade on tumor evasion. To achieve this, PD-1 and PD-L1 blocking antibodies were administrated to mice with MC38. The study shows that the immunosuppressive effect of the PD-1/PD-L1 pathway is sufficient to cause the evasion of the tumor.

The parameters of the model were calibrated to reproduce the experiments presented in **Figure 1B** of Juneja et al. (2017). The volume of the tumor was calculated using the formula V = 4 3 πr 3 , where r is the radius of the tumor. The tumor growth rate (k ∗ 3 ) was fitted to reproduce the growth of the wild-type MC38 tumor. Then, we determined the rate of immunosuppression (k ∗ <sup>11</sup>) in such a way that the administration of a PD-1 inhibitor corresponds to a reduction of the level of PD-1 by 85 % (Ksupp = 0.15). With these modifications, the model is able to accurately reproduce the experimental data as shown in **Figure 5**.

#### 3.2. Tumor Growth Dynamics in the Absence of Immune Response

After the calibration of the model, we proceed to study its dynamics in the absence of immune response (i <sup>0</sup> = 0). Then the system can be reduced to three Equations (1–3).

FIGURE 5 | The effect of PD-1 blockade on the growth of the tumor. Results of the numerical simulations using the continuous model are compared with experimental data in Figure 1B of Juneja et al. (2017) (shown in dots).

We solve them numerically for different parameter values. Since the EGFR pathway represents a possible target for cancer treatment, we quantify the effect of the EGF concentration in the lymph node (eg0) on the development of the tumor. Numerical simulations show the existence of two different scenarios. When the concentration of EGF is below the threshold value e ∗ <sup>g</sup><sup>0</sup> = 0.55, the tumor does not develop and will be eliminated within few weeks depending on the available concentration of the EGF (**Figure 6**).

When the EGF concentration in the lymph node exceeds the threshold value e ∗ g0 , the tumor cells increase their proliferating rate. Tumor growth is exponential in the beginning and linear after some time. The tumor expands in the form of a traveling wave and invades the neighboring tissue (**Figure 7**). The speed of wave propagation increases as the value of EGF in the lymph node grows. We conclude that there exist two possible regimes of tumor development in the absence of immune response: elimination and evasion. It is possible to obtain these two regimes by varying other parameters of the model, for example, depending on the proliferation and apoptosis rates of tumor cells determined by their phenotypes. The rate by which tumor cells enter and leave the quiescent state also determines the regime of tumor development and can vary depending on different conditions such as hypoxia and TGF-β signaling.

The two regimes of tumor development in the absence of immune response can also be obtained with the hybrid model.

To achieve this result, we consider an initial population of 20 cells and we run numerical simulations for different values of the EGF concentration in the LN (GF0). When this concentration is greater than or equal to the threshold value G ∗ <sup>F</sup><sup>0</sup> = 35 nM, the tumor grows and expands in the form of a traveling wave. When it becomes sufficiently large, the consumption of EGF at the core of the tumor will induce the dormancy of cells in this area (**Figure 8A**). Hence, a tumor spheroid organization composed of proliferating cells in the outer region and dormant cells at the core can be observed. In the regime of tumor escape, the ratio of dormant cells to proliferating cells becomes higher as the EGF concentration decreases (**Figure 8B**). When the EGF level is below the threshold value G ∗ F0 , the tumor cells will either die by apoptosis or will keep switching back and forth between the dormant and proliferating state until they die.

#### 3.3. The Regimes of Tumor-Immune Interaction in the Lymph Node in the Continuous Model

After studying the dynamics of tumor growth in the absence of the immune response, we now introduce the immune cells to the model (i <sup>0</sup> = 1), and we numerically solve the system (1–4) for different values of the model parameters. Immune cells affects the dynamics of tumor development by different mechanisms. First, they can eliminate tumor cells by secreting apoptosis-inducing cytokines such as FasL. Second, they can induce the dormancy of proliferating tumor cells by secreting type II interferon (IFN-γ ). Finally, they can prevent the reactivation of dormant cells with the same mechanism (Farrar et al., 1999). It is therefore important to study the outcome of the tumor-immune interplay under different conditions.

Depending on the parameters of the model, it is possible to observe three different regimes of tumor development. In addition to the previously described regimes that can be observed in the absence of immune response (tumor elimination and escape), one more regime can be obtained when the immune response is considered. This regime corresponds to the cancerimmune equilibrium state where the tumor and the immune cells coexist in the lymph node. In this case, immune cells cannot eradicate the tumor and they do not allow it to further develop and expand. This regime can be observed only if two conditions are fulfilled: immune cells should prevent tumor growth and the activation of dormant tumor cells. Under these two conditions, a pulse-shaped stationary solution is reached after several days of the tumor progression (**Figure 9**). The solution can still slowly evolve because of the low apoptosis rate of the tumor cells.

In addition to the EGFR pathway, PD-1 and PD-L1 constitute a potential target in cancer therapy. By inhibiting their immunosuppressive abilities, tumor cells become vulnerable to immune cells and therefore can die by apoptosis during the early stages of tumor development. While it is possible to study the effect of PD-1 expression on the dynamics of the cancerimmune interaction, it is more interesting to investigate the combined effects of PD-1 (Ksupp) and EGF (eg0) on this process (**Figure 10**). Numerical simulations reveal that the equilibrium regime can only be observed for the values of eg<sup>0</sup> between 0.1 and 0.9. The escape scenario can be avoided if the EGF concentration in the LN is below 0.7. The elimination of the tumor can be obtained if the immunosuppressive abilities of the tumor are very low. Interestingly, it is also possible to eradicate the tumor if the value of EGF concentration in the LN is above 0.9 and the Ksupp value is below 0.3. Thus, the model predicts that a possible combination of PD-1/PD-L1 inhibitors with exogenous EGF can be administrated

in order to overcome the drug resistance provoked by cell dormancy.

# 3.4. The Hybrid Model Reveals the Mechanism Regulating the Regimes of the Tumor-Immune Interplay in the Lymph Node

To elucidate the mechanisms underlying the interaction between secondary tumors and the immune response in the lymph node, we use the hybrid model to conduct numerical simulations under various conditions. We begin by studying the regime of tumor escape in the case where the concentration of EGF in the LN is equal to GF<sup>0</sup> = 100 nM and we introduce 50 tumor cells to the computational domain as an initial condition. At the beginning of the simulation, the tumor expands rapidly, and the tumor cells use their immunosuppressive abilities to escape the surveillance of the immune cells. However, some parts of the tumor are still eliminated due to the secretion of FasL by the surviving CD8<sup>+</sup> Tcells. As a result, the tumor grows faster toward the directions where the presence of effector T-cells is low and the shape of the tumor becomes irregular and nonspherical (**Figure 11A**). The ratio of dormant to proliferating cells becomes higher in the presence of immune cells than in their absence (**Figure 11B**). This can be explained by the induced dormancy or elimination of a part of the proliferating cells by the effector CD8<sup>+</sup> T-cells and released cytokines. Still, the tumor manages to dismantle the immune response and to develop using the immunosuppressive capacity of malignant cells. This is reflected by the low number of effector CD8<sup>+</sup> T-cells despite the high number of antigen-bearing APCs as shown in **Figure 11C**.

Next, we study the regime of tumor elimination by the immune response by maintaining the same settings as in the previous simulation and reducing the expression of PD-L1 per cell by decreasing PDe/cell from 0.5 to 0.02. During the first 4–5 days, the tumor grows and expands without being affected by the immune response. This period corresponds to the necessary time for the adaptive immune response to be activated following the exposure to tumor-antigens. The inhibition of immunosuppression prevents the tumor from evading the surveillance of immune cells. As a result, a relatively important number of effector CD8 T-cells will be produced and eliminate all tumor cells within several hours (**Figure 12A**). Effector T-cells eradicate the tumor by secreting apoptosis inducing cytokines such as FasL that diffuse in the ECM. They can also directly transfer these cytokines into tumor cells during cell-cell contact. During the whole simulation, tumor cells will be proliferating except for few cells that appeared at the beginning of the fourth day. These cells quickly leave the quiescent state because they are exposed to growth factors (**Figure 12B**). In the absence of immunosuppression, the number of effector CD8<sup>+</sup> T-cells will become high after the first 4 days while antigen-bearing APCs will extinct after the elimination of the tumor, as revealed in **Figure 12C**. It is important to note that the low expression of PD-L1 by malignant cells is not sufficient to observe the elimination of the tumor. With current model settings (PDe/cell = 0.02), this regimes can only be observed if the EGF concentration is higher than 70 nM or lower than 10 nM. This supports the conclusion drawn from simulations of the continuous model which implies that the administration of EGF can result in the elimination of the tumor by anti-PD-1/PD-L1 agents.

The regime of tumor-immune equilibrium can only be observed when the concentration of EGF in the LN is reduced. We set GF<sup>0</sup> = 50 nM and we suppose that tumor cells have a normal PD-L1 expression (PD<sup>e</sup> = 0.5). In this case, the cells start dividing, and then they enter the quiescent state due to the lack of EGF and the presence of type II IFN secreted by CD8<sup>+</sup> T-cells. They remain dormant for the rest of the simulation time (**Figure 13A**). As a result, the population of tumor cells remains constant (**Figure 13B**). Due to the low concentration of EGF, the cells can be maintained in a dormant state if they are constantly supplied by type II IFN. **Figure 13C** shows that a few numbers of effector CD8<sup>+</sup> T-cells remain in proximity of the tumor during the simulation. However, these cells cannot eliminate the tumor cells because of the resistance acquired due to dormancy. Hence, they only maintain the tumor cells in a dormant state by producing type II IFN. The role of CD8<sup>+</sup> T-cells in maintaining tumor cells in a dormant state was investigated in a previous in vivo experimental study (Farrar et al., 1999).

# 4. DISCUSSION

This study presents two models of the tumor-immune interaction inside the lymph node. The first model uses a population dynamics approach to study the spatial dynamics of the interplay between tumor and immune cells while the second model follows a multiscale approach for a more realistic representation of the mechanisms involved in this process. The continuous model was calibrated in order to reproduce the results of an experimental study on the effects of PD-1 in tumor evasion (Juneja et al., 2017). These experiments describe tumor-immune interaction outside of the LN. However, they were still useful for the identification of the model parameters for several reasons. First, the growth rates of primary and secondary tumors are approximately the same (Peng et al., 2018; Zhang and Niedermann, 2018). Second, the

parameters of the PDE model are kinetic constants that describe the rates of cellular processes such as division, dormancy, and apoptosis. These processes depend on the phenotype of the cell and not the location of the tumor. Finally, CD8 T-cells migrate to the site of the site of the tumor regardless if it is inside or outside the LN (Chheda et al., 2016). The hybrid model also qualitatively confirms the conclusions of this study by showing that PD-1 is sufficient to cause the evasion the evasion of the tumor. However, both models reveal that this conclusion is only valid in tissues with very low or very high concentration of EGF in the LN. Indeed, the higher the concentration of EGF in the LN, the higher the number of proliferating cells in the tumor, and therefore the more responsive the tumor will be to anti-PD-1/PD-L1 therapy. This result represents a testable hypothesis that can be considered in the design of future experimental studies and clinical trials. The two models were used in parallel to study an important question: what are the possible outcomes of the interaction between secondary cancer and the immune

system in the lymph node. To investigate this point, we began by studying the dynamics of tumor growth in the absence of immune cells and we have shown, using both models, that the resulting tumor will have a spheroid shape with two layers: a proliferating zone in the outside layer and a quiescent zone in the inside layer. This agrees with the previous in vivo, in vitro, and in silico studies (Weiswald et al., 2015; Sant and Johnston, 2017). This spheroid organization depends on the access of tumor cells to growth factors. In both continuous and hybrid models, we have demonstrated the existence of a threshold value for the EGF concentration in the LN that separates the regimes of tumor evasion and tumor elimination.

Next, we have introduced immune cells in the model and studied their interaction with the growing tumor. In particular, we have studied the role of the immuno-suppressive mechanism in the dynamics of tumor progression in tissues with different EGF concentrations. We have shown that immunosuppression plays an important role in the evasion of the tumor from immuno-surveillance. Furthermore, we observed three different regimes of tumor growth in the lymph node using both the continuous and hybrid models. These regimes consist of the elimination of the tumor, the cancer-immune equilibrium, and the evasion of the tumor. They are already reported in the biomedical literature, and they constitute the three main phases of the immunoediting process (Dunn et al., 2004). Therefore, the parameters of the model can be tuned to study the interaction between the immune response and LN metastases of other phenotypes of malignant cells in the normal conditions or during immunotherapy. Despite the differences between the two models, the same mechanisms were considered by both of them. In the continuous model, the kinetic rates of tumor dormancy and activation depend on the local concentration of EGF and the density of immune effector T-cells. In the hybrid model, these mechanisms were introduced explicitly by considering that the state of the cell depends on the ratio ERK/p38 as well as type II IFN secreted by mature CD8<sup>+</sup> T-cells. Overall, the two models allowed us to determine the conditions of each regime of tumorimmune interaction. This is important for the development of personalized immunotherapeutic strategies that consider the characteristics of individual patients and the site of metastatic tumors. In the future, we will use these two models to study the efficacy and safety of the treatment regimens applied to the therapy of secondary tumors in the lymph node. We will focus especially on the combination PD-1 and PD-L1 inhibitors with other chemotherapeutic agents.

The synergy between the continuous and hybrid model was essential for a proper representation of the cancer-immune interaction in the lymph node. On one hand, the continuous model is better suited for systemic parametric studies because numerical simulations are computationally cheap. Mathematical analysis of the model will be presented in a forthcoming work. On the other hand, the hybrid model, although computationally expensive, provides a more detailed description of the various mechanisms involved in the cancer immune interaction problem. Hence, it is possible to conduct more biologically accurate studies using this model. However, this model is less robust because it contains numerous sources of stochasticity such as low number

#### REFERENCES


of cells, their random motion, and the duration of the cell cycle. Thus, it is possible to observe two different regimes of tumor growth when repeating the same simulation twice and with the same parameter set.

In general, there exist few hypotheses that were considered while formulating the model. The interpretation of the obtained results was possible because of these considered a priori assumptions. First, the interaction between secondary tumors and the innate immunity in the LN was not studied in the present work. Tumor cells can evade the surveillance of natural killer (NK) cells by reducing the expression of major histocompatibility complex (MHC) class I molecules. Other considered assumptions include the simplifications that were introduced to the signaling pathways responsible for inducing the dormancy of tumor cells such as the PI3K-AKT cascade. In our models, we assume that only EGFR/ERK, p38 and Fas signaling pathways that are disturbed for tumor cells. Thus, we restrict the intracellular regulation of tumor cells to these three pathways. Furthermore, we do not include the other mechanisms that induce the dormancy and reactivation of tumor cells such as stress, hypoxia, angiogenesis, and other microenvironmental factors. Despite the recent advances in our understanding of the factors inducing the dormancy of cells, the decision mechanisms by which cells enter and leave the quiescent state remain poorly understood. Therefore, more experimental and modeling studies should focus on this particular area of research.

#### AUTHOR CONTRIBUTIONS

All authors contributed to the design of the study and the writing of the manuscript. AB and VV developed the mathematical models. MB and AB conducted the numerical simulations. All authors read and approved the final manuscript.

# FUNDING

The work was partially supported by the RUDN University Program 5-100, the Russian Science Foundation grant number 18-11-00171, and the French-Russian project PRC2307.


**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2019 Benchaib, Bouchnita, Volpert and Makhoute. This is an openaccess article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# APPENDIX

In this section, we present the numerical values that were assigned to the parameters of the two model. The values of the continuous model were identified by fitting the results with experimental data Juneja et al. (2017) (**Table A1**). For the hybrid model, the values of the immune response component of the model were taken from previous works Bouchnita et al. (2017b,c). The rest of parameters were fitted in order to obtain the desirable behavior of the model. Their values are given in **Table A2**.



TABLE A2 | Values of parameters used in simulations with the hybrid model. δ is an arbitrary length unit.


# Role of T Cell-To-Dendritic Cell Chemoattraction in T Cell Priming Initiation in the Lymph Node: An Agent-Based Modeling Study

#### Ivan Azarov <sup>1</sup> , Kirill Peskov 1,2, Gabriel Helmlinger <sup>3</sup> and Yuri Kosinsky <sup>1</sup> \*

<sup>1</sup> M&S Decisions, Moscow, Russia, <sup>2</sup> Computational Oncology Group, I.M. Sechenov First Moscow State Medical University of the Russian Ministry of Health, Moscow, Russia, <sup>3</sup> Clinical Pharmacology & Safety Sciences, AstraZeneca, Boston, MA, United States

#### Edited by:

Gennady Bocharov, Institute of Numerical Mathematics (RAS), Russia

#### Reviewed by:

Christian Kurts, University of Bonn, Germany Raffaele De Palma, Università degli Studi della Campania Luigi Vanvitelli Caserta, Italy

> \*Correspondence: Yuri Kosinsky yuri.kosinsky@msdecisions.ru

#### Specialty section:

This article was submitted to T Cell Biology, a section of the journal Frontiers in Immunology

Received: 12 February 2019 Accepted: 21 May 2019 Published: 11 June 2019

#### Citation:

Azarov I, Peskov K, Helmlinger G and Kosinsky Y (2019) Role of T Cell-To-Dendritic Cell Chemoattraction in T Cell Priming Initiation in the Lymph Node: An Agent-Based Modeling Study. Front. Immunol. 10:1289. doi: 10.3389/fimmu.2019.01289 The adaptive immune response is initiated in lymph nodes by contact between antigen-bearing dendritic cells (DCs) and antigen-specific T cells. A selected number of naïve T cells that recognize a specific antigen may proliferate into expanded clones, differentiate, and acquire an effector phenotype. Despite growing experimental knowledge, certain mechanistic aspects of T cell behavior in lymph nodes remain poorly understood. Computational modeling approaches may help in addressing such gaps. Here we introduce an agent-based model describing T cell movements and their interactions with DCs, leading to activation and expansion of cognate T cell clones, in a two-dimensional representation of the lymph node paracortex. The primary objective was to test the putative role of T cell chemotaxis toward DCs, and quantitatively assess the impact of chemotaxis with respect to T cell priming efficacy. Firstly, we evaluated whether chemotaxis of naïve T cells toward a nearest DC may accelerate the scanning process, by quantifying, through simulations, the number of unique T cell—DC contact events. We demonstrate that, in the presence of naïve T cell-to-DC chemoattraction, a higher total number of contacts occurs, as compared to a T cell random walk scenario. However, the forming swarm of naïve T cells, as these cells get attracted to the neighborhood of a DC, may then physically restrict access of additional T cells to the DC, leading to an actual decrease in the cumulative number of unique contacts between naïve T cells and DCs. Secondly, we investigated the potential role of chemotaxis in maintaining cognate T cell clone expansion. The time course of cognate T cells number in the system was used as a quantitative characteristic of the expansion. Model-based simulations indicate that inclusion of chemotaxis, which is selective for already activated (but not naïve) antigen-specific T cells, may strongly accelerate the time of immune response occurrence, which subsequently increases the overall amplitude of the T cell clone expansion process.

Keywords: T cells, dendritic cells, lymph node, chemotaxis, agent-based modeling

# INTRODUCTION

After maturation in the thymus, immunologically-naïve T lymphocytes (or T cells) continuously circulate between the blood and secondary lymphoid organs, including lymphatic nodes (LNs) and the spleen. Each one of the millions of T cell clones bear unique T cell receptors (TCRs), which define their antigen specificity. In LNs, naïve T cells may encounter dendritic cells (DCs) presenting cognate antigens as MHC-bound peptides (pMHC) on their surface. As a result of such a specific and durable contact T cell-to-DC contact, a naïve T cell may become activated and subsequently proliferate and differentiate into effector forms. This constitutes, in most simplified terms, the essence of the immune response. The fraction of naïve T cells that recognize a particular antigen can be as small as 10−<sup>5</sup> -10−<sup>6</sup> (1). Since most naïve T cells feature irrelevant specificities, the probability of an immediate contact between a DC bearing a particular antigen and a cognate T cell appears to be very low. Therefore, for efficient antigen recognition, each DC should be in a position to scan a large number of T cells with differing specificities.

Over the past two decades, experiments using two-photon microscopy (2PM) have been applied in the study of murine LNs in vivo and have provided a rich set of T cell migration characteristics, as well as information on T cell interactions with antigen-presenting DCs (2). Fibroblastic reticular cells (FRC) form a spatial network throughout the T zone, which is used by DCs as an adhesion scaffold, while T cells use this network as an overall routing system underlying their random migration process. As elucidated from 2PM observations, naïve T cells move with a mean free path of 20–30µm, interrupted by a change in direction every 2–3 min—a process which, over time, results in a migratory pattern which roughly resembles a "random walk" process (3). During their journey through the LN, naïve T cells are involved in short contacts with DCs, lasting several minutes on average (4–6). DCs migrate slower than T cells, and continuously expand and retract long thin dendrites, thereby significantly increasing the volume of the region they may efficiently scan (6).

Intravital LN observations have shown that cognate T cell interactions with antigen-presenting DCs can be categorized into several stages—with may possibly overlap over time (5, 7): (1) within the first 6 h: transient, serial encounters lasting 10–20 min and upregulation of T cell activation markers; (2) subsequently, and within 14 h: stable binding events lasting for hours, and initiation of cytokine production; (3) consequently, rapid motility followed by short contacts (10–20 min) with DCs, ultimately resulting in T cell proliferation. These observations point to processes of T cells integrating TCR signaling over serial DC contacts, with stage transitions occurring as signal thresholds are being reached. T cell priming in the lymph node spans over 3–4 days, a period after which clonally expanded T cells exit the LN via medullary sinuses (MS) and efferent lymphatics to disseminate in peripheral organs.

Despite such detailed observations, there is no comprehensive understanding, yet, of the detailed mechanisms and dynamics of immune cell interactions; in particular, the fate of individual cells is difficult to track for longer periods of time in vivo. As reviewed in (8), methods of computational biology can be used to integrate knowledge, to then simulate cellular dynamics which occur in the LN. In this modeling study, we explored factors influencing specifically the efficiency of T cell repertoire scanning and further expansion of rare cognate T cell clones in a LN. Toward this purpose, we developed a two-dimensional (2D) computational model of T cell–DC interactions and subsequent activation events. In this virtual lymph node, T cell migration, contact dynamics, signal integration and cell division were simulated while computationally tracking the contribution of multiple parameters influencing the properties and functional outcome of T cells, DCs, antigens. In particular, we sought to answer the following questions: (1) May local chemoattraction of naïve T cells toward the nearest DC accelerate the DC scanning process? We chose to quantify this process by modeling the number of unique T cell—DC contact events that occur per time unit, and tracked the evolution of that number over time; (2) May local chemoattraction of activated cognate T cells toward a DC influence T cell expansion efficiency? To this end, the time course of cognate T cell numbers in the virtual LN system was computationally tracked, as a measure of such immune response dynamics.

# MATERIALS AND METHODS

# Description of the Computational Agent-Based Model

Overall, an agent-based model (ABM) represents a system of interest, with a definition of key players and of relevant interactions among these players that influence the system's behavior. A typical ABM consists of a simulation space (world), stand-alone objects (agents), and rules to set the behavior of individual agents as well as interactions among them (rules). Thus, in our 2D ABM framework, we consider T cell motility and emerging interactions of immune cells within the LN T zone. The T zone was modeled as a lattice of 100 × 100 patches, resulting in an effective physical surface area of 500 × 500 µm<sup>2</sup> (i.e., 5 × 5 µm<sup>2</sup> per patch). This modeling framework also considers two types of agents: T cells and antigen-presenting DCs.

In order to reproduce interactions between agents present in the LN T zone, 2,000 naïve T cells were randomly placed in this square domain, along with 8 antigen-bearing DCs, randomly placed in 8 fixed positions, as shown in **Figure 1**. Each 5 × 5 µm<sup>2</sup> patch was set to contain, at most, one T cell. DCs are typically larger than T cells; thus, it was assumed that a given DC occupies 5 patches, thereby forming cross patterns as shown in **Figure 1**. Such an initial geometric design would allow each DC to interact with up to 11 neighboring T cells simultaneously.

For simplification, antigen-presenting DCs were assumed to be immobilized, given their migration speed in LNs vs. that of T cells is relatively slow (6), also recognizing that DCs may be more fully anchored onto the reticular network vs. T cells. Once a naïve T cell had found itself in contact with a DC within the neighborhood of a patch, it was allowed to remain in such a state of contact for 3 min (4, 5).

turned on in these simulations; consequently, formations of dense swarms of such T cells appear around DCs, as illustrated here.

# Definition of T Cell Motion Rules

The stochastic motion of a T cell was implemented in two ways, depending on the particular mechanism tested in the model:

#### (1) **T Cell Motion via a Random Walk Process:**

In the context of the present 2D ABM model, it was not possible to take explicitly into account interactions of T cells with the FRC network. Hence, empirical rules were introduced to describe T cell motion in this 2D space. To describe the random walk motion process, a T cell was allowed to move, at every discrete time step (30 s) of the simulation, to an unoccupied adjacent position (at a 5µm distance). This resulted in a modeled T cell velocity of 10 µm/min, in agreement with experimental observations (5).

To capture the short-term persistence character of T cell movement, T cell motion at each time step was set to be dependent on its previous direction. The new direction was thus calculated by defining a take-off angle from the previous direction, as the sum of two random angles sampled clockwise and counterclockwise from a uniform distribution in the range of (0, θmax) degrees. If the resulting adjacent lattice position happened to be already occupied (e.g., by another T cell, or a DC, or a boundary patch on the top or bottom side of the computational domain), then the T cell was computed to remain in its current position, and another direction calculation would be attempted at the next time step.

In our simulations, we aimed at reproducing not only realistic velocities of T cell movement, but also a physiologically-relevant T cell motility coefficient realistically reflecting the short-term persistence character of T cell movement. Motility coefficients used in such 2D simulations were calculated from time lapse microscopy records using the formula:

$$M = \frac{\left<\Delta r^2\right>}{4\pi} \tag{1}$$

where 1r 2 represents the mean squared displacement of an individual T cell from its initial position at time τ . Preliminary simulations were thus performed using different values for θmax. Motility coefficient estimates were averaged over 40 T cell trajectories, in each simulation. Simulations with an θmax of 80◦ resulted in a calculated average motility coefficient of 66 µm<sup>2</sup> /min, which nearly matched experimentally determined estimates of motility coefficients around 68 µm<sup>2</sup> /min (2).

#### (2) **T Cell Motion Toward a Neighboring DC via a Chemoattraction Process:**

Physiologically, local chemoattraction, or chemotaxis, may be mediated by a concentration gradient of specific chemokine molecules around DCs. In our ABM framework, the effective radius of a "chemokine cloud" around each DC was taken to be 5 patches (∼25µm). T cell motion would switch from a random walk process to a more-orless directed movement toward the DC center, once inside the "chemokine cloud." A chemotaxis strength parameter was introduced, representing the probability of performing a directed step instead of a random step. Typically, for simulation purposes, a probability value of 1/3 for a directed step can be used, in agreement with experimental estimates (9).

After some time spent in the "chemokine cloud," a T cell loses its sensitivity to the chemokine gradient and starts moving randomly again. The time of T cell de-sensitization to chemokine(s) was taken to be 10 min (10). Such desensitization potentially allows a T cell to leave the DC neighborhood. The time for a T cell to get re-sensitized to the chemokine gradient, once outside the cloud, was also assumed to be 10 min.

#### Definition of Boundary Conditions

Periodic boundary conditions were applied on the left and right sides of the computational domain: if a T cell were to leave the domain through one side, it would be allowed to immediately re-appear from the other side of the domain, moving in the same direction. In contrast, T cells were not allowed to randomly cross top and bottom boundaries of the computational domain. These boundaries, instead, contained "open patches," functionally corresponding to medullary sinuses (MS) and efferent lymphatics in a LN. Accordingly, if a T cell were to leave the computational domain through an MS patch at either the top or bottom side, a new T cell would be allowed to enter the computational domain, from a random position through its lateral borders. These settings allowed us to keep an overall constant T cell density in the system under study. The model was explored using a range of numbers of patches containing such MS escape structures (8–120 patches).

#### Cognate T Cell Activation and Proliferation

Relatively rare cognate naïve T cells, capable of recognizing DCpresented antigens, were included in the model. No difference in movements, between cognate vs. non-cognate naïve T cells before their first encounter with DC was assumed. After a first encounter, a cognate T cell was set to form a stable contact with a DC for ∼24 h; specifically, the duration of contact for each particular cognate T cell was a random value generated from a normal distribution with a mean of 24 h and a variance of 2 h. Upon completion of such a contact, a T cell became activated. The model subsequently simulated activated T cells which randomly migrated into a virtual LN and interacted with DCs bearing cognate antigen complexes. We also explored the option of chemoattraction of activated (but not naïve) T cells toward a neighboring DC.

Similarly to the models by Bogle et al. (11, 12), we assumed that the TCR stimulation signal could be summed over time, during the period of binding and also through sequential DC encounters. Milestones in the activation of a T cell were thus reached when the integrated stimulation were to exceed certain thresholds. In the present model, upon a DC encounter, activated T cells established a contact lasting for about 20 min (a random value generated from a log-normal distribution with a variance of 10 min). T cells were programmed to collect stimulation signals as long as the contact was maintained and to integrate, through summation, such collected stimulation signals upon additional DC encounters. During contact with a DC and the presenting cognate antigen, the stimulation level **S** of a lymphocyte was set to start increasing to a certain saturation level of a sigmoid curve, according to the following logistic equation:

$$\mathbf{S}\left(\mathbf{t}\right) = \mathbf{S}\mathbf{o} + \frac{\alpha}{1 + \mathbf{e}^{-\beta\mathbf{t}}},\tag{2}$$

where **S<sup>0</sup>** is the stimulation level at the beginning of the cognate contact. Parameters α = 2.0 and β = 0.005 min−<sup>1</sup> values were selected manually, as further detailed in the **Supplementary Materials**.

For activated T cells which ended up outside a DC contact zone, the stimulation level was set to decrease according to an exponential law:

$$\mathbf{S}\left(\mathbf{t}\right) = \mathbf{S}\_0 \cdot \mathbf{e}^{-\dot{\lambda}}\text{ \(\,^{\prime}\)}\tag{3}$$

where λ is the exponent indicator, corresponding to a half-life period of 24 h (12), and **S<sup>0</sup>** is the stimulation level at the start of decay. A typical trajectory of a cognate T cell stimulation signal is shown in **Supplementary Figure 2**.

Thus, a T cell was set to divide when the stimulation level **S** reached a defined threshold. The threshold value of the stimulation level **S<sup>n</sup>** is one critical model parameter which ultimately affects the proliferation intensity of cognate T cells. We therefore tested multiple threshold values, to estimate the sensitivity of the system to this parameter (**Supplementary Figure 3**). For the simulations presented here, we selected a value of **S<sup>n</sup>** = 3.5, which was close to the average **S** value for all cognate T cells represented in the computational domain.

Two factors were used in the model, to limit the proliferation of activated T cells: (i) a minimal time of about 8 h (random value generated from a normal distribution with a variance of 1 h for each newly formed T cell) was set between successive T cell divisions; (ii) a maximal number of successive proliferating T cell divisions was set. Following the last division of an activated T cell reaching effector status, no further division was allowed, and the cell was eliminated from the system within 24 h. Also, to avoid a strong increase in overall T cell density in the model (as a result of cognate T cell expansion), the following rule was added: if an activated T cell were to leave the computational domain via "open patches," no new T cell was allowed to come back in (in contrast to a naïve T cell leaving). This rule was set to take effect only if the overall number of T cells in the computational domain were to exceed the pre-set "equilibrium" value of 2,000.

#### Quantitative Outcomes Simulated via ABM Numerical Experiments

The following quantitative outcome measures were generated:


5. Cumulative outflux of cognate T cells: this was taken as the characteristic measure of the adaptive immune response intensity.

To calculate prediction intervals (90% PI) for each of the model outcomes, 45–100 independent ABM simulations were performed using identical model parameter values, yet different, randomly generated initial T cell and DC positions within the computational domain.

#### Software Packages

The NetLogo 5.0.1 software (13) was used as a computational tool for ABM. Additional details on model development and analysis, e.g., table with parameter values, results of sensitivity analysis and NetLogo 5.0.1 model scripts are given in the **Supplementary Material**. The NetLogo 5.0.1 model code was also uploaded to an open-source repository and is freely available under: https://github.com/Potamophylax/ABM\_ immune-response/.

## RESULTS

Using the 2D ABM model of a LN as described above, a large number of simulations were performed for various model parameter settings. For simplification purposes in these exploratory simulations, several model parameters were fixed using biologically reasonable estimates. In particular, the assignment of values to T cell and DC densities (respectively, 2,000 and 8 cells per 500 × 500 µm<sup>2</sup> ) and to the T cell movement parameter reflecting short-term persistence (θmax = 80◦ ) allowed us to reproduce, via simulations, a realistic motility coefficient of T cells in LN.

# Exploration of T Cell Repertoire Scanning Efficacy (Non-cognate Naïve T Cells Only)

Our first goal was to explore the impact of chemoattraction upon efficiency in the process of T cell repertoire scanning, as naïve T cells moved toward a DC. The measure of such efficacy was computed as the rate of accumulation of unique T cell— DC contact events. The chemotaxis strength estimate (P = 1/3) was identical to the one used by Riggs et al. (14). A value for an effective radius of the "chemokine cloud" around each DC was selected from the observed size of activated T cell swarms around a DC (∼25µm) (5). Time of T cell de-sensitization and time to T cell re-sensitization (values of 10 min taken for each) reflected characteristic times of cytokine receptor internalization and subsequent recycling.

We specifically explored the model behavior in relationship to the parameter reflective of the size of overall medullary sinuses (MS) and corresponding to the number of "open patches" at the top and bottom boundaries of the computational domain. This parameter is critical in the model, as it regulates T cell turnover rate and thus influences most of the quantitative outcomes. As shown in **Supplementary Figure 1**, setting the overall MS size in the range of 15–50 patches within the computational domain led to biologically realistic T cell transit times 10–20 h (15).

As shown in **Figure 2A**, the total number of contacts between T cells and DCs was only minimally affected by the overall MS size. In a "random walk" scenario, it is obvious that the total number of T cell-DC hits depended mainly on the density of T cells within the computational domain, which was kept constant and did not depend on the overall MS size. Interestingly, under a chemoattraction scenario, a much larger total number of contacts was obtained, as compared to a "random walk" motility process (respectively, 130,000 and 70,000 contacts counted during 3 days of simulations). Regions of high densities of naïve T cells ("swarms") formed locally, around DCs, in simulations under the chemoattraction scenario—which explains this substantial increase in the total number of T cell-DC contacts.

As shown in **Figure 2B**, the number of unique contacts sharply increased with an increase in overall MS size (which itself is an expression of increased T cell turnover rate through the computational domain). Hence, a higher turnover rate in T cells (from entering to leaving the computational domain) led to a larger number of T cells appearing de novo in the computational domain, thereby increasing the probability of new T cells to establish first-time unique contacts with DCs. Conversely, in the case of a slow T cell turnover rate, a larger proportion of T cells may have contacted single DCs multiple times, as they remained for longer times within the computational domain. Simulation results displayed in **Figure 2B** thus support the following important interpretation: a "random walk" motion scenario for T cells resulted in a substantially higher number of unique contacts between T cells and DCs—a number which is about twice higher vs. a chemoattraction scenario. These 2D ABM simulation results are in full agreement with those presented by Riggs et al. (14).

**Figure 2C** further displays the time evolution of this number of unique T cell-DC contacts, over 14 days of simulations. To explore the influence of the overall MS size upon unique T cell-DC contact dynamics, simulations were performed for two overall MS size values of 32 and 40 patches. The sensitivity of this number of unique T cell-DC contacts with respect to overall MS size, in the considered physiologically-reasonable range, was moderate. The resulting linear dynamics of this number, observed after 1–2 days of simulations, indicated that the system represented by the computational domain reached an equilibrium state by this time. The differing slopes of the two sets of curves on **Figure 2C** point to different rates of unique T cell-DC contact accumulation; this rate is indeed significantly higher in the case of a "random walk" scenario vs. chemoattraction.

# Exploration of Activation and Expansion of Cognate T Cell Clones

A second goal of this study was to explore simulations of cognate T cell priming and expansion under different model parameter settings. Here, we used small, albeit non-zero values for cognate T cell frequency parameters, i.e., the probability of a new incoming T cell to be cognate, which could be activated in contact with DCs and proliferate as described above. Also, we tested different simulations.

maximal numbers of activated T cell divisions, i.e., maximal numbers of T cell generations starting from naïve cognate T cells to finally differentiated effector T cell. Proceeding from the previous part, we fixed the overall MS size at 32 patches: it ensured a transit time through the LN T zone in agreement with experimental values. All simulations were performed under the assumption of a constant level of antigen stimulation in the LN: neither the number of DCs, nor their positions, nor their properties changed during the simulations.

In preliminary simulations not shown here, we sought to reproduce T cell immune responses using the same "random walk" scenario for both naïve and activated lymphocytes; under such conditions, cognate T cell expansion levels were low and not robust. By visual inspection of such simulation trajectories, we observed that most activated cognate T cells would leave the computational domain prior to any cell division occurring. To resolve this technical modeling issue, we allowed chemotaxis toward a neighboring DC to be selective for already activated (but not naïve) cognate T cells. As shown in **Figure 3**, such an approach allowed us to reproduce realistic T cell immune response dynamics, after varying values of key unknown model parameters over a wide range. **Videos S1**, **S2** (available in the online **Supplementary Material**) illustrate the kinetics of the system at, respectively, the start and Day 7 of representative simulations.

In particular, following days 3–5 of simulations (**Figures 3A,C,E**), a fast increase in cognate T cells numbers was computed, with distinct peak values around Day 7, in good agreement with the experimentally observed time of 5–7 days for an immune response to occur (16). Beyond 10 days of simulations, all the different trajectories exhibited numbers of cognate T cells which fluctuated around some "steady-state" values, due to a constant level of antigen stimulation which had been assumed in our model. As shown in **Figures 3B,D,F** simulations, cognate T cell outflux rates also became nearly constant, which correlated with the "steady-state" numbers of cognate T cells within the computational domain.

Model outcomes were highly sensitive to the activated T cell chemotaxis strength value (**Figures 3A,B**): stronger chemotaxis led to a larger cognate T cell number within the computational domain, via a facilitation of T cell proliferation. Taking into account activated T cell chemoattraction led to the accumulation of T cells in forms of swarms around DCs (**Figure 1B**), which further led to: (a) a longer half-life for these T cells in the LN, slowing down their elimination rate from MS patches; and (b) more frequent contacts with DCs, which favored the build-up of the activation signaling (S) to higher levels. Both factors allowed to effectively increase the number of overall cognate T cell divisions in the system.

Another important factor was the cognate T cell frequency, i.e., the probability for new incoming naïve T cells to recognize antigens presented by DCs (**Figures 3C,D**). In our simulation framework, a cognate frequency of 1/500 appeared sufficient to induce a robust outflow of cognate T cells (as a main characteristic measure of immune response intensity), however, this outflow rate was twice lower vs. a cognate frequency of 1/100. A further increase in the cognate frequency to up to 1/50 only slightly increased the rate of cognate T cell outflow. Additional simulations were performed, over a wide range of cognate frequencies; we determined a non-linear dependence, with a saturation of cognate T cell outflow vs. cognate frequency (see **Supplementary Figure 4**).

Only moderate increases in "steady-state" cognate T cell numbers and in the corresponding outflows were observed, when the parameter value reflecting the maximal number of T cell divisions was increased from 10 to 20 (**Figures 3E,F**). However, for a maximal number of divisions ranging from 15 and 20, the model predicted peak values of cognate T cell numbers (within the computational domain) which were twice as high

vs. when using a maximal number of divisions 10, on Day 10 of the simulations. This complex dynamic behavior reflected in the outcome of cognate T cell numbers was technically traced to a negative feedback loop included in the model, as described in the Methods section. If, during the drive of cognate T cell expansion, the overall T cell density became higher than 2,000 (pre-set T cell "equilibrium" density), the rate of new incoming naïve T cell inflow became smaller. The rate of cognate naïve T cell inflow decreased as well; this, consequently, led to a decrease in the number of cognate T cells within the system. Thus, in the absence of such a negative feedback loop, the domain would become over-populated with T cells, also resulting in "paralysis" of T cell motility.

# DISCUSSION

The original motivation driving this modeling study was to enable the exploration of dynamic spatial effects, in particular a detailed investigation of the relationship between T cell motility behavior and the timing and intensity of an immune response. Intravital microscopy (2PM) yields a wealth of information within a very restricted region of the LN and during a short period of time (hours), whereas histology provides complementary views, yet limited to two dimensions and with no dynamic time element. An ABM of the LN allows for integrative simulations which may help filling the gaps between these two experimental approaches. To explore several hypotheses on T

shaded areas represent 90% PI.

was set at 1/100, and chemotaxis strength was set at 1/3. All plots are based on measures from 45 simulations lasting 28 days each. Lines represent median values;

cell motility and their interactions with DCs, we independently developed a simplified, yet biologically reasonable version of a 2D ABM of the LN T cell zone. The model was developed and qualified using NetLogo (13), a freely available and flexible software tool.

The 2D ABM presented here, which includes T cell movement, activation and proliferation in a LN, allows for the integration of a number of processes and at the scale of the LN system, and illustrates the necessity to consider all essential processes simultaneously, in order to generate a realistic dynamic picture of the immune response. Because detailed experimental data required to characterize all these processes are not available, we made assumptions regarding several parameters embedded in the model; nonetheless, the model developed here realistically reproduced key temporal characteristics, such as the T cell motility coefficient (2), LN transit times (15) and kinetics of immune response development (16).

Operating characteristics of the model were supported by sensitivity analyses, whereby simulations were run over ranges of model parameter values, with model outcomes being compared against physiological values. Model simulations indicated that the generated T cell response was sensitive to factors, such as naïve cognate T cell frequency and the strength of the hypothetical chemoattraction of T cells toward neighboring DCs. Thus, despite a simplified, semi-empirical structure of the model, we obtained reasonable and robust simulations over a wide range of unknown parameter values.

The ABM LN model presented here was set as a twodimensional (2D) model, rather than a more physiological threedimensional (3D) model. A 2D setting of the model allowed us, obviously, to drastically reduce the computational cost of ABM simulations, while carefully estimating the impact of stochastic effects on simulation outcomes. In most of ABMs, T cell movement is implemented as a "random walk" in a nonguided biophysical domain; under such settings, advantages of 3D vs. 2D models may, in fact, not be obvious. The role of the fibroblastic reticular cell (FRC) network in guiding T cell motion in the LN has been studied over many years (17); such a network may adequately constrain T cell movement in a 3D domain. Thus, a realistic spatial structure of the FRC network, together with rules describing lymphocytes and DC interaction within the FRC network should be included in a 3D ABM of the LN. The development of such a detailed, highly parameterized and computationally intensive 3D model is a complex endeavor; the works from a number of such research groups have been reviewed (18). One of the limitations of the present 2D modeling work is the implicit consideration of the FRC network influence, via a short-term persistence description of T cell motility, since there would have been no other obvious way to take FRC network effects into account more realistically. In such a 2D context, it should not be expected to reproduce either realistic densities of T cells or realistic numbers of sites of T cells on the DC surface. Thus, the outcomes of our 2D ABM reported here, such as T cells—DC contact numbers and cognate T cell outflux from the LN should be viewed as qualitative measures of behavior, rather than absolute values of cell counts.

The numerical simulations reported here were focused on two pivotal questions. The first question focused on whether local chemoattraction of T cells toward DCs would promote or hamper the scanning efficiency of DCs, within a LN. We demonstrated that, for an effective DC scanning of the T cell repertoire, a T cell "random walk" motility scenario appeared to be the optimal strategy (vs. chemoattraction). We provided a physiological rationale, via simulations, as to why a chemoattraction motility scenario may actually lead to a non-optimal DC repertoire scanning of T cells: indeed, under chemoattraction, dense and relatively stable swarms of T cells may form around each DC. T cells within these swarms may experience repeated contacts with DCs; and owing to the higher cellular density within swarms, it may take significant time for a T cell to leave a swarm, even if it were to become insensitive to the local chemokine gradient. In addition, swarms may form a barrier for T cells outside the neighborhood, to make contact with DCs. Thus, under a chemoattraction motility scenario, our simulations demonstrated a large number of repeated T cell-DC contacts, while the number of unique T cell-DC contacts, reflective of T cell repertoire scanning efficacy, remained relatively small.

Such results are in full agreement with results from earlier 2D ABM research, in which T cell motility was accurately captured to help determine the impact of chemotactic attraction of T cells-to-DC on repertoire scanning (14). Accordingly, a T cell may move randomly, with a short-term persistence, until it encounters a chemokine gradient around a DC, at which point probabilities are updated so that a T cell is more likely to move toward a DC. Chemokine gradients were captured in a simplified manner, by assigning concentrations in the DC neighborhood (up to 20µm from the DC). Chemotaxis parameters included strength (related to the likelihood of moving toward a DC), duration (time before de-sensitization occurs), and recovery (time before a T cell may again detect a chemokine gradient). As strength and duration increased, the total number of T cells-to-DC contacts increased, yet the number of unique T cells-to-DC contacts decreased, suggesting that an increased competition of T cells for a DC, resulting from chemotactic-driven movement of T cells toward a DC, interfered with efficient repertoire scanning. In conclusion, a better strategy for efficient scanning is to briefly contact, then clear non-cognate T cells away from an antigenpresenting DC, to make scanning room for different, potentially cognate T cells.

The relevance of a chemo-attraction process on T cell scanning efficiency by DCs was also addressed in modeling work by Vroomans et al. (19), who developed a 2D model of the LN T zone, based on a Cellular Potts Model (CPM) formalism. The CPM is a grid-based spatial model, initially developed to describe the biophysics of cell sorting, based on differential adhesion properties (20). Within this formalism, cell motion is driven by the overall minimization of the energy of deformation and stretching of the cell membrane through stochastic fluctuations, in which global and local forces upon a cell edge are resolved (21). Extension of this CPM approach have been made to describe cell motion under control of a chemokine gradient, including movement under conditions of high cell density in clusters around a DC. In contrast to (14) and our conclusions, model simulations by Vroomans et al. (19) demonstrated that chemo-attraction of T cells does enhance DC scanning efficiency, leading to a greater probability for rare antigen-specific T cells to find DCs bearing the cognate antigen. Also, these authors found that de-sensitization of T cells following contact with a DC would further increase DC scanning efficiency, providing an improvement of nearly 3-fold, vs. a "random walk"-type migration. We here offer one interpretation for this apparent discrepancy: the CPM approach may not adequately reproduce T cell motility in the LN T zone (19). Indeed, in that work, motility was based on a cell adhesion process; also, very dense packing of T cells in the computational domain was assumed. Based on the experimental 2PM observations, a mean free length characteristic of T cell motility was estimated, in the range of 30–40µm (5, 7). Such a fast, intrinsic velocity of T cell motility would not be possible in the CPM-modeled system (19).

The second question which we sought to address here was about a potential role for chemotaxis in immune response initiation. For such a purpose, we simplified the description of cognate T cell activation, to minimize the overall number of parameters in the model. The concept of a TCR stimulation signal (S) accumulation and dynamics of individual cognate T cells was based on previous modeling work (12). However, the implementations of this concept, between the present 2D vs. the previously published 3D models were materially different: (a) for simplification, we considered a single cognate T cell clone instead of multiple cognate clones with varying affinities of their TCR to pMHC; and (b) we took into account local chemoattraction of activated cognate T cells toward a DC, as a factor which may accelerate, or even be critical for T cell immune response initiation.

Using our 2D ABM approach, we determined that a feature of selection for activated cognate T cells is required, to reproduce their pronounced expansion upon response to antigen stimulation. As mentioned above, activated T cell chemoattraction lead to T cell accumulation in swarms, around DCs. This effectively caused a longer half-life of these T cells in the LN, slowing down their elimination from MS patches, and also causing frequent contacts with DCs, thereby contributing to activation signaling (S) at a higher level. Both factors effectively increased overall numbers of cognate T cell divisions in the LN. The relative contributions of these two factors toward T cell immune response potentiation depended on specific model parameters, such as the activation threshold value (**Sn**), the overall MS size, the T cell motility coefficient, and the overall T cell density. If, indeed, a prolonged half-life of activated T cells in the LN is critical, then an explicit accounting for the effect of sphingosine-1-phosphate receptor down-regulation during T cell activation, leading to retention (for a number of days) of activated T cells in the LN, may be added to the model (22).

Similarly to previous ABM applications tailored to the LN, we showed that such a modeling technique proves to be a useful tool to integrate current knowledge and data on molecular and cellular interactions between immune cells, to then generate novel hypotheses which may guide further experimental studies, to overall improve our mechanistic understanding of the immune activation process that takes place in the LN. Many questions on this dynamical process in the LN remain open, in particular questions on the emerging role of the FRC network in regulating immune responses. Future developments of 3D models, with detailed stromal elements, may play an important role in further elucidating biological mechanisms (18).

# AUTHOR CONTRIBUTIONS

IA and YK designed, initiated and developed the modeling project. GH and KP participated in discussions and reviewed modeling design and concept. YK and IA wrote the computer program and translated the biological model into agent-based modeling language. IA performed model calculations. GH elaborated on paper outline and points of focus. All authors contributed toward manuscript writing and revisions.

# FUNDING

This work was funded by AstraZeneca Pharmaceuticals and the Russian Academic Excellence Project 5-100 program.

# ACKNOWLEDGMENTS

We are most grateful to Dr. Uri Wilensky (Northwestern University) for distributing the NetLogo 5.1.0 software. We thank Dr. Gennady Bocharov, the Institute of Numerical Mathematics, the Russian Academy of Sciences, and his team for most valuable discussions on immunological aspects of this work. We also thank Oleg Stepanov, M&S Decisions, for technical support.

# SUPPLEMENTARY MATERIAL

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fimmu. 2019.01289/full#supplementary-material

Video S1 | 2D agent-based modelling movie at the start of simulation.

Video S2 | 2D agent-based modelling movie at the 7th day post numerical experiment start.

Supplementary Information | Additional details on the model development and analysis, e.g. table with parameter values, results of sensitivity analysis and additional model simulations.

Source Code | NetLogo 5.0.1 scripts corresponding to the final model version presented in the manuscript.

# REFERENCES


**Conflict of Interest Statement:** The authors declare that this study received funding from AstraZeneca. IA, KP, YK are employed by and KP, YK are owners of M&S Decisions, a modeling consultancy which received research funding from AstraZeneca. GH is employed by, and shareholder of AstraZeneca.

Copyright © 2019 Azarov, Peskov, Helmlinger and Kosinsky. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# Spatial Lymphocyte Dynamics in Lymph Nodes Predicts the Cytotoxic T Cell Frequency Needed for HIV Infection Control

Dmitry Grebennikov 1,2,3 \*, Anass Bouchnita<sup>4</sup> , Vitaly Volpert 3,5,6, Nikolay Bessonov <sup>7</sup> , Andreas Meyerhans 8,9 and Gennady Bocharov 2,10 \*

<sup>1</sup> Moscow Institute of Physics and Technology, National Research University, Dolgoprudny, Russia, <sup>2</sup> Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia, <sup>3</sup> Peoples' Friendship University of Russia (RUDN University), Moscow, Russia, <sup>4</sup> Division of Scientific Computing, Department of Information Technology, Uppsala University, Uppsala, Sweden, <sup>5</sup> Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, Villeurbanne, France, <sup>6</sup> INRIA Team Dracula, INRIA Lyon La Doua, Villeurbanne, France, <sup>7</sup> Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, Saint Petersburg, Russia, <sup>8</sup> Infection Biology Laboratory, Department of Experimental and Health Sciences, Universitat Pompeu Fabra, Barcelona, Spain, <sup>9</sup> Institució Catalana de Recerca i Estudis Avançats (ICREA), Barcelona, Spain, <sup>10</sup> Sechenov First Moscow State Medical University, Moscow, Russia

#### Edited by:

Michael Loran Dustin, University of Oxford, United Kingdom

#### Reviewed by:

Jens Volker Stein, Université de Fribourg, Switzerland Martin Meier-Schellersheim, National Institutes of Health (NIH), United States Frederik Graw, Universität Heidelberg, Germany

#### \*Correspondence:

Dmitry Grebennikov dmitry.ew@gmail.com Gennady Bocharov bocharov@m.inm.ras.ru

#### Specialty section:

This article was submitted to T Cell Biology, a section of the journal Frontiers in Immunology

Received: 08 February 2019 Accepted: 13 May 2019 Published: 11 June 2019

#### Citation:

Grebennikov D, Bouchnita A, Volpert V, Bessonov N, Meyerhans A and Bocharov G (2019) Spatial Lymphocyte Dynamics in Lymph Nodes Predicts the Cytotoxic T Cell Frequency Needed for HIV Infection Control. Front. Immunol. 10:1213. doi: 10.3389/fimmu.2019.01213 The surveillance of host body tissues by immune cells is central for mediating their defense function. In vivo imaging technologies have been used to quantitatively characterize target cell scanning and migration of lymphocytes within lymph nodes (LNs). The translation of these quantitative insights into a predictive understanding of immune system functioning in response to various perturbations critically depends on computational tools linking the individual immune cell properties with the emergent behavior of the immune system. By choosing the Newtonian second law for the governing equations, we developed a broadly applicable mathematical model linking individual and coordinated T-cell behaviors. The spatial cell dynamics is described by a superposition of autonomous locomotion, intercellular interaction, and viscous damping processes. The model is calibrated using in vivo data on T-cell motility metrics in LNs such as the translational speeds, turning angle speeds, and meandering indices. The model is applied to predict the impact of T-cell motility on protection against HIV infection, i.e., to estimate the threshold frequency of HIV-specific cytotoxic T cells (CTLs) that is required to detect productively infected cells before the release of viral particles starts. With this, it provides guidance for HIV vaccine studies allowing for the migration of cells in fibrotic LNs.

Keywords: lymphoid tissue, cell motility, HIV infection, cytotoxic T cell scanning, multicellular dynamics, dissipative particle dynamics, stochastic differential equation

#### INTRODUCTION

The surveillance of host body tissues by cells of the immune system is central for mediating defense functions against invading pathogens and tumor cells (1, 2). The initial recognition of foreign antigens that leads to the induction of adaptive immune responses takes place in lymph nodes (LNs), which, by virtue of their location and structure, facilitate the interactions between immune cells (3). The motility of pathogen spread and immune cells represents relevant parameters

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controlling the fate of the pathogen–host interaction. In vivo imaging technologies have been used to quantitatively characterize target cell scanning and migration dynamics of lymphocytes within LNs (4, 5). The translation of these quantitative insights into a predictive understanding of immune system functioning in response to various perturbations critically depends on the availability of computational tools linking the individual immune cell properties with the systems response as a whole (6).

Multiscale models of the immune system provide the in silico tool to embed immune processes into their spatial context (7–9). A core module of the models is the mathematical framework used to describe individual cell migration in complex multicellular environments. One can distinguish two general types of modeling approaches, cellular automata-based models (CAMs), and physical models (PMs). CAMs consider a regular grid with cells that change their state in time and space according to some rules (functions of the system state). The respective computational algorithms can take the form of random walks (10) or cellular Potts models (11). Although CAMs incorporate experimentally defined characteristics of cell motion and, thus, simulate cell dynamics based on actual data, they lack quantifiable links to the underlying biophysical interactions between cells in multicellular environments and to intrinsic cell motility parameters (12). PMs of lymphocyte migration dynamics derived from the Newtonian second law offer the possibility to define cell motions in terms of the forces generated by the environment and the cell itself. Using the experimental data on cell movement, the potential functions underlying cell-to-cell interactions and intrinsic cell motility can be identified and can provide a deeper insight into the mechanical properties of cells. Thus, PMs of individual cells and coordinated cell migration represent a general and generic way to describe and predict the multicellular system dynamics for a broad range of cell numbers and external conditions (13, 14).

It is widely accepted in immunology that the physiological function of cytotoxic T cell (CTL) motility is to search for target cells, i.e., for virus-infected cells or cancer cells (15). Computational modeling studies have revealed that the search efficiency depends on the organization of the stromal environment of a tissue (16). In addition, the spatial behavior, for example, of HIV-infected target cells scanned for foreign antigens by CTLs strongly impacts the elimination efficiency of the infected targets (17, 18). Experimental investigation of live attenuated SIV vaccines clearly suggested that a robust protection against intravenous wild-type SIVmac239 challenge strongly correlates with the number and function of antigenspecific effector CTLs in LN rather than the responses of such cells in the blood (19). However, the quantitative effects of Tcell migration parameters in LNs on the efficiency of antiviral immune responses in vivo remain unknown.

In the current study, we have developed a physics-based description of spatial T-lymphocyte dynamics in the multicellular environment of LNs. A fundamental relationship between a cell motion and the forces acting on it is provided by Newton's second law. It is used to formulate, calibrate, and apply a generic mathematical model of coordinated T-cell migration dynamics in LNs. By choosing a first principles approach in formulating the governing equations in conjunction with published experimental data on T-cell motility in lymphoid tissues, we offer a broadly applicable generic mathematical tool linking individual and coordinated cell behaviors. The potential of the model is illustrated by an analysis of the combined effects of antigen-specific T-cell numbers and intrinsic T-cell motility parameters in LNs on the time needed to locate both mobile and non-motile HIV-infected target cells. Computed predictions of the ratio of effector CTLs to infected T cells in the LN paracortex needed for a timely detection of infected cells within 18 h postinfection, i.e., before the release of viral particles starts (20), provide a novel quantitative guide for an informed design of HIV vaccines.

# MATERIALS AND METHODS

# Programming Languages and Computing Resources

All algorithms were written in C++ and compiled using G++ (version 5.4.0). Pseudorandom numbers were generated using the PCG random library (version 0.98) and the PCG64-XSL-RR algorithm (21). The seed was either specified manually (for code development) or set based on the system's random device (for computational experiments). Simulations were run on a 2 core Xeon E3-1220 v5 @3.0 GHz × 4 processor. The wxWidgets library (version 2.8.12) was used for visualization purposes. The processing of the simulation results (i.e., calculating statistical motility profiles, comparing CDFs, and plotting) was implemented in Python and R scripts.

#### Model Equations of Multicellular Dynamics

According to a basic mechanics view, a system consisting of N cells of some mass located in a liquid milieu, interacting with each other and affected by some external field, is uniquely determined by their coordinates and velocities and is governed by the classical mechanics motion equations. In our model, each cell i, i = 1, N, is represented as the circle with certain mass m<sup>i</sup> , radius r<sup>i</sup> , and position of its center x<sup>i</sup> . The fundamental equation governing locomotion of cells is Newton's second law of motion. It can be expressed as follows:

$$m\_i \ddot{\mathbf{x}}\_i = F\_i = \sum\_{j \neq i} f\_{\mathbf{i}j}^{\text{int}} + f\_i^{\text{mot}} + f\_i^{\text{dis}}, \ i = \overline{1, N}, \tag{1}$$

where the first term on the right side specifies the net effect of the pairwise interaction forces with contacting neighbor cells, the second term stands for the cell intrinsic locomotion force, by which the cell establishes motility within the extracellular matrix (ECM) of the LN reticular network, and the last one takes into account the action of a dissipative force, taken to be proportional to the cell velocity f dis <sup>i</sup> = −µx˙<sup>i</sup> . We neglect the impact of gravity.

#### Random Motility Force Sampling

The random motility force f mot i for the ith cell is modeled as a stochastic vector f<sup>i</sup> sampled every 30 s from certain probability distributions analogously to the inverse homogeneous correlated

random walk (IHomoCRW) model (22). The motility magnitude fi <sup>=</sup> <sup>η</sup><sup>i</sup> ·K<sup>i</sup> is sampled from the following Gaussian distribution: K<sup>i</sup> ∈ N µ(K), σ 2 (K) . To obtain the motility magnitude fi , the sampled value K<sup>i</sup> is multiplied by the arresting coefficient ηi . The arresting coefficients are increased for both T cells and DCs if they establish a sufficiently long contact to temporarily arrest their inner motility as follows: (1) η<sup>i</sup> : = 10η default i for T cells and DCs when the duration of an uninterrupted contact exceeded 30 s, and (2) η<sup>i</sup> : = 100η default i if the contact outlasted 20 min. The cell inner motility is restored back to a default value if the contact lasted for a time longer than the sampled value tcontact ∈ N (2, 0.4) hours. The parameter η<sup>i</sup> is also used to decrease intrinsic motility when performing in silico simulations to study the effect of decreased T-cell motility on target cell location efficiency (see details in **Supplementary Text**).

The motility direction ˆ fi is turned from the previous direction on the angle θ<sup>i</sup> :

$$\alpha\_i \in N\left(0, \sigma^2\left(\alpha\right)\right), \quad \theta\_i = \alpha\_i \cdot \left(1 - \left(\frac{K\_i}{K\_{\text{max}}}\right)^{\beta}\right),$$

$$K\_{\text{max}} = \mu\left(K\right) + \mathfrak{J}\sigma\left(K\right). \tag{2}$$

Here, N 0, σ 2 (·) denotes a Gaussian distribution, and β is a scalar coefficient. The angle sampled from the normal distribution is multiplied by a factor depending on the sampled motility magnitude to reproduce the experimentally observed negative correlation between cell translational and turning angle speeds. Indeed, the cells do not simultaneously perform fast translational movements and large reorientations (22). Note that a similar feature was named "directional propensity" and modeled with trigonometric parameterization in a cellular Potts model to describe the motion of T cells (11). The Gaussian distribution for the motility magnitude is set so that the (µ − 3σ,µ + 3σ) range is positive. The absolute value is taken to ensure that the magnitude is non-negative. The parameter Kmax provides an upper boundary for sampled values K<sup>i</sup> (approximately 1 of 370 cases falls outside of the three-sigma interval). The hat above the vector denotes the normalized unit vector.

### Implementation of Contact Inhibition of Locomotion

After the stochastic vector f<sup>i</sup> is sampled, it is modified in accordance with the contact inhibition of locomotion (CIL) model, as described (23). The resultant vector f mot i is then used in the right-hand side of Equation (1). The modification consists of shifting the direction of vector f<sup>i</sup> away from the neighboring cells and decreasing the magnitude of vector f<sup>i</sup> proportionally to the number of neighboring cells:

$$f\_i^{\text{mot}} = \frac{|f\_i| \cdot \left(c\_{\text{inh}}\hat{f}\_i + \hat{R}\_i\right)}{c\_{\text{inh}} + n}, \quad \hat{R}\_i = \sum\_{j, h\_{\text{ij}} \le r\_i + r\_j} \frac{\varkappa\_i - \varkappa\_j}{h\_{\text{ij}}}, \tag{3}$$

in which fi is the magnitude and ˆ fi is the direction of the inner motility as it would be if unaffected by CIL, n is the number of neighboring cells in contact (such that the distance between cell centers hij ≤ r<sup>i</sup> + rj), and Rˆ <sup>i</sup> determines the net shift of the inner motility direction away from the neighboring cells, cinhis the weighting coefficient varying the level of CIL. The hat above the vector indicates that it is normalized.

#### Numerical Integration of the Equations of Cell Motion

To numerically integrate the equations of motion (Equation 1), we used the first-order semi-implicit (i.e., the cell coordinate at time t n+1 is computed using the velocity vector v n+1 i rather than v n i ) Euler method:

$$\nu\_{i}^{n+1} = \frac{m\_{i}\nu\_{i}^{n} + h \cdot \left(F\_{i}^{\text{int}}\left(t^{n}, \boldsymbol{\varkappa}\_{i}^{n}\right) + f\_{i}^{\text{root}}\left(t^{n}, \boldsymbol{\varkappa}\_{i}^{n}\right)\right)}{m\_{i} + h \cdot \mu} \tag{4}$$

$$
\boldsymbol{\alpha}\_{i}^{n+1} = \boldsymbol{\alpha}\_{i}^{n} + \boldsymbol{h} \cdot \boldsymbol{\nu}\_{i}^{n+1} \tag{5}
$$

in which x n i and v n i are the coordinate and velocity of cell i at the time t n after n steps t <sup>n</sup> = t <sup>0</sup>+h·n. We note that the second-order generalization of this method, i.e., the Störmer–Verlet method, could be developed. However, it will be computationally more demanding as the cell acceleration depending on velocity due to the presence of dissipative velocity-damping viscosity forces needs to be reevaluated at each time step t n+1 . We verified that the time step h = 0.02 min used in the simulations is sufficient for a stable integration of the initial value problem with the semi-implicit Euler method. To efficiently locate the neighboring cells (which is needed for intercellular force calculations and for determining the effect of CIL), we use a simple uniform-gridbased spatial neighbor search, which performs well for a densely packed multicellular environment. Note that the convergence of the integration scheme was verified by repeating simulations for a smaller time step.

#### Boundary Conditions

During the model calibration process, we used periodic boundary conditions for all boundaries of a square domain. To perform in silico simulations in a closed ellipse-shaped domain representing a LN, we implemented a biologically based boundary condition of cell repolarization. We do not model explicitly the interaction forces between cells and the boundary (i.e., the subcapsular sinus wall). At the stage of coordinate updates (in accordance with the numerical scheme specified in the section Numerical Integration of the Equations of Cell Motion), if the proposed coordinate of cell x n+1 i is outside the boundary, the current coordinate of the cell is preserved (x n+1 <sup>i</sup> = x n i ), while the direction of the motility vector Ef mot i is changed to be the opposite direction of vector v n+1 i , thus resulting in cell repolarization.

#### Generating the Initial Spatial Configuration for Simulations Within a LN

To generate the initial spatial configuration of the immune cells within a LN, we followed the descriptions from a LN imaging study (24). The following cell subsets were considered: CD4<sup>+</sup> T cells, CD8<sup>+</sup> T cells, and cross-presenting migratory CD8α intCD103hi DCs. Both T-cell subsets are distributed uniformly through the whole LN, while migratory DCs are found mainly deep in the paracortex area. To arrange cells in agreement with the experimental data, we approximated the DC-rich area as an ellipse Ωα=0.99 DC . The spatial positions for DC locations are iteratively sampled from the 2D Gaussian distribution with a 99 percentile ellipse Ωα=0.99 DC and accepted if the DC with sampled coordinates lies within the LN domain ΩLN and does not overlap with the other seeded DCs. After DCs are placed, the T cells are positioned uniformly through the remaining non-occupied space of ΩLN.

#### RESULTS

# Biophysical Parametrization of the Spatial Multicellular Dynamics

Multicellular systems dynamics can be accurately described by biophysical models as reviewed recently (14, 25). Here, we develop a physics-based mathematical model of coordinated immune cell motion that belongs to the class of self-propelled particle models (14) and, more generally, to the dissipative particle dynamics (26, 27) framework.

Immune cells in LNs are continuously interacting with each other and with stromal cells via forces of different origin, i.e., elastic (membranes), chemical (receptors), and electric. The respective forces in combination with cell intrinsic locomotion events act in concert to determine the basal intranodal motility of T cells. **Figure 1A** presents the overall summary of physical forces included in the model with some implementation details. The scheme of the forces exerted on cell i interacting with cells p and k is shown in **Figure 1B**. The quantitative features of the force functions are detailed in **Figure 1C**. Here, f int ij is the intercellular force acting on cell i due to interaction with cell j. The pairwise cell-to-cell interactions are assumed to have a finite cutoff distance and are considered to be elastic acting along the line of cell centers. The intercellular forces f int ij can be considered as the gradients of pairwise potentials, which are repulsive at short distances and attractive at larger distances, thus accounting for volume exclusion at the cell body and cell–cell adhesion near membranes. We consider the following cubic polynomial function to model the force exerted by cell j on cell i:

$$f\_{\rm ij}^{\rm int} = \frac{\mathbf{x}\_{i} - \mathbf{x}\_{j}}{h\_{\rm ij}} \cdot \left\{-a \cdot f^{\rm add} \cdot \frac{r\_{j} - \mathbf{x}}{r\_{j}} + b \cdot f^{\rm add} \cdot \left(\frac{r\_{j} - \mathbf{x}}{r\_{j}}\right)^{3}, \ h\_{\rm ij} \prec r\_{i} + r\_{j}, \tag{6}$$

where r<sup>i</sup> is a radius of the ith cell membrane, hij is the distance between cell centers (see **Figure 1C**), and x = hij − r<sup>i</sup> is the distance between the center of cell j and the surface membrane of cell i. The function a · f adh · rj−x rj describes the attraction force between two cells, and the function b · f adh · rj−x rj 3 corresponds to a repulsive force, both calibrated as shown in **Figure 1C**. The coefficients a and b are set such that the minimum of function f int ij is equal to f adh. Thus, the only remaining free parameter is f adh, the adhesive interaction strength. In the case of T cell/T cell interaction, it corresponds to weak nonspecific electrical forces (electrostatic and electrodynamic) that are expected to be present between all cells according to the model of Bell (28). We calibrate this parameter by the typical value of low-adhesive forces, with which integrins present on Tcell membrane bind to their ligands present on the other cells (29). For cognate T cell/APC interactions the attraction force is much stronger as it is determined by a broad spectrum of various adhesion molecules involved in T-cell activation clusters, i.e., the immunological synapse (30). The estimated values of the intercellular interaction forces are given in **Table 1**. For details on the data-based T-cell motility model calibration, see **Supplementary Text**.

The dissipative (friction) force acting on T cells describes the effect of viscous damping, which reduces the velocity of the cell. It is assumed to be proportional to the cell velocity f dis <sup>i</sup> = −µx˙<sup>i</sup> . The dissipative force acts along the line of the cell center and in opposite direction to the cell displacement. Consideration of viscous damping is appropriate for the highly viscous low-Reynolds-number environment of LNs (40). The viscous damping parameter estimate is listed in **Table 1**.

The random motility force f mot i determines the traction of self-propelled lymphocytes. It represents a stochastic process of receptor-mediated cell–ECM interactions regulated by either cytoskeletal or membrane reorganizations and governed by biomechanical and intracellular molecular mechanisms (4, 13). Basically, cells establish directed caterpillar-like movement by polarizing, forming contacts between their leading edge and collagen fibers of ECM, detaching their trailing edge from ECM, and contracting. However, T lymphocytes and dendritic cells (DCs) are characterized by low-adhesive integrin interactions with the microenvironment. This allows them to adapt their direction and morphology with no need to reorganize microstructure while effectively sliding along the stromal network of fibroblastic reticular cells (41, 42). As we do not model the reticular network and the ECM microstructure in this study explicitly, this motility behavior is considered implicitly in the stochastic nature of f mot i . Note that the autonomous cell motility can also be affected by external signaling, e.g., through chemotaxis, CIL, or immunological synapse formation. The cell trajectory in the model is characterized by three quantifiable values, i.e., the translational speed, the turning angle speed, and the meandering index as explained in **Figure 1D** and as described in Read et al. (22). The corresponding experimental data are shown in **Figure 1E**. To capture the experimentally observed patterns of T-lymphocyte migration in lymphoid tissues (see **Figure 1E**), the T-cell motility is modeled using a random variable f<sup>i</sup> with its magnitude and angle values updated every 1t seconds according to the IHomoCRW recently suggested and validated (22). The IHomoCRW model was shown to reproduce the experimentally measured statistical profiles of T-cell locomotion (22). In the present model, the magnitude and direction of the random vector f mot i are sampled from distributions provided by the experimental data (the specific rules are defined in section Materials and Methods). The key difference from the original IHomoCRW model is that it is the cell motility inducing force f mot i rather than the cell velocity x˙<sup>i</sup> that is sampled and then substituted into equation (Equation 1). In addition,

(Continued)

FIGURE 1 | including the repulsive–attractive interaction with neighbor cells p and k, respectively. (C) The parameterization of intercellular interaction force f int ij and formula definition. The calibrated force for non-specific interaction of two T cells with a radius of 3µm is depicted. By simulation, the parameters a and b are calculated at each time step depending on the radii ri , rj and the distances hij, x, so that the condition f ij λrj = f ij rj = 0, minfij (x) = −f adh i is satisfied. The parameter λ determines the relative deformation of the cells that separates the repulsive and attractive interactions between them. Parameter f adh i represents the adhesive strength between the membranes of cells i and j. (D) Schematic illustration and definition of the metrics characterizing T-cell motility: translational speed, turning angle speed, and meandering index. All metrics are measured for each cell every 1t seconds and pooled together to form statistical distributions. (E) Statistical profiles characterizing the T-cell locomotion consists of distribution histograms of translational speeds, turning angle speeds, and meandering indices. The histograms are derived from the corresponding empirical cumulative distribution functions (CDFs) available in Figure S17 from Read et al. (22), in which original in vivo data are presented. (F) The details of the 2D geometric setup for simulations used in the model calibration: spatial configuration, initial and boundary conditions, and the experimental protocol used to sample the statistical profile. (G) The statistical characteristics of T-cell motility coming from simulations of the calibrated model plotted against the in vivo histogram data (22). The statistical distributions of each metric are depicted as CDFs. The Kolmogorov–Smirnov statistics comparing the model and target CDFs are indicated with their respective p-values.

TABLE 1 | Set of calibrated model parameters used as a baseline for all simulations.


† Parameters obtained directly from experimental measurements.

\*Parameters estimated indirectly from experimental measurements.

‡Parameters derived from underlying computational models.

<sup>t</sup>Parameters tuned to fit cell motility profiles within the model calibration.

the random vector f mot i can be influenced by contact effects from neighboring cells, resulting in (1) a shift of the vector f mot i away from neighboring cells and (2) a decrease of its magnitude proportionally to their number, similar to the CIL model (23) (see details in section Materials and Methods). By default, the arresting coefficient for T cells is equal to one. For DCs, its value is estimated so that the resultant DC velocities do not exceed 5 µm/min (the estimated value is specified in **Table 2**).

Overall, the mechanistic model of the spatial multicellular dynamics is formulated as a system of N random ordinary differential equations (44) represented by Equation (1) and embedded into the 2D geometric domain as detailed in **Figure 1F**. Essentially, the system is a deterministic system of ordinary differential equations on each interval of 1t seconds, until the force f mot i becomes updated. The quantitative consistency of the computational model of multicellular dynamics with experimental data on translation speed, turning angle speed, and the meandering index is illustrated in **Figure 1G**. The relevant components of the numerical implementation of the model (computational domain, boundary conditions, integration algorithm) are described in Materials and Methods. The dynamics of the net forces and their contributions acting on a randomly selected T cell in a simulation of multicellular dynamics are shown in **Figure S1**.

# Calibration of T-Cell Motility

Our model mostly operates with biophysical parameters that are either directly measurable or can be estimated indirectly such as the mass m (wet weight) and the radius r of a cell, the adhesive strength between T-cell membranes f adh ij (measured by single cell force spectroscopy), the viscous damping coefficient µ, typical forces and velocities of T cells, and the location of demarcation between repulsive and attractive areas of a cell λ (nuclear-to-cytoplasma ratio). The other parameters that describe the random motility force or the contact inhibition of locomotion are derived using the information presented in the original IHomoCRW model (22) and the CIL model (23) with the underlying experimental data. To calibrate our model, we evaluated admissible ranges of parameters and tuned them manually to match the statistical characteristics of Tcell locomotion (22). The baseline sets of the estimated model parameters are presented in **Tables 1**, **2**. For details of the parameter estimation, see **Supplementary Text**.


#### Computational Domain, Immune Cell Subsets, and Initial Configuration

The computational domain was implemented as an ellipseshaped 2D approximation of the bean-like cross section of a murine skin-draining LN (see **Figure 2A**). At the beginning, both CD4<sup>+</sup> T cells (green) and CD8<sup>+</sup> T cells (blue) are evenly distributed throughout the domain. Some randomly chosen T cells are considered to be antigen specific and marked in light green and blue, respectively (their numbers are specified below). The antigen-presenting cells considered in this study represent the subset of cross-presenting migratory CD8α intCD103hi DCs, which are mainly involved in CD8<sup>+</sup> T-cell immune responses and which immigrate into LNs from the periphery (24). They are normally localized in the deep parts of the T-cell zone and leave LNs slowly with a turnover rate of 6 days. For initial configuration, these DCs are spatially placed according to a Gaussian distribution with 99-percentile ellipse Ωα=0.99 DC representing the T-cell zone (see **Figure 2A** and section Materials and Methods).

The numbers of antigen-specific DC and T-cell subsets are estimated using published data (24), which were rescaled according to the size of the computational domain. A total population of 12,469 immune cells was considered. The total number of non-antigen-specific T cells was estimated so that about 80% of the computation domain was filled up. The precursor frequency of antigen-specific T cells, that is, their proportion in the total amount of T cells, was set to be about 1%. We consider the inflow and outflow of immune cells to the region of interest to be negligible because of the short simulation time of 12 h. The closed boundary conditions used in the simulations are specified in section Materials and Methods. The overall geometrical scheme of the computational domain and the initial configuration of the multicellular system generated for simulations are presented in **Figure 2A**.

#### Data Assimilation and Model Validation

To assimilate the statistical data on the three T-cell locomotion measures (i.e., the translational speed, turning angle speed, and meandering index), the following numerical simulation protocol was used, which is close to the original experimental protocol (22). First, the same 2D 412 × 412µm<sup>2</sup> domain was used, in which we initialized 4,489 squarely tiled T cells with 3-µm radii and η ≈ 80% packing density. The initial direction of the intrinsic motility force was generated randomly for all cells. The positions of cells were saved every 30 s during 10 numerical experiments of 30-min simulation time after a 30-min pre-run to randomly mix the cells. Cells with total displacements <27µm were excluded as was done in the original experimental protocol. Likewise, cells that passed through the boundary and left the imaging volume were also excluded. The saved cell positions were post-processed to calculate the target metrics (defined in **Figure 1D**), which were pooled together to form three separate distributions. The pooled cell motility distributions were calibrated with in vivo data. The simultaneous adjustment of all distributions was computationally challenging due to the different uncorrelated aspects of cell migration captured in each of the motility metric as previously outlined (22). **Figure 1G** shows the best-fit cumulative distribution functions (CDFs) of the calibrated model with the baseline parameter set from **Table 1** and the target experimentally observed distributions with Kolmogorov–Smirnov statistics and p-values describing the discrepancy between CDFs.

The evolution of the above multicellular system was simulated over a 12-h period. The visualization of the systems spatiotemporal dynamics is presented in **Movie S1**. **Figures 2B,C** shows the kinetics of median velocities of antigen-specific CD4<sup>+</sup> T and CD8<sup>+</sup> T cells, and the median distances between the T cells and the centroid of their cognate antigen-presenting DCs throughout 12 h of an in silico experiment. The model demonstrates that antigen-specific CD8<sup>+</sup> T cells that interact with their cognate CD8α int DCs but not the CD4<sup>+</sup> T cells decrease their velocities, move closer to the area of DCs in the first 4–6 h, and remain there with low velocities afterward. **Figures 2B,C** is quantitatively consistent with experimental data shown in **Figures 1E,F**, and in Figure 2B from Kitano et al. (24).

## Quantitation of the DC and T-Cell Contact Interactions

The calibrated mathematical model of T-cell locomotion was validated by confronting its predictions with data from the intranodal spatiotemporal dynamics of different immune cell subsets after soluble antigen immunization presented in a recent experimental study (24). The data specify the evolution of the distances between the centroid of the migratory DC area and individual CD4<sup>+</sup> T and CD8<sup>+</sup> T cells. The model was adjusted to the functional configuration of skin-draining LNs specified

precursor frequency. (B) Twelve-hour kinetics of median velocities of antigen-specific CD8<sup>+</sup> T and CD4<sup>+</sup> T cells, and their distributions at the start and at the end of a 12-h simulation. (C) Twelve-hour kinetics of median distances from T cells to the centroid of DCs, measured for antigen-specific CD8<sup>+</sup> T and CD4<sup>+</sup> T cells, and their distributions at the start and at the end of a 12-h simulation. TC, T cell; DC, dendritic cell.

in the above study. A representative example of the numerical simulation of individual cell trajectories is shown in **Figure 3A**. An example of multicellular dynamics in a LN during 12 h is shown in **Figure 3B**.

#### Quantitation of the Forces Determining T-Cell and DC Motility and Their Interaction

To consider DCs in multicellular system simulations, we carried out a parameterization of their intrinsic motility forces and the intercellular forces for contacts between (1) two DCs, (2) antigen-presenting DCs and antigen-specific T cells, and (3) antigen-presenting DCs and polyclonal T cells. The values of the corresponding parameters are presented in **Table 2**. The physical forces driving the dynamics of individual cells in the LN and the respective velocities of the cells predicted by the model are shown in **Figure 3C**. We assume that the intrinsic motility of DCs can be represented by the same type of force function as that for T cells (**Figure 1C**); however, due to their much smaller average velocity, the respective DC force function value was of small magnitude. The adhesive force for cognate contacts

configuration specified in Figure 2A. Only cells with total displacement longer than 27µm are shown. (C) Values of forces and cell velocities driving the multicellular system dynamics in a square subdomain of a LN. In a center pane, the velocity field is represented as a contour plot of the field of cell velocity magnitudes linearly interpolated at uniform grid, as well as detected streamlines of possible cell flow patterns. (D) Kinetics of the numbers of cognate DC–T cell contacts at different stages of the simulation and distribution of durations of all cognate contact durations occurring within a 12-h simulation. DC, dendritic cell.

(i.e., of antigen-specific T cells with antigen-presenting DC) is around 100 times higher (∼1 nN) than the non-specific adhesion force for T cell/T cell contacts (36). We also implemented a computational procedure to temporarily arrest the motility for T cells in a sufficiently long cognate contact (see section Materials and Methods).

#### Ag-Specific CD8<sup>+</sup> T Cells Migrate Toward Cross-Presenting DCs and Form Cognate Contacts With Them

**Figure 3D** presents the model prediction for the kinetics of the number of cognate DC–CD8<sup>+</sup> T-cell contacts occurring at different time intervals during the in silico simulation. Antigen-specific CD8<sup>+</sup> T cells robustly increase the number of contacts with DCs over time in the process of T-cell zone scanning for antigen-presenting target cells. Although most of the cognate contacts are of short duration, i.e., they last for <5 min, the distribution has a heavy tail of stable more than 1-h length contacts. These predictions are in agreement with previous data (45).

# CTL Frequency Needed to Locate HIV-Infected Target Cells Before Viral Release

During viral infections, the induction of cellular immune responses takes place in secondary lymphoid organs such as LNs and spleen. Antigen-presenting cells such as DCs take up antigens and migrate to LNs to encounter specific lymphocytes, e.g., CD4<sup>+</sup> T and CD8<sup>+</sup> T cells, to induce their activation and differentiation into effector and memory cell subtypes (46). The low frequency of antigen-specific T cells in unprimed hosts turns the scanning of cognate DCs by specific T cells in a highly crowded cellular LN environment into a "needle-in-a-haystack" problem (47). It was revealed that optimal LN scanning depends on a combination of intrinsic T-cell motility, the chemokine milieu, and the microarchitecture of the LN (1). When virusinfected DCs reach the LN, the less the time needed to locate virus-specific T cells and to form stable DC–T cell contacts, the more likely is that the precursor CTL activation will happen before the viruses will be released from infected cells, therefore making the elimination of local clusters of infection spread more probable. This aspect of CD8<sup>+</sup> T-cell activity is crucial for a prompt activation of specific CTL immune responses and the elimination of viruses. The precursor frequency in blood can be as small as 0.0001% (48), reaching about 5–10% in the chronic stage of an HIV infection (49). The here-developed physicsbased model of T-cell dynamics can be directly used to study the efficiency of scanning the paracortical T-cell zone of the LN for target cells expressing cognate antigen as a function of the frequency of CTL and their motility.

Development of an effective AIDS vaccine remains a global priority, and there is a need for a vaccine to induce cellular immune responses capable of eradicating or efficiently containing virus replication (50). Experimental studies with attenuated SIV vaccines indicated that SIV-specific CTLs, if present in sufficient frequencies, can completely control and even clear an infection (19). Similar to SIV, HIV infection is sustained by the activation of CD4<sup>+</sup> T cells, which occurs in the form of transient bursts in the local microenvironment of lymphoid tissues (51, 52). The proximal activation and transmission involving latently infected cells represent locally propagating events (53). Therefore, we applied our calibrated model of spatial immune cell dynamics in LNs to study the necessary conditions for effector HIV-specific CTLs to promptly locate HIV-infected target cells before they can release viral progeny. We consider only one HIV-infected cell in the computational domain, which is consistent with the frequency of productively infected CD4<sup>+</sup> T cells of about 0.0001–0.001 (54). Specifically, the newly infected target cell should be located by the nearby effector cells before it can release viral progeny, i.e., before completion of the 18–24 h life cycle of HIV (20).

The overall simulation setup is the same as described in the model validation subsection above. Randomly chosen cells in the stochastically generated multicellular system configurations representing the LN cortex zone were marked as infected in yellow (see **Figure 4A**). Both the motile CD4<sup>+</sup> T cells and the non-motile DCs were considered as HIV-infected targets. In simulations, we varied the frequency of HIV-specific CD8<sup>+</sup> T cells and the intrinsic motility of T cells (searching CD8<sup>+</sup> T cells, infected- and uninfected CD4<sup>+</sup> T cells) (**Figure 4A**) to analyze the effect of variations on the target cell detection time. A 10-fold range of HIV-specific CD8<sup>+</sup> T-cell frequencies typical for HIV infection, i.e., from 0.4 to 5%, was examined. The intrinsic motility of T cell was varied within 100 and 50% relative to the calibrated baseline parameters of average T-cell velocity (see details in the **Supplementary Text**). A decreased intranodal T-cell motility (of searching CD8<sup>+</sup> T cells, infected- and uninfected CD4<sup>+</sup> T cells) is expected to take place during the chronic stage of an HIV infection when LN tissues become fibrotic, i.e., when collagen formation in T-cell zones takes place (55). Then, T cells have to move through increased collagen deposition with major consequences for search patterns (56). In addition, CD4<sup>+</sup> T-cell migration is also inhibited by the HIV-1 Nef protein as shown in chemotaxis assays (57). In our study, the motility of all considered types of T-cell subsets, i.e., the searching CD8<sup>+</sup> T cells, uninfected CD4<sup>+</sup> T cells, and infected CD4<sup>+</sup> T cells, is decreased uniformly.

**Figures 4B,C** illustrate the model predictions for the decrease of time to locate HIV-infected target cells with the increase of HIV-specific CD8<sup>+</sup> T-cell frequency. The modeling results imply that 5% is a sufficient effector CTL frequency for a timely detection of both types of target cells within 18 h post-infection, i.e., before the beginning of HIV release from productively infected cells (20). A stepwise increase by fivefold of the HIV-specific CTL frequency from 0.04 to 5% increases the probability of detection of HIV-infected cell within 24 h from 0.07 to 0.34 to 0.84 and to 1, respectively. In addition, the model shows that infected motile CD4<sup>+</sup> T cells are located faster than non-motile DCs with the probability of detecting them within 24 h increasing from 0.35 to 0.86 and to 1 with a CTL frequency rising from 0.04 to 0.2% and to 1%, respectively.

**Figure 4D** shows the increase of time to locate infected nonmotile DCs for a decrease of the T-cell motility from a basal level by 10 and 50% considering an HIV-specific CD8<sup>+</sup> T-cell frequency of 1%. If the average T-cell velocity is decreased by 50%, then the probability to locate DCs within 24 h is <0.5. **Figure 4E** depicts a similar dynamics for locating motile infected CD4<sup>+</sup> T cells. Note that the motile targets were located within 24 h even with a 50% decrease of the average CD8<sup>+</sup> T-cell velocity in all performed simulations.

target cell was introduced until it was located by effector HIV-specific CTLs was measured in 24-h simulations. The infected cell was either non-motile DC (B,D) or motile CD4<sup>+</sup> T cell (C,E). In (B,C), the precursor frequency, i.e., the frequency of effector T cells, was varied, from 0.04 to 5%. In (D,E), the effect of decreased intrinsic motility of T cells was studied. The average T-cell velocity was decreased up to 50%. In all plots, the fraction of cases with location time >24 h is indicated, thus providing the estimates for probability to locate target cells within 24 h. The time range between the start and the peak HIV release from infected T cells (20) is shown in pink. It is used to estimate the probability of a virus burst to escape effector CTLs and, thus, to contribute to the spread of HIV-infected cells within a LN. TC, T cell; DC, dendritic cell.

#### DISCUSSION

We have developed a biophysics-based computational model of T-lymphocyte motility that is calibrated using empirical in vivo data on T-cell migration in LN tissue representing three spatial metrics of multicellular systems behavior, i.e., translational speed, turning angle speeds, and meandering index. The model provides the tool to quantify the velocity and the driving force fields in the LN. It enabled us to predict frequency and motility parameters that are required for a timely detection of productively HIV-infected cells within LNs before they release viral progeny. As such, our study provides a quantitative guide for an informed design of HIV vaccines. Furthermore, as the immunological principles of antigen-specific T-cell activation and immune surveillance imbedded in our model also apply to other infections and cancers, our findings may be used to define the general requirements for any efficient immunotherapeutic intervention against pathogens or cancers in relation to disease-specific parameters and states of lymphoid tissue and T cells. Thus, our model has a significant potential to guide the search for better and more efficient immunotherapies in the near future.

Other processes, e.g., chemotaxis, haptotaxis, and others, can influence the efficacy of target cell search by CTL. The impact of chemotactic migration of T cells toward DCs has been computationally analyzed using a cellular Potts Model (58), an agent-based model (59), and a multicompartmental spatially resolved stochastic model of T-cell circulation (60). The results suggest that the chemoattraction toward target cells modestly speeds up the search process for T cells that successfully find the chemokine-producing DCs. However, a qualitative model presented in (59) suggested that with even weak chemotaxis, substantially lower numbers of CTL are required for sterilizing immunity. Further data-based model-driven research is needed to clarify the contribution of chemotaxis to T-cell migration under normal conditions and during inflammation (61).

Phenomenological Ordinary Differential Equation (ODE) models may also be developed to simulate the interactions between cell populations in the LN. However, these models are not suitable for the present study for three reasons. First, data on T-cell motility in the LN cannot be directly used to calibrate such models, thereby limiting the validity of their predictions. Second, the objective of our study, which is the early detection of HIVinfected T cells and DCs, requires the monitoring of the spatial density of T cells in the LN rather than the total number of T cells. Changes in the spatial distribution of T cells in the LN can be related to spatial mechanisms such as chemotaxis and migration. Therefore, it is crucial to consider spatial aspects in the model. Finally, ODE models based on "mass action"- or "predator– prey"-type parameterizations would require the parameter values specifying a per capita killing rate of target cells. The respective parameter can be determined by the mean time needed for a migrating CTL to locate infected cells. A priori estimates of this parameter are not available. It is the spatially resolved modelbased simulation that needs to be implemented in order to quantify the killing rate coefficient of the ODE model.

Moving from phenomenological models of spatiotemporal dynamics of immune processes (e.g., the compartmental models, CAMs) to a physics-based description of immune cell migration in complex multicellular tissue environments presents a challenge to mathematical immunology. Advances in the direct visualization of antigen-specific T-cell mobility during their search for and their interaction with antigen-presenting cells within LNs set the basis for diverse modeling approaches (7, 10, 11), which have been so far based on ad hoc postulated rules of cell behaviors. Our study gives a biophysics perspective on coordinated cell motility in lymphoid tissues, thus extending the range of modeling tools available for implementing integrative approaches to the exploration of the immune system.

CPMs have also been applied previously to study intranodal T-cell migration (58). The CPM framework is a valuable tool for a phenomenological description of multicellular patterning, providing realistic simulations of morphological changes for various cell types. The strength of this approach stems from its flexible energetic formalism that allows for extensions to incorporate various biological processes (62). Although the CPM framework has a richer potential for describing individual cell dynamics, including the cell shape, this comes at the expense of (i) a higher-dimensional representation of the cell configuration (e.g., the number of voxels or pixels), (ii) the use of phenomenologically rather than biophysically defined parameters, and (iii) a much higher computational cost to perform simulations required to explore T-cell search strategies. Besides, there is no direct correspondence of most of the CPM parameters with biophysical properties of cells, and the meaning of some CPM parameters is still under debate (62, 63). Moreover, CPM temporal kinetics obtained with the modified Metropolis algorithm does not preserve the detailed balance condition for the underlying stochastic process. This implies that the exact relation between forces of cell interactions and energy terms of CPMs cannot be obtained even for the overdamped dynamics approximation (62, 63).

Computational modeling of multicellular dynamics in lymphoid tissues provides a theoretical tool to be used for a better understanding of the determinants of efficient immune responses against pathogens with a final aim of an optimal manipulation of the immune systems performance (2, 15). Given that the quest for an effective HIV vaccine remains a global priority (64) and that the localization, migration, and frequency of CTLs in LNs determine the extent of virus elimination (17, 19, 36, 56, 65), we sought to use our modeling approach to define threshold frequencies of CTLs in LNs for protection against HIV. Since an HIV infection can influence (i) CD4<sup>+</sup> T-cell motility by a direct mechanism involving the HIV Nef protein and (ii) CTL locomotion via an indirect mechanism related to the induction of lymphatic tissue fibrosis, we considered both phenomena to predict the effect of reduction of T-cell motility. We estimated that the frequency of antigenspecific CTL should be about 5% to timely detect and completely eliminate productively infected DCs within 18 h. The time reduces to 4 h for productively infected CD4<sup>+</sup> T cells, which are motile. For an HIV-specific T-cell frequency of 1%, we computed that the inhibition of CTL locomotion by two-fold would reduce the probability of detection of infected target cells within 24 h post-infection from 0.84 to 0.42. Thus, the requirements for a prophylactic vaccine for seronegative individuals and an immunotherapeutic intervention of already HIV-infected individuals may differ significantly and are influenced by the state of the lymphatic tissue structure.

Understanding the spatiotemporal dynamics of immune cells globally in the lymphatic system and locally in LNs is considered to be a prerequisite for the development of novel immune interventions in the context of HIV cure strategies (15, 56). To this end, mathematical tools are being increasingly applied to predict the impacts of trafficking and motility parameters on the efficiency of immune surveillance in health and disease. For example, an optimal surveillance strategy for T cells was analyzed by compartmental modeling of their systemic recirculation and LN transit times using a multicompartmental consideration (66). The protective effect of increased CD4<sup>+</sup> T-cell trafficking on the dynamics of HIV infection has been recently shown using another compartmental model (67), thus providing a basis for considering cell trafficking as an adjunct therapy option. A multiscale model of Mycobacterium tuberculosis infection including an agent-based description of the cellular movement in a two-dimensional simulation grid representing the granuloma was developed and calibrated using non-human primates to derive the prediction of parameters underlying granuloma sterilization (8). However, such modeling attempts are still rare.

In conclusion, the large number of existing mathematical models based on low-resolution descriptions of immune functions has to be further extended and embedded into physiologically distinct compartments and 3D morphological constraints inherent to cells, tissues, and the whole organism. This will then allow the research community not only to get a better quantitative understanding of immune system functioning in infections such as HIV but also enable to build integrative models for antiviral and immunomodulatory drugs of various physical and chemical nature as well as the effects of adoptive cell transfer therapies. We believe that a comprehensive approach to combination therapies based on ART and immunomodulatory drugs affecting a range of processes, including LN fibrosis, the exhaustion of CTLs, and T-cell motility, should rely on formulation and implementation of hybrid spatially resolved multiscale mathematical models of virus infections (8, 9, 68). The here-developed model offers a broadly

# REFERENCES


applicable generic mathematical tool for linking individual and coordinated cell behaviors that can be used for in silico studies to embed immune processes into their spatial context. The physics-based computational model of multicellular dynamics of the immune response in lymphoid tissues provides a solid module that can be universally used in systems immunology studies (2, 6) for the benefit of patients suffering from chronic virus diseases.

## AUTHOR CONTRIBUTIONS

DG, AM, VV, and GB conceived and designed the study. DG, VV, AM, and GB wrote the paper. DG, AB, and NB performed the computational implementation of the model. DG performed the model calibration.

# FUNDING

This work is supported by the Russian Science Foundation (grant 18-11-00171). VV was supported by a grant from the RUDN University Program 5-100. AM is also supported by a grant from the Spanish Ministry of Economy, Industry and Competitiveness and FEDER grant no. SAF2016-75505-R (AEI/MINEICO/FEDER, UE) and the María de Maeztu Programme for Units of Excellence in R&D (MDM-2014-0370).

#### SUPPLEMENTARY MATERIAL

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fimmu. 2019.01213/full#supplementary-material


**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2019 Grebennikov, Bouchnita, Volpert, Bessonov, Meyerhans and Bocharov. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# Corrigendum: Spatial Lymphocyte Dynamics in Lymph Nodes Predicts the Cytotoxic T Cell Frequency Needed for HIV Infection Control

Dmitry Grebennikov 1,2,3 \*, Anass Bouchnita<sup>4</sup> , Vitaly Volpert 3,5,6, Nikolay Bessonov <sup>7</sup> , Andreas Meyerhans 8,9 and Gennady Bocharov 2,10 \*

<sup>1</sup> Moscow Institute of Physics and Technology, National Research University, Dolgoprudny, Russia, <sup>2</sup> Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia, <sup>3</sup> Peoples' Friendship University of Russia (RUDN University), Moscow, Russia, <sup>4</sup> Division of Scientific Computing, Department of Information Technology, Uppsala University, Uppsala, Sweden, <sup>5</sup> Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, Villeurbanne, France, <sup>6</sup> INRIA Team Dracula, INRIA Lyon La Doua, Villeurbanne, France, <sup>7</sup> Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, Saint Petersburg, Russia, <sup>8</sup> Infection Biology Laboratory, Department of Experimental and Health Sciences, Universitat Pompeu Fabra, Barcelona, Spain, <sup>9</sup> Institució Catalana de Recerca i Estudis Avançats (ICREA), Barcelona, Spain, <sup>10</sup> Sechenov First Moscow State Medical University, Moscow, Russia

#### Approved by:

Frontiers Editorial Office, Frontiers Media SA, Switzerland

#### \*Correspondence: Dmitry Grebennikov

dmitry.ew@gmail.com Gennady Bocharov bocharov@m.inm.ras.ru

#### Specialty section:

This article was submitted to T Cell Biology, a section of the journal Frontiers in Immunology

Received: 18 June 2019 Accepted: 19 June 2019 Published: 03 July 2019

#### Citation:

Grebennikov D, Bouchnita A, Volpert V, Bessonov N, Meyerhans A and Bocharov G (2019) Corrigendum: Spatial Lymphocyte Dynamics in Lymph Nodes Predicts the Cytotoxic T Cell Frequency Needed for HIV Infection Control. Front. Immunol. 10:1538. doi: 10.3389/fimmu.2019.01538 Keywords: lymphoid tissue, cell motility, HIV infection, cytotoxic T cell scanning, multicellular dynamics, dissipative particle dynamics, stochastic differential equation

#### **A Corrigendum on**

#### **Spatial Lymphocyte Dynamics in Lymph Nodes Predicts the Cytotoxic T Cell Frequency Needed for HIV Infection Control**

by Grebennikov, D., Bouchnita, A., Volpert, V., Bessonov, N., Meyerhans, A., and Bocharov, G. (2019). Front. Immunol. 10:1213. doi: 10.3389/fimmu.2019.01213

In the original article, there was a typo in **Figure 2A** color legend as published. The colored circles denoting Ag-specific and non-specific T cells should be swaped. That is, the dark green color should represent Ag-specific CD4<sup>+</sup> TCs, the light green color—non-specific CD4<sup>+</sup> TCs; the dark blue color should represent Ag-specific CD8<sup>+</sup> TCs, the light blue color—non-specific CD8<sup>+</sup> TCs. The corrected **Figure 2** appears below.

The authors apologize for this error and state that this does not change the scientific conclusions of the article in any way. The original article has been updated.

Copyright © 2019 Grebennikov, Bouchnita, Volpert, Bessonov, Meyerhans and Bocharov. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

FIGURE 2 | Heterogeneous dynamics of T cells in LNs. (A) The scheme of a LN and illustration of the initial configuration generated for simulations. DCs, CD4<sup>+</sup> T cells, and CD8<sup>+</sup> T cells are placed within a LN as described in the Supplementary Text with total cellularity of 12,469 cells, ≈80% packing density and ≈1% precursor frequency. (B) Twelve-hour kinetics of median velocities of antigen-specific CD8<sup>+</sup> T and CD4<sup>+</sup> T cells, and their distributions at the start and at the end of a 12-h simulation. (C) Twelve-hour kinetics of median distances from T cells to the centroid of DCs, measured for antigen-specific CD8<sup>+</sup> T and CD4<sup>+</sup> T cells, and their distributions at the start and at the end of a 12-h simulation. TC, T cell; DC, dendritic cell.

# Distributed Adaptive Search in T Cells: Lessons From Ants

Melanie E. Moses 1,2,3, Judy L. Cannon4,5,6 \*, Deborah M. Gordon3,7 and Stephanie Forrest 3,8

<sup>1</sup> Moses Biological Computation Laboratory, Department of Computer Science, University of New Mexico, Albuquerque, NM, United States, <sup>2</sup> Biology Department, University of New Mexico, Albuquerque, NM, United States, <sup>3</sup> Santa Fe Institute, Santa Fe, NM, United States, <sup>4</sup> The Cannon Laboratory, Department of Molecular Genetics & Microbiology, University of New Mexico School of Medicine, Albuquerque, NM, United States, <sup>5</sup> Department of Pathology, University of New Mexico School of Medicine, Albuquerque, NM, United States, <sup>6</sup> Autophagy, Inflammation, and Metabolism Center of Biomedical Research Excellence, University of New Mexico School of Medicine, Albuquerque, NM, United States, <sup>7</sup> Department of Biology, Stanford University, Stanford, CA, United States, <sup>8</sup> Biodesign Institute and School for Computing, Informatics, and Decision Sciences Engineering, Arizona State University, Tempe, AZ, United States

There are striking similarities between the strategies ant colonies use to forage for food and immune systems use to search for pathogens. Searchers (ants and cells) use the appropriate combination of random and directed motion, direct and indirect agent-agent interactions, and traversal of physical structures to solve search problems in a variety of environments. An effective immune response requires immune cells to search efficiently and effectively for diverse types of pathogens in different tissues and organs, just as different species of ants have evolved diverse search strategies to forage effectively for a variety of resources in a variety of habitats. Successful T cell search is required to initiate the adaptive immune response in lymph nodes and to eradicate pathogens at sites of infection in peripheral tissue. Ant search strategies suggest novel predictions about T cell search. In both systems, the distribution of targets in time and space determines the most effective search strategy. We hypothesize that the ability of searchers to sense and adapt to dynamic targets and environmental conditions enhances search effectiveness through adjustments to movement and communication patterns. We also suggest that random motion is a more important component of search strategies than is generally recognized. The behavior we observe in ants reveals general design principles and constraints that govern distributed adaptive search in a wide variety of complex systems, particularly the immune system.

#### Gennady Bocharov,

Institute of Numerical Mathematics (RAS), Russia

#### Reviewed by:

Edited by:

Frederik Graw, Universität Heidelberg, Germany Naomi Taylor, Institut National de la Santé et de la Recherche Médicale (INSERM), France

#### \*Correspondence:

Judy L. Cannon jucannon@salud.unm.edu

#### Specialty section:

This article was submitted to T Cell Biology, a section of the journal Frontiers in Immunology

Received: 06 February 2019 Accepted: 29 May 2019 Published: 13 June 2019

#### Citation:

Moses ME, Cannon JL, Gordon DM and Forrest S (2019) Distributed Adaptive Search in T Cells: Lessons From Ants. Front. Immunol. 10:1357. doi: 10.3389/fimmu.2019.01357 Keywords: T cells, ant foraging, adaptive search, collective search, ant inspired algorithms

# INTRODUCTION

T cells are key players in adaptive immunity, required for clearance of virally infected cells and tumor cells. Improved understanding of how T cells search may lead to more effective T cell vaccine design and cancer immunotherapies. Many types of immune cells search for pathogens or other targets, but T cell search is especially challenging because T cells are often responding to novel pathogens. Ant colonies are another distributed adaptive system in which individual agents search cooperatively, without centralized control, to find targets in unknown locations in a complex environment. However, ant colonies are simpler and, in some ways, easier to observe than immune systems. Here we propose new hypotheses about how T cells search suggested by successful search strategies in ant colonies.

T cells search for many kinds of targets, at many scales, and in many different tissues including lymph nodes, infected tissues and systemic infection in the whole body. Prior to infection, T cells migrate through the lymphatic network to search within lymph nodes for potential activating antigen presented by dendritic cells (1, 2). If a naive T cell finds a dendritic cell bearing its cognate antigen in the lymph node, the T cell then proliferates and migrates out of the lymph node through the cardiovascular network, extravasating at the site of infection in peripheral tissue, where activated T cells conduct a second search to find and eliminate target cells (3).

How do the interactions among T cells, host cells, target cells, and tissue architecture generate the remarkably rapid and effective immune response to a wide variety of pathogens and tumors? Computational and mathematical approaches have described aspects of immune responses (4, 5) including the T cell repertoire (6), development of the effector and memory responses (7–11), and T cell responses to infections including HIV (12), influenza (13), and anti-tumor responses (14–16) just to name a few. Mathematical models have been developed to study how T cell movement through lymph nodes impacts T cell activation (17–22). However, relatively few mathematical models have connected individual T cell movement and interactions during search to the broader outcomes of immune response to infection, particularly in complex tissue environments (19, 23, 24).

Foraging strategies in ants suggest a framework for understanding how collective search strategies emerge from the behaviors of individual agents (25–27). These questions are difficult to answer experimentally in immunology, especially at the scale of an entire organ or body. In ant colonies, we can simultaneously observe the small-scale behavior of individuals and the large-scale collective, such as shifts in the allocation of ants to various tasks (28), territorial interactions between different colonies, (29–31), and the recruitment of searchers to discovered food (32–34). Thus, extrapolating understanding about the search strategies of ants to immune responses can suggest general design principles that can then be tested in the immune system.

We use the understanding of ant foraging gained from experiments and models to provide insight into T cell search processes. We find that there are significant parallels between how ants forage for food and how T cells search for pathogens. First, both T cells and ants combine random movement with directed movement to produce an effective search strategy across a wide variety of environmental conditions. Both ants and T cells search for targets whose positions are unknown, dispersed, and can be both mobile and ephemeral, thus ants and T cells need random elements in their strategies to flexibly adapt to dynamic conditions and varied environments. Second, ants and T cells both use communication to improve search efficiency by following chemical signals to the locations of their targets; additionally, direct agent-agent interaction may provide a direct form of communication to increase search efficiency. Third, physical structures, such as nest and trail structure for ants and the lymphatic network and the stromal cell network in tissues, provide spatial networks embedded in the search space that can guide the movement of searchers.

Studies of ant foraging reveal that effective search strategies incorporate an appropriate balance of movement that is random, guided by signals and agent interaction, and mediated by traversal of physical structures. We focus on how the appropriate balance depends on two factors. First, the best search strategy depends on the distribution of targets in time and space. Second, the best strategy depends on whether the objective of the search is to be fast (finding targets as quickly as possible) or complete (finding all available targets), or some combination of the two. Search strategies from ant foraging suggest specific hypotheses that can be tested to reveal novel search strategies taken by T cells in complex tissue environments leading to more efficient immune responses.

#### ANT FORAGING AS A MODEL FOR T CELL SEARCH

Ant colonies are a canonical example of collective intelligence, demonstrating strategies for effective distributed search in varied ecological spaces in almost every terrestrial habitat on Earth. Each of the 14,000 species of ants has evolved in a particular environment, leading to diversity among species in how they move, interact with each other and use physical structures as they forage for food. Ecological and evolutionary studies show a correspondence between foraging behavior and the dynamics of the resources that a species uses (34–37). The Lanan review thoroughly catalogs a remarkable diversity of foraging strategies, including different forms of movement, recruitment and trail formation, and shows that different environmental conditions faced by different species generate predictable regularities in these foraging strategies.

A key feature that influences search strategy is the distribution of resources in time and space (36, 38–40). Targets can be patchy, clustered into one location, or dispersed uniformly at random through the entire search area. **Figure 1** shows examples of spatial distributions from highly clustered to highly dispersed, as well as a power law distribution with both clusters and dispersed targets. Models of ant foraging have demonstrated that the speed of target collection changes dramatically depending on how targets are distributed [**Figure 2**, discussed in section IV, (39, 41)]. For example, when resources are patchy, foragers recruit each other to the location where resources have been found. One mechanism for recruitment is chemical pheromone trails that, by inducing one ant to follow another, generate information about the location of food (34, 40, 41). By contrast, when resources are scattered or ephemeral, pheromone recruitment is pointless, and ants do not guide each other in any particular direction [instead regulating whether or not to forage at all, (42)].

The objective of the search also plays a role in determining the most effective search strategy. We highlight two such objectives. Fast detection weights detection speed of the first targets most heavily, while for complete detection the goal is to find all of the targets. Some searches combine these objectives, i.e., finding all targets as quickly as possible, but in many instances either speed or completeness is deemphasized. Our computational models show that the nature of the search problems matters: successful strategies for complete detection differ from those for fast detection (43–45).

Ant colonies provide several examples of fast detection. In foraging by desert seed harvester ants (one of most well-studied ant groups), the goal is to collect as many resources as possible in a fixed time window. The foraging window is limited because ants lose water rapidly while foraging in the hot sun (46), so the search must be fast. However, seeds remain on the ground and in the soil for a long time, so it is not important to collect all available seeds immediately, as they will be available later. Our models, described below, show that ants that recruit each other to a single pile may also achieve complete collection of that pile, but when resources are dispersed among many piles, the ant strategies we model fail to achieve complete collection, for example taking much longer to find the last 10% of targets than the first 90% (43).

The stability of targets over time also influences both fast and complete detection. For example, recruitment of agents to a particular location is useful only if targets persist long enough in one place for other searchers to find the targets when they arrive (34, 40). Consequently, ants that forage for resources that are both clustered in space and persistent in time evolve strategies for recruitment, and ants that forage for randomly dispersed resources do not.

#### T CELL SEARCH MUST BE FAST, AND OFTEN MUST BE NEARLY COMPLETE

Our characterization of ant search suggests new ways to interpret search behaviors of T cells. T cells in the lymph node search for antigen presented by dendritic cells. To succeed in initiating the adaptive immune response quickly, this search must be fast rather than complete. Speed is important because pathogen replication is an exponential process. However, because multiple antigens can be presented to multiple T cells, T cells do not need to interact with every possible DC. Instead, T cell search in the lymph node needs to detect only enough antigen to initiate activation. This search problem resembles our models of ants conducting fast, but not complete, searches for dispersed seeds.

In contrast, for T cell search in the periphery, thoroughness, or complete detection, is crucial in some cases. T cells must detect and eliminate virtually all pathogens. In influenza, for example, successful control requires finding and eliminating all, or nearly all, influenza virus in the lung. Similarly, in immunotherapy, the goal is for effector CD8 T cells to identify and kill all viable tumor cells. Our ant models quantify how this complete detection task becomes easier when targets are clustered in one or a few places and becomes more challenging when they are dispersed broadly.

Some immunological studies have described target distributions (10, 47, 48); however, models of the immune response rarely consider how effective different search strategies are at finding targets with different spatial distributions. Very little is known about how long different targets such as pathogens or tumor cells might persist in specific tissue locations or how mobile pathogens might be within tissues. Our ant foraging models suggest that understanding target distribution in tissues may be an important determinant of effective T cell search and immune responses.

### ANT-INSPIRED HYPOTHESES ABOUT T CELL SEARCH

#### Random and Directed Motion Combine to Produce Effective Search

Ants foraging for food often use a form of random search called a correlated random walk (CRW) (49–51). In a CRW, the angle of each successive turn is correlated with that of the previous one, and movement patterns are straighter and less convoluted than in Brownian motion. CRWs are more dispersive than Brownian motion, thus increasing the physical extent of the search area while decreasing its thoroughness by minimizing repeated sampling of the same area. In models designed to maximize the speed at which seeds are detected by foraging ants, a high degree of correlation among steps (leading to more straight-line ballistic motion) appears optimal for fast detection (39, 52, 53).

When the location of targets is known with sufficient probability, ants can move directly toward the location deemphasizing random movements. For example, ants use a process called site fidelity to return repeatedly to the location of a previously found a seed (54, 55). Beverly et al. found that when an ant finds a seed, it returns directly to that location with over 90% probability. Other species use site fidelity to search for resources that are clustered, even if those clusters are small or variable in size. Recruitment through olfactory interactions based on pheromones is a well-known mechanism by which ants attract other ants to locations where food is abundant or persistent.

fidelity or pheromone recruitment depending on the size of the piles they sense while searching. It is the most effective search strategy across all distributions, and it most clearly outperforms other strategies given intermediate pile numbers and sizes (e.g., 80 × 16) and the power law distribution which has mixed pile sizes.

The overwhelming complexity of the immune system reflects the different kinds of problems it is required to solve. For example, in some cases immune cells must search broadly for rare targets, and in other cases it must search thoroughly to find all targets. We used models of random and directed motion developed for ants to analyze how T cells move, building on existing studies of T cell motion in tissues. Initial work suggested that naive T cells in lymph nodes move randomly, and models of T cell movement assumed Brownian motion of T cells to estimate how many T cells are required to find DCs (8, 56). More recently, researchers hypothesized that T cell movement is characterized by Levy walks, another form of random motion in which cells move in random direction for multiple time steps drawn from a power law distribution (57). We demonstrated that prior to infection, both CD4 and CD8 T cell movement in the lymph node is random, but does not follow idealized Brownian or Levy movement patterns (58). Instead, our models show that T cells can disperse more quickly compared to Brownian motion, leading to more effective search for DC targets in lymph nodes. Our model predicts that the particular movement pattern we observe in T cells (a CRW with step lengths drawn from a lognormal distribution) balances thorough search in a small region with extensive search in a broader area. We hypothesize that these ant-like movement types affect how quickly T cells encounter rare vs. abundant antigen in the lymph node.

T cell motion in infected tissues varies according to the requirements of the task being performed. It is clear that migration of effector T cells into infected tissue is signal dependent and directional toward areas of inflammation, including in skin (24), brain (57), lung (59), vaginal tract, and gut (3). However, some studies suggest that once T-cells reach skin tissue their movement within the tissue is not highly directional toward the foci of infection (24). Our work found that effector T cells in inflamed lung also move in a CRW, similar to naive T cells in lymph node, suggesting random motion. In the lung, the T cell CRW is combined with a stop-and-go mode of intermittent motion, which enables effector T cells to search a larger area while also interacting with potential target cells (60). In contrast to cells in the lung, Harris et al. found that effector T cells in the brain of Toxoplasma infected animals move with a Generalized Levy walk (57). And, there is no evidence in brain or lung tissue that effector T cells move directionally toward sites of infection. Ant search models suggest one hypothesis to explain the lack of directional movement: if sites of infection are usually dispersed in space, for example, in tissues, T cells may have evolved search strategies to explore broadly for new sites of infection rather than focusing on exploiting already detected foci of infection.

#### Physical Structures for Effective Search

Networks provide physical structures that can increase search efficiency, minimizing the distance traveled to explore large spaces. Ants use environmental structures to extend their search. For example, turtle ants create trail networks within the network of vines and branches in the canopy of the tropical forest (26, 27). In some ant species, ants search along edges such as cracks in sidewalks and search from these main trails (61), apparently using environmental structures to explore the environment but not necessarily moving directly toward food sources. We suggest that one role of structural networks is to enhance the scalability of search to larger physical spaces.

Recent work imaging T cells in intact tissues suggests that T cells may also use structural networks to mediate motion. At the organism scale, cardiovascular and lymphatic networks disperse immune cells throughout all tissues to enhance response to infection anywhere in the animal. Effector cells in skin were shown to move along collagen fibers (62), and effector T cells in inflamed lung move along vasculature (60). Within the lymph node, T cells use fibroblastic reticular cells (FRC) as guidance cues (63, 64). The FRC network in the lymph node has the topological structure of a small world network, which likely enhances the robustness of T cell responses to damage to the network (65, 66). Small world networks with many local and a few long-distance connections significantly increase scalability, cohesion, and efficiency of exploration via the network (67).

T cell movement along tissue structures resembles that of ants traveling along branches. T cell movement along structures such as collagen, vasculature, and FRCs does not obviously lead to targets (64). There is also no evidence that effector T cells in lung and skin, where directional motion is important, use structural guidance to travel toward targets. Instead of providing directional guidance toward targets, we suggest that movement along networks may instead enhance scalability and maximize exploration of large spaces.

#### Distributed Communication: Soluble Signals and Direct Agent Contact

A striking similarity between ant colonies and immune systems is the use of chemical signals for communication. It is wellestablished that both systems use chemical cues to signal the presence of danger: alarm pheromone in the case of ants, cytokines in the case of immune systems. Both systems also use chemical signals to recruit other agents to search more effectively: immune cells can follow chemokine gradients to sites of infection, much like ants can follow pheromone trails to food.

Ants use pheromones to create dynamic maps. They do so by laying pheromone trails from locations with abundant food back to the nest, a form of communication through the environment, known as stigmergy (68). Such pheromone trails encourage other ants to travel directly to the food source, reinforcing the trail if they find food successfully. Once the food is depleted, the ants stop reinforcing the pheromone trail, and over time it dissipates and ceases to attract new ants to that location. This process is well-studied both experimentally in laboratory and field studies of various ant species, and in mathematical and computational models [as reviewed in (69)]. It is also the basis of a popular computational problem-solving heuristic called Ant Colony Optimization (70). These studies reveal the benefits and limits of pheromone communication in search problems, providing a roadmap for immunologists to understand how chemokines influence search.

A variety of chemical signals guide movement of immune cells, particularly to sites of infection. For example, chemokines provide migration and localization signals to dendritic cells, neutrophils, monocytes, T cells, and B cells. Other chemical cues including metabolic intermediates may also play a role. While it is clear that chemokines lead leukocytes to sites of infection, chemokines appear to have different effects on T cells. For example, neutrophils use the chemokine LTB4 as a signal to move directly toward a site of sterile injury (71). In contrast, the effect of chemokines on T cell movement seems to be less directional than LTB4 effects on neutrophils. In lymph nodes, T cells respond to the chemokines CCL21 and CCL19 by high speed random motion (chemokinesis) rather than directional movement (chemotaxis) (21, 58, 72, 73). Within infected tissue, chemokines appear to increase T cell speed (57, 60) but with only a slight bias toward infection foci (24). Interestingly, we found that the pattern of T cell movement in the lung, at least when infection is not present, does not appear to change when chemokine receptor signaling is inhibited (60).

In social insects, direct agent-agent interaction is an easy and effective way to transmit information. Ants use interaction networks to regulate behavior. Each ant can respond to the rate at which it experiences brief antennal contacts, in which one ant smells the other (74), and rates of brief olfactory interactions influence ant behavior (75). For example, we showed that in an active forager population, the rate of encounter with returning ants determines the probability that an outgoing forager leaves the nest to forage (76, 77). This feedback, based on direct ant-ant interaction, matches current foraging activity to the availability of seeds. Another example is ant-ant interaction leading to regulation of density. It seems that an ant can adjust its movement pattern in response to encountering another ant (78). We found that this change in motion regulates the density of ants in a specific area, enabling ants to spread out if they are too crowded. Rate sensing in ants through direct ant-ant communication provides an additional level of regulation to enhance foraging success. Similarly, direct bee-bee interaction has also been demonstrated to downregulate recruitment to less preferred food locations (79, 80).

It is currently not known whether T cells searching for pathogen infected cells use direct T cell-T cell contact as a mechanism to detect cell density or signal target location. Heterologous cell contacts in the immune response are clearly important, for example, direct contact between T cells and DCs, and T cell-B cell interactions are crucial for an immune response. However, a potential role for homologous cell-cell contact such as T cell-T cell interaction has not been carefully investigated. T cell-T cell interactions have been shown to be important for downregulation of the T cell response through fratricide: Fas-FasL interactions between effector T cells can lead to fratricidal T cell killing, effectively downregulating the T cell response as antigen load decreases (81). Direct T cell-T cell interactions were recently shown to be important in the first phases of T cell activation (82). In the context of T cell response in tissues, little is known about whether T cell—T cell interaction might impact T cell movement.

Thus, although T cells are capable of generating and responding to indirect communication via chemokines and cytokines and direct cell-cell contact with other T cells, it is unclear what the role of direct and indirect communication is in effector T cell search for infected cells (or tumors) in peripheral tissue. Our understanding of search in ants suggests that T cells might use both chemokine-cytokine communication as well as direct cell-cell communication to lead T cells to sites of infection, while also balancing this exploitation of known infection locations with exploration to find new sites of infection.

## Effective Search in Unknown Environments Requires Complex Search Strategies

We illustrate how ant search strategies may vary with the spatial location of resources with a computational model, comparing four foraging strategies in ants (**Figure 2**): (1) CRW alone (CRW-pink), (2) CRW combined with pheromone recruitment to previously found clusters (pheromone-orange), (3) CRW combined with site fidelity (each individual forager returns to the cluster that it previously found)(site fidelity-blue), and (4) an adaptive strategy known as the Central Place Foraging Algorithm, CPFA, (43) (CPFA-green). CPFA incorporates CRW, site fidelity, pheromone recruitment and the ability to choose among these behaviors based on the density of targets that the searcher senses in the locations immediately adjacent to the searcher.

The CPFA and the foraging model are described in more detail in Hecker and Moses (39). The model represents ants as points that move through space (without collisions and able to detect targets only in the cell in which it is located in and those directly adjacent), and seeds are represented as as points in a grid cell. All ants start at a central nest location, search using the specified strategy for 1 h, and each ant returns each individual seed that it finds directly to the nest (which is at a location known by every ant), carrying one seed at a time. The model uses unitless representations of velocity and length, and the size of the search area was chosen so that complete collection of all seeds is possible in the 1-pile case (**Figure 2**, column 1).

**Figure 2** shows the percentage of the 1,280 seeds that are collected for each spatial distribution. Each search strategy is tested on each of the spatial distributions shown in **Figure 1**. **Figure 2** shows the search performance of simulated ants using different strategies to search for different target distributions. The box plots show the median and interquartile range of 100 replicates for each target distribution, with the seeds placed at random locations drawn from the specified distribution. Where notches in the box plot overlap, the results are statistically indistinguishable (as is the case for pheromone and CPFA in the 1 pile case and randomly dispersed case; all other comparisons are statistically different at the p = 0.05 level). As the spatial distribution of targets varies from being concentrated in a single cluster (**Figure 1**, 1 × 1280) to being more dispersed (**Figure 1**, 1280 × 1), pheromones become less valuable (compare **Figure 2** "pheromone" vs. "CRW" from 1 × 1280 to 1280 × 1). When targets are dispersed at uniform random (1280 × 1), pheromones provide no benefit to foraging at all (and are actually detrimental as they attract ants to locations where a target once was but has been removed). Site fidelity is consistently more effective than random search alone unless resources are completely dispersed, in which case random search is the best strategy.

In the CPFA, searchers decide whether to use random search, site fidelity or pheromone recruitment to a location based on how many targets are there. In highly clustered situations, the CPFA and pheromone are similar in target identification efficiency (1 × 1280), because the CPFA selects a search strategy that relies almost entirely on pheromone search (39). However, the ability of agents to assess and adapt to the environment and choose the appropriate foraging strategy in the CPFA is particularly effective when resources are clustered in many intermediate size piles (16 × 80 or 80 × 16) or in piles with variable sizes such as the power law (compare pheromone and CPFA efficiency).

This search model supports the hypothesis that observed types of directed and random motion in searchers reflect differences in how targets are distributed in different environments. The results in **Figure 2** show how different search strategies perform in a fast detection task when searching for static targets. The CPFA has been shown to be effective at collecting up to ∼90% of static targets, but ineffective at complete collection (43), particularly when targets are dispersed. Although efficient strategies for complete search or search for mobile or replicating targets may be different (39, 52, 53), our model demonstrates that effective searchers require both a variety of search behaviors and the ability to sense the environment to determine which type of search behavior is best to use in a given time and place.

# CONCLUSIONS FOR T CELL SEARCH

Each ant species is tailored to the particular habitat in which it evolved, but T cells search in wide variety of tissues for a wide variety of targets. T cells demonstrate a variety of search behaviors, including directional movement using chemokine gradients, random motion using CRW, and movement along physical networks. As T cells do not know a priori about target distribution and require the capacity to counter unknown future threats, this adaptation and scalability for search in multiple tissues may be particularly important for maintaining effective immunity. Very little is currently known about how effector or memory T cell subsets move in infected tissue. Our observations and models of ants suggest the possibility that effector T cells move directionally toward infected areas in some circumstances (possibly following chemokine gradients, or more speculatively, responding to direct cell-cell contact) and move randomly in others, e.g., to search larger areas when infection has spread broadly or during the memory phase. We hypothesize that different classes of T cells (e.g., central memory cells, tissue resident memory cells and effector cells) have evolved different patterns of movement and responses to external signals and structures, varying with different search goals and target distributions in space and time.

In contrast to T cells moving randomly in tissues, neutrophils appear to move in a highly directed manner toward sites of infection (71). Neutrophils are rapidly recruited to sites of infection, so there is high fidelity between the actual location of infection and the signals, such as chemokines and cytokines, that are produced at sites of infection. Neutrophils moving directionally to foci of infection could be exploiting the close link between timing and spatial distribution early in an infection. The T cell response, on the other hand, develops over many days, with T cells often entering sites of infection 3–5 days post infection. Spatial distribution of the pathogen and related signals may no longer be spatially contained, as earlier in the infection cycle. We suggest that different immune cells (for example, T cells and neutrophils) respond differently to chemokine signals to promote effective immunity at different phases of the immune response with potentially different target distributions.

Understanding the parallels between search strategies in ants and T cells helps illuminate one of the central themes in immunology: how the enormously complex system of trillions of cells, signals, and structures are self-organized into a coherent immune response. As ants and ant search strategies have been studied in detail both experimentally and computationally, we have identified key concepts from ant foraging that suggest new concepts for understanding T cell search to clear infection. Like ants, T cells incorporate many strategies, including directional and random movement, direct agent-agent contact, and use of physical structures. Our proposal is similar to the "No Free Lunch" theorems (83), which posit that there is no single best search or optimization strategy for all computational problems, but that specific solutions can be tailored to specific types of search problems. We posit that there is no one best search strategy that can be used for all search problems in the immune system; instead searchers change how they move and interact with each other and the physical environment in response to specific search problems in specific environments.

Ant foraging strategies have served as inspiration for search heuristics in computer science (70) and as a model of search in a wide variety of complex adaptive systems (39, 84, 85). As we review here, there is increasing evidence that there is no single effective ant search strategy, but rather a repertoire of search behaviors that includes varied ways of moving, communicating, and using environmental structures to form an effective response to environmental conditions. Understanding

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the multiple components of effective ant search, and how they are combined into different strategies to respond to varied and dynamic environments can translate to new approaches for understanding the even more complex search processes of the immune system.

# AUTHOR CONTRIBUTIONS

MM, JC, DG, and SF conceived of the ideas for the article. MM, JC, DG, and SF contributed to the writing and editing of the article. MM generated the figures.

### FUNDING

This work was supported by funding from the following: DOD STTR Contract FA8650-18-C-6898 (JC and MM), grant from the UNM ADVANCE Women in Science STEM initiative (MM and JC), NIH 1R01AI097202 (JC), the Spatiotemporal Modeling Center (P50 GM085273), the Center for Evolution and Theoretical Immunology 5P20GM103452 (JC), and a James S. McDonnell Foundation grant for the study of Complex Systems (MM) and an LDRD grant from Sandia National Laboratories (MM and SF). JC is a member of the Center of Biomedical Research Excellence (CoBRE) Autophagy, Inflammation, and Metabolism (AIM) in Disease (P20GM121176). This project was supported in part by the Dedicated Health Research Funds from the UNM School of Medicine (JC). SF acknowledges partial support from DARPA (FA8750-15-C-0118), AFRL (FA8750-15- 2-0075), and the Santa Fe Institute.

#### ACKNOWLEDGMENTS

We thank Dr. Joshua Hecker for generating the data shown in **Figure 2** and Humayra Tasnim for generating the figure. We thank the Santa Fe Institute for workshops that fostered our collaboration and provided opportunities to discuss commonalities between ant colonies and immune systems.


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**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2019 Moses, Cannon, Gordon and Forrest. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# Computer Modeling of Clonal Dominance: Memory-Anti-Naïve and Its Curbing by Attrition

Filippo Castiglione<sup>1</sup> , Dario Ghersi <sup>2</sup> and Franco Celada<sup>3</sup> \*

1 Institute for Applied Computing, National Research Council of Italy, Rome, Italy, <sup>2</sup> School of Interdisciplinary Informatics, College of Information Science and Technology, University of Nebraska at Omaha, Omaha, NE, United States, <sup>3</sup> NYU School of Medicine, New York, NY, United States

Experimental and computational studies have revealed that T-cell cross-reactivity is a widespread phenomenon that can either be advantageous or detrimental to the host. In particular, detrimental effects can occur whenever the clonal dominance of memory cells is not justified by their infection-clearing capacity. Using an agent-based model of the immune system, we recently predicted the "memory anti-naïve" phenomenon, which occurs when the secondary challenge is similar but not identical to the primary stimulation. In this case, the pre-existing memory cells formed during the primary infection may be rapidly deployed in spite of their low affinity and can actually prevent a potentially higher affinity naïve response from emerging, resulting in impaired viral clearance. This finding allowed us to propose a mechanistic explanation for the concept of "antigenic sin" originally described in the context of the humoral response. However, the fact that antigenic sin is a relatively rare occurrence suggests the existence of evolutionary mechanisms that can mitigate the effect of the memory anti-naïve phenomenon. In this study we use computer modeling to further elucidate clonal dominance and the memory anti-naïve phenomenon, and to investigate a possible mitigating factor called attrition. Attrition has been described in the experimental and computational literature as a combination of competition for space and apoptosis of lymphocytes via type-I interferon in the early stages of a viral infection. This study systematically explores the relationship between clonal dominance and the mechanism of attrition. Our results suggest that attrition can indeed mitigate the memory anti-naïve effect by enabling the emergence of a diverse, higher affinity naïve response against the secondary challenge. In conclusion, modeling attrition allows us to shed light on the nature of clonal interaction and dominance.

Keywords: computer modeling, IMMSIM, memory-anti-naïve, attrition, CD8+ response

#### INTRODUCTION

Immunological memory, which appeared in the adaptive immune system roughly 600 million years ago, resulted in a substantial evolutionary advantage for vertebrates, whose immune systems acquired the ability to "remember" infectious agents and rapidly deploy effector cells in subsequent encounters with the same microorganisms or viruses.

#### Edited by:

Gennady Bocharov, Institute of Numerical Mathematics (RAS), Russia

#### Reviewed by:

Martin Meier-Schellersheim, National Institutes of Health (NIH), United States Giulia Russo, University of Catania, Italy

> \*Correspondence: Franco Celada franco.celada@nyumc.org

#### Specialty section:

This article was submitted to Viral Immunology, a section of the journal Frontiers in Immunology

Received: 13 December 2018 Accepted: 17 June 2019 Published: 03 July 2019

#### Citation:

Castiglione F, Ghersi D and Celada F (2019) Computer Modeling of Clonal Dominance: Memory-Anti-Naïve and Its Curbing by Attrition. Front. Immunol. 10:1513. doi: 10.3389/fimmu.2019.01513 However, the tendency of many infectious agents to mutate can reduce the efficacy of memory cells, whose affinity for mutated antigens can drastically decrease. Therefore, between the two extremes of the homologous challenge (with identical primary and secondary infections) and independent primary responses against two unrelated infectious agents, there exists a wide range of responses where the host has partial immunity against the new infection. Interestingly, partial immunity can exist not only between different strains of the same microorganism or virus, but also between apparently unrelated viruses, as shown by the pioneering work of Selin and Welsh in the field called heterologous immunity (1).

Cross-reactive immune responses against different viruses are believed to be ubiquitous, and can have beneficial, neutral, or detrimental effects for the host, in ways that are not easy to predict. Detrimental effects of partial immunity were described by Fazekas de St. Groth in 1966, when highest lethality rates were found among patients with a history of past encounters with far cross-reactive infectious agents. This phenomenon was studied in the humoral branch of the adaptive immune system and was labeled "original antigenic sin" (2). When cross-reactivity is too weak to cure the infection, the thwarting of naïve responses by memory is still blocking the development of the primary response, adding failure to failure: failure to cure and failing blocking the default defense.

The patterns of viral mutations and cross-reactive interactions are difficult to trace and define in vivo, making computational modeling highly beneficial. In a previous study we systematically studied the effect of a stepwise increase of the distance between two antigens subsequently injected in an in silico model (3). Unexpectedly, we identified an intermediate range of priming-challenge antigenic distances where memory is unable to mount an efficient defense, but it still outcompetes the primary response. Further in silico experimentation corroborated our first studies proving that the mechanism of memory anti-naïve (MaN) is fueled by the specific competition for antigen (4). Competition for antigen plays a key role in allowing high affinity clones to emerge in an immune response. However, memory has a faster dynamic than a primary response. In the early phases of an infection—while the primary response is still not ready to engage—the quantity of available antigen is growing but still limited. Thus, low affinity memory cells can potentially outcompete naïve cells, resulting in an immune response of lower quality.

Recently, Welsh et al. described a mitigating phenomenon named attrition, which is triggered by competition for space among clones of immunocytes at the time of antigen contact in a lymph node (5). Attrition is driven by short-distance effect of IFN-β that induces apoptosis on cytotoxic T-cells (Tc) by contact. The net effect is to reduce the growing lymphoid Tc population, and thus to favor the fittest cell lines in terms of affinity against the invader. The present in silico study is focused on modeling the mechanism of attrition and measuring its effects on the speed and on the affinity of the secondary response while systematically varying the degree of cross-reactivity.

## BACKGROUND

# Nature and Role of Computational Models

The biology of the immune response has been studied intensively in the few decades before and after the turn of the century and we witnessed an extraordinary growth in the number of researchers worldwide. As a result, we witnessed an exponential increase in the data being generated, resulting in the need for computational models to help make sense of it.

Computational modeling of the immune system experienced a strong burst in the 1980s, when several interdisciplinary collaborations brought together immunologists and mathematicians of various shades. These collaborations were fostered by two breakthrough events in the theoretical immunology community that had been engaged in adaptive immunology for some decades: the first solved the genetic problem of immune diversity (6); the second explained the formation of synapses between lymphocytes, allowing cell cooperation in most actions of the immune system (7). These achievements increased the size and the complexity of the field. At the same time, they created space for computational modeling.

Agent-based modeling is a relatively novel paradigm of modeling that satisfies the requirements of simplicity and parsimony in the description of a phenomenon by emphasizing first principles. It is a general modeling paradigm for complex systems inspired by von Neumann's "cellular automata" (8). Agent-based models consist of discrete dimensional space and time scales, where agents are, in our case, the relevant cells (or molecules) equipped with virtual receptors and capabilities, which reflect experimental observations.

The computational model C-IMMSIM, as well as the pioneering IMMSIM (9, 10), has been conceived to allow the dynamic representation of hypotheses and their preliminary in silico testing. These may further elicit ideas and new hypotheses to be eventually tested in vivo. In several applications over recent years, the model has generated emergent, sometimes surprising, data that shed light on the mechanisms and interactions of the model itself and on their counterparts in the biological immune system. For example, during the simulation of the affinity maturation of the humoral response, the varying density of cells and availability of antigen were shown to cause the shift from the bottleneck of the primary response, obtaining the help of CD4<sup>+</sup> cells, to the secondary bottleneck, winning the competition for antigen (11).

The model offers the possibility to manipulate the elements of virtual runs like experimental biologists do, by using the computational equivalent of knock-out mice or cell transfer (4, 12). Stratagems of this kind were applied in parallel experiments comparing the response of the humoral branch only, the cellular branch only, and both branches, to relate the efficiency of responses to different viral features (13). In a study about crossreactive memory, the silencing of one or the other of two suspected kinds of attrition, active or passive, revealed interesting cooperative effects of the combined mechanisms (1). In another study, selective "freezing" of humoral cross-reactive responses was obtained by increasing the bit distance in epitopes but not in peptides, while the antibody lifetime was artificially shortened or extended over a 50-fold range in order to reveal antibodymediated competition against cellular responses (3).

# MATERIALS AND METHODS

#### The Computational Model Polyclonality

In the present computational model, the specific recognition in adaptive immunity is simulated by borrowing ideas from binary calculus (14). Epitopes and paratopes are represented by strings of zeros and ones. When an epitope meets a paratope the strings are checked for complementarity at each position and a match (or equivalently a mismatch) is scored. Thus, the match is a number between 0 and N where N is the length of the binary strings representing the two binding regions. The model is polyclonal since it equips cells and molecules (e.g., lymphocytes receptors, B-cell receptors, T-cell receptors, Major Histocompatibility Complexes (MHC), antigen peptides and epitopes, immuno-complexes, etc.) with specific bit strings to represent the "binding site."

This minimalistic definition allows a diversity of 2<sup>N</sup> for each immunocyte (CD4+ or Th, CD8+ or TC, B). Such a setup can model cross-reactivity with remarkable smoothness, and accuracy in predicting the effect of competition among cross reactive cells.

#### Binding Affinity

In vivo, the paratope-epitope attraction is the sum of weak electrostatic and hydrophobic interactions when juxtaposed. In the simulation, two entities interact with a probability that is a function of the Hamming distance between the binary strings representing the entities' binding site. We indicate with m = <sup>r</sup>, <sup>p</sup> ∈ {<sup>0</sup> . . . <sup>N</sup>} the distance or the match between <sup>r</sup>, <sup>p</sup> <sup>∈</sup> {0 . . . 2 <sup>N</sup> − 1}. A good and widely used analogy is the matching between a lock and its key. If more than a threshold value m<sup>c</sup> over N bits matches (i.e., 0–1 or 1–0) occur, the interaction is allowed with a certain probability that is a function of the number of matches between the bit-strings. This attraction force (called affinity or affinity potential) is equal to one when all corresponding bits are complementary. Specifically, if m = <sup>r</sup>, <sup>p</sup> is the Hamming distance between the two strings r and p, the affinity potential f (m) ∈ [0, 1] defined in the range 0, ..., N is

$$f\left(m\right) = f\left(\left\|r, p\right\|\right) = \begin{cases} e^{\log(A\_L)\frac{m-N}{m\_c-N}} & m\_c \le m \le N\\ 0 & m < m\_c \end{cases} \tag{1}$$

where A<sup>L</sup> is a free parameter which determines the slope of the function, whereas m<sup>c</sup> ∈ {N/2 . . . N} is the cut-off (or threshold) value below which no binding is allowed.

#### Humoral and Cellular Responses

The model simulates a very simple form of innate immunity and an elaborate form of adaptive immunity (including both humoral and cytotoxic immune responses).

In the case of innate immune response by "exogenous signal" (e.g., Pathogen-Associated Molecular Pattern, PAMP or PAMPagonist, used for specific adjuvants) the activation sequence will begin with antigen presenting cells stimulation. The only mechanisms of this kind which is embedded in the model accounts for the presence of lipopolysaccharides in pathogens as in Gram-negative bacteria.

#### Working Assumptions

In the model, a single human lymph node (or a portion of it) is mapped onto a three-dimensional Cartesian lattice. The primary lymphoid organs thymus and bone marrow are modeled apart: the thymus (15, 16) is implicitly represented by the positive and negative selection of immature thymocytes before they enter the lymphatic system, while the bone marrow generates already mature B lymphocytes. Hence, only immunocompetent lymphocytes are modeled on the lattice.

The C-IMMSIM model incorporates several working assumptions or theories, most of which are regarded as established immunological mechanisms, including: (i) the clonal selection theory of Burnet (17); (ii) the clonal deletion theory (i.e., thymus education of T lymphocytes) (18); (iii) the hypermutation of antibodies (19); (iv) the replicative senescence of T-cells, or the Hayflick limit (i.e., a limit on the number of cell divisions) (20); (v) T-cell anergy (21) and Ag-dose induced tolerance in B-cells (22); (vi) the danger theory (23); (vii) the idiotypic network theory (24). Variations on the basic model have been used to simulate different phenomena ranging from viral infection [e.g., Human Immunodeficiency Virus (25) or Epstein-Barr Virus (26)] to cancer immunoprevention and type I hypersensitivity (27, 28).

Each time step of the simulation corresponds to 8 h. The interactions among the cells determine their functional behavior. Interactions are coded as probabilistic rules defining the transition of each cell entity from one state to another. Each interaction requires cell entities to be in a specific state choosing from a set of possible states (e.g., naïve, active, resting, duplicating) that is dependent on the cell type. Once this condition is fulfilled, the interaction probability is the effective level of binding between ligand and receptor.

Unlike many other immunological models, the present one not only simulates the cellular level of the inter-cellular interactions but also the intra-cellular processes of antigen uptake and presentation. Both the cytosolic and endocytic pathways are modeled. In the model, endogenous antigen is fragmented and combined with MHC class I molecules for presentation on the cell surface to CTLs' receptors, whereas the exogenous antigen is degraded into smaller parts (i.e., peptides), which are then bound to MHC class II molecules for presentation to the T helpers' receptors.

#### Stochasticity

The stochastic execution of the algorithmic rules, as in a Monte Carlo method, produces a logical causal/effect sequence of events culminating in the immune response and development of immunological memory. The starting point of this series of events is the injection of antigen (the priming). This may take place any time after the simulation starts. In general, the system is designed to maintain a steady state of the global population of cells if no infection is applied (homeostasis). Initially the system is "naïve" in the sense that there are neither T and B memory cells nor plasma cells and antibodies. The various steps of the simulated immune response depend on what is injected, i.e., virus or bacteria.

#### The Virus

Virus is the "foreign agent" in the model. It is constructed with B-cell epitopes and T-cell peptides. In addition, it replicates, simulating a living entity, and the combination of three factors (speed of duplication, infectivity, and lethal load level) results in its "fitness" which is independent of antigenicity. Any infection begins with the penetration of virus into an epithelial cell, though this could be any designated target cell. Whether the infection is cured or becomes persistent or even kills the virtual mouse depends on the virus dose, its fitness, and the strength of the immune response it has elicited. All these variables determine whether—and to what degree—the immune system's success requires the cooperation of both the cellular and humoral branch, as has been shown in several simulation studies (13).

#### Modeling Active Attrition

Active attrition is enacted in the present version of the model by describing the release of IFN-β by macrophages in the presence of high concentrations of danger signals, e.g., in infection sites. This lymphokine diffuses locally and then "causes" the death of cytotoxic memory T-cells by contact. The locally-limited bystander effect of this cytokine is set to be dependent on the cell's age but also on its affinity to the viral peptide. Specifically, the death of cytotoxic cells is modeled as a stochastic event whose probability is proportional to the cell's age and inversely proportional to the affinity between TCR and the peptide attached to class 1 HLA (1, 5) of infected cells, i.e.,

$$\Pr\left[die\right] = \frac{a^{n\_1}}{a^{n\_1} + k\_1} \times \frac{i^{n\_2}}{i^{n\_2} + k\_2} \times \left(1 - f\right) \tag{2}$$

where a is the age of the T-cell (in units of days), f the affinity of its TCR to the viral peptide as defined in Equation (1) and i is the local concentration of IFN-β (in pg/ml). In the experimental setup that we are going to describe in the following section, the parameters of Equation (2) have been chosen as follows: k<sup>1</sup> = 10<sup>6</sup> × days−<sup>1</sup> and k<sup>2</sup> = 10<sup>9</sup> × (pg/ml)−<sup>1</sup> were taken to obtain a probability of killing which was much stronger for memory compared to naïve cells; parameter n<sup>1</sup> = 3 > n<sup>2</sup> = 2 were chosen to make age the limiting factor in the killing. The last term in Equation (2), 1 − f ∈ [0, 1], stands for a protective factor for cells able to establish a stronger immunological synapse during peptide recognition on the membrane of infected cells and f therefore is the same function in Equation (1).

#### Experimental Setup

The model represents both paratopes and epitopes by N = 16 bit binary strings. A successful paratope-epitope interaction is limited to a match m greater than or equal to the cut off m<sup>c</sup> = 13 over the 16 allowed. This setup results in a diversity of 2<sup>16</sup> for each lymphocyte and gives N − m<sup>c</sup> = 4 matching classes thus allowing to model the immune recognition and predicting the effect of competition among cross-reactive cells with reasonable accuracy. In vivo, the diversity among epitopes and that among paratope are mind boggling (conservatively, 10<sup>10</sup> to 1012). Simulating those numbers, though theoretically possible by enlarging the repertoires which is obtained by elongating the strings, is practically not viable for computational reasons.

#### The Antigenic Distance Experiments

In studying memory, it is important to quantify the degree of cross-reactivity between related antigens. While in vivo this appraisal is difficult to attain, the following modeling setup allows us to measure the effect of cross-reactivity on a secondary immune response quite effectively.

The series of simulations we perform mimic a prime/challenge experiment in a virtual mouse (or individual) where successive injections carry equal or different antigenic determinants (see **Supplementary Figure 2**). The priming infection is performed always with the same virus, but the challenge or secondary infection performed later is done with a different virus whose peptide is at a defined distance d from the priming one. We use N/2 = 8 bits to represent a virus peptide thus we have d = 0 . . . 8 levels of cross-reaction by suitably choosing the prime/challenge couple. Viral peptides are presented to T-cell receptors bound to the major histocompatibility complex molecule (MHC) and indeed in the model the match is an N-bit match. However, for simplicity, the contribution to the affinity given by the portion of the cell receptor binding the portion of the MHC molecule is set to a constant value so not to influence the overall match to the virus. In other words, the affinity between receptors and MHCbearing virus peptides depends only on a N/2 = 8 bit match rather than an N bit match.

Let's call V I the virus injected first (i.e., the primer at time tI), V II the virus injected subsequently (the challenger at time tII) and d the "bit distance" between V I and V II, that is, d = V I ,V II . The experiments realize the protocol consisting in a priming injection that is always performed with the same virus V <sup>I</sup> = V<sup>0</sup> and a challenge injection consisting of a certain saturating dose of one of the nine viruses reported in **Table 1** which also includes V0. Therefore V II = V<sup>k</sup> for k = 0 . . . 8. Note that the set of chosen viral peptides is such that d = Vi ,V<sup>j</sup> <sup>=</sup> i − j , for all choices of i, j ∈ {0 . . . 8}. Following this description, it is convenient to name the experiments on the basis of the distance between priming with V<sup>0</sup> and challenge with V<sup>k</sup> . For instance, we call d = 3 the experiment in which V <sup>I</sup> = V<sup>0</sup> and V II = V<sup>3</sup> because d = V I ,V II <sup>=</sup> <sup>k</sup>V0,V3<sup>k</sup> <sup>=</sup> 3. While <sup>d</sup> <sup>=</sup> <sup>0</sup> realizes the homologous response, and can indeed be considered the control, d = 1 to d = 6 represent cases of cross-reactivity, with progressively fewer matches. Finally, d = 7 and d = 8 are heterologousresponses (i.e., no match at all). We note that all viral peptides are chosen to be distant with respect to self-peptides, to avoid having to deal with autoimmune responses, which are outside the scope of this work.

The simulated space is equivalent to a fraction of the lymphatic system represented at once. This simulated volume is 10 micro liters or, equivalently, 10 cubic millimeters. Both priming and challenge consist in injecting a saturating viral dosage of 10<sup>3</sup> viral particles per microliter. For all experiments, the setup is identical except for the two viruses injected, V I and

TABLE 1 | Viruses used in the experiments are numbered from zero to eight.


The injection protocol comprises two viruses V<sup>I</sup> and V II presented one after the other at time steps separated by a time sufficient to fully develop an immune response. The Hamming distance between the two viruses injected determines the level of crossreactivity hence the degree of exploitation of the immune memory to V<sup>I</sup> in the response to VII. Viruses to are equipped with a peptide string of length N such that d = V I , V II <sup>ǫ</sup>{<sup>0</sup> . . . <sup>8</sup>}. In the antigenic distance experimental protocol V<sup>I</sup> is always equal to V0.

V II. Thus, the simulated space is populated with the same initial number of cells (i.e., no variability allowed), the viruses share the same infectivity and replication rates, etc. Moreover, since the model is stochastic, for each setup d ∈ {0 . . . 8}, we repeat the experiments 100 times for each protocol and calculate statistics (averages and standard deviations) afterwards.

#### Useful Definitions

With the aim of defining two quantities which help in measuring the effect of cross-reactivity, we now need to introduce some formalism.

We call diversity D the set of possible bit strings of length N in the base-ten system, that is, D = {0...2N/<sup>2</sup> − 1}. We indicate by nr(t) the number of cytotoxic T-cells with specificity r ∈ D at time t. For each virus V the Hamming distance creates the equivalent classes in the set of cell receptors D. In other words, two receptors r<sup>1</sup> and r<sup>2</sup> are in the same matching class for V if kr1,Vk = kr2,Vk = m. We can therefore define qm(t) as the total number of cells matching the virus V with m bits, that is, ∀m ∈ {0 . . . N}

$$q\_m(t) = \sum\_{r \in D, \|r, V\| = m} n\_r(t).$$

Then we call Am(t) the affinity of the response to the peptide of virus V relative to the matching class m ∈ {0...N}, that is, all the lymphocytes that are equivalent in terms of affinity. This quantity is calculated by summing the number of cells with receptor matching with m bits the virus peptide and multiplied by the affinity value f(m), that is, ∀m ∈ {0 . . . N}

$$A\_m(t) = f(m) \cdot q\_m(t). \tag{3}$$

Finally, we define the total affinity to virus V as

$$TA\left(t\right) = \sum\_{m=0}^{N} A\_m\left(t\right). \tag{4}$$

Note that since we are interested in quantifying the effects of cross-reactivity on the secondary immune response, all the quantities qm(t), Am(t) and TA(t) should be considered relative to V II and be written, for instance, A II <sup>m</sup>. However, to simplify the representation we just avoid using the superscripts and write Am, etc.

Furthermore, we call V II (t) the number of viral particles at time t of the challenge virus and define

$$t\_c = \min\_t \left\{ t \ge t\_{II} : V^{\Pi}(t) = 0 \right\} - t\_{\Pi}.$$

the time-to-clear the virus V II, that is, a measure of how quickly the response eradicates the virus injected at time t II .

We can now finally define two almost complementary measures. The first one is the efficacy of the immune response to the second virus injected V II. The efficacy E d is a function of the distance

$$d = \left\| \left| V\_0, V^{\mathrm{II}} \right\| \right\|$$

to the first injected virus V <sup>I</sup> = V<sup>0</sup> and is defined as the peak value of TA (t) for t ≥ tII divided by the time-to-clear the virus tc . In formula

$$E\left(d\right) = \frac{1}{t\_{\mathcal{L}}} \cdot \max\_{t \ge t\_{\mathcal{II}}} \left\{ TA\left(t\right) \right\}.\tag{5}$$

The efficacy measures how good the immune response to V II is in terms of how many cytotoxic T-cells are developed by clone expansion and how quickly the virus is eliminated. Clearly the maximum value of the efficacy is achieved for d = 0 because of the immune memory developed to respond to V <sup>I</sup> = V0, but decreases for increased distance d between prime V<sup>0</sup> and challenge injection V II .

The maximum value attained by the sum of all Tc counts qm(t) for m = 0 . . . N averaged over a number of simulations (h·i indicates averages) can be designated as

$$
\langle \tilde{M} \rangle = \left\langle \max\_{t \ge t\_{\text{II}}} \left\{ \sum\_{m=0}^{N} q\_m(t) \right\} \right\rangle. \tag{6}
$$

Cell counts are calculated for each antigenic distance experiments. We can therefore use superscripts to indicate a specific experiment and refer to this quantity in the case d = 8 as M˜ <sup>d</sup>=<sup>8</sup> . This value measures the magnitude of the cytotoxic immune response to V II = V8. Since it corresponds to the completely heterologous response, the effect of the MaN is zero and the quantity in Equation (6) is maximal with respect to d. The other extreme case is found when d = 0, corresponding to a homologous immune response for which the immune memory is so perfectly fit to the second injected virus V II = V<sup>0</sup> = V I that the latter is eliminated without the need for a clonal expansion of cytotoxic T-cells. The measure that we call compression is then defined as

$$C\left(d\right) = \left<\tilde{M}\right>^{d=8} - \left<\tilde{M}\right>^{d}\tag{7}$$

and is the difference of the maximum number of cytotoxic T-cell count attainable in the absence of memory. In other words, this measure quantifies the degree of hindrance (or reduction, hence the name compression) of the naïve response due to the presence of cross-reactive memory cells against past infections. The compression is maximal for d = 0 and diminishes for larger d reaching its minimum for d = 8.

#### RESULTS

# The Memory Anti-naïve (MaN) Phenomenon

We first illustrate the MaN phenomenon by studying the primary and memory responses against viruses with different antigenic distance. The results of three cross-reacting viral infections are shown in panels A, B, and C of **Figure 1**, where we can track the primary and memory responses of proliferating individual Teffector memory clones. In each panel we learn the composition of the naïve response, represented by filled markers, and of the memory response, represented by empty markers present only in panels A and C. Panel A shows the case of priming with V <sup>I</sup> = V<sup>0</sup> and challenge V II = V2, that is, a virus with antigenic distance d = 2 that elicits a cross-reactive memory response. The result is a strong dominance of memory over primary clones. In fact, no new primary clone emerges after tII. Panel B has a priming identical to that of panel A but is challenged by a virus with d = 8 from the priming, thus a heterologous virus. As predicted, the secondary response does not trigger memory cells, but elicits a naïve response specific to the challenge V8. Panel C shows the case with all three viral infections: V<sup>0</sup> at time t<sup>I</sup> , producing a primary stimulation, and both V<sup>2</sup> and V<sup>8</sup> at time tII. The two latter viruses are, by virtue of their antigenic distance, not interfering with each other. Any primary anti-V<sup>2</sup> is silenced by cross reacting memory previously elicited against V0, while the naïve anti-V<sup>8</sup> mounts, as expected, an undisturbed primary response. Taken together, these results allow us to conclude that the force underpinning MaN is specificity, and the mechanism is competition for antigen. Note that both panels A and C show a clear advantage of memory over naïve: most memory clones are higher than naïve ones at the peak.

# MaN Has Two Different Effects

Another way of showing MaN is to extend the range of distances between the priming and the secondary infection, that is,

response E(d) (i.e., a measure of cytotoxicity in unit of cell counts) vs. the antigenic distance d. In panel (B), the compression C(d) (i.e., the blocking effect of the cross-reactive memory, also in unit of cell counts). The difference of the curves indicates that for d ∈ {2 . . . 5} the blocking of the primary response is not justified by an efficient secondary response, that is the phenomenon that constitutes the MaN. Panel (C) presents the same data of panels (A,B) but respectively normalized by the Min-Max method (i.e., y ′ = (y − ymin)/(ymax − ymin)) so to fall within the same range [0, 1]. It visualizes at once the effect of MaN as the area between the two curves assuming the shape of an "eye".

following the antigenic distance experiment schema described in section The Antigenic Distance Experiments. In this set of experiments the attrition plays no role as it has been disabled, allowing us to study the MaN in isolation.

**Figure 2** shows the efficacy E(d) and the compression C(d) defined in section Useful Definitions [respectively in Equations (5) and (7)] as a function of the viral distance d. The figure shows values before and after normalization. Clearly, the efficacy of the immune response decreases by increasing d (panel A) simply because it is a measure of the efficiency of the cross-reactive immune memory. We can point to d = 2 as the critical value for the greatest reduction in efficacy. On the other hand, the compression also decreases by increasing d for the same reason, but changes more "slowly" compared to the efficacy: the critical value is d = 5. Based on these results, we can say that the range d between 2 and 5 is the domain of the MaN.

#### The Effect of Attrition in Alleviating the MaN

In another set of experiments, we turned the attrition on and studied its effect on the immune response in general and on the MaN in particular. The attrition effect was modulated by modifying Equation (2) by multiplying the interferon concentration by a factor α, that is, taking α × i n2 , with α equal to 1,2, . . . 5. For formal correctness, the case of no attrition designated as α =0 needs to be made explicit in the new definition of Pr- die of Equation (2) as follows

$$\Pr\left[\text{die}\right] = P(\alpha) = \frac{a^{n\_1}}{a^{n\_1} + k\_1} \times \frac{\alpha i^{n\_2}}{\alpha i^{n\_2} + k\_2} \times \left(1 - f\right) \tag{8}$$

All runs exhibit identical primary response to d = 0 virus (green). Analyzing the memory responses those challenged with V<sup>0</sup> (i.e., d = 0 thus homologous) are the strongest while all others are cross-reactive and expected to turn out progressively weaker with increasing d. For d = 7 and d = 8 the viral challenging epitope is so different from the priming epitope that memory fails to recognize it, thus there is no cross-reactive memory and the response is a primary response directed against the second virus (V<sup>7</sup> or V8). Another point that may seems counterintuitive is to see a very weak immune response representing the fact that when the memory is at its strongest (e.g., d = 0 and d = 1) the virus is eliminated very efficiently. This happens because memory is "speedy in deployment" and eliminates the growing population of pathogens when they are still few in numbers. This has two important effects: (i) the lack of further stimulation keeps the effectors low, and (ii) no stimulation of naïve response takes place. This is the same competition for antigen already seen in Cheng et al. (3). The results in **Figure 3** quantitatively confirm these early observations and provide further insights into the mechanisms. Either large antigenic distances or high levels of attrition will counter the MaN effect but, as expected for synergistic actions, smaller antigenic distances or levels of attrition result in a balance between memory and naïve total affinity. This is visible in several cases of the grid in **Figure 3**. To simplify the study of these "ties," and the eventual takeover by naïve responses, the position of the critical runs of **Figure 3** are pinpointed in **Figure 4**. In eight cases ties between primary and memory occur, at the point where the primary curve is about to surpass the memory response. Since all primary responses are identical against any virus, in these eight cases we know that the total affinity [i.e., TA(t) of Equation (4)] of memory is comparable to the total affinity of the primary response. Surpasses are easy to spot in **Figure 3**, following the coordinate marked on **Figure 4**, where distance and attrition are color coded in red and blue, respectively.

Results for B cells approximately follow those shown for T cells (see **Supplementary Figure 1**). This was expected since in the definition of viruses V<sup>0</sup> . . .V<sup>8</sup> we have purposefully followed the same logic (i.e., distance) in the definition of the viral epitopes with the aim of not favoring the humoral response of one virus in particular. The conclusion is that, with regards to the balance between MaN and attrition, the humoral response is consistent (i.e., is very similar) to the cytotoxic response and, in substance, it does not prevent or limit the latter but adds to it instead.

**Figure 5** shows the effect of attrition on the Tc total affinity TA(t) defined in Equation (4). Panel A refers to how attrition influences the total affinity of a homologous response, i.e., when to V <sup>I</sup> = V II = V<sup>0</sup> resulting in d = 0. Only three levels of attrition are shown: α = 0 corresponding to absence of attrition, α = 3 considered the intermediate "optimal" level and α = 5 deemed an excessive value for attrition. The highest α = 5 does not affect the peak of memory since in the homologous response

FIGURE 3 | Cytotoxic cell counts (percentage) vs. time. This is the result of different simulations obtained varying the level of attrition α = 0 . . . 5 (α = 0 is control case of no attrition) and the antigenic distance d between the two viral infections, for a total of 6 × 9 = 54 panels, each containing average ± standard deviation results of Tc counts in simulated primary viral infections, followed by a second challenge infection by an identical, or by a selected mutant virus. Color codes: green: response by naïve T effector cells to primary virus V I injected at t I = 0; orange: cross-reactive memory response primed by the first virus V <sup>I</sup> and challenged by V II; blue: response by naïve T effector cells to V II injected at t II = 1000 time steps. See text for further explanation.

the second peak is due to memory recall but trims the curve via its effect on aged cells. As expected, attrition facilitates the emergence of higher affinity cells thus increasing TA(t) especially in the primary response. This is better shown in **Figure 6**. Panel B unveils the heterologous response (i.e., V II= V<sup>0</sup> thus d = 4), the total affinity to the V <sup>I</sup> = V<sup>0</sup> is equivalent to the one in panel A. Panel C shows the same total affinity to V <sup>I</sup> = V<sup>0</sup> but calculated at the time of the challenge, namely, during the competing presence of anti-V<sup>4</sup> naïve cells. Here, by increasing α, the memory to V <sup>I</sup> = V<sup>0</sup> disappears. This effect is striking when compared to the case α = 0 (green curve). Panel D shows the total affinity in a d = 4 experiment but this time relative to V II = V4. The comparison

be trimmed down by turning two independent knobs: the first increases d and affects affinity, the second through attrition, thins the memory cell population, by inducing apoptosis. If both knobs are turned up in the same run a synergic effect is observed. The equal counts of memory and naive cells are due to the fact that in the clearing of the virus that in the clearing of the virus the total affinity required is contributed equally by the two populations.

of TA(t) should be made for t > 1000 in panel A. The overall message is that attrition favors the emergence of higher affinity clones (blue and purple curves in panel D corresponding to α = 3 and 5 respectively) with respect to the green curve (α = 0).

**Figure 6** shows the peak values of Am(t) as defined in Equation (3) per match class, from the lowest value of match m<sup>c</sup> = 13 to the highest N = 16. As in **Figure 5** we have three attrition levels, α = 0, 3, 5 and we run the d = 4 experiment meaning that V <sup>I</sup> = V<sup>0</sup> and V II = V4, because this case is the one in which the MaN and the attrition display the greatest effects. The first observation is that the class m = 15, although not the best match, reaches the highest affinity peak value. This is expected given the relatively short time to develop a complete affinity maturation during either the primary response (panel A) and the secondary heterologous response (panel B). Another observation pertaining to the lower panel is that the attrition helps improving the affinity maturation in all match classes. Moreover, clearly visible in panel B, while attrition α = 3 helps the maturation of m = 16 thus the high matching clones, the case α = 5 knocks them down. This comparison of levels of attrition suggests that the α = 3 has been correctly dubbed the moderate or optimal level.

# DISCUSSION AND CONCLUSIONS

Viral infections and pandemics are prime examples of the dynamic between evolution and mutability of viruses on one side and cross-reactivity of antibodies and cytotoxic cells on the other. Pandemics are often the result of recurrent infections with distant cross-reactive agents. The "original antigenic sin" (2) hypothesis, that is, the case of patients whose memory responds to a previous priming and whose primary response is blocked by memory, still lacks an explanation for why cross-reactive anti-virus cells that unable to clear the virus, are still able to outcompete the naïve cells. More recently, Monsalvo et al. (29) found signs of antigenic sin in non-protecting antibodies and low affinity immune complexes.

In this study we obtained quantitative data that enables us to propose a plausible mechanistic explanation for this phenomenon. We measured the ability of the immune system to deal concurrently with two viruses, one cross-reactive that is eliminated by memory cells that, as a side effect, block the naïve response, and another that is not cross-reactive and is eliminated by a naïve independent response (**Figure 1**). This shows that the two processes do not interfere with each other.

It is therefore the degree of cross-reactivity that determines the engagements of different concurrent immune responses. When the antigenic distance between the priming and the challenging virus is increased, memory efficacy E(d) falls immediately while the compression of naïve response C(d) through antigen deprivation is affected only later (see **Figure 2**). This result is expressed in the combination of the two curves of C(d) and E(d) as a function of antigenic distance, producing the iconic image of an "eye," a representation and measure of MaN.

The efficacy E(d) is sensitive to decreased cross-reactive affinity, immediately from the first step and continues the descent as a concave curve, as expected in a cellular response where each cell's affinity contributes individually to the final result.

The compression C(d) shows no effect whatsoever in the first three steps decrease of affinity, while the fourth step causes only partial block of the naïve engagement. This resistance to severe decrease of affinity gives a measure of the dominance exerted by memory over naïve responses by depriving them of the antigen required for their growth. The display of strength by memory is certainly sustained by its speed of deployment, and this experiment detects a second concurrent mechanism that materializes as a cooperative action. Affinity is the energy displayed by a single paratope, but in a competition for the antigen, the presence of many paratopes nearby may decrease the chances of the antigen to "escape" from another memory cell. The best example is the higher "catching" ability of bivalent and pentavalent antibodies compared to Fab monovalent antibodies. To mark this difference the serologists of the last

century invented a new strength inclusive of affinity and a "cooperation bonus" and called it avidity. Mechanistically two Fabs will bind two monovalent epitopes, but the weak forces will alternate periods of sticking together with period of detachment. Chances are that the two couplets will not stay in this conformation for long. On the other hand, the bivalent Fab has a definitely higher chance of staying with one epitope bound or at least, at short distance, quite stably. Avidity enhances binding and allows low quality memory cells to still dominate.

Any immune response, and particularly the fast, cellular memory is always in need of space (e.g., physical space, metabolic space, etc.), which like other resources are subject to competition. Active attrition (5, 30) consists of timely secretion of IFN-β operated by the same vectors that signal danger, and has the effect of eliminating crowding of cells by allowing a selection of more efficient young clones, at the expense of dominant clones. In this case, the action of the attrition has the specific connotation of helping the specific response as the thinning of clonal population will favor clonal expansion randomly. Based on the results shown in **Figure 2**, we propose that the difference in strength between antigen biding and compressing naïve cells depends on the advantage in favor of the latter: avidity is affinity enhanced by intra-clone paratope synergisms.

We have shown that the memory anti-naïve effect, the "necessary" byproduct of memory, can be mitigated by the attrition signals produced during the early stages of an infection. These signals kill a fraction of the population of effectors (**Figures 5**, **6**). While resulting in a decreased speed of the response, this mechanism gets rid of low affinity cross-reactive cells thus allowing naïve clones to emerge and eventually achieve better affinity maturation.

In conclusion, the trimming effect of attrition which mitigates the MaN effect is well-documented in the present work and corroborates earlier studies (1, 4, 31). The present data is more precise as it independently monitors memory and naïve cells thus facilitating the detection of a phenomenon that affects different cellular compartments in opposite directions.

The results of this study add new predictions on the mechanism underpinning memory's clonal dominance on naïve

FIGURE 6 | This plot show the peak values of the Am(t) per match classes, from the lowest m = mc = 13 to the highest m = N = 16. As in Figure 5, we have three attrition levels α = 0 (corresponding to absence of attrition), α = 3 (considered the "optimal" level) and α = 5 (deemed a excessive value for attrition) and we run the d = 4 experiment meaning that V <sup>I</sup> <sup>=</sup> <sup>V</sup><sup>0</sup> and <sup>V</sup> II <sup>=</sup> <sup>V</sup>4, because this case is the one in which the MaN and the attrition display the greatest effects. The first observation is that m = 15 is the match class which reaches the highest peak values. This is expected given the relatively short time to develop a complete affinity maturation during either the primary response (panel A) and the secondary heterologous response (panel B). Another observation pertaining the lower panel is that the attrition helps improving the affinity maturation in all match classes. Moreover, while attrition α = 3 helps the maturation of m = 16 high affinity clones, the case α = 5 knocks them down (panel B).

responders: the competition is based on affinity to viral antigen, enhanced in the case of memory, by two factors, speed of action and intra clonal cooperation, resulting in the deprivation of antigen for naïve cells. We predict that clonal competitions are at the core of many pathologies that will not be understood and treated properly without explaining all causative forces.

#### REFERENCES


In conclusion, results produced by computational models, however reasonable they may look, must be confirmed by in vivo or in vitro experiments before being considered scientific truth. However, their value may be realistically appraised if they trigger new hypotheses, and help guiding wet lab research. In this regard we believe that the presented modeling study has indeed provided a clearer picture of the complex relationship between MaN and attrition.

#### AUTHOR CONTRIBUTIONS

FrC contributed conception and design of the study. FiC performed numerical simulations and statistical analysis. DG and FrC contributed ideas during analysis and interpretation of the results. FrC wrote the first draft of the manuscript. DG and FiC wrote sections of the manuscript and revised the first draft. All authors contributed to manuscript revision, read and approved the submitted version.

### FUNDING

This study is partially funded by the European Commission under the 7th Framework Programme (MISSION-T2D, No. 600803). Partial funding from the H2020 Programme (iPC– Pediatric Cure, No. 826121) is also acknowledged.

# ACKNOWLEDGMENTS

R. Strom is kindly acknowledged for helpful comments.

#### SUPPLEMENTARY MATERIAL

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fimmu. 2019.01513/full#supplementary-material


**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2019 Castiglione, Ghersi and Celada. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# Estimating Residence Times of Lymphocytes in Ovine Lymph Nodes

#### Margaret M. McDaniel <sup>1</sup> and Vitaly V. Ganusov 2,3 \*

*<sup>1</sup> Department of Immunology, University of Texas Southwestern, Dallas, TX, United States, <sup>2</sup> Department of Mathematics, University of Tennessee, Knoxville, Knoxville, TN, United States, <sup>3</sup> Department of Microbiology, University of Tennessee, Knoxville, Knoxville, TN, United States*

The ability of lymphocytes to recirculate between blood and secondary lymphoid tissues such as lymph nodes (LNs) and spleen is well established. Sheep have been used as an experimental system to study lymphocyte recirculation for decades and multiple studies document accumulation and loss of intravenously (i.v.) transferred lymphocytes in efferent lymph of various ovine LNs. Yet, surprisingly little work has been done to accurately quantify the dynamics of lymphocyte exit from the LNs and to estimate the average residence times of lymphocytes in ovine LNs. In this work we developed a series of mathematical models based on fundamental principles of lymphocyte recirculation in the body under non-inflammatory (resting) conditions. Our analysis suggested that in sheep, recirculating lymphocytes spend on average 3 h in the spleen and 20 h in skin or gut-draining LNs with a distribution of residence times in LNs following a skewed gamma (lognormal-like) distribution. Our mathematical models also suggested an explanation for a puzzling observation of the long-term persistence of i.v. transferred lymphocytes in the efferent lymph of the prescapular LN (pLN); the model predicted that this is a natural consequence of long-term persistence of the transferred lymphocytes in circulation. We also found that lymphocytes isolated from the skin-draining pLN have a 2-fold increased entry rate into the pLN as opposed to the mesenteric (gut-draining) LN (mLN). Likewise, lymphocytes from mLN had a 3-fold increased entry rate into the mLN as opposed to entry rate into pLN. In contrast, these cannulation data could not be explained by preferential retention of cells in LNs of their origin. Taken together, our work illustrates the power of mathematical modeling in describing the kinetics of lymphocyte migration in sheep and provides quantitative estimates of lymphocyte residence times in ovine LNs.

# Edited by:

*Gennady Bocharov, Institute of Numerical Mathematics (RAS), Russia*

#### Reviewed by:

*Rob J. De Boer, Utrecht University, Netherlands Masayuki Miyasaka, Osaka University, Japan*

#### \*Correspondence:

*Vitaly V. Ganusov vitaly.ganusov@gmail.com*

#### Specialty section:

*This article was submitted to T Cell Biology, a section of the journal Frontiers in Immunology*

Received: *15 February 2019* Accepted: *14 June 2019* Published: *16 July 2019*

#### Citation:

*McDaniel MM and Ganusov VV (2019) Estimating Residence Times of Lymphocytes in Ovine Lymph Nodes. Front. Immunol. 10:1492. doi: 10.3389/fimmu.2019.01492* Keywords: mathematical model, lymphocyte migration, lymph nodes, residence time, sheep

# 1. INTRODUCTION

One of the peculiar properties of the mammalian adaptive immune system is the ability of its lymphocytes to recirculate between multiple tissues in the body; that is lymphocytes in the blood are able to enter the tissues and after some residence times in the tissues, they return to circulation (1). The pattern of lymphocyte recirculation in general depends on the lymphocyte type (e.g., B or T cell), status of the lymphocyte (resting vs. activated), and perhaps tissues via which lymphocytes are migrating. Naive, antigen-unexperienced lymphocytes, primarily recirculate between secondary lymphoid tissues such as lymph nodes, spleen, and Peyer's patches (2–4). Following activation after exposure to an antigen, naive lymphocytes become activated and differentiate into effector lymphocytes which are able to access non-lymphoid tissues such as the skin and gut epithelium (2, 3, 5–7).

Why lymphocytes recirculate is not entirely clear (8, 9). Because the frequency of lymphocytes specific to any given antigen is in general low and the place of entry of any pathogen is unknown by the naive host, recirculation of lymphocytes may increase the chance to that pathogen-specific cell will encounter its antigen (10, 11). Experimental evidence that impairing lymphocyte recirculation influences the ability of the host to respond to infections is very limited. For example, the use of the drug FTY720 (fingolimod) in humans has been associated with a higher incidence of severe infections (12, 13). FTY720 prevents lymphocyte exit from lymph nodes, thus, reducing their ability to recirculate (14–16). However, whether the side-effects of FTY720 is exclusively due to its impact on lymphocyte recirculation is unknown.

The ability of lymphocytes to recirculate between blood and lymph have been nicely demonstrated in now classical experiments by Gowans in rats and later by Hall and Morris in sheep (17, 18). Interestingly, subpopulations of lymphocytes can migrate preferentially to different regions of the body, based on their origin as well as their type (19–24). Molecular interactions between receptors and associated ligands corresponding to the selective entry of lymphocytes to both lymphoid and nonlymphoid tissue have been relatively well-characterized (25–28). However, the actual kinetics of lymphocyte recirculation have been characterized mostly qualitatively, and we still do not fully understand how long lymphocytes reside in the spleen, LNs, and Peyer's patches, and how such residence times depend on lymphocyte type and animal species.

Understanding lymphocyte migration via lymph nodes (LNs) may be of particular importance for larger animals, including humans, where LNs constitute the majority of the secondary lymphoid tissues (29, 30). Lymphocytes may enter the LNs via two routes: from the blood via high endothelial venules (HEVs) or from the afferent lymph draining interstitial fluids from surrounding tissues (1). Experimental measurements suggest that under resting, noninflammatory conditions most lymphocytes (about 80–90%) enter the LNs via HEVs (31). Lymphocytes in LNs exit the nodes with efferent lymphatics which either passes to the next LN in the chain of LNs, or via right or left lymphatic ducts return to the circulation (1, 9, 32). In contrast, cells can only enter the spleen from the blood and cells exiting the spleen directly return to circulation (33, 34).

In the past, to study lymphocyte recirculation via individual LNs sheep or cattle have been used (18, 35–40). In such experiments, lymphocytes are collected from specific tissue of the animal, e.g., blood, a removed LN, or efferent lymph of a LN, labeled with a radioactive or fluorescent label, and then reinfused back into the same animal (e.g., intravenously, i.v.). The dynamics of the labeled cells is then monitored in the blood, or more commonly, in the efferent lymph of different LNs over time (37, e.g., see **Figure 1A**). The dynamics of the labeled cells in the efferent lymph of various LNs follow a nearly universal pattern—the number of labeled cells increases initially, reaches a peak and then slowly declines over time (20, 31, 42, 43, and see **Figures 2C,D**). Given that in many such experiments, the peak of labeled cells in the efferent lymph is reached in 24 h and declines slowly (e.g., **Figure 2C**), it can be interpreted that the average residence time of lymphocytes in the ovine lymph nodes is about 48 h. To our knowledge, the actual residence time in ovine LNs has not been regularly reported. Interestingly, with the use of mathematical modeling an accurate quantification of how long lymphocytes spend in LNs in sheep has been recently performed (44).

In their pioneering study Thomas et al. (44) analyzed data on migration of lymphocytes via individual ovine LNs. To quantify these dynamics, the authors developed a mathematical model which considers cell migration via a LN as a random walk between multiple sub-compartments in the LN. In the model cells entering the LN start in the first sub-compartment, progress via the series of subcompartments by a random walk and eventually exit the node by leaving the last subcompartment (44). As their model allows for the possibility for cells to spend variable lengths of time in the LN, it can naturally explain the long duration of labeled lymphocytes exiting the LN. By fitting the model to several different sets of data, the authors concluded that the average residence time of lymphocytes in ovine LNs is about 31 h (44).

In addition to the average residence time the distribution of residence times may be important. In particular, if lymphocytes that just entered the LN have the same chance of exiting it as lymphocytes that already spent some time in it (i.e., distribution of residence times is exponential), this could suggest that exit of lymphocytes from LN is a simple Poisson-like stochastic process. Indeed, one recent study suggested that residency time of naive CD4 and CD8 T cells in LNs of mice is exponentially distributed (45). In contrast, if lymphocytes require some time to be spent in the LN, for example, to acquire the ability to exit the node, then the distribution of residence time cannot be exponential. Recent work from our group suggests that residence time of thoracic duct lymphocytes in LNs of rats is not exponential and is best described by a gamma distribution with the shape parameter k = 2 or 3 (41). Our analysis of data on kinetics of lymphocyte exit from inguinal LNs of photoconvertable Kaede mice did not allow to firmly establish the shape of the residence time distribution (9). Whether the distribution of residence time of lymphocytes in ovine LNs is exponential or more complex is unknown.

In this paper, we formulated a series of mathematical models aimed at describing the kinetics of recirculation of lymphocytes in sheep. All models take into account basic physiological constrains on lymphocyte migration, for example, that lymphocytes enter the LNs continuously from the blood and lymphocytes that exit LNs return back to circulation. The models were fitted to a series of experimental data from previously published studies on lymphocyte migration in sheep. Our results suggest that the distribution of residence times of lymphocytes in ovine LNs is best described by a nonexponential distribution with estimated average residence times being 12–22 h depending on the type of lymphocytes used in experiments. The long-term presence of labeled lymphocytes in efferent lymph of cannulated LNs was explained by a continuous entry of new cells from the blood to the LN, thus, simplifying a previous modeling result (44). Overall, our analysis provides a quantitative framework to estimate kinetics of lymphocyte recirculation using measurements of lymphocyte numbers in the blood and efferent lymph of ovine LNs. Such a framework may be useful to understand the efficacy and potential limitations of immunotherapies involving adoptive transfer of T cells in humans (46–48).

#### 2. MATERIALS AND METHODS

#### 2.1. Experimental Data

For our analyses we digitized the data from three publications (20, 42, 49). We describe in short design of these experiments and how the data have been collected. For more detail the reader is referred to the original publications.

#### 2.1.1. Lymphocyte Dynamics in Blood and Efferent Lymph (Dataset #1)

Experiments of Frost et al. (42) have been performed with 4– 24 month old Alpenschalf or Schwarzkopf sheep (**Figure 1A**). Lymphocytes were collected from the efferent lymph of different LNs (e.g., prescapular) then labeled with <sup>51</sup>Cr and resuspended in buffer. A surgical procedure to install an indwelling cannula into the jugular vein was performed to allow for i.v. infusions and collection of blood samples. The efferent duct of specific lymph nodes was also cannulated to allow for collection of lymph that passed through the node. The number of labeled lymphocytes in efferent lymph was determined by washing cells in a buffer and counting with a gamma scintillation counter. Labeled lymphocytes in venous or peripheral blood were counted similarly by a gamma scintillation counter then compared to the activity of plasma from the same volume of blood.

Labeled lymphocytes were injected i.v. and the efferent lymph of prescapular LN (pLN) was collected. Lymph was collected as described above at 20-min intervals for 3 h or daily for about 2 weeks. The number of labeled lymphocytes in the lymph node was expressed as cpm per 10<sup>7</sup> cells collected. Peripheral blood samples were only collected for certain experiments and they were taken at 10 min, 30 min, 1 h, and 3-h intervals thereafter. The number of labeled lymphocytes in peripheral blood was expressed as cpm/ml of whole blood minus the activity in the plasma. Cells exiting the lymph node were measured for 120 h, while cells in the blood were measured for 90 h, then once more at the 120th h. For this reason, we discuss lymphocyte migration kinetics in terms of short- and long-term migration experiments. The short-term experiments include the dynamics of the labeled lymphocytes in both blood and efferent lymph for the first 90 h, and the long-term dataset considers all available data (measurement up to 120 h).

According to "Blood Volume of Farm Animals," Hampshire sheep less than a year to 3 years old have on average 6.3–5.8 ml blood per 100 g weight and an average weight of 92–156 lbs (50). This results in about 2.63 L of blood for Hampshire sheep less than a year old and about 4.1 L for Hampshire sheep 2–3 years old. The breed of sheep used in this study weight from 60 kg (Alpenschaf) to 100 kg (Schwarzkopf) at maturity, so it is reasonable to assume that these total blood estimates are less than those calculated for Hampshire sheep. We assume the volume of blood in the sheep is an average of these given volumes, specifically V = 3.4 L. This estimate was used to convert the estimate of the number of labeled lymphocytes per ml of blood to the total number of lymphocytes in the whole blood.

In one set of experiments [Figure 1 in (42)] 2.5 × 10 9 labeled cells (representing 12.6 × 10<sup>6</sup> cpm) were injected i.v. into sheep corresponding on average RL<sup>o</sup> = 2.5 × 10<sup>9</sup> cells/(12.6 × 10<sup>6</sup> cpm) = 200 cells/cpm. Because in the original data, the cells in the blood at time t (RLt) were measured in cpm/ml of blood, the total number of cells in the blood B at time t was calculated using the formula

$$B = RL\_o \times V \times RL\_t,\tag{1}$$

where RL<sup>o</sup> = 200 cells/cpm, V = 3.4 L and RL<sup>t</sup> was digitized from Figure 1 of Frost et al. (42). In the same experiments labeled cells exiting the pLN were measured as cpm/10<sup>7</sup> cell with each data point summing 3 h of cell collected every 20 min. Therefore, the amount of labeled cells exiting the pLN per hour (Ct) is one third of this number. The total output of the pLN is given in Figure 3 in Frost et al. (42) and was estimated to be f = 10.85 × 10<sup>7</sup> cell/hr. We calculate total cells exiting the pLN per hour at time t as

$$m\_{LB}L = RL\_o \times f \times \mathcal{C}\_l,\tag{2}$$

where RL<sup>o</sup> = 200 cells/cpm. The final data (dataset1.csv, given as **Supplement** to the paper) includes changes in the total number of labeled lymphocytes in the peripheral blood and the number of labeled lymphocytes exiting pLN per hour over time.

#### 2.1.2. Migration of Lymphocytes From Afferent to Efferent Lymph (Dataset #2)

To further investigate how lymphocytes migrate via ovine LNs we digitized data shown in Figure 4 of Young et al. (49). In these experiments different subsets of lymphocytes (CD4 T cells, CD8 T cells, γ δ T cells, and B cells) were collected from efferent prescapular lymph and labeled with PKH-26 or CFSE to distinguish between different cell subsets. A maximum of 2 × 10<sup>6</sup> cells of each type were infused into two popliteal afferent lymphatics over 1 h. Cells were collected as they exited the cannulated efferent lymph of the popliteal LN (poLN), and phenotyped. The final data (dataset2.csv, given as **Supplement** to the paper) includes the percent of labeled cells found in the efferent lymph of the poLN over time.

#### 2.1.3. Migration of T Lymphocytes via Skin-Draining and Gut-Draining Lymph Nodes (Dataset #3)

The final set of experimental data we used come from recirculation experiments of Reynolds et al. (20) with young sheep (**Figure 6**). Lymphocytes were isolated by cannulating the skin-draining pLN or the ileal end of the gut-draining mesenteric lymph node chain (mLN). Collected lymphocytes were enriched for T cells and labeled with FITC or RITC based on origin. Labeled cells were re-infused into the same animal i.v. and cell frequency (given in labeled cells per 10<sup>4</sup> cells) was reported for 240 h [Figure 1 in (20)]. Because the authors did not report the overall cell output in efferent lymph of pLN and mLN, we used the provided numbers of the frequency of labeled cells in the lymph for fitting the models. Existing data suggest similar output of lymphocytes from pLN (1 − 5 × 10<sup>8</sup> cells/h) or mLN [1 − 10 × 10<sup>8</sup> cells/h, (37)] which in part justifies our approach. The final data (dataset3.csv, given as **Supplement** to the paper) includes the number of labeled cells from skin or intestinal lymph per 10<sup>4</sup> cells found in efferent lymph of pLN or mLN over time during cannulation.

The raw data from the cited papers were extracted using digitizing software Engauge Digitizer (digitizer.sourceforge.net).

#### 2.2. Mathematical Models

#### 2.2.1. Basic Assumptions

In our models we assume that blood is the main supplier of lymphocytes to other tissues and that when exiting these tissues, lymphocytes return to the blood (41, 51). We also assume that lymphocyte have the same exit kinetics for all LNs (except in the analysis of the dataset #3), and that cell infusion does not impact lymphocyte migration via individual LNs. There is a disagreement whether residency time of lymphocytes depends on the LN type which may be due to differences in animal species or lymphocyte subsets used, or in experimental techniques (41, 45).

#### 2.2.2. Models to Predict Lymphocyte Dynamics in Efferent Lymph of LNs

#### **2.2.2.1. Recirculation model**

To predict the dynamics of i.v. transferred labeled lymphocytes in the blood and efferent lymph of pLN we extended a previously proposed compartmental model describing recirculation kinetics of lymphocytes in the whole body (41). The model (**Figure 1B**) predicts the number of i.v. transferred lymphocytes in the blood (B), spleen (S), lymph nodes (L), and other non-lymphoid tissues (T). In the model we ignored migration of lymphocytes via vasculature of the lung and liver since previous work suggested that resting lymphocytes pass via these tissues, at least in rats, within a minute (41). This is different from migration of activated lymphocytes via these tissues which could take hours (9). Yet, it should be emphasized that because in experiments that we have analyzed migration of lymphocytes via lung and liver vasculature has not been measured, it is possible that lung/liver may represent "spleen" or other "non-lymphoid tissues" in our model. We explore the impact of including lung/liver in our recirculation model on estimates of lymphocyte residency time in the LNs in section 4.

In the model, cells in the blood can migrate to the spleen, lymph nodes, or other tissues at rates mBi and cells can return to circulation from these tissues at rates miB where i = S, L, T. When exiting spleen or non-lymphoid tissues lymphocyte follow the first order kinetics, so the decline of cells in the tissues in the absence of any input is given by an exponential function. This in part is based on our previous work suggesting of migration of thoracic duct lymphocytes via spleen can be described as first order kinetics (41). In contrast, migration of lymphocytes via LNs may not follow the first order kinetics [e.g., (41)] and thus was modelled by assuming k sub-compartments in the nodes with equal transit rates mLB. Such sub-compartments may represent different areas in the LNs, for example, paracortex and medulla.

can be measured, while remaining cells (1 − λ) migrate to other lymph nodes (*L*). Exit of lymphocytes from the spleen or peripheral tissues follows first order kinetics at rates *mSB* and *mTB*, respectively. In contrast, residency times in LNs are gamma distributed and are modeled by assuming *k* sub-compartments (in the figure *k* = 3) where exit from each sub-compartment is given by the rate *mLB*. (C) The alternative "blood-LN dynamics" model allows for cells in the blood (*B*) to enter the cannulated LN at a rate λ*mBL* and exit the LN by passing via *k* sub-compartments at a rate *mLB*. Cells in the blood also leave the blood at a rate α (in case of a single exponential decline).

Mathematically, we used the sub-compartments to model nonexponential residency time of lymphocytes in the LNs.

To describe accumulation and loss of labeled lymphocytes in the cannulated lymph nodes we assume that a fraction of lymphocytes λ migrating to lymph nodes migrate to the cannulated node Lo<sup>1</sup> (e.g., pLN, **Figure 1B**) while 1 − λ cells migrate to other LNs (L1). We assume that cells do not die but the process of migration to non-lymphoid tissues with no return back to circulation is equivalent to cell death. We did consider several alternative models in which death rate was added to the model (see section 4). Taken together, with these assumptions the mathematical model for the kinetics of lymphocyte recirculation in sheep is given by equations:

$$\frac{\mathrm{d}B}{\mathrm{d}t}\_{\mathrm{loc}} = m\_{\mathrm{SB}}\mathrm{S} + m\_{\mathrm{LB}}L\_{\mathrm{k}} + m\_{\mathrm{TB}}T - (m\_{\mathrm{BS}} + m\_{\mathrm{BL}} + m\_{\mathrm{BT}})B, \text{(3)}$$

$$\frac{d\mathcal{S}}{dt} = m\_{\rm BS}\mathcal{B} - m\_{\rm SB}\mathcal{S},\tag{4}$$

$$\frac{\mathrm{d}L\_1}{\mathrm{d}t} = (1 - \lambda)m\_{\mathrm{BL}}B - m\_{\mathrm{L}B}L\_1,\tag{5}$$

$$\frac{\text{d}L\_i}{\text{d}t} = m\_{LB}(L\_{i-1} - L\_i), \qquad i = 2, 3, \dots, k,\tag{6}$$

$$\frac{\mathrm{d}L o\_1}{\mathrm{d}t} = \lambda m\_{BL} B - m\_{LB} L o\_1,\tag{7}$$

$$\frac{\text{d}Lo}{\text{d}t}\_{\text{new}} = m\_{\text{LB}} (Lo\_{i-1} - Io\_i), \qquad i = 2, 3, \dots k,\tag{8}$$

$$\frac{\mathrm{d}T}{\mathrm{d}t} = m\_{\mathrm{BT}}\mathrm{B} - m\_{\mathrm{TB}}T,\tag{9}$$

where cells exiting the cannulated lymph node, mLBLo<sup>k</sup> do not return to the blood because they are sampled, and mLBLo<sup>k</sup> is the rate of labeled lymphocyte exit from the sampled lymph node (which is compared to experimental data, e.g., column 2 in the dataset #1, and see **Figures 2C,D**).

Our experimental data were restricted only to labeled lymphocytes found in the blood and efferent lymph (or only efferent lymph). It may be argued that a minimal recirculation model to describe such data should only involve blood and LNs (i.e., n = 1). Also, the number of sub-compartments k in LNs is also unknown. Therefore, in our analyses we fitted a series of models assuming different values for k and n to the data from (42) (dataset #1) and compared the quality of the model fit to data using AIC (see section 3).

The average residence time of lymphocytes in the spleen or non-lymphoid tissues is 1/mSB and 1/mTB, respectively. The residence time of lymphocytes in the LNs is given as RT = k/mLB. The initial number of labeled cells in the blood varied by experiment and is indicated in individual graphs. In some fits parameter λ could not be identified from the data and thus was fixed (indicated by absent predicted confidence intervals). The rest of the parameters were fit.

#### **2.2.2.2. Blood-LN dynamics model**

Many studies of lymphocyte recirculation that reported kinetics of accumulation and loss of labeled lymphocytes in efferent

FIGURE 2 | Dynamics of recirculating lymphocytes (RLs) in the blood naturally explains kinetics of accumulation and loss of RLs in pLN of sheep. (A,B) Dynamics of RLs in the blood and (C,D) in the efferent lymph of the pLN for the first 90 h after cell transfer (short-term migration) are shown by markers. (A,C) The recirculation model (Equations 3–9, Figure 1B) with *n* = 3 tissue compartments and *k* = 3 sub-compartments in the LNs resulted in the best fit. The average residence times (*RT*1 and *RT*2) estimated for the two first compartments are shown. (B,D) The blood-LN dynamics model (Equations 10–12, Figure 1C) with *j* = 2 and *k* = 3 resulted in the best fit. The estimated average *RT* of lymphocytes in the pLN is shown. Fits of models that assume different numbers of tissue compartments, different number of sub-compartments in LNs, or different numbers of exponential functions are shown in Table 1 and Table S1. Parameters for the best fits of these models are given in Table 2.

lymph of ovine LNs did not report dynamics of these cells in the blood which is the major limitation of such studies. Therefore, to gain insight into whether LN cannulation data alone can be used to infer lymphocyte residence times in the LNs we propose an alternative model which only considers lymphocyte dynamics in the blood, one cannulated LN, and the efferent lymph of the cannulated LN. In this "blood-LN dynamics" model (**Figure 1C**) the dynamics of lymphocytes in the blood is described by a phenomenological function B given as a sum of j declining exponentials. The rationale to use such a model stems from the kinetics of labeled lymphocyte dynamics in the blood observed in (42) and other studies [**Figures 2A,B** and (44)] even though actual dynamics in other experimental systems may not follow the same pattern. The dynamics of labeled cells in the sampled lymph node is thus driven by the continuous entry of labeled cells from the blood into the LN. The model is then given by following equations

$$B = \sum\_{i=1}^{j} X\_i e^{-\alpha\_i t},\tag{10}$$

$$\frac{\mathrm{d}L o\_1}{\mathrm{d}t} = \lambda m\_{\mathrm{BL}} \mathrm{B} - m\_{\mathrm{L}B} L o\_1,\tag{11}$$

$$\frac{\text{d}Lo\_{i}}{\text{d}t} = m\_{\text{LB}}(Lo\_{i-1} - Io\_{i}), \qquad i = 2, \ldots, k,\tag{12}$$

X<sup>i</sup> and α<sup>i</sup> are the initial values and the rate of decline in the i th exponential function, λmBL is the overall rate at which lymphocytes from the blood enter the LN, and mLB is the rate at which lymphocytes move between k sub-compartments in the LN and exit the LN. The rate at which lymphocytes exit the LN and thus are sampled in the LN efferent lymph is mLBLo<sup>k</sup> . It should be noted that parameters λ, mBL and X<sup>i</sup> in many cases are not identified from the data on lymphocyte dynamics in the efferent lymph, thus, the main parameter that we are interested in is the average residence time of lymphocytes in the LNs given by RT = k/mLB. Because the dynamics of cells in the blood is generally unknown when fitting the model predictions to data, we varied the number of exponential functions j = 1, 2, 3 and compared the quality of fits of different models using AIC.

#### 2.2.3. Migration When Cells Are Injected Into Afferent Lymph

In one set of experiments migration of labeled lymphocytes via LNs was measured by directly injecting lymphocytes into the afferent lymph of a LN and observing accumulation and loss of these cells in the efferent lymph of the LN. To use these data to estimate the lymphocyte residence time in the LN we assume that cells injected into afferent lymph A migrate into the lymph node at a rate mA, and then the cells migrate via each of k subcompartments in the LN at a rate mLB. With these assumptions the dynamics of cells in the afferent lymph and the LN are given by equations:

$$\frac{\text{d}A}{\text{d}t} = -m\_A A,\tag{13}$$

$$\frac{\mathrm{d}L o\_1}{\mathrm{d}t} = m A - m \mathrm{l}s L o\_1,\tag{14}$$

$$\frac{\text{d}Lo\_i}{\text{d}t} = m\_{\text{LB}}(Lo\_{i-1} - Io\_i), \qquad i = 2, 3, \dots, k,\tag{15}$$

where initially all labeled cells were in the afferent lymph. As previously stated, the rate of lymphocyte exit from the LN via efferent lymph is given by mLBLo<sup>k</sup> . The average residence time of lymphocytes in the LN is then RT = k/mLB.

#### 2.2.4. Homing to Different Lymph Nodes

In the final set of experiments Reynolds et al. (20) collected lymphocytes from efferent lymph of pLN or mLN, labeled and then re-infused the collected cells i.v. into the same animal. The labeled cells were then collected in the efferent lymph of the pLN and mLN. Because the authors did not report the dynamics of labeled cells in the blood, we extended the "blood-LN dynamics" model (see Equations 10–12) to describe cell migration from the blood to the efferent lymph of two LNs. The number of labeled lymphocytes found in the i th sub-compartment of the pLN and mLN are given by Lo1,<sup>i</sup> and Lo2,<sup>i</sup> , respectively:

$$B = \sum\_{i=1}^{j} X\_i e^{-\alpha\_i t},\tag{16}$$

$$\frac{\mathrm{d}L o\_{1,1}}{\mathrm{d}t} = \lambda m\_{\mathrm{BL}1} \mathrm{B} - m\_{\mathrm{L1B}} L o\_{1,1},\tag{17}$$

$$\frac{\mathrm{d}Lo\_{1,i}}{\mathrm{d}t} = m\_{L1B}(Lo\_{1,i-1} - Lo\_{1,i}), \qquad i = 2, \ldots, k,\tag{18}$$

$$\frac{\mathrm{d}L o\_{2,1}}{\mathrm{d}t} = \lambda m\_{\mathrm{BL}2} \mathrm{B} - m\_{\mathrm{L2B}} L o\_{2,1},\tag{19}$$

$$\frac{\text{d}Lo\_{2,i}}{\text{d}t} = m\_{\text{L2B}}(Lo\_{2,i-1} - Lo\_{2,i}), \qquad i = 2, \dots, k,\tag{20}$$

where mL1<sup>B</sup> and mL2<sup>B</sup> are the rate of lymphocyte exit from the pLN and mLN, respectively, mBL<sup>1</sup> and mBL<sup>2</sup> are the rates of lymphocyte entry from the blood to pLN and mLN, respectively, and j = 1, 2 in fitting models to data. Because the data clearly showed the difference in accumulation of lymphocytes in different LNs, we considered two alternative explanation for this difference. In one model we assume that the difference in kinetics is due to differences in the rate of lymphocyte entry into specific LNs (mBL<sup>1</sup> 6= mBL2) while residence times are identical in the two LNs (mL1<sup>B</sup> = mL2B). In the alternative model, the rate of entry into the LNs are the same but residence times may differ (mBL<sup>1</sup> = mBL<sup>2</sup> and mL1<sup>B</sup> 6= mL2B).

#### 2.2.5. Statistics

The models were fitted to data in R (version 3.1.0) using modFit routine in FME package (version 1.3.5) by log-transforming the data (single or two different measurements) and model predictions and by minimizing the sum of squared residuals. For example, when fitting the recirculation model to the data on lymphocyte numbers in the blood and efferent lymph from Frost et al. (42) the SSR was calculated in the following way:

$$\text{SSR} = \sum\_{t\_i=1}^{l} \left[ \log\_{10} \left( \frac{RL\_o \times V \times RL\_{t\_i}}{B(t\_i)} \right) \right]^2$$

$$+\sum\_{t\_i=1}^{l} \left[\log\_{10}\left(\frac{RL\_o \times f \times C\_{t\_i}}{m\_{LB}Lo\_k(t\_i)}\right)\right]^2,\tag{21}$$

where experimental measurements are given in Equations (1) and (2) and model predictions are from Equations (3) and (8), respectively, for l measurements at times t<sup>i</sup> , i = 1 . . . l.

Numerical solutions of the system of equations were obtained using ODE solver lsoda (from the deSolve package) with default absolute and relative error tolerance. Different algorithms such as BFGS, L-BFGS-B, or Marquart in the modFit routine were used to find parameter estimates. Discrimination between alternative models was done using corrected Akaike Information Criterion, AIC (52)

$$\text{AIC} = N \log \left( \frac{\text{SSR}}{N} \right) + 2p + \frac{2p(p+1)}{N-p-1},\tag{22}$$

where SSR is the sum of squared residuals, N is the number of data points, and p is the number of model parameters fitted to data. The model with the minimal AIC score among all tested models was viewed as the best fit model, but a difference of AIC score of 1–3 between best fit and second best fit models was generally viewed as not significant (52). Predicted 95% confidence intervals for estimated parameters were calculated as ±2σ with standard deviation σ provided for each parameter by the modFit routine.

#### 3. RESULTS

#### 3.1. Estimating Lymphocyte Residency Time in the LNs Using Lymphocyte Dynamics in Efferent Lymph

To gain quantitative insights into the kinetics lymphocyte migration via sheep lymph nodes we first fitted our recirculation model (given by Equations 3–9) to the "short-term migration" data on lymphocyte dynamics in the blood and efferent lymph of the prescapular LN from Frost et al. (42) (see section 2 for more detail on the data).

While the overall structure of the recirculation model was defined by the number n of different compartments through which lymphocytes could recirculate (**Figure 1B**), we first investigated how many such compartments are in fact necessary to describe the experimental data by varying n between 1 and 4. Additionally, we tested how many sub-compartments k in the LNs are needed for best description of the data (see Equations 3– 9). The analysis revealed that n = 3 tissue compartments and k = 3 sub-compartments in the LNs are needed to adequately describe the dynamics of labeled cells in the blood and efferent lymph (**Table 1**). Such a model could accurately describe simultaneously the loss of labeled cells in the blood and accumulation and loss of labeled cells in the efferent lymph (**Figures 2A,C**). The model predicted the existence of two recirculation compartments with average residence times of 2.4 and 19.5 h, with the latter compartment corresponding to LNs in the sheep. The nature of the first compartment is unclear but given the estimated residence time it is likely that it represents the spleen [e.g., see (41)]. The final third compartment was needed to explain the long-term loss of labeled cells from the blood at a rate of about d = 0.02/h. The predicted rate of lymphocyte migration to tissues (mBS + mBL + mBT ≈ 1.4 − 1.6/h ≫ d, see **Table 2**) is higher than the observed rate d because of the return of lymphocytes that had migrated to lymphoid tissues back to circulation. The model also naturally explains the long-term decline in the number of labeled lymphocytes found in the efferent lymph of the cannulated pLN which is simply driven by the decline of labeled cells in the blood. The analysis also suggests that none of the obvious characteristics of the distribution of the lymphocyte exit rate from the LN such as the time of the peak or the average of the overall distribution (e.g., see **Figure 2C**) accurately represent the average residence time. This result strongly suggests that to accurately estimate lymphocyte residence times from LN cannulation experiments it is critical to use appropriate mathematical models.

Assuming a smaller (k = 1) or a larger (k = 4) number of sub-compartments in the LNs resulted in poorer fits of the data (**Table 1**). The intuitive reason of why the model in which lymphocyte residence times in LNs are exponentially distributed (k = 1) does not fit the data well follows from the rapid loss of labeled lymphocytes in the blood within the first hours after lymphocyte transfer (**Figure 2A**). Rapid decline in the number of labeled lymphocytes in the blood reduces the rate at which new labeled cells enter the pLN which would have resulted in a relatively rapid exit of cells from the pLN for exponentially distributed residence time. Similarly, the model in which there are too many sub-compartments would force the distribution of cells in the efferent lymph to be even broader, thus, also resulting in poorer fit. Thus, this analysis suggests that migration of lymphocytes via LNs is not described by a simple exponential function and there is a requirement for lymphocytes to spend some minimal time in LNs before exiting into circulation.

It is interesting to note how the dynamics of labeled lymphocytes in the blood may be used to infer recirculation kinetics of cells. Indeed, the initial rapid decline of the number of labeled lymphocytes is explained in the model by migration to secondary lymphoid tissues and change in the decline rate at 2–3 h after lymphocyte transfer is naturally explained by the exit of initially migrated cells from one of the compartments (most likely spleen) back to the blood. Thus, lymphocyte kinetics in the blood suggests residence time in first compartment of about 2–3 h (**Table 2**).

The recirculation model makes a strong assumption that the dynamics of labeled lymphocytes in the blood and efferent lymph are due to migration of lymphocytes into and out of different tissues (**Figure 1B**). When the experimental data is provided for the dynamics of labeled cells in the blood as in dataset #1, we are able to use Equations (3)–(9) to accurately describe the data and estimate lymphocyte residence times in various tissues. However, when experimental data do not contain measurements of the dynamics of labeled cells in the blood, predictions of the recirculation model remain speculative. Therefore, to estimate residence times of lymphocytes in LNs in the absence of such data, we developed an alternative mathematical model. This model involves a smaller number of assumptions, the major of TABLE 1 | Comparison of different recirculation mathematical models fitted to the data on RL dynamics in blood and prescapular LN.


*Mathematical models assuming recirculation of lymphocytes via n different tissue compartments with LNs having k sub-compartments (Equations 3–9) were fitted to experimental data (shown in* Figures 2A,C*). We tested n* = *1…4 different tissue compartments with k* = *1…4 sub-compartments in LNs. The bold AIC value shows the model of best fit with n* = *3 and k* = *3. Parameters for the best fit model are shown in* Table 2 *and the best fit is shown in* Figures 2A,C*.*

which is the physiological constraint that most lymphocytes enter lymph nodes from blood (2). Intuitively, however, this model allows us to estimate the time taken by cells to migrate from the blood to the efferent lymph of a LN irrespective of the specific route of this migration.

In the"blood-LN dynamics" model, the dynamics of labeled lymphocytes in the blood is described phenomenologically as a sum of several exponential functions, and by fitting a series of such models we found that the dynamics of labeled cells in the first 90 h after cell transfer is best described by a sum of two exponentials (**Table S1** and **Figure 2B**). The model predicted a rapid initial loss of lymphocytes in the blood at a rate of 2/h (halflife time of about 21 min) and a slower loss rate of 0.02/h after the first 4 h (half-life time of 35 h, **Figure 2B** and **Table 2**).

By fitting a series of mathematical models in which the number of sub-compartments in the pLN was varied, to the data on dynamics of labeled lymphocytes in the efferent lymph we found that k = 3 sub-compartments provided fits of the best quality (**Table S1** and **Figure 2D**). Importantly, the model predicted the average residence time of lymphocytes in the pLN of 19.6 h which is nearly identical to the value found by fitting recirculation model to the same data (**Table 2**).

Both recirculation and blood-LN dynamics models were then fitted to the long-term migration data in which the dynamics of labeled cells in efferent lymph was measured continuously for 120 h while in the blood there was an extra measurement at 120 h (**Figures 3A–D**). The recirculation model with n = 3 tissue compartments and k = 3 sub-compartments was able to accurately describe the data (**Figures 3A,C**) although the model underestimated the number of labeled lymphocytes in the blood at 120 h post-transfer (**Figure 3A** and **Table S2**). Interestingly, the model required a positive rate of lymphocyte return from the "third" tissue back to circulation (**Figure 3A** and **Table 2**). In the absence of lymphocyte return from the tissue compartment TABLE 2 | Parameters of the best fit recirculation model (Equations 3–9) or the blood-LN dynamics model (Equations 10–12) fitted to either short-term migration data (*t* < 90 *h*) or the long-term migration data of dataset #1 from Frost et al. (42).


*In the recirculation model the three tissue compartments are suggested to be spleen, LNs, and other non-lymphoid tissues and the migration rates from the blood to these compartments are denoted as mij with i*, *j* = *B*, *S*, *L*, *T. In the blood-LN dynamics model it was not possible to estimate accurately the initial number of labeled lymphocytes in the blood (X*1*), so that parameter was fixed to X<sup>1</sup>* = *2.5*×*10<sup>9</sup> cells. Residence times in LNs were calculated as RT* = *k*/*mLB and as 1 miB for other compartments (i* = *S*, *T).*

the model poorly matches the number of labeled cells in the blood (results not shown). Importantly, the recirculation model predicted similar average residence times of lymphocytes in the first two compartments (representing spleen and LNs) to that of the model fitted to short-term dataset (**Table 2**).

Perhaps unsurprisingly, to describe the dynamics of labeled lymphocytes in the blood over 120 h the sum of three different exponential functions was required (**Table S1**). Furthermore, the model with k = 3 sub-compartments in the pLN was able to describe the dynamics of labeled lymphocytes in efferent

lymph with best quality (**Table S1** and **Figures 3B,D**). The model predicted the average residence time of lymphocytes in LNs to be 19.6 h which is consistent with results from the recirculation model fitted to the same data or the models fitted to short-term migration data.

Taken together, analysis of data from Frost et al. (42) on recirculation of lymphocytes via prescapular LN in sheep suggests non-exponentially distributed residence times of lymphocytes in the LNs with the average time being approximately 20 h. Our mathematical models naturally explain the long-term presence of labeled lymphocytes in efferent lymph node of a cannulated LN by continuous input of new labeled cells from the blood to the LN.

# 3.2. Migration of Lymphocytes From Afferent to Efferent Lymph Suggests Non-exponentially Distributed Residence Time in LN

Analysis on the dynamics of labeled lymphocytes transferred i.v. into sheep suggested that migration of lymphocytes via LN follows a multi-stage process which can be described as cell migration via identical sub-compartments (**Figures 1B,C**). Since the time it takes for lymphocytes to cross the endothelial barrier and enter LNs is very short [few minutes, (53)], the finding that distribution of lymphocyte residence times are not exponential could still be due to some unknown processes. Therefore, to further investigate the issue of the distribution of residence times of lymphocytes in ovine LNs we analyzed experimental dataset #2 (49). In these experiments, Young et al. (49) isolated lymphocytes from the efferent lymph of the pLN, labeled and injected the cells into the afferent lymph of the popliteal LN (poLN), and then measured exit of the labeled cells the efferent lymph of the poLN (**Figure 4** and see section 2 for experimental data detail). Cells, injected into the afferent lymph, cannot move to any other tissue but the draining LN, and thus, such data allow to directly evaluate kinetics of lymphocyte migration via individual LN.

To describe these data, we adapted the blood-LN dynamics model to include migration of labeled lymphocytes from the afferent lymph to the LN and then to the efferent lymph (Equations 13–15). The model has 3 unknown parameters that must be estimated from the data (A(0), mA, mLB). Unfortunately, the original data for cell dynamics for individual animals were not available, and the digitized data only included 3 time points which does not allow accurate estimation of all model parameters (results not shown). Therefore, to investigate the dynamics of labeled cells in the efferent lymph we fitted a series of mathematical models with a varying number of subcompartments k in the LN and average residence times RT = k/mLB fixed to several different values to the experimental data (**Table S3**). Analysis revealed that several sub-compartments are needed for accurate description of the data and the actual number of sub-compartments varied for different cell subtypes, but was never less than k = 3 (**Table S3**). The expected residence times also varied with the cell type but overall were within 18–20 h range which is consistent with the previous analysis of Frost et al. (42) data (**Figure 4**).

By fitting the data with the model in which the number of sub-compartments k was varied we found that the estimated residence time of lymphocytes in the poLN was dependent on the assumed k (**Table S4**). This is consistent with our recent result on estimating residence time of T and B lymphocytes in LNs of mice using the data from photoconvertable Kaede mice (9). Interestingly, the model fit predicted a relatively slow movement of lymphocytes from the afferent lymph to the LN, which is determined by the parameter m<sup>A</sup> (1/m<sup>A</sup> ≈ 5 h) and was dependent on the number of sub-compartments. These results also support our conclusion that residence times of lymphocytes in ovine poLN are not exponentially distributed and the average residence time for different lymphocyte subsets is around 20 h.

# 3.3. Impact of Lymphocyte Kinetics in the Blood on Estimates of the Lymphocyte Residency Time in the LNs

Our analysis of the Frost et al. (42) data demonstrated the usefulness of having measurements of the dynamics of labeled lymphocytes both in the blood and efferent lymph of a specific cannulated LN. Unfortunately, many published studies that we reviewed lacked measurements of lymphocyte counts in the blood and only recorded cell numbers in the efferent lymph, often as the percent of labeled cells in the overall population. An important question is whether the estimates of the residency time of lymphocytes in LNs, found by fitting mathematical models to the data on lymphocyte counts in efferent lymph only, depend on the assumed lymphocyte dynamics in the blood.

In several different experiments the decline of i.v. injected labeled lymphocytes in the blood is bi-exponential with the rapid decline in cell numbers within a few hours and slower decline in the next days (44, see **Figure 2B**). By fitting the blood-LN dynamics model (with k = 3) to the data on the dynamics of labeled lymphocytes in efferent lymph (shown in **Figure 2D**) we found that fits of similar quality could be obtained independently whether the dynamics of labeled cells in the blood follow either a single or bi-exponential decline (results not shown). However, the estimates of the average residence time of lymphocytes in LNs was dependent on the assumed model of lymphocyte dynamics in the blood; namely, assuming a bi-exponential decline resulted in longer average residence times (results not shown).

To investigate this issue further we fitted the blood-LN dynamics model (with k = 3) assuming that the number of labeled cells in the blood follows an exponential decline, to the data on labeled cell dynamics in efferent lymph. In this analysis we either fitted the rate of cell decline in the blood (α1) or fixed it to different values (**Figure 5**). We found that the decline rate of labeled cells in the blood has a dramatic impact on the quality of the model fit of the data as well as on estimates of the average residence times (**Figure 5B**). In particular, assuming that the number of labeled cells remains constant in the blood (α<sup>1</sup> = 0) predicts a constant output of labeled cells in efferent lymph, and as the result, failed to accurately describe the data (**Figure 5B**). Similarly, assuming that the loss of labeled cells occurs relatively rapidly (α<sup>1</sup> = 0.05/h) also results in poor fits of the data and longer average residence time of lymphocytes in the LN (**Figure 5B**). However, allowing the rate of lymphocyte loss α<sup>1</sup> to be fitted resulted in good fits of the data further suggesting that the long-term dynamics of labeled cells in the efferent lymph is the consequence of cell dynamics in the blood.

# 3.4. Lymphocytes Migrate More Rapidly to LNs Which the Cells Recently Exited

All data analyzed so far have been for lymphocytes isolated from pLNs which migrate back to pLN or poLNs. An important question is whether the average residence time of lymphocytes varies across types of LN. Indeed, previous analysis of migration of naive T cells in mice suggest that lymphocytes spend less time in gut-draining mesenteric LNs (mLNs) than in skin-draining pLNs (45). In contrast, another study suggested similar residency times of thoracic duct lymphocytes in pLN and mLN of rats (41). To address this issue we analyzed experimental dataset #3 (20).

In their studies, Reynolds et al. (20) isolated T lymphocytes from efferent lymph of pLN or mLN, labeled them with different fluorescent dyes, re-injected the cells, and measured their exit in the efferent lymph of pLN and mLN (**Figure 6** and see section 2 for experimental data detail). The data showed that T cells isolated from pLN accumulate to higher numbers in the efferent lymph of pLN as compared to cells from mLN and vice versa (**Figure 7**). There could be at least two alternative explanations for such differential accumulation of cells in LNs of their origin: preferential migration or preferential retention. According to the preferential migration hypothesis, cells from pLN have a higher rate of entry into pLN than the rate at which cells from mLN enter the pLN (and vice versa). In contrast, in the preferential retention hypothesis, cells from pLN have a longer residence time in pLN as compared to cells from mLN (and vice versa).

To discriminate between these alternative hypotheses, we fitted the blood-LN dynamics model to these data. The blood-LN dynamics model was chosen because of its relative simplicity and because Reynolds et al. (20) did not report dynamics of transferred lymphocytes in the blood, which was needed for accurate estimation of parameters of the recirculation model. Specifically, we assumed that the T cell kinetics follow biexponential decline (j = 2 in Equation 10) and that lymphocytes must traverse via k = 3 sub-compartments in the LNs. In the "preferential migration" model we fixed the average residence times of lymphocytes in LNs for cells from pLN and mLN (determined by the parameter mLB) but allowed different rates of entry into the LN from the blood (determined by the parameter mBL, see Equations 16–20). This model can accurately describe experimental data (**Figures 7A,B**). Interestingly, the model predicted 2 fold higher entry rate into pLN by cells of pLN origin as compared to cells of mLN origin, and 3 fold higher entry rate into mLN by cells of mLN origin, as compared to

FIGURE 4 | Mathematical modeling suggests non-exponentially distributed residency time of different subsets of lymphocytes in the LNs. (A) CD4 T cells, (B) CD8 T cells, (C) γ δ T cells, or (D) B cells were labeled and injected into afferent lymph of ovine poLN. The percent of labeled cells was measured in efferent lymph over time (see section 2 for experimental data detail). A series of mathematical models assuming migration of injected lymphocytes into the LN was fit with a variable number of sub-compartments *k* in the LNs (Equations 13–15). The best fits of the model leading to the lowest AIC values with the noted number of sub-compartments in the LN (*k*) are shown by lines and parameters of the models are given in Table S4.

FIGURE 5 | Kinetics of lymphocyte loss in the blood influences the estimate of the lymphocyte residence time in LNs. Assuming that lymphocyte dynamics in the blood follows an exponential decay at a rate <sup>α</sup><sup>1</sup> (A), the blood-LN dynamics model (Equations 10–12 with *<sup>k</sup>* <sup>=</sup> 3, *<sup>X</sup>*<sup>1</sup> <sup>=</sup> <sup>5</sup> <sup>×</sup> <sup>10</sup><sup>8</sup> , and λ = 0.01 being fixed) was fit to the data on accumulation and loss of labeled lymphocytes in the efferent lymph of the pLN of sheep (panel B; see section 2 for experimental data details). In contrast with other analyses, fits of the model to data in this case were done without log10 transformation. The decline rate α1 was either fixed to several different values (α1 = 0, α1 = 0.01/*h*, or α1 = 0.05/*h*) or was estimated by fitting the model to data in (B) (α1 = 0.021/*h*). The data on labeled lymphocyte dynamics in the blood from Frost et al. (42) was not used in model fitting and is only displayed by markers for illustrative purposes in (A). The estimated residence time (*RT*) of lymphocytes in the LN are shown in (B).

cells of pLN origin (**Table S5**). Importantly, assuming identical average residence time of T cells from pLN in skin-draining or gut-draining LNs resulted in fits of excellent quality suggesting the average residence time of T cells from pLN does not depend on the LN type. However, T cells from mLN migrated via LNs nearly 2 fold faster than T cells from pLN suggesting that the average residence time does depend on the origin of T cells.

In the alternative "preferential retention" model we fixed the rate of lymphocyte entry from the blood to the LNs and allowed the residence times (or more precisely, the rate of exit of T cells from the LNs) to vary depending on the LN type. This model failed to accurately describe the data (**Figures 7C,D**) suggesting that the data cannot be explained solely by increased retention of cells in the LN of their origin. Importantly, allowing both entry

and exit rates to depend on the LN type did not improve the model fit of the data for lymphocytes from pLN [F-test for nested models, F(1, 24) = 0.26, p = 0.62] but marginally improved the fit of the data for lymphocytes from mLN [F-test for nested models, F(1, 24) = 6.3, p = 0.02].

#### 4. DISCUSSION

It is well understood that some lymphocytes are able to recirculate between blood and secondary lymphoid tissues such as lymph nodes. In part, this understanding came from multiple experiments on lymphocyte migration from the blood to efferent lymph of various LNs in sheep. Yet, while the data on the kinetics of lymphocyte migration via individual LNs have been published, quantitative interpretation of these data has been lacking until recently. In particular, the average residence time of lymphocytes in the ovine LNs remained largely unknown and there was incomplete understanding of why labeled lymphocytes persisted in the efferent lymph of cannulated LNs.

The first attempt known to us to explain lymphocyte dynamics in efferent lymph during cannulation experiments in sheep was by Thomas et al. (44) who modeled lymphocyte migration via the LN as a random walk. The model suggested that longterm detection of labeled lymphocytes in efferent lymph was due to inability of some lymphocytes to exit the LN. Here we formulated several alternative mathematical models, based on the basic understanding of lymphocyte recirculation in mammals which accurately explain the cannulation data, and proposed an alternative explanation for long term detection of labeled cells in efferent lymph. Namely, because labeled cells persist in the blood, continuous reentry of such cells into the LN can naturally explain long-term persistence of labeled cells in the efferent lymph.

Our mathematical modeling approach allowed to provide novel estimates of lymphocyte residence time in ovine lymph nodes which vary between 12 and 20 h depending on lymphocyte type and being approximately independent of the type of LN (e.g., skin- or gut-draining, **Figures 2**, **7**). Furthermore, the combination of data and mathematical model predicted an existence of a compartment with a shorter residence time, about 2–3 h, which we propose is likely to be the spleen. Indeed, we recently published a similar estimate of lymphocyte residence time in the spleen of rats (41). However, this prediction remains to be tested experimentally, and it remains possible that the unobserved compartment with the residency time of 2–3 h represents vasculature of the lung or liver. We consider this alternative to be unlikely given that we previously estimated that resting lymphocytes spend a rather short time in the lung/liver vasculature of rats [less than 1 min, (41)].

Parameter estimates also suggest a relatively short residence time of lymphocytes in the blood (e.g., for long-term migration data 1/(mBS + mBL + mBT) ≈ 1.6/h or RT<sup>B</sup> = 28 min). This is relatively similar to previous estimates (9, 41).

While we did not specifically model lymphocyte migration via lung or liver vasculature, it is well understood that lymphocytes in the blood do pass via these tissues (41, 54, 55). To determine whether inclusion of the lung/liver vasculature impacts our estimates of the lymphocyte residence time in LNs, we extended the recirculation model (given in Equations 3–9) by adding an equation for lymphocytes in the lung/liver vasculature V (which would be identical to Equation 4). The migration of lymphocytes from the blood to the vasculature was given by rate mBV and exit of lymphocytes from the vasculature into the circulation was given by rate mVB. Then we varied the rates mBV = mVB from 0/h to 30/h and fitted other parameters of the model to the short-term recirculation data. Analysis showed that the estimate of the lymphocyte residence time in the LNs varied relatively little with changes in lymphocyte migration rate via the lung/liver vasculature (from 19.5 to 17.6 h) suggesting that the estimate of the lymphocyte residency time in LNs is robust to exclusion of the lung/liver vasculature from the model. Interestingly, however, increasing the rate of lymphocyte migration via the lung/liver vasculature reduced the quality of the model fit to data (measured as SSR or AIC) which is in line with our result that the models with more than n = 3 tissue compartments describe the cannulation data with poorer quality.

Another important conclusion from our analyses is that the data on lymphocyte dynamics in efferent lymph is not described well by a model in which residence times of lymphocytes in LNs are exponentially distributed (**Table 1**). In part, this is because of the wide distribution in the exit rates of labeled lymphocytes in efferent lymph over time. However, describing cell migration via LNs as a simple one directional process and ignoring the ability of lymphocytes to remain in the LN for longer [e.g., by including a "backflow" in cell movement as was done by Thomas et al. (44)] may be an over-simplification. Yet, because the model in which lymphocyte residence times are gamma distributed describes the experimental data with acceptable quality (e.g., see **Figure 2**), introducing additional details/parameters contradicts the fundamental "Occam's razor" principle.

Our analysis suggests difficulty with interpreting data from ovine LN cannulation experiments in which the dynamics of transferred lymphocytes is not tracked in the blood. In particular, we found that estimates of lymphocyte residence times in LNs do depend on the assumed model for lymphocyte dynamics

in the blood (e.g., single vs. double exponential decline and slow or rapid decline). Therefore, future studies on lymphocyte recirculation kinetics in sheep should always attempt to measure and report concentration and total numbers of transferred cells in the blood.

One of the fundamental questions of lymphocyte recirculation is whether lymphocytes in the blood have some "memory" of the specific LN they recently came from, and if such memory exists, whether it comes from preferential entry into a specific LN or from preferential retention in the LN. Several experimental studies have addressed the question qualitatively. For example, activated lymphocytes, or lymphoblasts, collected from the intestinal lymph of sheep were shown to accumulate preferentially in tissues associated with the gut (56), and a similar finding was reported for lymphoblasts isolated from intestinal lymph of rats (57, 58). In contrast, lymphoblasts isolated from peripheral lymph preferentially accumulated in peripheral lymph nodes (57). There has also been a distinction in migratory preference based on cellular subset as it has been observed that small lymphocytes accumulate in mucosal sites such as Peyer's patches (59, 60).

We used mathematical modeling to investigate whether preferential accumulation of lymphocytes in the LN of their origin is due to preferential entry or preferential retention for one specific dataset (20). Our analysis showed that a model with preferential retention was not able to accurately describe the experimental data, while the model in which cells could preferentially enter a LN was able to describe the data well (**Figure 7**). Intuitively, this may be because the earliest increase in the number of cells found in the efferent lymph seems to be driven by rate of cell entry into the node and the data clearly indicate difference in cell accumulation in the efferent lymph depending on the cell's origin. While we have not addressed it formally, it is possible that the overall distribution of residence times (determined by the parameter k) may be different in LNs of different types.

There are a number of limitations with experimental data and our modeling analyses that need to be highlighted. In particular, in all of our experimental data, the dynamics of labeled lymphocytes in the efferent lymph was reported as a frequency of total cells, which required recalculation to determine the total number of cells exiting a specific LN per unit of time [e.g., (42)]. Similarly, calculation of the total number of lymphocytes in the blood requires the knowledge of the total blood volume of animals which was not reported. The required recalculations may introduce errors (e.g., due to incorrectly assumed blood volume in the animals) and thus may influence the values for some estimated model parameters. For example, a smaller assumed blood volume in animals would naturally lead to a lower number of transferred lymphocytes in Frost et al. (42) experiments detected in the blood which should directly impact the estimate of the rate of lymphocyte migration from the blood to the LN. Thus, the absolute values of estimated rates at which lymphocytes are predicted to migrate to LNs from the blood should be treated as approximate. However, estimates of lymphocyte residency time in LNs should be robust to changes in the scaling of lymphocyte numbers in the lymph.

One of the major assumptions we made in the models was that all labeled cells have identical migratory characteristics, e.g., all cells are capable of entering and exiting the LNs and do so at the same rates. In many previous studies the types of lymphocytes used in recirculation experiments (e.g., naive or memory lymphocytes, B or T cells) were not specified, and it is very well possible that migratory properties vary by cell type [e.g., see (9)]. It is clear however that including multiple cell subpopulations will increase complexity of the models, making them unidentifiable from the data we have available. Also, the models based on kinetically homogeneous cell populations could describe the data reasonably well, which suggests that there is no need to introduce a more complex model for such data. Yet, comparison of predictions found by the best fit models for the data did indicate some discrepancy, for example, the best fit model was not fully capable of capturing the peak of the exit rate of labeled lymphocytes in efferent lymph (e.g., **Figure 5B**). It is possible that including slow and fast recirculating cell sub-populations may be able to fully capture the peak in labeled cells even though we were not able to improve fit of these data by extending the model to two sub-populations with different migration kinetics (results not shown). Additional data that includes variability in lymphocyte dynamics in efferent lymph between different animals may be useful to further show the need for more complex models. Another major assumption of our modeling approach is that lymphocytes in circulation enter LNs via HEVs and not via afferent lymph of the tissues. While there is some experimental support for this assumption for lymphocytes migrating in noninflammatory conditions (2), it is clear that during inflammation in the skin, many cells may enter the skin-draining LNs via afferent lymph (19).

At its core, the combination of experimental data and our models allowed us to estimate the time it takes for lymphocytes to migrate from the blood to the efferent lymph of specific LNs and our results suggest that this time is gamma distributed. For lymphocytes migrating to LNs via HEVs, this distribution is likely to be related to lymphocyte residence time in LNs as lymphocytes pass via HEV rather quickly (53). However, if lymphocytes migrate from the blood to efferent lymph by first entering non-lymphoid tissues (e.g., skin), then exiting the tissue into afferent lymphatics, and then passing via the LN—then our estimates of the average residence time of lymphocytes in LNs are upper bound values. It is also possible that the rate at which lymphocytes leave the final, k th sub-compartment in the LN may be different from that for other sub-compartments. However, there was no need to increase model complexity as the model with a constant rate mLB was sufficient to accurately describe the data.

In most of our models we ignored the possibility of cell death. When describing labeled lymphocyte dynamics during short-term (< 90 h) migration experiments with the recirculation model (**Figure 2**) we found the need to have a tissue compartment which acts as a sink and thus may represent a death process (**Table 2**). However, there appears to be an equilibrium reached by recirculating lymphocytes in the blood by 120 h of the experiment (**Figure 3A**) suggesting a limited role of death process in determining overall dynamics of labeled lymphocytes. Still, we performed some additional analyses by adding death rate to all tissue compartments and found that the best fit is found when such death rate is small or non-existent (**Table S6**). It is possible, however, that the early loss of lymphocytes in the blood as was observed in Frost et al. (42) data (**Figure 2A**) may be due cell death. It is important to highlight that high rates of cell death may influence interpretation of the data and estimates of the model parameters so future studies should attempt to quantify total cell numbers in as many tissues as possible.

Even with all limitations in the data and assumptions of the models, we provided a quantitative framework to analyze data from LN cannulation experiments in sheep. The models, developed in the paper, may need to be tailored to explain kinetics of lymphocyte recirculation in specific experiments. As illustrated in this work, greater insights into mechanisms regulating lymphocyte migration in large animals such as sheep and humans may thus be obtained by combining the use of quantitative experiments and mathematical modeling.

#### DATA AVAILABILITY

All datasets analyzed for this study are included in the **Supplementary Files**.

# AUTHOR CONTRIBUTIONS

The study was originally designed by VG. Data were digitized from cited publications by MM. All major analyses were performed by MM. The paper was written by MM and VG.

# ACKNOWLEDGMENTS

We would like to thank the immunology community for discussion over this research, especially Michio Tomura, Gudrun Debes, and David Masopust. Two reviewers gave constructive suggestions on the previous version of the paper. This work was in part supported by the NIH grant (R01 GM118553) to VG. This manuscript has been released as a Pre-Print on BioRxiv at https:// doi.org/10.1101/513176 (61). Partial funding for open access to this research was provided by University of Tennessee's Open Publishing Support Fund.

#### SUPPLEMENTARY MATERIAL

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fimmu. 2019.01492/full#supplementary-material

Data Sheet 2 | Three datasets analyzed in the paper.

# REFERENCES


**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2019 McDaniel and Ganusov. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# Principles of Effective and Robust Innate Immune Response to Viral Infections: A Multiplex Network Analysis

Yufan Huang<sup>1</sup> , Huaiyu Dai <sup>1</sup> \* and Ruian Ke2,3 \*

*<sup>1</sup> Department of Electrical and Computer Engineering, North Carolina State University, Raleigh, NC, United States, <sup>2</sup> Department of Mathematics, North Carolina State University, Raleigh, NC, United States, <sup>3</sup> T-6, Theoretical Biology and Biophysics, Los Alamos National Laboratory, Los Alamos, NM, United States*

The human innate immune response, particularly the type-I interferon (IFN) response, is highly robust and effective first line of defense against virus invasion. IFN molecules are produced and secreted from infected cells upon virus infection and recognition. They then act as signaling/communication molecules to activate an antiviral response in neighboring cells so that those cells become refractory to infection. Previous experimental studies have identified the detailed molecular mechanisms for the IFN signaling and response. However, the principles underlying how host cells use IFN to communicate with each other to collectively and robustly halt an infection is not understood. Here we take a multiplex network modeling approach to provide a theoretical framework to identify key factors that determine the effectiveness of the IFN response against virus infection of a host. In this approach, we consider the virus spread among host cells and the interferon signaling to protect host cells as a competition process on a two-layer multiplex network. We focused on two types of network topology, i.e., the Erdos-Rényi (ER) network and the ˝ Geometric Random (GR) network, which represent the scenarios when infection of cells is mostly well mixed (e.g., in the blood) and when infection is spatially segregated (e.g., in tissues), respectively. We show that in general, the IFN response works effectively to stop viral infection when virus infection spreads spatially (a most likely scenario for initial virus infection of a host at the peripheral tissue). Importantly, we show that the effectiveness of the IFN response is robust against large variations in the distance of IFN diffusion as long as IFNs diffuse faster than viruses and they can effectively induce antiviral responses in susceptible host cells. This suggests that the effectiveness of the IFN response is insensitive to the specific arrangement of host cells in peripheral tissues. Thus, our work provides a quantitative explanation of why the IFN response can serve an effective and robust response in different tissue types to a wide range of viral infections of a host.

Keywords: immune response, interferon, viral infection, mathematical modeling, multiplex network

# INTRODUCTION

Virus infections and the resulting diseases are major challenges that our society faces today (1). One important determinant of the outcome of an infection is the innate immune response, particularly the type-I interferon (IFN) response ("the IFN response" for short). The IFN response is a highly optimized and general response that provides a critical first line of defense against a wide variety

*Institute of Numerical Mathematics*

Specialty section: *This article was submitted to Viral Immunology, a section of the journal Frontiers in Immunology*

*United States Food and Drug Administration, United States* \*Correspondence: *Ruian Ke rke@lanl.gov Huaiyu Dai hdai@ncsu.edu*

Received: *30 November 2018* Accepted: *09 July 2019* Published: *24 July 2019*

#### Citation:

Edited by: *Gennady Bocharov,*

*(RAS), Russia* Reviewed by: *Nicholas Funderburg, The Ohio State University,*

> *United States Aikaterini Alexaki,*

*Huang Y, Dai H and Ke R (2019) Principles of Effective and Robust Innate Immune Response to Viral Infections: A Multiplex Network Analysis. Front. Immunol. 10:1736. doi: 10.3389/fimmu.2019.01736*

**180**

of virus infection (2). Failure to mount an effective IFN response against virus leads to systematic infection, while excessive IFN production leads to pathogenicity, severe symptoms or even fatality (2–4). It has been shown that the ability to evade host IFN response is an important determinant of viral replication (5–7), transmission (8), and host species range of viral infection (9). Viruses that lack the ability to evade the innate immune response are not able to infect and replicate in a host (7, 8). This demonstrates that the IFN response plays a crucial role in protecting hosts from virus invasion.

IFN molecules belong to a group of signaling proteins, known as cytokines, used by the immune system for cellto-cell communication and induction of protective response. Upon infection, detection of viral RNA/DNA in the host cell triggers a signaling cascade and gene regulation, resulting in the production of IFNs (10). These IFNs then exit the infected cell and act as signaling molecules to bind to surface receptors located on the membranes of host cells (a process termed the IFN signaling), leading to induction of antiviral genes and thus an antiviral state in those cells (11). If an IFN molecule reaches an uninfected cell, i.e., paracrine signaling, this anti-viral state renders the cell refractory to viral infection. If an IFN molecule binds to the receptor of the infected cell that produces it, i.e., autocrine signaling, it inhibits viral replication and decreases the quantity of viral progeny being shed from that cell (6). Although the molecular mechanisms of the IFN response in individual cells have been well characterized (12), the collective dynamics of the host cell response arising from communications through IFN signaling and how the IFN response can effectively and robustly stop or suppress viral infections especially during the initial period of viral exposure in different peripheral tissues and different types of host cells are not understood.

To address these questions, we take a mathematical modeling approach using multiplex networks. Previous modeling works on virus dynamics and the IFN response focused on interpreting in vitro experiments and in vivo systematic infection dynamics (6, 13–17). For example, several elegant studies combining both single-cell experiments and mathematical modeling showed the importance of the timing of the IFN response in determining the outcome of an infection of a population of cells (6) and the importance of the IFN signaling in regulating the population response despite stochasticity in the single-cell level IFN response (16, 17). Two modeling works incorporated the IFN response into within-host viral dynamic models and showed that the IFN response can reduce the peak viral load during an influenza infection and explain the viral load plateau observed after peak viremia (13, 15). In this work, we introduce a multiplex network approach to understand virus invasion of a host and the immediate IFN response. In this framework, we assume in the multiplex network that virus and IFN molecules mediate contacts between cells through the infection layer and the protection layer, respectively. By considering different types of network topologies, i.e., reflecting host cell contact patterns, we show how the IFN response can effectively and robustly respond to virus infection especially in the initial site of viral exposure/infection where host cells are likely arranged spatially in the peripheral tissue.

#### METHODS

#### The Multiplex Network Model Framework

In general, the multiplex network is modeled by a family of graphs n G<sup>m</sup> , (Vm, Em) oM m=1 where all graphs share the same set of nodes i.e., V<sup>1</sup> = V<sup>2</sup> = ... = V = [n]. In our network models, we consider two layers of networks, i.e., the infection and the protection layers, and four types of cells, i.e., susceptible/target cells (S), infected cells (I), protected cells (P), and recovered/dead cells (R). The two layers share nodes (representing host cells) in the network; however, the two layers may have different edges that represent the infection or the protection of susceptible cells in the infection layer and the protection layer, respectively. The nodes have average degrees of k<sup>I</sup> and k<sup>F</sup> in the infection and the protection layer, respectively. Viruses and IFN molecules are not explicitly considered; instead, we assume that the contacts between infected cells and susceptible cells are mediated by viruses and IFNs through two layers in the network (**Figure 1**).

In this work, we consider two types of graphs for the two layers of a network. The first type is a well-mixed intralayer topology modeled by the Erdös-Rényi (ER) graph G(n, p) (18) in which a link exists between any two nodes with a uniform probability p. Then, the average degree of the ER graph is k = (n − 1) p ≈ np. The second type is a spatial graph modeled by the 2-dimensional Geometric Random (GR) graph G(n,r) (19), in which a link exists between two nodes only when their 2-dimensional Euclidean distance is smaller than the prefixed range r, which we term the radius of diffusion. The radii of diffusion are r<sup>I</sup> and r<sup>F</sup> in the infection layer and the protection layer, respectively. The average degrees in the infection layer and the protection layer are calculated as k<sup>I</sup> = (n − 1) πr<sup>I</sup> <sup>2</sup> ≈ nπr<sup>I</sup> 2 and k<sup>F</sup> = (n − 1) πr<sup>F</sup> <sup>2</sup> ≈ nπr<sup>F</sup> 2 , respectively. Simulation procedures of the network models are described in Huang et al. (20).

The following ordinary differential equations (ODEs) describe the mean field model of the infection and protection processes we consider in the networks:

$$\begin{aligned} \frac{dS}{dt} &= -\beta SI - \,\varphi SI\\ \frac{dI}{dt} &= \beta SI - \,\gamma I\\ \frac{dP}{dt} &= \varphi SI\\ \frac{dR}{dt} &= \,\gamma I \end{aligned}$$

In this model, susceptible cells (S) are infected at rate β or become protected at rate ϕ. Since we mainly focus on the initial infection dynamics, generation and death of susceptible cells are ignored. Infected cells (I) die at per capita rate γ to become cells in the R class. We assume that protected cells remain protected for simplicity, although anti-viral response in protected cells can be switched off over time (2, 15). Again, since we are mostly interested in the initial infection dynamics, ignoring the transition from protected cells to susceptible cells is a reasonable assumption. Here, we mainly focus on how the

topology of a network impacts on the effectiveness of IFN to halt an infection through protecting susceptible cells, i.e., the paracrine IFN signaling. The impact of IFN on already-infected cells can be considered by extending the model with another infected class, i.e., infected cells that are at an antiviral state, and assume that infected cells in this class have a reduced viral production. However, this makes many analytical derivations impossible. Note that as a common practice in the network modeling approach, we rescale the four state variables against the total population size, such that S + I + P + R = 1. Then, S, I, P, and R in our network models represent the fraction of cells that are in their corresponding states.

#### Analytical Derivations

To evaluate the impact of IFN on the infection threshold in the mean field/ODE model, we first define R<sup>I</sup> as the reproductive number of the virus in the absence of IFN. We also refer this quantity as a measure of virus infectivity. It can be calculated as:

$$R\_I = \frac{\beta}{\nu}$$

We then define a quantity R<sup>F</sup> for IFN similar as R<sup>I</sup> for virus as:

$$R\_F = \frac{\varphi}{\nu}$$

Then, R<sup>F</sup> is the average number of cells that an infected cell protects over its life time. Note that, protected cells do not further generate IFN and thus IFN signaling does not propagate in the absence of further infection. Thus, R<sup>F</sup> is a single step measure of the effectiveness of the IFN signaling for individual cell response, and we refer this parameter as the individual-cell effectiveness of the IFN signaling.

The infection threshold β<sup>c</sup> of the ODE model can be derived as: β<sup>c</sup> = γ , i.e., as long as the infectivity parameter β is greater than the rate of recovery γ , the virus can cause sustained infection. Note the expression is independent of parameter ϕ, i.e., the parameter for the impact of IFN on protecting target cells. The infection threshold β<sup>c</sup> of the network with two ER graphs and how it depends on the similarity between the two layers are derived previously in Huang et al. (20).

## Heterogeneity in the Susceptibility of Host Cells

To evaluate the impact of heterogeneity in the susceptibility of host cells, e.g., due to heterogenous receptor expressions, we assign each cell with a specific rate of infection, β, and this rate is drawn from a gamma distribution:

$$\mathcal{P}(\beta) = \frac{1}{\Gamma\left(k\right)\theta^k} \beta^{k-1} e^{-\frac{\beta}{\theta}}.$$

where k and θ are the shape and scale parameters, respectively, and Ŵ is the gamma function. In this way, the extent of heterogeneity is determined by the shape parameter k. The smaller k, the more heterogenous.

We follow the derivations in Huang et al. (20) to calculate the values of R<sup>I</sup> for the simulations with heterogenous infection rate. First, we calculate the probability that a susceptible cell become infected when it is connected to an infected cell in the infection layer. Because infected cells die after a fixed period of time τ = 1 day in the simulation, this probability can be calculated as ζ = 1 − e <sup>−</sup>βτ = 1 − e β , whose mean, ζ , is given by:

$$\begin{aligned} \overline{\xi} &= \int\_0^\infty \xi P(\beta) d\beta = \int\_0^\infty \left( 1 - e^{-\beta} \right) \frac{1}{\Gamma\left( k \right) \theta^k} \theta^{k-1} e^{-\frac{\theta}{\theta}} d\theta \\ &= 1 - \frac{1}{\left( 1 + \theta \right)^k} . \end{aligned}$$

Then, the value of R<sup>I</sup> is the product of ζ and the average degree of the infection layer: R<sup>I</sup> = kIζ .

#### RESULTS

#### A Well-Mixed Model and a Network Model With Two Random (ER) Graphs

We first focused on multiplex networks where both layers are ER graphs as baseline models. In this framework, contacts between

FIGURE 2 | The effectiveness of the IFN response under different assumptions and topologies of the network. In general, protection of susceptible cells by IFN signaling, i.e., the IFN response considered in this study, works most effectively when viruses spread in a spatial manner (i.e., in the GR network). (A) The final sizes (fractions) of cells that are infected (and ultimately dead) at the end of the infection, *R*(∞), in the homogenous mixing model (blue line; partly overlaid by the red line) and the network models (in red, yellow, and black). Results are average of 1,000 simulations. (B) The final sizes (fractions) of cells that are protected at the end of infection, *P*(∞), in the homogenous mixing model (blue line) and the network models (in red, yellow and black). The individual-cell effectiveness of the IFN signaling, *RF* is set to seven. The network model with two ER graphs (results of the model with two independent layers are in red; results of the model with two identical layers are in yellow), and the network model with two GR networks (in black). Lines denote analytical results derived in Huang et al. (20), whereas dots denote simulation results.

host cells (through viruses and IFNs) are random and there is no spatial structure in the contacts. These assumptions are reasonable for infections where cells move and contact with other cells (through viruses and IFNs) roughly randomly, for example, HIV infection in the blood. In our multiplex network model, the topologies of the graphs in the two layers, i.e., the contact structure between cells, can be explicitly modeled, in contrast to well-mixed models or single-layer network models. This allows us investigate how the IFN signaling through the protection layer competes with virus infection through the infection layer at the level of individual infected cells.

We considered two scenarios of the relationship between the two layers, i.e., the topologies of the two layers are independent of or identical to each other. We simulated the model and analyzed how the fractions of infected and then dead cells (a measure of the size of total infected cells) and protected cells (R(∞) and P(∞), respectively) changes with the infectivity of the virus (measured as R<sup>I</sup> ; see Method). When the two layers are independent of each other, the subset of target cells that an IFN molecule can reach is independent from the subset that a virus (produced from the same cell as the IFN molecule) reaches, and thus there is no direct competition for target cells between viruses and IFNs at the individual infected cell level. We found that the predicted infection threshold value for virus infectivity, β<sup>c</sup> , i.e., the threshold value that viruses can cause sustained infection in a host, is independent of the parameter that governs the IFN protection of target cells, i.e., ϕ. On the other hand, when the two layers are identical (i.e., a more biologically relevant assumption), IFN molecules will reach to the same subset of target cells as the viruses produced from the same infected cell. In this case, the infection threshold becomes much larger than the threshold in the absence of IFN response, suggesting that IFN can prevent virus infection (the green line in **Figure 2A**). As we showed previously, IFNs inhibit viral spread effectively when IFNs reach the same subset of cells as viruses and thus reduce the number of susceptible cells that an infected cell can infect (20). Interestingly, these conclusions are similar to those in a previous network modeling work analyzing the impact of the spread of epidemic awareness on the transmission of infectious diseases (21). Further, we found that when viruses can cause infection, i.e., β > β<sup>c</sup> , there is a sharp increase in the number of protected cells (**Figure 2B**). This increase in protected cells prevents susceptible cells from being infected and thus the proportion of infected cells increases slowly with increases in R<sup>I</sup> (**Figure 2A**).

### A Network Model With Two Spatial (GR) Graphs—IFN Can Effectively Halt Infection When Infection Is Spatial

For most viruses, initial viral infection events at the site of viral entry are expected to occur at the peripheral tissue where host cells are spatially structured. Spatial infection spread has also been shown to be a prominent infection mode of many viruses, especially for virus infections in the tissue (22–24). To evaluate the effectiveness of the IFN response in tissue, we constructed a multiplex network where the two layers are assumed to be GR graphs (see Methods). In the GR graph, we define nodes on a twodimensional space and a maximal distance (i.e., radius of virus or IFN diffusion) such that an edge exists between two nodes

cells by the IFN signaling leads to an outer layer of protected cells that contain the infection at the local area of the initial infected nodes, i.e., the initial site of viral entry.

only if the distance between two nodes is shorter than the radius of diffusion.

We simulated the model and found that strikingly, over a large parameter range of virus infectivity (measured by R-I), IFN protection of susceptible cells works much more effectively in the GR network than in the ER network. As shown in **Figure 2A**, the IFN response halts infection such that the total number of infected cells are kept at very low levels for a much wider range of virus infectivity. IFN protection also leads to a much lower total number of protected cells in the GR network than in the ER network (**Figure 2B**). This conclusion holds true as long as the individual-cell effectiveness of the IFN signaling (measured as RF; defined in Methods) is sufficiently high, e.g., when R<sup>F</sup> > R<sup>I</sup> (**Figure S1**).

To understand why IFN protection of target cells works well in the GR network, we show two simulation realizations using networks assuming two ER graphs and two GR graphs in **Figures 3A,B**, respectively. In the network with two ER graphs (**Figure 3A**), connections/links between nodes are random. As a result, infection can propagate until most cells are either protected or infected/recovered. In contrast, cells in the GR network are connected only to neighboring cells in space. If the IFN response is strong enough, the IFN signaling can build up an outer layer of protected cells which effectively contains the infection near the site of initial infection. As a result, most of the cells (outside of the area of infection) stay susceptible without being infected (**Figure 3B**). Overall, the results suggest that the IFN response, i.e., the IFN signaling to protect susceptible cells, works extremely effectively when the virus spread spatially, a likely scenario for infections in tissues.

#### Robustness of the IFN Response to Virus Infection in Tissue

The IFN response is a general response strategy employed by different types of host cells to prevent or suppress infections of a variety of viruses. This suggests that the IFN response works efficiently and robustly in a wide range of host cell or tissue environments. Here, we evaluated the robustness of the IFN response against variations in two assumptions in our model to understand how this collective host cell response work effectively despite heterogenous host environments.

We first focused on one particular parameter that relates to the host tissue environment in our model: the diffusion coefficient of viruses and IFNs, i.e., the radius of the cell-cell edges (contacts) in the GR network. Due to differences in the viscosity of the fluid in the tissue and the layout of target cells, the ratio of the IFN diffusion over the virus diffusion and thus the ratio of the numbers of target cells they reach may differ in different tissue compartments. Below, we evaluate how the effectiveness of the overall IFN response changes with changes in these ratios. In the analysis, we varied the radius of the IFN diffusion in the protection layer (rF; defined in Methods), and assumed that the individual-cell effectiveness of the IFN signaling, RF, is constant. In this way, when the radius of IFN diffusion increases, the average degree of nodes in the protection layer (kF) increases; however, the protection rate per contact decreases. We explored how the final fraction of infection R(∞) changes with the ratio of the radius of IFN diffusion over the radius of virus diffusion, rF/r<sup>I</sup> . We found that there exists an optimal ratio, such that the total fraction of infection is minimized (**Figure S2**). Although the exact optimal ratio is parameter dependent, generally it occurs when the ratio is >1, i.e., the radius of IFN diffusion is similar or larger than the radius of virus infection. In general, when R<sup>F</sup> > R<sup>I</sup> , there exists a wide range of ratios of IFN diffusion over virus diffusion that the IFN can suppress the virus infection below a very small fraction (blue areas in **Figure 4**). This suggests that as long as the IFN response is effective and diffuses similarly or faster than viruses, the IFN response is in general robust against variations in the IFN diffusion.

response effectively suppresses virus spread (low *R*(∞) values; blue areas in the plots) as long as *RF* > *RI*

FIGURE 5 | Heterogeneity in host cell susceptibility reduces the total size of infected cells. (A) The final sizes (fractions) of infected cells at the end of infection [Average *R*(∞)], in a model using two identical layers of ER graphs. Results are average of 1,000 simulations. Colored lines show simulations assuming different levels of heterogeneities in host-cell susceptibility. The heterogeneity is characterized by a gamma distribution with the shape parameter *k* and scale parameters θ. Note, the lower the value of *k*, the more heterogeneous the host cell susceptibility. (B) The corresponding final sizes (fractions) of protected cells at the end of infection, *P*(∞), in simulations shown in (A). (C,D) Similar plots as in (A,B), respectively, except that the model assumes GR graphs in the network. The individual-cell effectiveness of the IFN signaling, *RF* is set to seven. The average degree of the networks is set to 40, such that the value of *RI* reaches 10 in simulations assuming *k* = 0.1 and θ = 20.

In the analysis above, we assumed in the model that the host cells are a homogenous population of cells; whereas in reality, viruses typically infect a wide range of host cells and the host cells likely exhibit widely different levels of susceptibility to infection, e.g., as a result of heterogenous expression of receptors for viral infection (25–28). To evaluate the consequences of heterogenous host cell susceptibility to infection, we modified our model simulation to assume that each cell has a susceptibility drawn from a gamma distribution (instead of being the same), while keeping the rate of protection by IFNs, ϕ, constant (see Methods). The simulation results using ER and GR networks show that in general, the more heterogenous the host cell susceptibility (i.e., lower k values), the lower the final fraction of infection R(∞) (**Figures 5A,C**). This is because when host cell susceptibility is extremely heterogenous (e.g., the shape parameter k = 0.1 in the gamma distribution in **Figure 5**), the infection is driven a small fraction of highly susceptible cells. For the remaining large fraction of cells, they are much less likely to be infected than protected. Overall, this leads to a small fraction of cells being infected, yet the fraction of protected cells P(∞) remains similar across simulations (**Figure 5**). Therefore, the IFN response is effective to suppress viral infection when the susceptibility of host cells is heterogenous.

.

# DISCUSSIONS AND CONCLUSIONS

Here, we use a multiplex network approach to show how the collective host cell IFN response can effectively and robustly halt/suppress virus spread especially when viruses spread spatially. For a wide variety of viral infections, including influenza infection (22), HIV infection (29), mosquito borne viral infection, such as dengue (30) and zika (31), the site of entry is at the epithelium where target cells for infection are spatially arranged. The spread of viruses is thus expected to be a spatial process, i.e., infected cells only further infect a finite number of neighboring cells. We found that in this case, IFNs diffuses and signal to susceptible cells further away from infection, which builds up an outer layer of protected cells to contain infection locally. We also found that the collective IFN response is highly effective and robust against variations in parameter values that represent heterogenous host environments. This we argue is a property that allows the IFN response to be a general response employed by different types of host cells in peripheral tissues to respond to a wide variety of viruses to prevent viral establishment and invasion of a host at the initial site of the infection.

During systematic infection, viral infection process can be spatial or non-spatial. For viruses like HIV, infection in the blood and in the lymph nodes occurs among host cells that move around and contact each other randomly, the infection process may be better modeled using a random (ER) network. We show that in this case, the critical parameter that determines the effectiveness of IFN protection of target cells is the similarity between the infection layer and the protection layer (20). The higher the similarity, the more effective the IFN response. The IFN response can halt/suppress infection by directly competing with viruses at each individual cell level such that the number of target cells that each infected cell can infect is reduced. For many other viruses, e.g., influenza virus (22, 24) and HCV (23, 32), spatial viral spread may be prevalent throughout the infection course, if not the only infection mode.

The findings of our study, especially that IFN response is effective when infection spreads in a spatial manner, are consistent with a wide range of in vivo and in vitro observations. For example, imaging of liver biopsy from patients chronically infected with hepatitis C virus (HCV) showed that HCV infected cells form clusters and that IFN stimulated genes are highly expressed in infected cells as well as the surrounding susceptible cells. This strongly suggests effective IFN response to constrain cell-to-cell spatial spread in the liver (23). In another study (33), to understand the evolutionary trade-off of viral suppression of the IFN response, Domingo-Calap et al. compared the spread of a wild-type strain of the vesicular stomatitis virus to a mutant strain that stimulates stronger IFN response than the wild-type. Realtime fluorescence microscopy showed that in contrast to a faster and homogenous spread of the wild-type virus in monolayer host cells, the mutant viruses spread slower and infected cells form clusters. This again suggests that the IFN response triggered by the mutant acts to constrain infection. Interestingly, when the monolayer spatial structure of host cells is disrupted, the mutant grew faster than the wild-type in well-mixed culture. This is consistent with the results we show in this study that spatial structure is a key determinant of the effectiveness of the IFN response. Overall, these experimental observations support our model predictions, and thus, our model serves a useful tool to understand the quantitative principles of the IFN response. These understandings may lead to development of effective therapies/vaccines to prevent virus transmission and infection (5–8).

Overall, our results suggest that considering the topology of the spreading process is critical to the understanding and prediction of the impact of collective IFN response arising from

#### REFERENCES

1. Fauci AS, Morens DM. The perpetual challenge of infectious diseases. N Engl J Med. (2012) 366:454–61. doi: 10.1056/NEJMra 1108296

host cells. Therefore, experimental studies that examine the contact structure and topology for an infection process would help to parameterize the model to make precise predictions. Here our work considered two distinct scenarios of the topology of the spreading process, i.e., the random (ER) network and the spatial (GR) network. An actual infection in vivo may involve both spatial and non-spatial contacts. For example, it has been shown HCV mostly spread to neighboring cells, forming clusters of infected hepatocytes in the liver; while it is also able to have a long-range dispersal to hepatocytes through blood flow (23, 32). Similar patterns of foci of infection are also observed for influenza virus (22). Further work is warranted to consider network structures that incorporate both spatial spread and random spread, and evaluate the effectiveness of IFN response in those settings.

Given that the IFN response is a highly optimized and highly effective general response against viruses (2), we argue that the strategies employed by IFN and the results derived from this work could shed light on or lead to solutions to problems in other disciplines. For example, network models are frequently used in the modeling of epidemics to understand how infection dynamics or control strategies are impacted by network topologies (34– 36). Furthermore, we speculate that the understanding of the population IFN response may lead to bio-inspired strategies for controlling rumor spreading in social networks or cyberattacks in computer networks.

#### AUTHOR CONTRIBUTIONS

YH and RK derived and analyzed the theoretical models. YH conducted the computer simulations. All authors analyzed the data, wrote and edited the manuscript and contributed to the development of ideas.

#### FUNDING

This work is funded by the DARPA INTERCEPT program (Contract No. W911NF-17-2-0034) and the Army Research Office under Grant W911NF-17-1-0087.

#### ACKNOWLEDGMENTS

We thank Barbara Sherry for extensive discussions and inputs throughout the project.

#### SUPPLEMENTARY MATERIAL

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fimmu. 2019.01736/full#supplementary-material


pathogenic influenza virus. Proc Natl Acad Sci USA. (2009) 106:3455–60. doi: 10.1073/pnas.0813234106


**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2019 Huang, Dai and Ke. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# From Discrete to Continuous Modeling of Lymphocyte Development and Plasticity in Chronic Diseases

Jennifer Enciso1,2,3, Rosana Pelayo<sup>1</sup> and Carlos Villarreal 3,4 \*

<sup>1</sup> Centro de Investigación Biomédica de Oriente, Instituto Mexicano del Seguro Social, Mexico City, Mexico, <sup>2</sup> Programa de Doctorado en Ciencias Biomédicas, Universidad Nacional Autónoma de México, Mexico City, Mexico, <sup>3</sup> Centro de Ciencias de la Complejidad, Universidad Nacional Autónoma de México, Mexico City, Mexico, <sup>4</sup> Departamento de Física Cuántica y Fotónica, Instituto de Física, Universidad Nacional Autónoma de México, Mexico City, Mexico

#### Edited by:

Gennady Bocharov, Institute of Numerical Mathematics (RAS), Russia

#### Reviewed by:

Y-h. Taguchi, Chuo University, Japan Sylvain Cussat-Blanc, Université de Toulouse, France

> \*Correspondence: Carlos Villarreal carlos@fisica.unam.mx

#### Specialty section:

This article was submitted to Molecular Innate Immunity, a section of the journal Frontiers in Immunology

Received: 12 December 2018 Accepted: 30 July 2019 Published: 20 August 2019

#### Citation:

Enciso J, Pelayo R and Villarreal C (2019) From Discrete to Continuous Modeling of Lymphocyte Development and Plasticity in Chronic Diseases. Front. Immunol. 10:1927. doi: 10.3389/fimmu.2019.01927 The molecular events leading to differentiation, development, and plasticity of lymphoid cells have been subject of intense research due to their key roles in multiple pathologies, such as lymphoproliferative disorders, tumor growth maintenance and chronic diseases. The emergent roles of lymphoid cells and the use of high-throughput technologies have led to an extensive accumulation of experimental data allowing the reconstruction of gene regulatory networks (GRN) by integrating biochemical signals provided by the microenvironment with transcriptional modules of lineage-specific genes. Computational modeling of GRN has been useful for the identification of molecular switches involved in lymphoid specification, prediction of microenvironment-dependent cell plasticity, and analyses of signaling events occurring downstream the activation of antigen recognition receptors. Among most common modeling strategies to analyze the dynamical behavior of GRN, discrete dynamic models are widely used for their capacity to capture molecular interactions when a limited knowledge of kinetic parameters is present. However, they are less powerful when modeling complex systems sensitive to biochemical gradients. To compensate it, discrete models may be transformed into regulatory networks that includes state variables and parameters varying within a continuous range. This approach is based on a system of differential equations dynamics with regulatory interactions described by fuzzy logic propositions. Here, we discuss the applicability of this method on modeling of development and plasticity processes of adaptive lymphocytes, and its potential implications in the study of pathological landscapes associated to chronic diseases.

Keywords: lymphocytes, chronic diseases, boolean, fuzzy logic, computational modeling

# 1. INTRODUCTION

The extensive accumulation of data from short and large-scale experiments involving a wide spectrum of biological functions of B and T lymphocytes in both, normal and pathological scenarios, has inspired an intensive research on molecular events leading to their early development, plasticity and emergency differentiation. As a result, the construction of regulatory networks has become a resourceful tool for the systemslevel analyses of cell fate decisions through interconnection of molecular elements, such as biochemical signals provided by the microenvironment (e.g., cytokines, growth factors, transmembrane ligands, antigens, etc.) and transcriptional modules underlying the regulation of lineage-specific gene expression. Getting insights into the dynamical behavior of regulatory networks in biology requires simulation as continuous or discrete models (1). Discrete modeling, represented by Boolean and multi-valued network models, has been useful in differentiation processes of adaptive B and T lymphocytes (2– 8), for molecular switching in cellular specification (9), for the prediction of microenvironment-dependent cell plasticity (6, 10), and for the analyses of signaling events occurring downstream activation of antigen recognition receptors (11, 12). Moreover, Boolean algebra has been used in cytometry to create combined gates for the identification and selection of cellular subsets and lymphoid phenotyping (13). Nevertheless, the utility of discrete models is limited as they cannot predict outcomes from quantitative biological experiments when working on phenomena sensitive to graded expression of transcription factors or biochemical gradients. This is the case of most diseases where lymphocytes are involved and nondiscrete fluctuations in the microenvironment may influence cell differentiation and plasticity, affecting immune responses at the progression of chronic pathologies, such as lymphoproliferative disorders, tumor growth, diabetes, cardiovascular, and chronic respiratory diseases, among others. Discrete models might be then transformed into differential equations to allow a dynamical analyses of regulatory networks, as transformed continuous models, with potential implications in lymphoid cell- associated pathologies (14–17).

Here we propose the fuzzy logic transformation of a discrete model into a continuous model to compensate their disadvantages and to simulate biological systems with a well-known network architecture strongly influenced by concentration-dependent cues (**Table 1**).

# 2. DISCRETE MODELING OF LYMPHOID DIFFERENTIATION LANDSCAPE

#### 2.1. Boolean Interpretation of Molecular Data

To deeply understand the gene regulatory processes involved in cellular development, C. H. Waddington introduced in 1957 the metaphoric concept of epigenetic landscape (18). He proposed a unique perspective of cellular development as a ball rolling down within a landscape formed by peaks and valleys. Following its trajectory, the ball may finally fall into a valley, representing its final position that defines a steady-state -and a cellular fate-, also known as attractor. Waddington's epigenetic landscape was formalized, among others, by S. A. Kauffman, who studied the behavior of large networks of randomly interconnected binary "genes" with a dichotomous (on-off) behavior, establishing the principles of Boolean modeling (19). The assumption of a discrete transcriptional regulation was further investigated in Drosophila embryogenesis, showing that the gradient of Bicoid morphogen resulted from averaging binary states of transcriptional activity, active or inactive, at individual nuclei level (20).

The general system's behavior and the number of attractors of a Boolean or multi-valued regulatory network depends on topological characteristics, such as the number of components and the degree of interconnectivity among them. It is now recognized that biological networks are scale-free systems, which means that the nodes have a high diversity of number of edges, including few elements with many links and many elements with few links (21, 22). Scale-freeness provides, among other attributes: network robustness, better information spreading performance, and the property that the number of attractors is almost independent from the number of nodes (23, 24).

Mathematical modeling based on Boolean regulatory networks (BRN) provides meaningful qualitative information on the basic topology of relations that determine alternative cell fates and may be used for the analysis of biological circuits without requiring explicit values of the network parameters. In this type of approach, the network nodes represent genes, transcription factors, proteins mediating signaling cascades, RNA, environmental factors, etc., and links representing positive or negative regulation between pairs of nodes. The state variable of each node takes a discrete value of 0 (inhibited, or inactive) or 1 (expressed or active) (1). The state of each node at time t + 1 is specified by a dynamic mapping that depends on the state of its regulators at a previous time t:

$$q\_k(t+1) = F\_k\left(q\_1(t), \ldots, q\_n(t)\right) \tag{1}$$

where F<sup>k</sup> is a discrete function representing a logical proposition, also known as Boolean rules, constituted by elementary terms related by the logical connectives: AND (∧), OR (∨), and NOT (¬). Logical propositions satisfy Boolean's axiomatics, which complies associativity, commutativity, distributivity, absorptivity, and identity. The discrete nature of the truth values involved in Boolean logic propositions implies that this approximation is not always enough to investigate the enormous variability inherent to biological processes.

The dynamics induced by the Boolean mapping is completely determined once a set of initial expression values of the network components is specified. From a given initial set, the network nodes iteratively update their value based on the Boolean transfer rules until eventually reaching a steady-state determined by condition q<sup>k</sup> (t + 1) = q<sup>k</sup> (t). This latter condition specifies a fixed-point attractor. Then, the dynamics of a model is evaluated by tracking the trajectories from all the possible initial configurations in the states space toward the attractors. The size of the states space of a model is given by = 2<sup>n</sup> where n is the number of nodes in the network. Alternatively, a cyclic attractor associated to the condition q<sup>k</sup> (t + N) = q<sup>k</sup> (t) may also arise after the simulation of some regulatory networks, where the integer number N signals the period of the attractor. Cyclic attractors are generally interpreted as oscillatory behaviors and are sustained by at least one negative feedback circuit in the network topology, which involves an odd number of inhibitory interactions (25). This type of attractors can be directly associated



to biological events, for example, in models predicting cell cycle oscillations (26–28) or, sometimes they can be interpreted as intermediate or oscillatory activations in multi-valued and Boolean differentiation models, as has been reported with T cell attractors (7, 29). Each fixed-point and cyclic attractor is reached from a number ω of different initial conditions. The parameter ω denotes the size of the attraction basin which may be visualized as a ratio of areas in the epigenetic landscape. Consequently, the probability that a steady state is expressed is given by p = ω/.

To briefly exemplify how a Boolean model is constructed we used the information compilated by Bhattacharya et al. (30) of the transcriptional core orchestrating the terminal differentiation of B cells into antibody-secreting plasma cells upon antigenic stimulation. The transcription factors to be considered were Pax-5, Bcl-6, and Blimp-1. Construction of the gene regulatory network and the Boolean transfer rules are based on evidence showing the existence of a mutual repression by Bcl-6 and Blimp-1, as do Blimp-1 and Pax-5, establishing a system with two double-negative feedback loops. Pax-5 and Bcl-6 are two transcriptional factors of high expression in B cells, downregulated by Blimp-1 after its AP-1 mediated activation. In turn, AP-1 is phosphorylated downstream B cell stimulation with lipopolysaccharides. Beside the direct inhibition of Blimp-1, Bcl-6 can also act as a passive repressor through its binding to AP-1, blocking its transcriptional activity (**Figure 1A**). Such information is sufficient to predict two fixed-point attractors interpretable as B-cell and plasma cell configurations. The presence of at least one positive loop containing an even number of inhibitory regulations is necessary for the generation of multiple steady states (25). This type of models has been useful to merge independently published data from different molecular circuits involved in cellular specification, to probe how these circuits orchestrates differentiation, and to generate new testable hypothesis on missing interactions or cellular transitions.

# 2.2. Genetic Regulatory Networks Underlying Lymphoid Specification

As of the discovery of HSCs by Ernest A. McCulloch and James E. Till in the 1960s, the hematopoietic system has served as the most recurrent biological model for the study of stem cell biology and differentiation. For many years, the differentiation process was represented as a hierarchical dichotomic model of strict myeloid/lymphoid branching. However, multiple observations mostly based on single cell experiments have challenged this classical view and introduced cell differentiation as a process of continuous transitions directed by two events running in parallel: the gradual commitment through the acquisition of lineagespecific features and the gradual lost of potential to generate cells of a different lineage (31–36) (**Figure 1B**).

stable cellular states, however other cellular phenotypes may be represented as transitory stages. HSC, hematopoietic stem cell; MPP, multipotent progenitor; CMP, common myeloid progenitor; LMPP, lymphoid-primed multipotent progenitor; MEP, megakaryocyte/erythroid progenitor; MegP, unipotent megakaryocyte progenitor; GMP, granulocyte/macrophage progenitor; CLP, common lymphoid progenitor; NK/ILC, natural killer/innate lymphoid cell; ALP, all-lymphoid progenitors; BLP, biased lymphoid progenitors; ETP, early thymic progenitor; DN, double negative; DP, double-positive.

In the metaphorical Waddington's view, the cell type positioned in the "summit" of the hematopoietic epigenetic landscape is the hematopoietic stem cell (HSCs) population, which resides in specialized niches within the bone marrow. Early specification begins upon "ball rolling" from the HSCs to the multipotent progenitor (MPP) attractors, either committing to myeloid or lymphoid lineages by differentiating into common myeloid progenitors (CMPs) or lymphoid-primed multipotent progenitors (LMPPs), respectively (37–39). As more is deciphered on the transcriptional network underlying the lymphoid differentiation, more is discovered about intermediate steps and novel transitional cell subpopulations. It is now wellknown that LMPPs contain a mixture of myeloid and lymphoidrestricted progenitors, including early lymphoid progenitors (ELPs), giving rise to common lymphoid progenitors (CLPs), endowed with the ability of generating all types of adaptive and innate lymphocytes without noticeable myeloid potential, and some categories of dendritic cells (DCs) (40–50). The CLP population bisects into all-lymphoid progenitors (ALPs) and B-cell-biased lymphoid progenitors (BLPs) (44) that predominantly generate T and B lymphocyte precursors, respectively. From the ALP pool, some circulating progenitors reach the thymus and differentiate into early thymic progenitors (ETPs), progress to CD4/CD8 double-negative 2 cells (DN2) and DN3 cells. CD4/CD8 double-positive (DP) cells are then produced before differentiation toward CD4 or CD8 singlepositive (SP) T effector cells (51). B cells reach also a partial maturation in the bone marrow (BM), following a series of sequential differentiation steps from prepro-B, pro-B, early pre-B and pre-B stages, where the rearrangement of immunoglobulin heavy-chain (IgH) genes takes place and results in the expression of the pre-B-cell receptor (pre-BCR). Downstream the pre-BCR activation and the signaling cascade deriving in a clonal expansion and the subsequent cell cycle arrest, a second wave of recombinases Rag1 and Rag2 expression induces the rearrangement of the immunoglobulin light-chain (IgL), marking the transition from pre-B-cell to immature B cells (44, 52, 53). Upon migration to the secondary lymphoid organs, T and

B lymphocytes are exposed to antigens and signals provided by a number of immune cells in the microenvironment.

Even though differentiation transitions are now recognized as continuous processes, commitment to stable phenotypes is dependent on molecular switches that act as lineage-determining steps, what has made the differentiation process a target for its simulation through discrete models. More specifically, hematopoietic differentiation has been approached with discrete models at many levels (**Figure 2**), from the top of the epigenetic landscape hill sloping down to the final stages of mature cells production. The main type of information provided by the construction and simulation of BRN is obtained after the confirmation of the functional integration of the proposed components. This generally occurs validating the attractors and transitions with previous experimental observations. To compare with experimental data, Boolean models are subjected to different types of perturbations including permanent mutations (e.g., gene knock-out or overexpression), or temporal changes in the nodes activation value which can be understood as triggering cues for network state transitions. An example of this type of evaluations is the case of the hematopoietic stem/progenitor (HSPC) network model generated by Bonzanni. The HSPC model contains ten genes expressed in the immature stem cell population besides GATA1, which is expressed in the early progenitor MPP (2). The dynamic simulation of the HSPC network generated two single state attractors, one with an erythroid cell profile, and one with a non-hematopoietic cell profile with all genes turned-off, as well as a periodic attractor composed of 32 interconnected states with oscillatory activation values for four genes (Gata2, Zfpm1, Erg, and Eto2a) compatible with single cell gene expression data from HSPCs (54). The activation state of one or more genes in the states comprising the HSPC complex attractor were modified to compute the dynamic transitions and mapping the developmental route from HSPC toward erythrocyte, granulocyte, monocyte, natural killer (NK), B cell, CD4, or CD8 T cells profiles (55). This type of evaluation provided information about the stability of the HSPC attractor, the type of genes involved in the developmental route considering those that trigger differentiation, and the suggestion that there were missing interactions or components that avoid differentiation reversal.

Furthermore, the myeloid/lymphoid branching has been addressed through the assembly of a GRN integrating 23 nodes that, when computed using a logical multi-valued formalism, produced four stable stages corresponding to CLPs, B-lineage cells, granulocyte-monocyte progenitors (GMPs) and macrophages (9). As previously discussed, even the network assembling may constitute a useful mechanism to propose novel interactions. This was the case with this model, by envisioning three missing regulations: negative regulation of C/EBP(α) transcription by Foxo1, E2A activation by Ikaros, and Gfi1 positive regulation by Pax-5. Moreover, the model was useful to explore molecular mechanisms of transient induction of the transcription factor C/EBP that down-regulates the transcriptional core of B cell specification and promotes an irreversible trans-differentiation toward macrophages. The theoretical findings complemented the results of a previous experimental report where B cells were transdifferentiated into macrophages by the enforced expression of C/EBP and C/EBP, but without a full understanding on the molecular steps leading to the loss of early and late lymphoid markers and acquisition of myeloid-specific genes (56). Predictions from these models and their perturbations might be useful to unravel the pathobiology of diseases where neoplastic cells concomitantly express myeloid and lymphoid markers (57–61). Also, early branching models may help to deepen the research on plasticity-related processes, such as those suggested to be involved on leukemia lineage switching and relapse (62, 63). It has become of particular interest the integration of microenvironmental cues capable of influencing and regulating transcriptional cores, particularly to approach the two-way feedback between cells and their surrounding microenvironment.

# 2.3. Microenvironmental Modulation of Lymphoid Differentiation and Plasticity

The applicability of discrete models seems to be simplistic but their scopes are expanding in parallel with the knowledge on cellular heterogeneity and plasticity. Their flexibility for the analysis of biological systems integrated with multiple types of molecular events makes them a useful tool for evaluation of different microenvironments that consider the modulation of genetic and signaling networks. Molecules, such as integrin, cytokine, or antigen receptors, might be included in computational models as they are involved in maintaining particular hematopoietic compartments, enhancing proliferation, regulating apoptosis or migration, or guiding differentiation to either one phenotype or another. As previously mentioned, some of these processes become discrete cellular decisions with a bi-modal behavior as a result of the combined effect of their connectivity in molecular networks and noise (64–66).

Early logical mathematical approaches for modeling lymphocyte behavior upon antigen exposure preceded the development of networks that connected intracellular events regulating the cellular fates of hematopoietic progenitors and lymphocytes (67, 68). However, as the different subtypes of lymphocytes were discovered, efforts focused on the reconstruction of the GRN underlying the emergence of mature phenotypes in response to variable microenvironmental factors under normal and pathological conditions. The first model of lymphoid differentiation branching using a discrete perspective resulted from the transformation of a previous continuous model based on Hill functions describing the polarization of naive Th cells (Th0) into Th1 or Th2 cells (69). The Boolean version proposed by Mendoza (3) integrated 17 nodes and replaced the transcription factor Gata-3 positive self-feedback loop in Yates' model (70) with a more refined functional feedback circuit engaging Gata-3 and interleukin-4 (IL-4) (71). The activation of this functional circuit characterizes the Th2 cell subtype (72, 73). Besides recovering the Th0 polarization into Th1 and Th2, the model was able to describe the transition between Th1 and Th2 attractors by the stimulation with IFN, IL-4, or the combination of IL-12 and IL-18. Later on, the model was extended to include novel transcription factors, cytokines, and signal transduction molecules to describe additional fates to T regulatory (Treg) and Th17 cells (29). More refined molecular data has resulted in the reconstruction of larger versions and their simulations, predicting a larger repertoire of Th cell subsets including Tfh, Th9, Treg, iTreg, Th9, Th17, Th22, and T regulatory Foxp3 independent cells (6, 7, 74).

B and NK cells have been less studied by mathematical modeling. During terminal B cell differentiation in the germinal centers of secondary lymphoid organs, the exposure to particular environmental factors, including the antigen-mediated activation of the B-cell receptor (BCR), defines the transition of the naïve B cell to a memory cell or an antibody-producing plasma cell. This terminal differentiation of B cells has been simulated as a Boolean model that recovered four cellular profiles: naive B cell before and after arriving to the germinal center (GC), memory cell (MC) and plasma cell (PC) (8). The B cell model reproduces not only the expected cellular attractors, but also the transitions with biological significance. Of note, it predicts four interactions that have not been declared experimentally but are suggested through indirect mechanisms: two self-feedback loops involved in Pax5 and Bcl6 activation, the positive regulation of Bcl6 by Pax5, and the inhibition of Pax5 by Irf4.

On the other hand, NK cell biology has been recently approached by a Boolean model providing a CLP attractor that transits toward pro-B, early T progenitor, or three different subtypes of NK attractors, depending on the activation pattern of IL-7, IL-15, and Delta ligand (75). NK cell subsets are characterized by differential expression of the transcription factors T-bet and Eomes. The NK attractor reached after CLP is stimulated with IL-15 activates both transcription factors and correlates with highly cytotoxic NKs both, in humans and mice periphery. On the other hand, perturbation of the CLP attractor with combined activation of IL-15/IL-7 or IL-15/Delta ligand, leads to a T-bet- Eomes<sup>+</sup> profile correlating with BM NKs or T-bet<sup>+</sup> Eomes<sup>−</sup> compatible with liver NKs (76). The incorporation of more transcriptional regulators may lead to new hypothesis about the branching step between NK cells and the more recently described, innate lymphoid cells (ILCs). It has been purposed that CLPs transition to NK lineage may have an intermediate step of a common progenitor for NKs and ILCs with a probable expression of transcription factors shared by both lineages, such as Nfil3 and TOX (77). In contrast to adaptive lymphocytes, the knowledge on transcriptional circuits controlling ILC development remains limited, although their role in the orchestration of immune responses has become of particular interest. ILCs are enriched in mucosal tissues and have been correlated with the progression of allergic, gastrointestinal, and central nervous system inflammatory diseases, like inflammatory bowel disease (IBD) and multiple sclerosis (78, 79). Similar to T lymphocytes, ILCs show plasticity under microenvironmental challenges modifying their cytokine secretion patterns and in consequence, the response exerted by other cells of the immune and adaptive branches (80).

The continuous integration of data, an inevitable process to improve computational modeling of biological systems, leads Enciso et al. Modeling Lymphocytes in Chronic Diseases

to the generation of large and complicated networks. To facilitate their analysis, large networks can be subjected to model reduction, a process of iterative removal of particular nodes and redirection of the logical rules that ideally, preserve the reachability of the attractors while keeping the main dynamical properties (29, 81, 82). Model reduction considers that, in a number of cases, a central core of nodes drives the dynamics of other dependent nodes. One of the simplest methodologies to drive model reduction (16) consists in excluding from the steadystate computation those nodes that follow linear downstream pathways. For example, the consecutive rules q3(t + 1) = q2(t) and q2(t + 1) = not q1(t) may be transformed into q<sup>3</sup> = q<sup>2</sup> = not q1, so that the state of q<sup>1</sup> automatically determines q<sup>2</sup> and q3. A more elaborate example would be q5(t + 1) = q4(t) and - q4(t) or q3(t) or not q2(t) which leads to q<sup>5</sup> = q<sup>4</sup> and - q<sup>4</sup> or q<sup>3</sup> or not q<sup>2</sup> ; using the Boolean absorption rule a and (a or b) = a, this expression is finally transformed to q<sup>5</sup> = q4. In this latter case, the steady state of q<sup>5</sup> is merely determined by q4, independently of the state of q<sup>3</sup> and q<sup>2</sup> which appear in the original rule. Furthermore, model reduction is a useful tool to identify regulatory cores or redundant signal transduction pathways, reduce the states space and obtain qualitative data comparable to experimental results (83–85). An alternative to deal with networks whose size complicates the exhaustive analysis of their state space, consists in the evaluation of cellular transitions assessed with a computational technique known as model checking (6). Model checkers are based on the transformation of states space into graphic or symbolic structures that facilitate verification of properties and trajectories, allowing fate mapping of all possible cell transitions and emerging as a potent predictive tool for cellular plasticity under multiple microenvironmental contexts.

The role of the microenvironment in lymphoid differentiation is successfully implemented in the reviewed models by considering the hypothesis that cytokines are either absent or present, and do not care about graded availability. Other models integrate assumptions to simulate signal processing and propagation using a discrete model, such as the models of the downstream events occurring after the activation of the T-cell receptor (TCR) (11, 12). Saez-Rodriguez et al., based a Boolean model in a large network of 94 nodes and considered that some signaling events occur in a different timescale, so that logical rules were updated in a first and second wave depending on the molecular nature of the event. From the attractors resulting after the simulation, the authors made predictions about the signaling cascade activated by the receptor engagement and confirmed them experimentally. The implementation of two updating waves is a way to recognize that the cellular events occur in different timescales, for example biochemical reactions occurring in the cytoplasm (e.g., molecular inhibition by phosphorylation) are faster than the transcriptional modulations (e.g., transcription factor translocating to the nucleus and binding to a gene promoter that will be activated). Even though their utility, it is necessary to recognize that Boolean models are sometimes insufficient, particularly when there is enough data about the continuous concentration of a biomolecule determinant for the process that is being modeled.

The study of chronic diseases has strongly influenced the understanding of how slight changes derive in the complete perturbation of complex biological systems. If it were desired to simulate the way in which the progressive accumulation of pro-inflammatory factors in the intestinal tract perturb the proportions of T cell populations, the use of Boolean models would be of very limited use to investigate the transitory stages between the healthy attractor and a pathological attractor, like in IBD (78).

# 3. MODELING OF CONTINUOUS VARIABLES TO STUDY LYMPHOCYTE DIVERSITY

Modeling lymphoid cells production or activation may require the integration of molecules involved in dosage-dependent effects, as is the case of ligand-receptor affinities, cytokine gradients and even some transcription factors like C/EBP and PU.1 (9, 56). As suggested by the number of publications, continuous mathematical models are the most recurrent tool for the study of lymphocyte development and response and are useful tools to evaluate population dynamics and receptor repertoire (86–89).

However, most time parameters are fitted to experimental data without a deep understanding of molecular mechanisms, unless enough kinetic and biochemical information is available. Some cellular processes involving dosage variations may still be simulated with discrete approaches using multi-valued models or probabilistic Boolean networks, but there exist other alternatives to integrate discrete and continuous molecular events like the construction of hybrid and fuzzy models. On one side, hybrid models have been applied to simulate the activation of Th and B lymphocytes by DCs, and their subsequent departure from the lymph node. The cellular entities and the replication steps were modeled in terms of discrete variables, while the migration was simulated by means of differential equations involving continuous variables and parameters (e.g., chemokines concentration and diffusion, cellular velocity) (90). On the other side, Boolean models may be transformed to continuous systems using fuzzy logic (5, 8, 15–17, 91, 92). These approaches may be useful to use existing GRN of lymphoid differentiation and activation to model complex scenarios that involve intercellular communication among immune cells, interaction of immune cells with normal or pathologic tissue, and immune cell population transitions in response to microenvironment remodeling.

# 3.1. Dosage Variations in Multi-Valued and Probabilistic Models

The molecular pathways participating in TCR signaling have been successfully modeled with a set of differential equations. The first step for T lymhpocytes activation involves a process known as ligand discrimination that differentiates between weak and strong binding antigens. After TCR engages with peptides processed and expressed on the surface of antigen-presenting cells, a well-regulated discrimination between self and nonself antigens is triggered. The simulation of TCR activation as a continuous model suggested that the MAPK cascade is the responsible for this discriminatory engagement process. A negative feedback loop that modulates the TCR response until an ERK activation threshold is reached may take place, resembling a bimodal behavior (93). The model was expanded to answer the question of how stochastic variations of protein expressions among a clonal population of CD8 T cells could affect their responsiveness. Variations on the expression of CD8 and two components of the MAPK signaling pathway, ERK-1 and SHP-1, generate dispersion in responsiveness among individual cells, but the co-regulation of CD8 and SHP-1 restrain the phenotypic variability (94). It was later discovered that the ligand discrimination process influences T cell differentiation to Treg or Th phenotypes through the downstream modulation of PTEN and Akt/mTOR signaling pathways (95, 96). To represent a weak or strong ligand affinity, a multi-valued model was useful allowing three possible activation levels of TCR and PI3K nodes (off = 0, low = 1, and high = 2). The computational simulations of the model corroborated that low TCR signal favors Treg differentiation, while a stronger signal result in the induction of Th profile (97). Additionally, varying the number of rounds or time-steps for TCR activation, as an approach for ligand binding lifetimes, showed that the Th phenotype is more rapidly stabilized than a Treg profile, suggesting that the transition from naïve to Treg cells is less direct than the Th differentiation. The generation of Treg cells goes through intermediate stages during which the secretion of IL-12 is promoted and activates the PTEN signaling pathway that enables Foxp3 permanent activation (97). Under high TCR signaling, Foxp3 is transiently activated but further turned off by mTOR pathway, while the Aktdependent regulation of T cell fate choice is also dependent on the differential phosphorylation of additional proteins (98). There are ongoing studies focused on the blockage of TCR signaling by some pathogens like Yersinia pseudotuberculosis (99).

To deepen in the composition of the microenvironmental patterns affecting the diversity of T lymphocytes, a probabilistic Boolean control network (PBCN) was developed for simulation of all possible microenvironments combining nine external signals including TCR activation, TGF-β and IFNγ cytokines, and six interleukines. In contrast with conventional Boolean models, PBCNs contemplate activation probabilities as an approach to input dosages, increasing the range of testable microenvironments (74). Experimental research on T lymphocytes diversity has led to the discovery of intermediate phenotypes that co-express lineage-specific transcription factors from more than one T cell subset, such as Th1-Th2 and Th1-Th17 cells identified on bacterial and parasitic infection (100–102). Through a sensitivity analysis of the PBCN the minimum microenvironment requirements have been identified, on composition and dosage, for the description of each of the 10 T cell profiles. In addition, they have been used to predict the way in which different input patterns influence the internal balance determining the phenotype of canonical and complex cellular profiles, such as cells with mixed phenotypes. With a continuous model constructed to simulate iTreg-Th17 differentiation, Hong and collaborators reported a double expressing phenotype with either regulatory or dual (regulatory and proinflammatory) functions in vivo. This mixed phenotype is suggested to be a stable state reached from the transition of single-expressing cells, iTreg and Th17. Th17 and iTreg cells are able to produce TGF- which may either increase the percentage of both types of cells, or induce the transition from single-expressing to double-expressing cells. The iTreg-Th17 model was also used to analyze how different concentrations of TGF- influence the rate of co-existing cellular subtypes, making evident that priming factors not only drive differentiation events, but also promote cell heterogeneity (103).

The models presented in this section have different limitations. The multi-level and probabilistic models do not allow the integration of temporal hierarchies in the events involved in the biological system of interest, particularly important when modeling more than one type of cellular processes. The continuous model includes a limited number of components depending on the availability of kinetic parameters or enough information to establish assumptions. As an alternative, fuzzy logic can merge large transcriptional regulatory networks participating in cell differentiation and plasticity, with qualitative knowledge about the kinetics of signaling pathways involved in the transduction of microenvironmental variations, for example, events proceeding relatively faster than others, or ligands binding to receptor above other ligands.

# 3.2. Continuous Simulation of Discrete Differentiation Networks

Extracellular signals and some intracellular components are continuous variables and their adequate representation in mathematical models may determine the simulation of lymphoid cellular fates like differentiation, phenotypic transitions and activation. The transformation of discrete models to a set of differential equations is useful to identify additional attractors and unstable states with biological relevance. In a comparison between Boolean and continuous simulation of a B cell terminal differentiation network, the continuous counterpart provided three additional stable states with intermediate values of Bcl-6 and/or Irf4; however, only one of them was comparable with a previous reported phenotype that may correspond to the centrocytes found during the germinal center selection (8, 104). This intermediate phenotype together with centroblasts, are particularly important in the study of follicular lymphomas characterized by an accumulation of cells unable to reach terminal differentiation stages.

In comparison with Boolean models, the computational simulation of continuous fuzzy models is simpler and in consequence faster, thus allowing the integration of independently developed BRN without caring a lot about the number of resultant equations. An example is the T/B lymphoid differentiation model of 81 equations representing cytokines and transcription factors that lead to ten attractors with Th0, Th1, Th2, Th17, Treg, cytotoxic T lymphocyte, DP T lymphocytes, CD8 T naive, naive B cell, and PC profiles (105). The attractors obtained by the continuous model show a higher compatibility with experimental data than previous discrete models. In this case, all the attractors display intermediate values for Ikaros, Gfi1, and PU1. For each of the three transcription factors there exists strong evidence that associates this intermediate expression with the delimitation toward lymphoid lineage during hematopoietic differentiation (106–108). The intermediate modulation of PU1 and Ikaros was also reproduced with a different continuous model of B-lymphocyte lineage commitment, evidencing their participation in the transcriptional core that reproduces the irreversible transition from LMPP to lineage restricted progenitors expressing IL-7R (109).

An additional application of fuzzy logic models is the simulation of virtual cultures where independent GRNs, representing multiple cells, may interact with a microenvironment expressing graded and dynamic concentrations of cytokines. A virtual culture of T lymphocytes was proposed by Mendoza to evaluate the evolution of 100 cells with an initial Th0 configuration after being stimulated with IFN, I-4, TGF alone, or TGF in combination with IL-6. The phenotype of each cell was determined by the activation state of each of the 36 nodes integrating the internal Th differentiation network, in turn, regulated by 11 cytokines produced depending on each cellular profile (Th0, Treg, Th1, Th2, and Th17). The produced cytokines involved endocrine and paracrine signaling to evaluate the final balance of the T lymphocyte subpopulations arising from different types of stimulus (91). This particular implementation is computationally expensive, but represents a more realistic approach to analyze the interaction between heterogeneous populations of immune cells susceptible to transit among phenotypes, including dynamic secretion patterns that influence the composition of the microenvironment.

# 3.3. From Discrete to Continuous Using Fuzzy Logic

A more realistic approach must considerate that the expression levels, concentrations, and parameters of biological systems may take any value within a continuous range limited only by functionality constraints. In this case, the discrete dynamic mapping given by Equation (1) may be generalized by introducing a set of ordinary differential equations (ODEs) for the rate of change of the expression level of the network components. For k-th node, this is written as

$$\frac{dq\_k}{dt} = \mu \left[ \omega\_k (q\_1, \dots, q\_n) \right] - \alpha\_k q\_k. \tag{2}$$

Here, µ[w<sup>k</sup> ] is an input function that expresses a continuous realization of the Boolean rule w<sup>k</sup> (see below), while α<sup>k</sup> is a decay rate. In this scheme, the equilibrium states of the system are defined by the steady-state condition dqk/dt = 0, which leads to

$$q\_k^s = \frac{1}{\alpha\_k} \,\mu\left[\boldsymbol{\nu}\_k(q\_1^s, \dots, q\_n^s)\right],\tag{3}$$

where the superindex s denotes the steady-state value. A straightforward consequence of this is that the expression level of node k is strongly dependent on its decay rate. In the case α<sup>k</sup> > 1, a node will be under-expressed with respect to the value attained for α<sup>k</sup> = 1; in particular, for α<sup>k</sup> ≫ 1, the expression of that node will be completely inhibited: q s <sup>k</sup> → 0. The converse also holds: if α<sup>k</sup> < 1, a node will be relatively over-expressed [it must be noticed that a decay rate α<sup>k</sup> < 1 may lead to a steady expression value q<sup>k</sup> > 1. Although in fuzzy logic the values of the variables are assumed to be constrained to the interval 0 ≤ q<sup>k</sup> ≤ 1, values >1 are not excluded by the formalism, and it is a matter of convenience the range in which the variables are defined (110)]. It follows that modifications of the characteristic decay rates of network components may alter the steady expression patterns arising from the nodes interactions. This may be interpreted as a modulation of the metaphorical or Waddington's epigenetic landscape which eventually may lead to transitions between attractors associated to different cell fates. This approach has been formerly employed, for example, to model plastic phenotype changes in T CD4<sup>+</sup> lymphocytes (92).

The translation of the interactive Boolean rules to the continuous domain may be accomplished by considering an approach based on fuzzy logic. Fuzzy logic is a theory aimed to provide formal foundation to approximate reasoning with applications in physical, biomedical, and behavioral sciences. It is characterized by a graded approach (110–112), so that the degree to which an object exhibits a given property is specified by a membership (or characteristic) function µ[w<sup>k</sup> ], with truth values ranging from total falsity, µ[w<sup>k</sup> ] = 0, to totally true, µ[w<sup>k</sup> ] = 1. For example, the property of "being a good person" implies that there is a set of persons that share certain characteristics with no definite boundary. Fuzzy logic satisfies an axiomatic similar to the implied in Boolean logic, except for the identity principle, meaning that the principle of no-contradiction does not hold. Thus, although seemingly paradoxical, a proposition w and its negation 1 − w may be simultaneously true. For example, the assertion "he was not a good, but not bad guy" has a meaning in language theory. In biological systems, fuzzy propositions may describe cases in which a cell displays an intermediate expression pattern that does not necessarily belong to a specific phenotype. That is the case of individuals with food allergies, in which Treg cells produce IL-4, which is a characteristic usually ascribed to Th2 cells. Similarly, diseases like rheumatoid arthritis or colorectal cancer are associated to the expression of IL-17+Foxp3<sup>+</sup> Treg cells or RORγ t + Foxp3<sup>+</sup> Treg cells, respectively. The absence of no-contradiction is formally expressed by the equation w = 1 − w, with solution w = 1/2. It follows that the value w ≡ wthr = 1/2 may be interpreted as a threshold between falsity and truth.

Similar to the Boolean approach, in the continuous regime the network regulatory interactions are characterized by fuzzy logic propositions denoted here as w<sup>k</sup> [q1(t), ..., qn(t)]. They are either inferred from experimental observations or suggested by inner consistency requirements. In fact, a translation scheme from the discrete to the continuous scenario may be straightforwardly implemented translation by replacing the Boolean connectives AND, OR, and NOT, for its fuzzy counterparts. In fact, the definition of fuzzy connectives is not unique, and a number of different alternatives not entirely equivalent, have been proposed. In the following table we present Zadeh's original proposal (111) and a probabilistic-like scheme (110):


Both schemes satisfy the modified Boolean axiomatics discussed above. However, the probabilistic-like scheme leads to continuously differentiable expressions if q and p are differentiable. This is a desirable condition when dealing with ODEs systems. Furthermore, it shows the same properties as joint probabilistic distributions for independent variables, so that probabilistic statements may be directly translated into fuzzy propositions.

An example of translation from the Boolean to the fuzzy framework is

$$\begin{array}{rcl}W[p,q,r] &= \left(q \lor p\right) \land \neg r \to & w[p,q,r] \\ &= \left(q+p-q \cdot p\right) \cdot \left(1-r\right). \end{array}$$

Continuous logical propositions can be used to construct an explicit expression of the characteristic function µ[w<sup>k</sup> ]. In the discrete Boolean approach, this function would be equivalent to a step 2 function:

$$
\mu[\boldsymbol{w}\_k] \to \Theta[\boldsymbol{w}\_k - 1/2] = \begin{cases} 0 & \text{if } \boldsymbol{w}\_k < 1/2; \\ 1 & \text{if } \boldsymbol{w}\_k > 1/2. \end{cases}
$$

In the continuous approach this behavior may be approximated by a characteristic function with a sigmoid structure that gradually changes from a null to a unit value. Many functions share this property. An expression employed in a number of applications of fuzzy logic in systems biology is the logistic function:

$$\mu\left[\boldsymbol{w}\_{k}\right] = \frac{1}{1 + \exp\left[-\beta \left(\boldsymbol{w}\_{k}(q\_{1},...,q\_{n}) - \boldsymbol{w}\_{thr}\right)\right]} \tag{4}$$

Here, wthr is a threshold value such that if w<sup>k</sup> > wthr, then wk tends to be true (or expressed). Usually wthr = 1/2. The parameter β is a saturation rate that measures the pace of the transit from an unexpressed to an expressed state, as displayed in **Figure 3**. We observe in the figure that this pace is gradual for small β, and steep for large β. In the case β ≫ 1, µ[w<sup>k</sup> ] → 2 [w<sup>k</sup> − wthr]. This latter result, together with the steady-state condition given by Equation(2), implies that in the case is another manifestation of the robustness of the qualitative predictions generated by the fuzzy approach. A related result is that in the limit β ≫ 1 and α<sup>k</sup> = 1 for every network interaction, then the steady-state condition given by Equation (7), guarantees that the set of fixed-point attractors resulting in the Boolean and fuzzy approaches coincide by construction. On the contrary, the corresponding sets of periodic attractors usually differ.

It may be argued that the predictions obtained in the fuzzy formalism may depend on the specific form of the characteristic function µ[w<sup>k</sup> ]. In fact, there are multiple expressions employed for example, in engineering applications and control theory, such as triangular, trapezoidal, or Gaussian functions (113). However, the logistic structure of µ[w<sup>k</sup> ] considered in this review may be derived, rather than postulated, by introducing the concept of Shannon's information entropy (work in preparation). This is related with the number of independent ways in which a logical proposition may acquire partial values of truth for fixed values of the parameters β and wthr. In other words, the more general expression involving the least number of assumptions concerning a graded approach to truthiness of a fuzzy proposition is the logistic distribution. Interestingly, the mathematical formalism associated to fuzzy regulatory networks including the description of logical rules with a logistic structure is formally equivalent to that employed in the computation of neural circuits in the theory of neural networks (114).

Another useful (and equivalent) representation of the characteristic function may be derived by considering that the expression levels of biological variables, such as the concentrations or the affinities of a given molecule, may show variations involving several orders of magnitude. In that case, it may be convenient to introduce in the description the logarithms of the corresponding quantities. This is easily performed by means of the change of variable w<sup>k</sup> = ln x<sup>k</sup> and substituting this into Equation (8), leading to the well-known expression for the Hill function:

$$
\tilde{\mu}[\mathbf{x}\_k] = \frac{\mathbf{x}\_k^{\beta}}{\mathbf{x}\_{thr}^{\beta} + \mathbf{x}\_k^{\beta}},
\tag{5}
$$

where the parameter xthr represents the value at which µ˜[x<sup>k</sup> ] acquires half its maximum value. The Hill function and its negation 1 − ˜µ[x<sup>k</sup> ] display both a sigmoid shape and have been widely employed in the modeling of biochemical, physiological, and pharmacological processes. A paradigmatic example is the set of non-linear differential equations

$$\frac{d\mathbf{x}\_k}{dt} = \frac{a\_k \mathbf{x}\_k^{\beta}}{\mathbf{x}\_{thr}^{\beta} + \mathbf{x}\_k^{\beta}} - \frac{b\_k \mathbf{x}\_{thr}^{\mathcal{V}}}{\mathbf{x}\_{thr}^{\mathcal{V}} + \mathbf{x}\_k^{\mathcal{V}}},\tag{6}$$

describing, for example, the simultaneous binding (unbinding)of β (γ ) ligands to (from) a single receptor. This latter representation has been employed in the construction of a GRN that characterizes fate decisions and reprogramming signaling pathways of pancreatic cells (115). Although this latter model was not built within the fuzzy logic approach, we observe that in this and numerous instances a formal equivalence may be established by a convenient re-scaling of the variables and parameters involved in the description.

#### 3.4. Self-Organization and Time Ordering

To describe the transitions between distinct steady states, in conjunction with fuzzy logic elements, general concepts of theory of non-equilibrium phase transitions and self-organization are highly relevant to consider. The adaptation of that theory to the fuzzy logic modeling scheme allows a sound description of the transitions between the different disease stages. In the

FIGURE 3 | Fuzzy networks. (A) Characteristic distribution µ h <sup>w</sup><sup>k</sup> <sup>−</sup> <sup>w</sup>thr<sup>i</sup> for a threshold value <sup>w</sup>thr <sup>=</sup> <sup>1</sup>/2 of the logical proposition <sup>w</sup><sup>k</sup> , and different values of the saturation parameter β. In the case β ≫ 1, the characteristic distribution becomes a step-like distribution. (B) Fuzzy modeling of the transcriptional core regulating B cell to plasma cell (PC) differentiation using three different saturation values (β = 8, 15, 60), wthr = 1/2 and the decay rate for each component α = 1. For the initial state of network all nodes were considered inactive, except AP-1, the promoter of the PC differentiation. When β = 60, a full B cell attractor is reached with no final expression of AP-1 or Blimp-1.

description the transitions between steady states it is important to contemplate that differentiation from a multipotent stem or progenitor state to a mature cell is an essentially irreversible process, and that the associated changes in gene expression patterns exhibit time-directionality. Whereas, in equilibrium systems time-irreversibility is a direct reflection of the second law of thermodynamics, the cell's gene regulatory network represents a system far from thermodynamic equilibrium. These problems have been contemplated by the theory of cooperative phenomena, non-equilibrium phase transition and self-organization (116). Accordingly, cooperative phenomena arise from non-linear interactions of a large number of elementary subsystems (represented here by the fuzzy logic rules), leading to the emergence of organized patterns or phases. The theory relies upon two main concepts, the existence of ordering and control parameters. The order parameters are those variables that mainly drive the system organization, while the control parameters are variables whose value determines which of the possible organizations is actually realized. In the case of thermodynamic systems, an order parameter would be the density, which defines an aggregation state, such as liquid, solid, or gas. These states may somehow "compete," in the sense that one or other may prevail depending on the value of an external control parameter, such as the temperature of the system, for fixed values of pressure and volume. In the context of fuzzy GRNs, the order parameters are the activation patterns that specify the different cell phenotypes, determined in turn, by the activation state of central nodes or functional moduli of the GRN, while the control parameters are those involved either in the logic rules, or those characterizing the decay rates {α<sup>k</sup> }. This latter set is of prime importance. Given that α<sup>k</sup> = 1/τ<sup>k</sup> , where τ<sup>k</sup> is a characteristic expression time, the set {α<sup>k</sup> } implicates a hierarchy for the temporal expression of the GRN components. By assuming that an ordering α<sup>1</sup> > α<sup>2</sup> > ... can be constructed, this procedure induces an associated ordering τ<sup>1</sup> < τ<sup>2</sup> < .... As in the thermodynamic example, the phenotypic landscape (or state space) may be explored by varying each of the control parameters α<sup>k</sup> , while maintaining fixed the rest. As a consequence, transitions between different ordered phases may be induced by modifications of the control parameters. This is similar to the description of chemical reactions in the reaction coordinate space, where the substrate and product states are

separated by an activation energy barrier; when an enzyme is added the activation barrier is lowered, and the chemical reaction ensues. In Waddington's landscape context, this mechanism may be interpreted as alterations of the peaks separating the valleys, allowing the exploration of the landscape and transit between valleys. This kind of description has been employed in the modeling of the long-term progression of type-2 diabetes based on a GRN for pancreatic beta cells. In this case, the organization patterns correspond to states identified with health, metabolic syndrome, or manifest diabetes. The alteration of decay rates of key cellular components involved in inflammatory and metabolic pathways lead the transitions between different disease stages.

An important consequence of establishing a time ordering, is that the system dynamics may discriminate among "slow" and "rapid" variables and it may be shown that the main dynamics is driven by "slow" variables, while the "rapid" variables adapt almost instantaneously to the environment defined by the "slow" ones. It turns out that the relaxation times of the order parameters are usually much longer than those of irrelevant variables and thus work like control parameters of the system. Irrelevant variables decay rapidly to a steady state, so that they may be effectively eliminated from the overall dynamics. In this view, the order parameters define the general features of the system, including the final expression patterns associated to a set of initial conditions, while less relevant variables adapt their values to the instructions dictated by the order parameters. This property may be relevant in the study of multifactorial diseases, since it could help in the identification of variables that constitute a target for the development of therapies.

Another element that may be relevant in the study of transitions between steady states is the consideration of extrinsic and intrinsic noise, i.e., the existence of random interactions inherent to every biological system. Depending on its intensity, the existence of noise may drastically alter the predictions yielded by the deterministic formalism considered before, especially at bifurcation points of the landscape, where noise may accelerate a transition rate between neighbor attractors. In the chemical reaction analogy, this is equivalent to adding heat to the process. The action of noise may be incorporated in the fuzzy logic approach by assuming that this is characterized by a stochastic variable ξ (t), with zero mean ξ (t) = 0, and statistical dispersion given by ξ (t) ξ (t ′ ) = DG(t − t ′ ). Here, D is a diffusion coefficient, and G(t − t ′ ) is a function that characterizes the duration of the self-correlation of the variable ξ . The case in which G is a Dirac delta, i.e., a sharply peaked distribution only for t = t ′ , and null elsewhere, corresponds to a white noise with no-memory effects.

The fuzzy stochastic dynamics (16) can be described by a Langevin equation (116, 117):

$$\frac{dq\_k}{dt} = \mu \left[ \left. \nu\_k(q\_1, \dots, q\_n) \right| - \alpha\_k q\_k + \xi\_k(t), \tag{7}$$

with steady states given by the mean value q s k = µ - w s k /α<sup>k</sup> . In the same way as in the deterministic approach, the parameters α<sup>k</sup> control the relative heights of peaks and valleys in the mean epigenetic landscape. In the case of small noise (D ≪ 1) the time-dependent solutions are composed by the mean path q s k (t) plus random fluctuations around this path, similarly as dust particles driven by a gentle breeze. The Langevin formalism was implemented by Zhou et al. (115) by means of a GRN addressed to study the processes of differentiation and cell fate reprogramming in pancreatic cells. They show that it is possible to recapitulate the observed attractors of the exocrine and β, δ, α endocrine cells and to predict which gene perturbation can result in a desired lineage reprogramming.

A related approach rests upon a probabilistic or quasi-potential landscape (118, 119). In this case, it is not the ensemble of stochastic trajectories q<sup>k</sup> (t), but their probability distribution P[q<sup>k</sup> (t)] what constitutes the central concept. One may envisage an epigenetic landscape in which the maximal expression probabilities lie over the deepest (or wider) attraction basins, while the minimal probabilities lie over the hills' tops. Thus, the probabilistic landscape corresponds to an inverted realization of the epigenetic landscape. It can be shown that the probability distribution P[q<sup>k</sup> (t)] satisfies the Fokker-Planck (FP) diffusion equation (116, 117), and that the information contained in this formalism is equivalent to that inherent to the Langevin approach. It has been proposed by Wang et al. that the genetic circuitry connections in the quasi-potential landscape imposes the arrow of time in stem cell differentiation, so that the generic asymmetry of barrier heights indicates that the transition from the uncommitted multipotent state to differentiated states is inherently unidirectional.

#### 4. LYMPHOCYTE INVOLVEMENT IN CHRONIC DISEASES: CELLULAR DIVERSITY AND PATHOLOGICAL FEEDBACKS

The logical framework has also been applied to the simulation of signaling pathways involved in lymphoid related-diseases, like acute lymphoblastic and T cell large granular lymphocyte (T-LGL) leukemia. In the first case, it was predicted that a proinflammatory microenvironment may induce instability in two molecular axes responsible for the retention of hematopoietic progenitor cells within regulatory bone marrow niches (120). In the second case, the model helped to decipher some of the molecular mechanisms that promote survival in T-LGL leukemia cells (121). Both models integrated microenvironmental factors with signaling pathways participating in cellular fate decisions, and in both cases the role of the pro-inflammatory NFkB pathway emerged as important player in the pathogenesis.

Few mathematical models have managed to simulate the dynamic communication between lymphocytes and microenvironment, considering that the feedback loops between both systems are key to modulate immune responses, although the in vivo regulation of both systems is more complex due to influence of neighbor tissues and endocrine signals. The perturbation or inadequate coupling of the regulatory interactions among systems have been suggested to trigger inflammation in multiple chronic diseases. For many years the study of pro-inflammatory conditions was focused on the identification of cytokines as biomarkers or target for adjuvant therapies. With the advances on immunotherapy, the study of immune cells as active participants in chronic diseases development and progression has become of great importance because they represent therapeutic targets with less co-lateral effects than conventional therapies.

Recently, it has been probed that epigenetic landscape approach is useful for the in silico analysis of health to pathogenic progression (122), such as the epithelial-mesenchymal transition and the induction to migratory phenotype induced after chronic pro-inflammatory conditions, offering a tool to delve deeper into transition stages important for early diagnosis (123–126). Computational modeling of epithelial-mesenchymal transition induced by pro-inflammatory cues has suggested an intermediate stage with a senescent profile (125). The process of epithelial malignant transformation is promoted, among other factors, by TGFβ secreted by CD8 and Treg cells, and TNFα produced by macrophages and pro-inflammatory T cells (127, 128). Importantly, CD8 T lymphocytes have been purposed as players in the promotion of aggressive features in breast cancer tumorigenesis (129); but using CD8 T cells as therapeutic targets implicates affecting one of the most important defenses toward infections, so research about the regulatory networks underlying T cell polarization in dynamic feedback with epithelial cells open new opportunities for the development of more precise therapies by simulating multiple or all the possible perturbations in an integral network as also suggested for breast cancer therapy (130).

The same approach is applicable for the study of emergent attractors from many other networks associated to chronic diseases, for example, type 2 diabetes described in terms of beta-pancreatic cell (115) and T lymphocyte interacting networks, based on evidence of the participation of different T subpopulations as inductors of local and systemic inflammation (131). A first approach targeting CD4 T cell plasticity in metabolic diseases showed that hyperinsulinemia, a condition associated with metabolic syndrome and early stages of type 2 diabetes, inhibits the generation of T regulatory Foxp3 cells and stabilizes the Th17 attractor (10). Besides type 2 diabetes, the increase of Th17 subpopulation and decrease of T regulatory cells have been linked with the destruction of beta-pancreatic

FIGURE 4 | Fuzzy models to study signaling pathways activation. (A) NFkB network where IKK is stimulated by microenvironmental TNFα. IKK phosphorylates IkBa unrepressing the dimer p65\_RelA to allow its translocation to the nucleus. In the nucleus p65\_RelA promotes the transcription of IkBA closing a negative feedback loop of the NFkB pathway. (B) The Boolean simulation of the network generates three attractors, two of them are cyclic attractors with TNFα activation. Here, green = 1, red = 0. (C) Activation value of the node in the NFkB network obtained by fuzzy logic simulation. In this case, β = 3, and α varies depending on the type of biochemical event in which each node is involved. Node tIkBa represents a transcriptional event, with decay rate α = 0.2. (D) Figure taken from (138) showing the nuclear to the cytoplasmic GFP intensity (NCI) of three single GFP-p65 cells stimulated with a constant flow of 10 ng/ml of TNFα.

cells in the pathogenesis of type 1 diabetes, an increased risk of breast cancer recurrence in diabetic patients and increased susceptibility to develop colitis (132–134). The modulation of the Th17 subpopulation as a promising treatment of colitis was predicted by computational simulations of a continuous model. With in silico perturbations of the GRN underlying CD4 T cell differentiation it was predicted that the increase of PPARγ in Th17 cells derives in its transition toward an iTreg profile characterized by the upregulation of Foxp3. The in vivo effect of transplanting PPARγ null Th0 lymphocytes was an increased severity and earlier development of colitis in mice. In contrast, pharmacological activation of PPARγ resulted in the induced shift from Th17 to iTreg phenotype that favored colonic tissue reconstitution (135).

The use of integral models of regulatory networks can be also applied to chronic infections. Existing models of infectious diseases and their interconnection with lymphoid regulatory networks are very limited. Even though, one group has reconstructed a logical network to study the intracellular pathways in CD4 T cells affected by the viral proteins during HIV infection. By considering a model composed by 16 viral and 121 CD4 T cell nodes, they predicted new viral-human molecular interactions and obtained conclusions on the signaling flow affecting cellular fate decisions (136).

All chronic processes mentioned above involve multiple developmental stages where different changes in the microenvironment and the cellular composition take place, depending one on the other through feedback loops. With discrete models we can clearly map the stable stages and the transition between them in the presence or absence of particular nodes, while in conventional continuous models it is quite complicated to include as much as transcription regulators are required to simulate cellular transitions of more than one type of cell. So, the transformation of genetic Boolean models using fuzzy logic, is a promising approach to integrate differentiation networks of lymphoid cells and cells from other tissues to construct more accurate models for the study of chronic diseases, as it becomes important the consideration of temporal evolution and graded changes in molecular compositions. Additionally, less frequent inflammatory cells participating in chronic diseases can be included, like in chronic allergic lung disease, where the progressive accumulation of B cells in the lung promotes Th2 responses by the antigen presentation process (137).

Moreover, intermittent or persistent rapid perturbations in chronic and complex diseases, but not during steady states, provoke small and sometimes, cumulative variations within the cells or their environment, including modifications in cytokines secretion patterns, cellular populations proportions, miRNAs expression, etc., that mostly become visible until there is an abrupt transition of the whole system. Of note, fuzzy

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logic continuous models permit an easy simulation of such periodical and transient signals that are transduced by cell signaling pathways.

The utility of fuzzy models may apply to a small network composed by some of the main components of the NFkB signaling pathway that behave as a damped oscillator during activation with TNFa (**Figure 4**). The Boolean simulation of the NFkB network generates two different cyclic attractors when TNFa is activated. However, when simulating the network as a fuzzy logic and varying the parameter of the "slow" reaction corresponding to the genetic transcription of IkBa, damped oscillations as observed in Zambrano et al., are recorded (138). Without introducing any specific kinetic information of receptor affinity, phosphorylation kinetics or translocation velocity, the fuzzy model shows the transition from an initially perturbed system toward a stable state with a controlled or regulated NFkB activation. As suggested by Zambrano et al., these approaches may aid to understand normal cell responses but also their behavior in diseases like cancer, where NFkB activity is usually disregulated and out of control, driving to multiple biological consequences including hyperproliferation, cell survival or migration.

# 5. CONCLUSIONS

Lymphocytes are active participants of many biological processes involved in homeostasis and can evolve concomitantly to tissues transiting through a pathogenic transformation, due to their responsiveness to a large diversity of biochemical signals and their plasticity. In silico experimentation with regulatory networks has shown the potential to identify the underlying mechanisms of feedback loops that participate in the promotion of disease progression or in the establishment of chronic inflammation. Additionally, the adaptation of existing models for the study of lymphocytes diversity in pathogenic contexts using powerful tools like fuzzy logics represents an approach to visualize the global effect of potential immunotherapeutics.

#### AUTHOR CONTRIBUTIONS

JE: analysis of published data, discussion of the topic-related information, drafting, and writing the paper. CV and RP: analysis of published data, discussion of the related information, drafting, writing the paper, critical review of the intellectual content.

#### ACKNOWLEDGMENTS

JE is scholarship holder from CONACyT and IMSS. The authors acknowledge financial support from Instituto de Física, Universidad Nacional Autónoma de México.

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**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2019 Enciso, Pelayo and Villarreal. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# T-Cell Receptor Cognate Target Prediction Based on Paired α and β Chain Sequence and Structural CDR Loop Similarities

#### Esteban Lanzarotti <sup>1</sup> , Paolo Marcatili <sup>2</sup> and Morten Nielsen1,2 \*

1 Instituto de Investigaciones Biotecnológicas, Universidad Nacional de San Martín, Buenos Aires, Argentina, <sup>2</sup> Department of Health Technology, Technical University of Denmark, Lyngby, Denmark

T-cell receptors (TCR) mediate immune responses recognizing peptides in complex with major histocompatibility complexes (pMHC) displayed on the surface of cells. Resolving the challenge of predicting the cognate pMHC target of a TCR would benefit many applications in the field of immunology, including vaccine design/discovery and the development of immunotherapies. Here, we developed a model for prediction of TCR targets based on similarity to a database of TCRs with known targets. Benchmarking the model on a large set of TCRs with known target, we demonstrated how the predictive performance is increased (i) by focusing on CDRs rather than the full length TCR protein sequences, (ii) by incorporating information from paired α and β chains, and (iii) integrating information for all 6 CDR loops rather than just CDR3. Finally, we show how integration of the structure of CDR loops, as obtained through homology modeling, boosts the predictive power of the model, in particular in situations where no high-similarity TCRs are available for the query. These findings demonstrate that TCRs that bind to the same target also share, to a very high degree, sequence, and structural features. This observation has profound impact for future development of prediction models for TCR-pMHC interactions and for the use of such models for the rational design of T cell based therapies.

Keywords: MHC, TCR, CDR, epitope, structure

## INTRODUCTION

A central checkpoint to unleashing a cellular immune response is the recognition of peptides presented by major histocompatibility complexes (pMHCs) by T cell receptors (TCRs). T cells undergo thymal selection. During this selection, T cells with TCRs that either cannot bind pMHCs (negative selection) or bind MHC molecules presenting self-peptides (positive selection) are removed. This process results in a repertoire of T cells with highly specific and selective TCRs, and it is estimated that each TCR can only bind a few thousand (1, 2) distinct pMHC complexes (of a total of more than 20<sup>6</sup> possibilities, assuming up to 3 MHC anchor positions). TCRs are composed of two subunits: α and β. Each subunit has three loops called complementary determining regions (CDRs) that directly interact with pMHCs. Structural studies from the last 30 years have shown that CDR3 loops usually present the most discriminative interactions with peptides, meanwhile CDR2 loops interact mainly with the MHC and CDR1 loops tend to present soft interactions with both peptide and MHC (3–5).

#### Edited by:

Burkhard Ludewig, Kantonsspital St. Gallen, Switzerland

#### Reviewed by:

Raffaele De Palma, University of Campania Luigi Vanvitelli, Italy Hansjörg Schild, Johannes Gutenberg University Mainz, Germany

#### \*Correspondence:

Morten Nielsen mniel@bioinformatics.dtu.dk

#### Specialty section:

This article was submitted to T Cell Biology, a section of the journal Frontiers in Immunology

Received: 28 March 2019 Accepted: 16 August 2019 Published: 28 August 2019

#### Citation:

Lanzarotti E, Marcatili P and Nielsen M (2019) T-Cell Receptor Cognate Target Prediction Based on Paired α and β Chain Sequence and Structural CDR Loop Similarities. Front. Immunol. 10:2080. doi: 10.3389/fimmu.2019.02080 The vast diversity of TCRs allows the recognition of an immense number of different antigens. In the last few years, highthroughput profiling of TCRs have become of routine use and it has been shown that some signatures can be used to describe in general terms the interaction between TCRs and the cognate pMHC complex (6–11). Some studies have demonstrated changes in T-cell populations after several stages of vaccination or exposure to diseases using TCR sequencing (12–16). The specificity of a TCR is most often described using only CDR3 β loop sequences. CDR1 and CDR2 β loops can be included by sequencing TCR β V and J germline regions, thus the full β sequence has also been used to describe the set of TCR signatures (8, 17). Further, the pairing of β with α sequences can be used to allow for more accurate description of the TCR binding specificities (10, 11). This pairing can be obtained through statistical or single cell techniques allowing the most complete modeling of TCR:pMHC restrictions (18–22).

Knowing which pMHC a TCR would bind is a key component toward understanding the mechanisms of T cell immunity. While this can be achieved experimentally, it is an expensive, timeconsuming, and low-throughput procedure (23–26). Given this, it would be of great interest to develop means to predict the cognate pMHC target(s) of a TCR based on its sequence alone. At present, however, resolving this task remains a substantial challenge (10, 11, 27). Recently, machine learning approaches have been described (28, 29) that use sequence-based strategies to infer TCR cognate target, but the performance of these methods is severely limited by the very small volume of existing data associating TCRs with their cognate pMHCs target.

In addition to sequence-based methodologies, approaches based on structural information have also been suggested (30– 32). As the protein structure often is conserved despite of sequence divergence (33), TCR structure modeling could be helpful to compare binding specificities between TCRs with limited sequence similarity. Some studies have shown how 3D models of the structure of the TCR dimer can be used to complement sequence similarity information and in this way improve our understanding of TCR binding specificities (34–36). Several studies have also achieved promising results in modeling structurally TCR:pMHC complexes and using force field energy functions to assess binding between TCRs and their cognate pMHCs (37–41).

Here, we seek to expand these analyses to further address the issue of TCR similarity and the potential impact on this similarity by the different sequence and structural properties of the TCR and CDR loops. We do this in the context of predicting the cognate pMHC target of a TCR using a simple inference-based approach: for a given TCR query, we search a database of TCRs with known pMHC target(s), rank each entry using a measure of similarity, and finally predict the TCR target based on the most similar pMHC in the database. To develop and benchmark this approach, we define a training set using mouse TCRs binding peptides presented by H-2Db and H-2Kb molecules. Next, the model is applied to an independent evaluation dataset of TCRs that bind peptides presented by HLA-A ∗ 02:01. We analyze the effect of predicting TCR targets using only CDR3 β loop sequences compared to using both CDR3s, all CDR loops from the β chain and CDR loops from both the α and β chains in the similarity measure. We explore the effect of combining differentially the CDR sequence similarities to boost the prediction performance of our method. Exploiting the fact that full-length paired TCR sequences allow the construction of TCR homology models, we also build TCR dimer structures and predict TCR binding by the means of CDR loops structural similarity. Next, we investigate how such structural information can complement sequence information to improve TCR target prediction, in particular when no reference sequence with high similarity is available for the target annotation.

# MATERIALS AND METHODS

#### Benchmarks for Mouse and Human Alleles

A data set of TCRs with known binding target and peptide MHC restriction to HLA-A<sup>∗</sup> 02:01, H-2Db, or H-2Kb was obtained from VDJdb (42). Only entries with paired α and β CDR3 loop sequence and corresponding V and J regions annotations were included. Next, to construct full length α and β TCR sequences, V and J sequences were downloaded using their accessions codes from IMGT/GENE-DB (http://www.imgt.org/ genedb/) and CDR3 segments extended by aligning the four residues of the C-terminal end of V region to the four N-terminal residues of CDR3 loop and aligning the four residues of the Nterminal end of J region to the four C-terminal residues of CDR3 loop, for both α and β chains. Next, cross-reactive TCRs (the same α and β sequences assigned to bind multiple and distinct pMHCs) were removed. Redundant entries were removed by clustering at threshold of 99% over the average sequence identity between α and β subunits, and selecting the centroid of each group. An overview of the benchmark construction is shown in **Table 1** and the number of TCRs for each pMHC is detailed in **Table S1**. Starting from 3,112 entries, the final benchmark consisted of 984 TCRs binding to H2-Db and H2-Kb, and 520 that bind HLA-A<sup>∗</sup> 02:01. We used these two datasets for different purposes. The mouse data set was used to develop the best prediction setup, and the human data set was used to evaluate the quality of the model.

### TCR Structural Modeling and Loop Detection

The structure of each TCR was modeled using LYRA (35). For each TCR, templates with more than 70% average sequence


identity between α and β were included in the blacklist form field of the LYRA server to exclude them from the modeling process. Next, the LYRA output was parsed to detect CDR1, CDR2, and CDR3 loops for both α and β chains.

#### TCR Similarity Measures

Three similarity measures were used to identify the cognate pMHC of each TCR: (i) For the global sequence similarity, the sequence identity (SeqID) was calculated separately for the α and β sequences using blast2seq to align the sequences, and was defined for each chain by dividing the number of identical residues by the minimum length between the two aligned chains. (ii) For the CDR sequence similarity, the similarity was calculated by comparing two TCRs using the CDR loops as defined by LYRA annotation. We used the CDR1, CDR2, and CDR3 loops from the α and β subunits. We calculated a similarity between CDRs using the alignment-free Kernel function defined by Shen et al. (43), based on the similarity between all k-mers contained within the sequence of two loops. Briefly, this function is defined as follows: Let B be a BLOSUM62 based similarity measure between two amino acids, as defined by Shen et al. (43) appendix, a similarity between two amino acid sequences u and v of the same length k can be defined as:

$$K\left(\boldsymbol{u},\boldsymbol{\nu}\right) = \prod\_{i=1}^{k} B\left(\boldsymbol{u}\_i,\boldsymbol{\nu}\_i\right)$$

Based on this, the sequence similarity between two CDR loops f and g possibly with different lengths as can be defined as:

$$\operatorname{cdr}\left(f,\emptyset\right) = \sum\_{\substack{\mu \subset f,\,\nu \subset \emptyset\\|\mu| = |\nu| = k\\k = 1,\dots,\min\left(|f|,\lfloor\varrho\rfloor\right)}} K\left(\mu,\nu\right)$$

Then, we normalized this relation as follows:

$$\text{CDR}\left(f,\mathfrak{g}\right) = \frac{cdr\left(f,\mathfrak{g}\right)}{\sqrt{cdr\left(f,f\right)cdr\left(\mathfrak{g},\mathfrak{g}\right)}}$$

This CDR similarity measure is normalized between 0 and 1 and gives higher values for similar sequences. Finally, (iii) for similarity at structure level, we computed the Root Mean Square Deviation (RMSD) between LYRA detected CDR loops. To do this, Superimposer module of Biopython library was used to structurally aligned all the α and β CDR loops simultaneously using the LYRA numbering scheme to match alpha carbon of the loops. After the alignment, the RMSD between pairs of CDR loops was computed using the following procedure:

$$\begin{array}{rcl} \mathtt{proc}\,\,\mathsf{comput}\,\mathsf{enMSD}(\mathsf{cdr1}\,\mathsf{copl}\,\mathsf{1},\,\mathsf{cdr1}\,\mathsf{copl}\,\mathsf{2}): \\ \mathtt{RMSD}\,\,\mathsf{N}=\mathtt{0}/\,\mathsf{0} \\ \mathtt{for}\,\,\mathsf{alpha\\_carbon}\,\,\mathsf{in}\,\mathsf{cdr1}\,\mathsf{copl}\,\mathtt{1}: \\ \mathtt{alph\\_carbon}\,\,\mathsf{2} = \\ \mathtt{lookup\\_nearest\\_ca(\mathsf{alph\\_c}\,\mathsf{l})} \\ \mathtt{carbon1},\,\,\mathsf{cdr1}\,\mathsf{o}\,\mathsf{p}\,\mathtt{2}) \end{array}$$

$$\begin{array}{rcl} \text{alpha\\_carbon\\_prime} & = & \text{alpha\\_nearest\\_ca(alpha\\_} \\ \text{lochup\\_nearest\\_ca(alpha\\_)} & = & \text{alpha\\_carbon\\_1} \\ \text{if.\\_alpha\\_carbon\\_prime} & = & \text{alpha\\_carbon\\_prime} \\ \text{d} & = & \text{euc1.idean\\_distance\\_1} \\ & = & \text{alpha\\_carbon\\_1} \\ & \text{alpha\\_carbon\\_2} & = & \text{d}^2 \\ & \text{NSD} & += & \text{d}^2 \\ & \text{N} & += & 1 \\ \text{return } & (\text{RMSD/N})^{1/2} & 
\end{array}$$

# TCR Target Prediction and Pipeline Validation

As depicted in **Figure 1**, a TCR query is defined as a pair α and β chains. The target of a query TCR, is predicted from the most similar TCR in a database of TCRs with known binding targets. Both query and database TCRs were first modeled using Lyra to identify the CDR loops and the structure of the folded TCR. As shown in **Figure 1B**, we tested the performance of the proposed pipeline in scenarios of varying difficulty when no similar TCRs are available to infer the target of the query. To achieve this, before searching in the database, we removed entries having more SeqID (averaged between α and β chains) with the query than a given cutoff. In order to analyze the performance as a function of the maximum SeqID allowed, we vary this threshold from 70 to 99%. After removal of similar entries, TCRs are ranked with alternative loop weighting schemes with the syntax (1:1:1–1:1:1), where the values in parentheses define the relative weight of each loop. The first triplet identifies the three CDR alpha loops and the second triplet the CDR beta loops. Finally, we assign a pMHC target to the query using the top ranked TCR. We evaluated the pipeline performance at each configuration using Adjusted Rand Index (ARI). ARI is a corrected-by-chance generalization of Matthew's Correlation Coefficient for cases where the data has more than two labels (44, 45). ARI has a value of 1 for perfect predictions, and a value of 0 for the random model. In situations with many labels, the ARI value will often drop substantially below 1, even if a minor subset of predictions is misclassified. Calculations of ARI index were performed using scikit-learn python library.

# RESULTS

In this work, we describe a framework to predict the peptide-MHC (pMHC) binding target of a TCR query based on inference from TCRs with known pMHC binding preference (**Figure 1A**). A query TCR is scored against a database of TCRs with known binding preference, and the pMHC target is inferred from the top-scoring hit. In a first approach, the scoring is based on sequence similarity over the six CDR loops (for details see methods), and in a second model, structure similarity is added to complement TCR linear sequence information.

To assess the impact of the different loops on the predictive power of the model, a series of different weighting schemes were

evaluated (**Figure 1B**). In the simplest scheme, only the CDR3 β loop was included in the model [i.e., weighting scheme (0:0:0– 0:0:1)]. Secondly, we included the full β sequence by adding the CDR1 and CDR2 β loops with weights (0:0:0–1:1:1). In the third model, both α and β subunits were included using either an equal weighting scheme (1:1:1–1:1:1), a scheme with increased CDR3 loops relative weight [(1:1:2–1:1:2) or (1:1:4–1:1:4)], or a scheme with differential weighting between β and α subunits [(1:1:1–2:2:2) or (1:1:1–4:4:4)]. In the case of the global sequence similarity (see methods), a weighting scheme combining α and β subunits was used where SeqID(0:1) stands for using only β subunit, SeqID(1:1) for using both α and β subunits and SeqID(1:2) for doubling the β weight over α.

The results of benchmarking these different models on the mouse benchmark data set are shown in **Figure 2A**. Here, the performance measured in terms of the Adjusted Rand Index (ARI) of each model is shown as a function of the maximum sequence identity (Max SeqID) allowed between the query TCR and TCR database (for details see methods). An example of this is given in **Figure 2B**. Here, the confusion matrix underlying the calculation of ARI is shown for the model CDR(1:1:1–1:1:1) in the situation allowing Max SeqID of 99% corresponding to the extreme right point in the performance curve. The corresponding ARI value is 0.35 and the accuracy 66%.

The performance of each model was tested for a range of maximum sequence identity allowed between the query TCR and TCR database (Max SeqID%) from 70 to 99%. As shown in **Figure S1**, the minimum SeqID% for each TCR to other TCRs binding the same pMHC is below 32%, which means that even if we filter out TCRs that share more than 70% SeqID when we search the TCR database, we will always, for each query, find at least one other TCR sharing the given target. Predicting the correct cognate target should therefore be possible in all cases. Additionally, we evaluated the performance of a random model, assigning a random TCR in the database search and obtained, as expected, an ARI value close to zero for all Max SeqID thresholds (**Figure S2**).

First, we investigated how the predictive performance of the framework was improved as the sequence information included in the model was increased. The prediction model defined by only including the CDR3 loop of the β chain [model CDR(0:0:0– 0:0:1)] had improved performance compared to the model using SeqID with the whole β sequence [SeqID(0:1)]. Adding the CDR1 and CDR2 loops from β subunit to the model [CDR(0:0:0–1:1:1)] led to a general drop in performance compared to using the CDR3 alone (**Figure 2A**). Only for very high similarities (Max SeqID>97%) the performance improved when adding these loops in addition to CDR3, suggesting that incorporation of CDR1 and CDR2 loop similarities might be detrimental to the model. This is further illustrated in **Figure S3**, where we show the confusion matrices for the two models model CDR(0:0:0– 0:0:1) and CDR(0:0:0–1:1:1) evaluated at a Max SeqID threshold of 92%. This figure clearly demonstrates that the fraction of cases with wrongly predicted MHC target is increased for the model including the CDR1 and CDR2 loop information.

Next, we added the paired α sequences to the model. Using the complete α and β sequences [model SeqID(1:1)] led to an improved performance compared to using only the β sequences [model SeqID(0:1)]. Likewise, the model using the α and β CDR3 loops together (model CDR(0:0:1–0:0:1) outperformed the model including only CDR3 β model [CDR(0:0:0–0:0:1)]. This model also outperformed the model including the two full length sequences [model SeqID(1:1)]. When including the CDR1 and CDR2 from both α and β subunits using a (1:1:1–1:1:1) weighting scheme, we observed a general improvement of performance compared to using only the paired CDR3 loop sequences, but also here, we observe a small drop in performance around a Max SeqID of 91% suggesting that a differential weighting would be needed over the CDR3 loop similarity.

Up to this point, we have analyzed the predictive performance as a function of maximum SeqID% allowed between the query TCR and any entry in the TCR database. This approach could clearly be unfair to models based on full length sequence identity such as SeqID(1:1), since we exclude possible database entries based on the same measure used to define the best database target. To assess to what degree this is the case, we assessed the prediction outcome also as a function of CDR3 similarity, incrementally including more similar CDR3 α and β loops while predicting using different weights (**Figure S4**). This benchmark confirmed the earlier conclusions that model CDR(1:1:1–1:1:1) outperformed all other models including SeqID(1:1).

#### Adjusting Weights to CDR Loop Similarity

To further investigate the relative contribution of each CDR loop, we investigated differential weighting schemes for CDR3 over CDR1 and CDR2 loops (**Figure 3**). The schemes are defined using a (1:1:X−1:1:X) scheme varying the relative weight on the CDR3 loop or a (1:1:1–X:X:X) scheme varying the relative weight of the β over the α chain.

We found improvements in the prediction when different weights were applied to the CDR3 loop, and the optimal performance was found for the model CDR(1:1:4–1:1:4). This model outperformed both the flat model [CDR(1:1:1–1:1:1)],

the model with double relative weight on CDR [CDR(1:1:2– 1:1:2)], and demonstrated a monotonic increased in performance from low to high sequence identities. Moreover, doubling and quadrupling the β subunit weight over the α subunit was investigated [models CDR(1:1:1–2:2:2) and CDR(1:1:1–4:4:4)] but these weighting schemes consistently led to decreased predictive power compared to the flat model [CDR(1:1:1–1:1:1)]. Other weighting schemes were investigated but did not lead to consistent improvements in the prediction accuracy (data not shown).

#### Adding Structural Modeling Improves TCR Cognate Target Prediction

We next extended the models to also include structural information. We constructed TCR models using LYRA with templates sharing no more than 70% SeqID with the target to avoid the effect of overfitting in the modeling process. Then, we calculated CDR loops structural similarity by computing the RMSD between two given TCRs and used these loops similarities to predict each query (for details see **Figure 1** and methods). By itself, the structure-based model performed worse than the sequence-based approach described above (**Figure S5A**). Furthermore, the flat model RMSD(1:1:1–1:1:1) outperformed the model RMSD(1:1:4–1:1:4) with differential CDR loop weighting (**Figure S5A**). This observation is most likely due to the fact that CDR3 loops in general are modeled with relative low accuracy, as shown previously by Gowthaman et al. (36), limiting the predictive signal contained within the structure of these loops. Finally, we screened relative weights for combining structural and sequence information in a single model. We integrated sequence and structural similarities with a weight W in the linear model defined below:

$$\begin{aligned} \text{CDR} + \text{RMSD} &= \, ^\ast [1 - \text{CDR}(1:1:4 - 1:1:4)] \\ &+ (1 - \text{W}) ^\ast \, \text{RMSD}(1:1:1 - 1:1:1) / 5.0 \end{aligned}$$

Screening different values of W, the optimal performance was W = 0.9 (**Figure S5B**). The performance of this combined model was only slightly better than the best sequence based model CDR(1:1:4–1:1:4), with a gain more pronounced for lower values of Max SeqID (**Figure 3**). We assessed the significance of this performance gain using bootstrapping, and we found the gain to be statistically significant only at SeqID = 70% (**Figure S6**).

#### Independent Model Evaluation on Human TCR Targets

We now turned to the HLA-A<sup>∗</sup> 02:01 data sets to validate the prediction pipeline and the conclusions obtained from the mouse data. As also observed in the mouse benchmark, the performance using SeqID(1:1) was lower than using CDR similarities (**Figure 4**). Consistently, the differential weighting scheme (1:1:4–1:1:4) resulted in better predictions compared to using the (0:0:1–0:0:1) and (1:1:1–1:1:1) schemes. We assessed the CDR+RMSD model combining sequence and structural information using the relative weight W = 0.9 optimized on the mouse data, and found a significantly (p < 0.04, bootstrap test) improved performance for Max SeqID<72% compared to the CDR(1:1:4–1:1:4) model (**Figure S6**). For Max SeqID in the range 75%<SeqID<90%, model CDR+RMSD slightly outperformed the sequence based CDR(1:1:4–1:1:4) model, but this difference was not statistically significant (p = 0.4, bootstrap test). As expected, the addition of structural information at higher value of Max SeqIDs (Max SeqID>90%), did not improve the predictive power of the model.

As a final remark, we investigated the distribution of prediction accuracy for each peptide at Max SeqID=70% for the combined CDR+RMSD model (**Figure 4B**). It is apparent that the prediction quality varies substantially between peptides. This variation is, to a very high degree, related to the number of TCRs sharing the given peptide target. For instance, the model performs rather poorly for the peptides CVNGSCFTV and YVLDHLIVV, both characterized by a very small number of TCRs sharing them as target. The CINGVCWTV, ELAGIGILTV, GLCTLVAML, and NLVPMVATV entries all share 20 or more TCR entries and the model obtained accuracy values between 40 and 60%. Consistently, for the most populated cases GILGFVFTL and LLWNGPMAV with more than 100 TCRs sharing each peptide, the model obtained an accuracy of 72% (103/144) and 85% (120/142), respectively. These observations underline, as expected, the very high dependency of the accuracy of the proposed modeling framework to the number of TCRs in the database known to bind a given peptide. It also suggests that increasingly accurate predictions will be achievable as the space of pMHC-TCR sequences becomes populated by new experimental data documenting these interactions.

clarity. In parentheses is displayed the average number of TCRs that bind the same peptide and remain after removing entries with Max SeqID > 70%.

# DISCUSSION

The activation of T cells depends on specific interactions between TCRs recognizing peptides presented by MHC. These interactions depend almost exclusively on CDR loops. Generally, analyses of T cell repertoires have been oriented to TCR β chains because obtaining the paired α sequence is more difficult and costly. Further, clonal expansion is often analyzed by the means of sequencing only the CDR3 loop of the TCR β sequence (11, 33). While these constrains on the TCR sequence being generated and analyzed might be justifiable seen from a cost perspective, it is clear that focusing only on the TCR β chain, and in most cases only of the CDR3 β loop potentially has large and limiting implications for the conclusions drawn and information harvested from such TCR sequence data.

We found the predictive power of the model to improve substantially when including the α in addition to the β chain. We also showed that, as expected, focusing on CDR loops rather than the full-length protein sequence led to improved performance. Investigating the relative importance of the different CDR loops for the predictive power of the model, we found an increased performance for models with higher relative weight on the CDR3 loops compared to CDR1 and CDR2. Finally, we demonstrated that the inclusion of structural similarities in the model improved, modestly but consistently, the accuracy of the target prediction, in particular in situations where no sequence with high similarity is available in the TCR database. While being statistically significant, gain in predictive performance obtained by including structural information was limited. We expect this to change, as the accuracy of TCR structural modeling tools improve (in particular for the two CDR3 loops) and the number of available TCR structures (to be used as templates) increases. However, as data available is limited in terms of the diversity and the number different epitopes involved, we find it impossible to draw conclusions about how these interactions mediate different T cell responses. Also, we neither have enough data to tackle the importance of each loop in the recognition of different MHC alleles as we only have enough information about HLA-A<sup>∗</sup> 02:01 for human, and H2-Kb and H2-Db for mouse. As well, we have only MHC class I data, and it would be of great importance to have more MHC class II binding TCRs to get better insights on the difference between CD4 and CD8 T cell interactions with antigens. We hope some day would be more data and more diverse in all of these aspects in order to learn more about the regulation of the immune response.

Predicting TCR cognate targets is a very difficult challenge and the main limit is imposed by the lack of data availability on this huge sequence space. This puts some barriers in our understanding of TCR binding specificities and, the issue gets even more complicated if we try to predict unknown binding specificities. If this problem would be solved, our capability to predict T cell responses would be dramatically improved, but we are still far from achieving it. In the present work, we present a framework to predict specificities to known cognate targets of TCRs using an inference-based model, seeking to understand the importance of using paired TCR sequences.

Despite the very simple modeling approach taken here, these findings clearly demonstrate both that paired full length sequence information is essential for the accurate assessment of TCR function, and that given such information, simple structural, and sequential properties that are common between TCRs that share cognate binding target can be identified. This observation not only underlines the need for the generation of large TCR data sets containing the full information about the triad involved in the TCR:pMHC synapse, using for instance single cell based methods (46), but also suggests that prediction of TCR:pMHC interactions is feasible and thus lays the foundation for the development and application of such models to rational design of T cell based therapies.

It is however critical to stress that due to the availability of data the work and results presented here are limited to the MHC class I and CD8 TCR system. While MHC class II and CD4 TCRs share large structural and functional similarities to this system, several important properties sets them apart—in particular imposed by the longer peptide resituating in the MHC class II binding cleft. Likewise, are the analyses presented limited to cover only three different MHC class I molecules, and certain caution should be taken when extrapolating the conclusions to all class I molecules. However, as more data become available, the framework proposed here can readily be applied to investigate if the presented conclusions are indeed applicable to the general TCRpMHC system.

Finally, it is essential to reiterate that we here have presented a framework to predict cognate targets of TCRs using an inference-based model, seeking to understand the importance of using paired TCR sequence and structural information. Using an inference-based model imposes very large limitations on the applicability of framework for the task of general prediction of the cognate target of TCRs since it depends on the availability of other TCRs sharing the same target, and hence does not allow for true ab initio predictions.

This said, our findings demonstrating an improved predictive power when including information from the α chain in addition to the β chain hold consistently true throughout our benchmark calculation. This important observation not only underlines the need for the generation of large TCR data sets containing the full information about the triad involved in the TCR:pMHC synapse, using for instance single cell based methods (46), but also demonstrates that TCRs with a common cognate target share tractable common sequence and structural properties suggesting that prediction of TCR:pMHC interactions is feasible and thus lays the foundation for the development and application of such models to rational design of T cell based therapies.

#### DATA AVAILABILITY

The datasets for this manuscript are not publicly available because they are already public access data. Requests to access the datasets should be directed to Esteban Lanzarotti, elanzarotti@dc.uba.ar.

# AUTHOR CONTRIBUTIONS

EL performed analysis and drafted the paper. PM and MN participated in the design. MN wrote the final version of the manuscript.

## FUNDING

This work was funded in parts by the Federal funds from the National Institute of Allergy and Infectious Diseases, National Institutes of Health, Department of Health and Human Services, under Contract No. HHSN272201200010C.

## ACKNOWLEDGMENTS

The authors would like to thank Massimo Andreatta, Ph.D. for providing in-depth critics to manuscript and the results presented.

# SUPPLEMENTARY MATERIAL

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fimmu. 2019.02080/full#supplementary-material

Figure S1 | Minimum inner SeqID similarities between the TCRs that bind the same target. (A) Mouse alleles: H-2Kb, H-2Db. (B) Human allele HLA-A∗02:01.

Figure S2 | Prediction pipeline random performance. Random prediction is performed picking a random TCR when we search the database for the TopHit. Error bars are estimated using bootstrap with 1,000 iterations on the final prediction outcome.

Figure S3 | Contingency matrix for the prediction of mouse pMHC binders at a 92% Max SeqID threshold. (A) Prediction performed using only CDR3 β loop [model CDR(0:0:0–0:0:1)] with an Adjusted Rand Index (ARI) equal to 0.14. (B) Prediction performed using CDR1, CDR2, and CDR3 loops weighted equally [model CDR(0:0:0–1:1:1)] with an ARI = 0.04.

Figure S4 | TCR prediction performance as a function of maximum CDR3 similarities. Using increasing CDR similarities from using only CDR3 β, CDR(0:0:0–0:0:1) to all α and β CDR loops, CDR(1:1:1–1:1:1). When we use only β chain we filter sequences using CDR(0:0:0–0:0:1) and when we predict using both chains we filter sequences using CDR(0:0:1–0:0:1). Error bars are estimated using bootstrap with 1,000 iterations on the final prediction outcome.

Figure S5 | Looking for the best weights to combine structural similarity using the mouse benchmark. (A) RMSD prediction performance as a function of maximum SeqID allowed in the database for different weights. (B) Grid search for combined model weight between sequence and structural similarities different SeqID% cutoffs.

Figure S6 | Bootstraping p-values comparing CDR+RMSD against CDR(1:1:4–1:1:4) as a function of maximum SeqID%. Tests were performed with 1000 iterations bootstrapping on the final prediction outcome for both models and p-value is obtained dividing by 1,000 the number of times the ARI value of CDR(1:1:4–1:1:4) resulted better than CDR+RMSD.

Table S1 | Number of TCRs in datasets used for model discovery (mouse) and validation (human).

# REFERENCES


**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2019 Lanzarotti, Marcatili and Nielsen. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# Mechanistic Models of Cellular Signaling, Cytokine Crosstalk, and Cell-Cell Communication in Immunology

#### Martin Meier-Schellersheim<sup>1</sup> \*, Rajat Varma<sup>2</sup> and Bastian R. Angermann<sup>3</sup>

*<sup>1</sup> Laboratory of Immune System Biology, National Institute of Allergy and Infectious Diseases (NIAID), National Institutes of Health (NIH), Bethesda, MD, United States, <sup>2</sup> Xencor Inc., Monrovia, CA, United States, <sup>3</sup> Translational Science and Experimental Medicine, Early Respiratory, Inflammation and Autoimmunity, BioPharmaceuticals, AstraZeneca, Gothenburg, Sweden*

The cells of the immune system respond to a great variety of different signals that frequently reach them simultaneously. Computational models of signaling pathways and cellular behavior can help us explore the biochemical mechanisms at play during such responses, in particular when those models aim at incorporating molecular details of intracellular reaction networks. Such detailed models can encompass hypotheses about the interactions among molecular binding domains and how these interactions are modulated by, for instance, post-translational modifications, or steric constraints in multi-molecular complexes. In this way, the models become formal representations of mechanistic immunological hypotheses that can be tested through quantitative simulations. Due to the large number of parameters (molecular abundances, association-, dissociation-, and enzymatic transformation rates) the goal of simulating the models can, however, in many cases no longer be the fitting of particular parameter values. Rather, the simulations perform sweeps through parameter space to test whether a model can account for certain experimentally observed features when allowing the parameter values to vary within experimentally determined or physiologically reasonable ranges. We illustrate how this approach can be used to explore possible mechanisms of immunological pathway crosstalk. Probing the input-output behavior of mechanistic pathway models through systematic simulated variations of receptor stimuli will soon allow us to derive cell population behavior from single-cell models, thereby bridging a scale gap that currently still is frequently addressed through heuristic phenomenological multi-scale models.

Keywords: computational models, cellular signaling, cytokine crosstalk, multi-scale modeling, rule-based modeling

## INTRODUCTION

Immune cells have been found to play important roles for processes ranging from embryogenesis to tumor clearance to host defense against pathogens (1). What allows them to perform such diverse tasks is the ability to respond to a great variety of different signals, many of which reach them simultaneously (2, 3), and adjust their behavior through communication with other, immune and non-immune, cells (4–6). When their response mechanisms fail to induce the appropriate action, clearance of pathogens or tumor rejection may fail and immune-pathologies such as autoimmune, or inflammatory diseases may develop.

#### Edited by:

*Gennady Bocharov, Institute of Numerical Mathematics (RAS), Russia*

#### Reviewed by:

*Evgeni V. Nikolaev, Rutgers Cancer Institute of New Jersey, United States Michael Loran Dustin, University of Oxford, United Kingdom*

> \*Correspondence: *Martin Meier-Schellersheim mms@niaid.nih.gov*

#### Specialty section:

*This article was submitted to Molecular Innate Immunity, a section of the journal Frontiers in Immunology*

Received: *08 March 2019* Accepted: *09 September 2019* Published: *25 September 2019*

#### Citation:

*Meier-Schellersheim M, Varma R and Angermann BR (2019) Mechanistic Models of Cellular Signaling, Cytokine Crosstalk, and Cell-Cell Communication in Immunology. Front. Immunol. 10:2268. doi: 10.3389/fimmu.2019.02268*

In our efforts to understand immune cell function, the challenge of understanding multi-signal cellular responses or multi-cellular communication and how these integrate at the tissue level is perhaps the most daunting since it seems to go directly against the paradigm of reductionism that has brought forth most of the insights science, not just immunology, is based upon. Indeed, approaching this challenge requires more comprehensive data than classical one-condition-one-readout assays. In model organisms, such as mice, lack of approaches to generate data elucidating cellular behavior under various conditions is no longer the main problem, though. Highly multiplexed assays can be employed and allow us to glimpse into cellular protein expression levels including post-translational modifications (7) and genomic states, increasingly also at the single-cell level (8, 9). Multi-parameterin-vivo microscopy shows us where cells are, where they go and with whom they interact (10–12). However, such data are dots waiting to be connected into mechanistic hypotheses: Even though we may be able to use the data directly to predict disease progression probabilities through artificial intelligence based informatic approaches we need to understand mechanisms to devise therapeutic interventions. Moreover, the invasiveness of many assays prevents us from generating similar data in humans, both in clinical practice or in a research setting. Thus, we are facing the conundrum that we are able to generate highly detailed data but cannot be certain which of the predictions we derived from the data will translate to humans.

Here, we will discuss how the complexity of some of these challenges may be addressed using mechanistic computational models using an approach that can clearly state even complex biological hypotheses involving multiple overlapping signals and, sometimes, may permit to test them directly through simulations. We will first explain how such models can be concise and flexible representations of knowledge and hypotheses and, along the way, demonstrate that these representations can be fully accessible to researchers without modeling experience. Then, we will use the modeling approaches we introduced to investigate multiparametric manipulations of a simple example pathway (a simple model of G-protein coupled signaling and its "pharmacological" manipulation) before illustrating how computational models can be used to explore possible mechanisms of crosstalk in immune signaling pathways. Finally, we will briefly discuss how to extrapolate from single-cell models to models of communicating cell-populations that could serve, for instance, as a basis for more realistic pharmacokinetics-pharmacodynamics (PK-PD) simulations (13) to improve practical applications of basic immunological research.

#### MODELING SIGNALING PATHWAYS BASED ON MOLECULAR INTERACTIONS

Cellular responses toward stimuli they receive emerge from interactions among proteins, lipids, and sugars mediated through specific binding sites. Sequences of such interactions are frequently depicted as networks or pathway diagrams linking, for instance, phosphorylation of a given protein domain to the recruitment of other proteins that subsequently induce or undergo further biochemical modifications. Mathematical and computational models translate such scenarios into quantitative predictions by describing how the abundances of the involved molecules or their post-translational modifications change over time as a result of the interactions among the network's molecules. Depending on the modeling approach, those predictions are generated by solving differential equations or other, sometimes stochastic, algorithms. Many excellent reviews have been written on computational modeling of cellular behavior. See, for instance (14, 15), or (3, 16) for a focus on mathematical modeling approaches in immunology. Ideally, whatever the approach, the mathematical descriptions should not contain more assumptions than the underlying biological hypotheses. One way to achieve this is, perhaps counter-intuitively, to try to model directly the components and interactions within those biological hypotheses, rather than use abstract elements that are introduced for simplification. This avoids introducing properties that do not follow directly from the modeled biology and that may be difficult to spot for non-modelers, in particular when they are formulated in mathematical terms. Moreover, given that cell-biological, immunological, and biochemical research has assembled a wealth of mechanistic insights, it would be unwise not to take as much as possible advantage of prior knowledge about the constituents and interactions that shape signaling pathway behavior. Finally, building models that incorporate details considered important by experimental biologists allows us to convert model behavior directly into experimental assays for validation since model components have real biological counterparts.

Yet, we are typically lacking many of the parameters required for detailed models, such as protein abundances or kinetic interaction rates, and more experimental data than are usually collected would be needed to determine the values of the unknown parameters through fitting (17). This problem is frequently taken as a motivation to resort to models that incorporate pathway structure but not kinetics [for instance in Boolean models (18)] or abandon prior pathway knowledge altogether in favor of extracting only as much information as the data being modeled provide directly (19). Both such approaches have their merits given the problem of "overfitting" in models with large numbers of parameters. But we wish to argue that we can use detailed models in spite of parameter uncertainty simply by asking what kinds of behaviors the models can have when taking into account the possible range of their parameters. Exploring crosstalk among cytokine signaling pathways in T cells, we will show that, in contrast to what many theorists would assume, pathway models based on the description of molecular binding sites can have surprisingly little flexibility in their behavior. Thus, the frequently cited von Neumann quote about the four parameters that can fit an elephant and five that can make him wiggle his trunk does not always apply.

Another potential hurdle when creating detailed, mechanistic models is that they can be rather large and assembling or maintaining them (i.e., adapting them to new hypotheses) can be laborious and error prone. However, the translation of a pathway diagram (which is, in a way, a model) into a formal language can be done automatically nowadays and in a manner that does not modify the biological content. A number of tools have been developed that can perform such automated translations into computer simulations (see, for instance, http://sbml.org/SBML\_Software\_Guide). Among them, "rule-based" approaches permit specifying details such as the binding sites that mediate the molecular interactions (20–22), thereby incorporating aspects that can, for instance, help identify molecular binding motives as potential targets for pharmacological modulation through small molecule inhibitors. Finally, constructing models step-by-step by specifying the interactions among its components and then letting algorithms assemble the computational representations of the resulting networks will allow us to consider models that would be too complex for manual construction because of the number of components or because they span several scales (23) or utilize as experimental input very large data sets, for instance based on proteomic studies (24).

## A SIMPLE EXAMPLE: MODULATING A MODEL OF G-PROTEIN COUPLED RECEPTOR SIGNALING

Cells use G-protein coupled receptors (GPCR) for a wide range of extracellular stimuli, among them such that guide immune cells to and within lymphoid structures (25). While the GPCR themselves have been a frequent target of potential pharmacological manipulation based on molecular structural studies (26) the downstream signaling events present many not fully explored opportunities for modulation (27). GPCR mediated signaling follows a simple common principle (28): A ligand binds to the receptor's extracellular binding site, thereby inducing changes in the accessibility or affinity of intracellular binding sites that can recruit heterotrimeric G-proteins. In complex with the receptor, the α subunit (Gα) of the G-proteins will more readily exchange a GDP (Guanine-diphosphate) group for a GTP (Guanine-triphosphate) group, and as a result, will lose its high affinity for the Gβγ subunit that subsequently will be released and can activate downstream signaling proteins such as, for instance, Ras. Avoiding the need to write equations or computer scripts, we can use an iconographic representation (see **Figure 1A**) to represent the sequence of reactions in a "formal" way just as precisely as differential equations would. In fact, the diagram contains additional information about the interacting binding sites. The modeling software Simmune (22, 29) and a recent extension to the Virtual Cell platform (30) permit using such graphical symbols to specify molecular interactions and illustrates how these interactions are linked in the resulting signaling network. These approaches expand the network beyond the manually specified complexes by determining which complexes can form based on the user-specified bi-molecular interactions. Then, they generate computational representations that can be explored through computer simulations and display time courses for the concentrations of these complexes.

# Simulating and Modulating the GPCR Pathway

Once we have a computational representation of a signaling pathway we can not just simulate the kinetics of the concentrations of the involved molecular complexes and, typically, their post-translational modifications. We can systematically vary the parameters in the model to analyze their influence on the behavior of the modeled system. We might, for instance, ask how the affinity of the G-proteins for the activated receptor or the rate at which Gα switches back to its GDP state (the auto-GTPase activity of Gα) shape the characteristics of the response. Experimenting with these rates in the computational model is far easier than altering molecular properties experimentally in the lab.

The possibility to vary reaction rates and molecular concentrations easily in a computational model becomes particularly interesting when starting with a well-established model, such as the GPCR model here, and adding signaling components to identify which combinations of such additions can be used to achieve a desired type of response—a recurring question for pharmacological research on "small molecule inhibitors." In this example, we added a receptor kinase that phosphorylates the activated receptor and an inhibitor that can associate with the receptor binding site used by the kinase ("RecKin\_Inhib"). Furthermore, we added a molecule ("Gbg\_Inhib") that competes with Gα for binding to Gβγ (and thus interferes with the reassembly of the activatable heterotrimeric G protein complex) and a molecule that competes with Gα for binding to the receptor ("Rec\_Inhib," **Figure 1D**). Varying the concentrations of the three inhibitors, the single response curve shown in **Figure 1C** turns into a series of time courses (**Figure 1F**) for the concentration of free Gβγ, each corresponding to a particular set of inhibitor concentrations. Now, we can analyze which features of the curves are compatible with which ranges for the inhibitor concentration parameters. In the diagram, we selected a region (green square) that corresponds to a strong sustained generation of free Gβγ and find that the inhibitors interfering with the association of Gα and Gβγ need to have a low concentration to allow for efficient activation of the G proteins. On the other hand, the concentration of the inhibitor interfering with the kinase phosphorylating the receptor must be high since phosphorylation deactivates the receptor. In this sense, the inhibitor of the receptor kinase actually strengthens the output (see the figure legend for more details). Whereas, these results are simply what we would have predicted intuitively, they illustrate how features can constrain parameter ranges and how we can map between the two.

# COMPUTATIONALLY EXPLORING CYTOKINE CROSSTALK IN T CELLS

In the previous section, we showed how the features of a simulated model can constrain the ranges of its parameters. In this section, we take advantage of the parameter mapping technique to show that we can identify the limits of what the pathway can do by varying model parameters over a broad range

FIGURE 1 | released from the receptor. (B) Network diagram of a simple GPCR signaling network. Lines connecting different molecules represent possible association or dissociation events. Loops indicate possible state changes, such as the auto-GTPase activity of Gα. Partially filled boxes indicate the presence of states in the model without specifying their values. (C) Simulated response of the signaling network shown in (B) to exposure with the Ligand. The initial increase of the free Gβγ concentration is due to the model equilibration to a homeostatic state. After 50 s the ligand is added to the model and concentration of Gβγ increases as the G-protein rapidly dissociates. After 120 s a virtual wash of the cell is performed, removing the ligand from the simulation. This leads to a recombination of the G protein subunits and thus a reduction of the concentration free Gβγ. (D) Iconographic representation of an inhibitor competing with the recruitment of the G-protein complex to the receptor. (E) Expanded network model including receptor phosphorylation by a kinase and inhibitor molecules interfering with receptor-kinase interaction (dark green molecule), formation of the heterotrimeric G-protein (dark brown molecule) and recruitment of the G-protein to the receptor (orange molecule). (F) 500 simulated responses of the model in (E) to varying inhibitor concentrations. Red lines indicate simulations matching the selection of high Gβγ concentration (green square in upper panel). Empirical cumulative distributions function (Ecdf) of simulation parameters for selected simulations (red), unselected simulations (blue), and total distribution (black). The Ecdf curves are automatically constructed based on the selected curves. The red Ecdf curve increases whenever a parameter value (x-axis) is part of a parameter set that contributes to the selected curves in the upper panel. (G) Network representation of a JAK-STAT signaling network downstream of the IL-4 and IL-7 receptors (IL4Rα and IL7Rα) sharing the common gamma-chain. (H) Simulated behavior of STAT6 phosphorylation of the model in (I) following different doses of IL-7 pre-treatment. Red lines show experimentally observed values and their corresponding parameter distributions in the matching simulations. The inset focuses on the parameter determining the rate of dissociation of the common gamma (CG) chain from the IL7-bound IL7 receptor. The selected phospho-STAT6 levels (red ranges in upper panel) impose clear constraints, ruling out parameter sets with high off rates for the binding between CG and cytokine-bound IL7Rα. (I) Expanding the model in (G) by a JAK1 induced phosphatase acting on both STAT3 and STAT6. (J) The hypothesis of a signal induced phosphatase is inconsistent with experiments, which observed a signal independent decay of STAT6 phosphorylation (indicated by the range between the red lines). In contrast, the simulations predicted at least 10-fold induction of phosphatase activity, as indicated by the lines connecting low and high IL7 stimulus for pairs of simulations that match all other experimental constraints. (K) Predicted dissociation constants for the private receptor chains with the γ-chain in the affinity conversion (light gray) and the ruled-out phosphatase induction models (medium gray and black).

of physiologically plausible values. Exploring these possibilities becomes useful when we want to test whether a model can explain experimentally observed features even if we have only rough estimates for many of its biochemical parameters.

We have recently used this approach (31) to study elements of the common gamma cytokine signaling pathways that are highly important for many aspects of lymphocyte activation, differentiation and survival (32). The cytokine receptors in these pathways all require the common gamma chain (GC) to initiate downstream signals after binding to their specific cytokines, hence the name "common." The fact that GC is shared among multiple receptor systems means that, depending on the amount of GC and the combined number of receptors that can interact with it, the behavior of the downstream signaling pathways that lead to activation of STATs may be affected when several cytokine signals have to be processed simultaneously by the responding cell. Indeed, stimulating CD4 T cells with the common gamma cytokine IL-7 reduced their responsiveness toward IL-4 and IL-21, two other CG dependent cytokines. Experimental determination of cell surface receptor abundances revealed a limited abundance of CG relative to other private receptor chains. Intuitive first explanations for this crosssuppression would thus posit that the limited abundance of CG leads to competition for this rate limiting signaling component. Paradoxically, however, the observed cross suppression was asymmetric as neither IL-4 or IL-21 were able to suppress IL-7 signaling. Further, only a few ligated IL-7 receptors were required to cause suppression of IL-4 signaling leading us to question whether CG was truly limiting. To explore this quantitative riddle and determine whether CG sequestration can, nevertheless explain the crosstalk, we simulated the model with private IL-4 and IL-7 receptor chains and a shared common gamma chain as well as receptor-associated JAKs and STAT6 and STAT5 as downstream targets of IL-4 and IL-7 signaling, respectively (**Figure 1G**). Mapping back from the various experimentally observed suppression strengths for different IL-7 doses we found that the private IL-7 receptor chain needs to have an order of magnitude higher affinity for CG than the private IL-4 chain (**Figure 1K**). Importantly, we found that the IL-7 private chain was required to have a high affinity for CG even before cytokine stimulation (**Figure 1K** light gray bar for unligated IL7Rα), a result which we confirmed experimentally. Previous hypotheses alternatively suggested that CG is associated with private receptor chains prior to cytokine binding (33, 34) or assumed that private receptor chains gain high affinity for CG only subsequent to cytokine binding (35, 36). Our explorations suggested that both are probably true: CG has a substantial preassociation with some private receptor chains that is further increased upon cytokine binding (**Figure 1K** light gray bar for ligated IL7Rα).

# Competing Computational Models of CG Pathway Crosstalk

Being able to modify models with less effort than would be required when writing equations or scripts by hand, we computationally explored other mechanisms that could potentially explain IL-7 induced cross suppression. In particular, we explored the hypothesis whether IL-7 induced phosphatases acting on the JAKs at the receptor level or on the STATs further downstream would be compatible with the experimental data on cytokine induced responses and IL7 mediated suppression. **Figure 1I** shows the model modification that includes such a phosphatase acting on the STATs. Assessing phosphatase activity prior to and after IL-7 stimulation in quantitative experimental assays, we found a much narrower range of activities than would be required for the degree of suppressive crosstalk we had observed (see **Figure 1J**). In summary, combining multi-dose stimulation data with a detailed model and mapping back from simulations that reproduced the data to parameter ranges we identified quantitative relationships between receptor-ligand affinities and were able to reject alternative models that relied on IL-7-induced phosphatases.

# FROM PATHWAYS TO CELLULAR BEHAVIOR AND CELL POPULATIONS

Here, we discussed two strategies for using parameter scans: (i) to explore how model features depend on parameters such as molecule concentrations (how can a model be compatible with the data?) and (ii) to test whether a model can reproduce data at all when allowing the model parameters to vary over a physiologically plausible range (is the model compatible at all with the data?). Both strategies can be used to calibrate or select models at the single cell level (**Figure 2A**). Building on such calibrated models we can sample the input-output behavior of the single-cell models for such combinations of inputs and cellular states (e.g., abundance of cytokine receptors) that would occur in a multi-cellular system with cells that exchange signaling molecules such as cytokines (**Figure 2B**). Such a strategy has recently been explored (37). These input-output patterns could then be considered as a collection of simplified models themselves and thus could be employed on the multi-cellular scale to allow for large numbers of cells within a single simulation. Importantly, however, these simplified models would be based on systematic explorations of mechanistic detailed models as opposed to having been designed as simplified models from the beginning. If a combination of stimuli is outside of the range of the previously performed detailed single cell simulations, the collection simplified models could be extended automatically. Using a similar sampling strategy, we will soon be able to extrapolate toward simulations that can generate realistic behavior of compartments comprising many cells of different

types while exchanging cells and or molecular messengers (**Figure 2C**), as is the case for the lymphatic system. This kind of stepwise coarse-graining will be required to link behavior of cell populations to the scale of single-cellular mechanistic models that not only incorporate the current state of biological knowledge but also will allow us to link pathway modulation (e.g., through small molecule inhibitors) to cell population or even tissue behavior.

#### AUTHOR CONTRIBUTIONS

BA developed the Simmune Analyzer discussed in this manuscript. RV planned and supervised the cytokine crosstalk experiments. MM-S and BA developed and simulated the

#### REFERENCES


computational models. All authors contributed to writing the manuscript.

#### FUNDING

This research has been supported, in part, by the intramural program of the National Institute of Allergy and Infectious Diseases (NIAID), NIH.

#### ACKNOWLEDGMENTS

The authors thank Fengkai Zhang, Thorsten Prustel, and all other members of the Computational Biology Section of the Laboratory of Immune System Biology for helpful discussions.


**Conflict of Interest:** BA is an employee of AstraZeneca. RV is an employee of Xencor.

The remaining author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2019 Meier-Schellersheim, Varma and Angermann. This is an openaccess article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# An Integrated Pipeline for Combining in vitro Data and Mathematical Models Using a Bayesian Parameter Inference Approach to Characterize Spatio-temporal Chemokine Gradient Formation

#### Edited by:

*Gennady Bocharov, Institute of Numerical Mathematics (RAS), Russia*

#### Reviewed by:

*H. T. Banks, North Carolina State University, United States Rory R. Koenen, Maastricht University, Netherlands*

\*Correspondence:

*Bindi S. Brook bindi.brook@nottingham.ac.uk*

*†Joint first authors*

#### Specialty section:

*This article was submitted to Molecular Innate Immunity, a section of the journal Frontiers in Immunology*

Received: *08 April 2019* Accepted: *06 August 2019* Published: *11 October 2019*

#### Citation:

*Kalogiros DI, Russell MJ, Bonneuil WV, Frattolin J, Watson D, Moore JE Jr, Kypraios T and Brook BS (2019) An Integrated Pipeline for Combining in vitro Data and Mathematical Models Using a Bayesian Parameter Inference Approach to Characterize Spatio-temporal Chemokine Gradient Formation. Front. Immunol. 10:1986. doi: 10.3389/fimmu.2019.01986*

#### Dimitris I. Kalogiros 1†, Matthew J. Russell 1†, Willy V. Bonneuil <sup>2</sup> , Jennifer Frattolin<sup>2</sup> , Daniel Watson<sup>2</sup> , James E. Moore Jr. <sup>2</sup> , Theodore Kypraios <sup>1</sup> and Bindi S. Brook <sup>1</sup> \*

*<sup>1</sup> Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, Nottingham, United Kingdom, <sup>2</sup> Department of Bioengineering, Imperial College London, London, United Kingdom*

All protective and pathogenic immune and inflammatory responses rely heavily on leukocyte migration and localization. Chemokines are secreted chemoattractants that orchestrate the positioning and migration of leukocytes through concentration gradients. The mechanisms underlying chemokine gradient establishment and control include physical as well as biological phenomena. Mathematical models offer the potential to both understand this complexity and suggest interventions to modulate immune function. Constructing models that have powerful predictive capability relies on experimental data to estimate model parameters accurately, but even with a reductionist approach most experiments include multiple cell types, competing interdependent processes and considerable uncertainty. Therefore, we propose the use of reduced modeling and experimental frameworks in complement, to minimize the number of parameters to be estimated. We present a Bayesian optimization framework that accounts for advection and diffusion of a chemokine surrogate and the chemokine CCL19, transport processes that are known to contribute to the establishment of spatio-temporal chemokine gradients. Three examples are provided that demonstrate the estimation of the governing parameters as well as the underlying uncertainty. This study demonstrates how a synergistic approach between experimental and computational modeling benefits from the Bayesian approach to provide a robust analysis of chemokine transport. It provides a building block for a larger research effort to gain holistic insight and generate novel and testable hypotheses in chemokine biology and leukocyte trafficking.

Keywords: chemokine transport dynamics, microfluidic device, model validation, Bayesian parameter inference, sequential Bayesian updating, MCMC methods, partial differential equations

# INTRODUCTION

The precisely orchestrated migration of leukocytes plays a key role in all immune and inflammatory responses, including those that take place in infectious diseases. Their guidance to key destinations in tissues such as lymph nodes is coordinated by a group of small, secreted proteins called chemokines. Despite major recent advances in understanding chemokine functions (1–3), it is not yet clear how chemokine gradients are formed, maintained and regulated in tissues. A wide range of transport and biological processes contribute to the establishment, stabilization and regulation of chemokine gradients in interstitial tissue. These include e.g. chemokine production by endothelial cells in lymphatic vessels, chemokine diffusion and advection via interstitial fluid flow, chemokine binding to the extracellular matrix, scavenging of extracellular matrix-bound chemokine by atypical chemokine receptors expressed by macrophages or truncation of chemokines by dendritic cells. Dendritic cells exhibit both chemotaxis (by migrating up gradients of soluble chemokine) and haptotaxis (by migrating up immobilized chemokine gradients). Chemokine truncation or scavenging likely modifies the gradients as the leukocytes migrate, with the potential to affect subsequent leukocyte migration. Multiple cell types, competing interdependent processes and considerably uncertainty in both animal and in vitro models make for a system of such complexity that it cannot be understood using experiments alone (4–6). Mathematical models in combination with experiments can provide a way forward.

A full mathematical model represented by a system of partial differential equations [based on the original models of Keller and Segel (7)] accounting for all of the relevant processes results in a very large number of parameters, most of which have not been estimated from experiments. The predictive power of such mathematical and computational models relies critically on accurate estimates of these parameters. We have thus formulated a strategy to systematically estimate the parameters for the system. This requires the reduction of both mathematical model and corresponding experimental set-up to limit the number of parameters to be estimated at any one time. In this paper we have chosen to focus only on the transport processes associated with chemokine gradient formation. We present an integrated pipeline demonstrating the use of an advection-diffusion mathematical model in combination with measured spatio-temporal chemokine concentration profiles from microfluidic chambers in order to estimate the key transport parameters underlying the formation, development and establishment of chemokine gradients.

To provide a physiologically relevant environment for quantifying chemokine concentration profiles, we have designed a microfluidic chamber enabling the imaging and quantification of the diffusion of fluorescently tagged molecules from sources of low concentrations, similar to those measured in vivo for chemokines of 10–100 nM (8). Microfluidic chambers constructed of Polydimethylsiloxane (PDMS) provide a functional framework for both experimentally forming chemokine gradients and testing their effects on cultured cells. The devices can be imaged microscopically in real time. They feature a central hydrogel region lined by trapezoidal posts, which separate it from fluid channels into which chemokines are pumped. Previous designs have featured a space for deployment of extracellular matrix (ECM) bounded on either side by channels through which fluids containing cytokines can be pumped (9). Pressure differences across the hydrogel can be modulated to generate and control advection. The fluid velocity field across the hydrogel and diffusivity of chemokines within it need to be precisely known for model specification.

The purpose of this paper is to build a Bayesian framework that enables the estimation of these model parameters incorporating an assessment of the uncertainty in parameter estimation. In contrast to the classical frequentist inference approach, Bayesian methodology treats experimental data as a fixed quantity and parameters as random variables drawn from a probability distribution. This allows us to determine the probability of the parameters taking certain values given the observed data. Within this framework, we are able to incorporate prior knowledge about the probability distribution of the parameters which can then be updated through experimental observations. In addition, it allows for the assessment of the reliability of the parameter estimate through quantification of the uncertainty. This is a robust alternative to the traditional frequentist approach which deals with a single "best-fit" and confidence intervals based on potentially unrealistic assumptions in real experimental settings. Employing the Bayesian paradigm also facilitates the design of further experiments by demonstrating which experimental parameters have the greatest uncertainty. The suggested framework is validated by analyzing three datasets (hereafter referred to as DextranI and DextranII and CCL19), which capture the development of gradients of Dextran and CCL19 in microfluidic chambers.

# MATERIALS AND METHODS

# Experimental Set-Up

The experimental data in this paper were obtained by microscopy imaging of Dextran and CCL19 transport in a polydimethylsiloxane (PDMS) microfluidic chip (**Figure 1A**). This chip enables the observation of the transport of fluorescently tagged solutes through a porous hydrogel (10). Here, the solutes were 10 kDa Dextran (ThermoFisher Sci., U.K.), which is of a similar molecular weight as the chemokines CCL19 and CCL21, and the chemokine CCL19 (Almac, U.K.). Both were labeled with the fluorophore Alexa <sup>R</sup> 647 at one fluorophore per diffusing molecule and the hydrogel is collagen type I (Corning, U.S.A.) at 2.0 mg/mL. The fluorescent solution was supplied to an openended channel on one side of the hydrogel by means of a syringe mounted on a precision linear displacement mechanism (World Precision Instruments, model AL4002X). It was transported orthogonally to the supply flow direction into the hydrogel and was washed away by phosphate-buffered saline (PBS) on the opposite side of the hydrogel channel (**Figure 1B**). Dextran was supplied at a concentration of 100 nmoles/L, which is within the range of the concentration of bound CCL21 in lymph nodes in vivo and CCL19 was supplied at 25 nmoles/L, which is

also within its concentration range in lymph nodes (8). The fluorescent intensity across the hydrogel was recorded at intervals of 30 or 120 s from an initial state of no fluorescence and averaged orthogonally using Fiji (11) with a custom Matlab code (MathWorks, Inc., U.S.A.). The fluorescence was also recorded across the source and sink fluid channels (**Figure 1B**) to provide boundary conditions for the posterior analysis.

# The Mathematical Advection-Diffusion Model

In this experimental set-up, the distance between the source and buffer (sink) of the microfluidic device (depicted in **Figure 1B**), is much larger than the gap between the trapezoidal structures at the side of each channel. Thus, we model the transport of Dextran and CCL19 in a one-dimensional domain 0 < x < d denoting the concentration of the solute by C(x, t) where x indicates the distance between the source and buffer with time denoted by t > 0. We assume that the supply of the solute at the source is approximately uniform along the channel, so that longitudinal variations are neglected. The transport of Dextran and CCL19 can, therefore, be described mathematically by the 1D unsteady advection-diffusion equation,

$$\frac{\partial C}{\partial t} = D \frac{\partial^2 C}{\partial x^2} - \mu \frac{\partial C}{\partial x}, \quad 0 < x < d,\tag{1}$$

where D is the effective diffusivity (assumed uniform in the hydrogel) and u is the uniform advection velocity in the x direction, referred to as "advection" for the rest of the paper. Initial conditions for the concentration are extracted from the experimental data such that:

$$\mathcal{C}\left(\mathfrak{x},t\_{0}\right) = \mathcal{C}\_{0}(\mathfrak{x}).\tag{2}$$

We apply the following boundary conditions at the source and buffer:

$$\mathcal{C}\left(0,t\right) = \mathcal{C}\_{\mathfrak{s}}(t) \text{ and } \mathcal{C}\left(d,t\right) = \mathcal{C}\_{b}\left(t\right),\tag{3}$$

with Cs(t) and C<sup>b</sup> (t) specifying the measured time-varying concentration of solute (Dextran and CCL19) at the source and buffer, respectively. We solve Equations (1 − 3) numerically using a finite difference scheme. Central differences are used to discretize the diffusive terms of the equations and secondorder upwinding is used for the advective terms. Time-stepping is performed using the implicit Euler method.

### Integration of Mathematical Model and Experimental Data in a Bayesian Framework

A key objective of this study is to quantify the parameters of diffusivity and advection from the available concentration profiles at each time step (**Figures 2A,B**). Estimation of model parameters consists of evaluating those values of the parameters which maximize the ability of the model (**Figure 2C**) to capture the experimentally observed concentration profiles (**Figure 2B**). We also aim to provide robust, quantitative information on the uncertainty associated with the estimated parameter values (**Figure 3**).

#### Experimentally Measured Initial and Boundary Conditions Incorporated in the Model

The crucial first step was to extract concentration profiles at each time point (**Figure 2B**) from time-lapse image data (**Figure 2A**) using Fiji (11). They were averaged over 300µm orthogonal to the main direction of diffusion and assimilated to fluorophore concentration using an assumption of proportionality between both values. The gray-scale profiles in the dataset at the first time step were used to determine the initial condition (Equation 2) for the mathematical model and the averaged gray-scale values closest to the source and buffer (sink) were used to generate the two boundary conditions (Equation 3) required for the mathematical model. However, the spatial grid and numerical time steps used to solve the discretized model equation do not necessarily coincide with the data points extracted from the imaging data. Therefore, it is convenient to find continuous approximations of the initial and boundary conditions from experimental data. We used linear interpolation for the initial conditions and fitted polynomials for the boundary conditions. Then, these are sampled at the relevant grid points and time steps used in the numerical method to provide the initial and boundary conditions for the model simulations. For each dataset, we evaluated polynomial fits for a range of orders and in each case we chose the lowest-order polynomial that gave a suitable qualitative fit to the experimental data.

For DextranI and DextranII, the initial conditions are derived from the experimental data at t<sup>0</sup> = 120 s (**Figures 4A,C**); for CCL19 they are derived from the data at t<sup>0</sup> = 0 s (**Figure 4E**). The time-varying boundary conditions are given by 5th order polynomials for DextranI (**Figure 4B**) and 7th order polynomials for DextranII and CCL19 (**Figures 4D,F**).

#### The Bayesian Paradigm

The main idea underlying the fitting of the model to data is to identify the parameters that best describe the observed concentration profiles (**Figures 2B,C**). If one were to use a traditional frequentist approach, the best estimates for the model parameters are those for which model and data outputs match as closely as possible, based on some objective function such as the sum of squared differences in the widely used "least squares" optimization technique. The frequentist approach asks the question—given a particular set of model parameters how well do the model solutions fit the experimental data? The Bayesian approach turns this question around: given the experimental data, what are the model parameters that best fit the observations? In addition, assessment of goodness-offit using frequentist approaches relies only on considering whether the data lie within some confidence intervals (with an underlying assumption that the model parameter estimates have an asymptotic Normal distribution). In contrast, the Bayesian approach enables the assignment of a probability distribution to the model parameters (which may or may not be a Normal distribution) and a quantification of the uncertainty associated with the fit (12).

We, therefore, adopt the Bayesian paradigm which enables us to (i) directly and satisfactorily assess the estimates of the model parameters given the observations already made in experiments

FIGURE 1 | (A) Schematic representation of the polydimethylsiloxane (PDMS) microfluidic chip used for obtaining the experimental data. (B) Enlarged representation of the imaged hydrogel section between two open-ended channels. The Dextran diffuses from the one open-ended channel (source) to the other open-ended channel (buffer) and the fluorescent intensity across the distance *x*, with 0 ≤ *x* ≤ *d*, between the source and buffer (sink) fluid channels is recorded at fixed time steps. Based on the design of Farahat et al. (9).

and (ii) quantify the uncertainty of our estimates in a consistent, sound and intuitive probabilistic manner (13, 14). In order to fit the model described in Equation (1) to the fluorescence images at each time step, we assume additive Gaussian noise ε, independent for the experimental observations at each time step, with mean zero and standard deviation σ, i.e. ε ∼ N(0, σ 2 ), so that:

$$
\overline{C}\left(\mathbf{x},t\right) = C\left(\mathbf{x},t\right) + \varepsilon,\tag{4}
$$

where C (x, t) indicates the model-based concentration and C (x, t) denotes the experimental data-based concentration at position x and time t.

Thus, at each time step both transport parameters of diffusivity D and advection u are considered random variables and our prior beliefs about them are formulated into probability distributions, referred to as prior distributions (**Figure 3A**). Based on Bayes' theorem, the experimental data are used

to improve upon our prior belief by multiplying the prior distribution for each of the transport parameters by the likelihood, which describes the probability of a specific parameter value describing the observed data (**Figure 3B**) (15). After normalizing, this leads to the posterior distribution π θ data , i.e.,

$$\pi \text{ (}\theta \left| \left. \theta \right| data) = \frac{\pi \left( \text{data} \middle| \theta \right) \pi \left( \theta \right)}{\int\_{\theta} \pi \left( \text{data} \middle| \theta \right) \pi \left( \theta \right) d\theta} \propto \pi \left( \text{data} \middle| \theta \right) \pi \left( \theta \right), \text{for} \,\theta \in \{D, \ u\} \,, \tag{5}$$

where π(θ) signifies the prior distribution and π(data|θ) indicates the likelihood for each of the model transport parameters, i.e. the diffusivity D and advection u. However, in this study the uncertainty inherent in the experimental data, primarily caused by random error and its associated sources, was not measured directly in the observations and therefore the standard deviation σ of the noise ε also had to be estimated. This leads to the updated version of Equation (5), i.e.

$$\pi \left( \theta \middle| \text{data} \right) = \frac{\pi \left( \text{data} \middle| \theta \right) \pi(\theta)}{\int\_{\theta} \pi \left( \text{data} \middle| \theta \right) \pi(\theta) d\theta} \propto \pi \left( \text{data} \middle| \theta \right) \pi(\theta), \text{for} \,\theta \in \{D, \ u, \ \sigma\} \,. \tag{6}$$

#### Sequential Bayesian Inference of the Model Parameters

In order to accommodate the additional information provided by concentration profiles at different time points, we employ a sequential Bayesian approach. At the first time step, we assume no prior knowledge for the transport parameters of diffusivity D (mm<sup>2</sup> /s) and advection u (mm/s), while for the fluorescence imaging experimental noise some prior knowledge can be assumed. Specifically, at the start we assign a noninformative uniform prior distribution to both non-negative parameters of diffusivity D and advection u (**Figure 3B**) with 0 and 1 as their lower and upper bounds respectively, and a folded Normal distribution with mean zero (Half-Normal) to the non-negative standard deviation σ (arbitrary units based on fluorescence intensity). Thus, for the first time step:

$$D \sim \pi\_1(D) = \ U(0,1) \,,\tag{7}$$

$$
u \sim \pi\_1(\boldsymbol{u}) = \boldsymbol{U}(0, 1)\tag{8}$$

and

$$\sigma \sim \pi\_1 \left( \sigma \right), \text{with } \sigma = \left| \sigma' \right| \text{ and } \sigma' \sim N\left( 0, 1 \right). \tag{9}$$

By updating the prior distributions π<sup>1</sup> (θ) through the likelihood function, which incorporates the information from the

FIGURE 4 | (A) The initial conditions for the model simulations extracted from DextranI (A), DextranII (C) and CCL19 (E) through piecewise linear interpolation of the experimental concentration profile at each point along the channel of width 0.91 *mm* at the initial time *t*0 = 120 *s* for DextranI and DextranII and along the channel of width 0.496 *mm* at the initial time *t*0 = 0 *s* for CCL19. The concentration at the boundaries of the channel (the source and the buffer) was derived from DextranI (B), DextranII (D) and CCL19 (F) (data points marked with crosses) by fitting polynomials of degree 5, 7 and 7 respectively (solid lines) to experimental data before being used as input to the model simulations.

experimental data E<sup>1</sup> = C (x<sup>i</sup> , t = t1): 0 ≤ x<sup>i</sup> ≤ d at the discrete points x<sup>i</sup> at t = t1, Equation (6) leads to the posterior distribution π<sup>1</sup> (θ|E1) which summarizes the information for each parameter θ ∈ {D, u, σ} at the first time step, i.e.

$$D \sim \pi\_1\left(D|\mathcal{E}\_1\right),\tag{10}$$

$$
u \sim \pi\_1 \left( \boldsymbol{u} | \mathcal{E}\_1 \right), \tag{11}$$

and

$$
\sigma \sim \pi\_1 \left( \sigma \left| \mathcal{E}\_1 \right| . \right. \tag{12}
$$

At every subsequent time step n, with n ≥ 2, our knowledge of the parameter of diffusivity D, which is a characteristic quantity of the solute, is mathematically formulated in the prior distribution πn(θ) at the current time step n but it is also included in the posterior distribution πn−<sup>1</sup> (θ|En−1) at the previous time step n − 1. We also assign a uniform prior distribution to advection u, which denotes the advection velocity, as we did for the first time step. Therefore, with the available experimental data En−<sup>1</sup> = C (x<sup>i</sup> , t = tn−1): 0 ≤ x<sup>i</sup> ≤ d at t = tn−<sup>1</sup> we start afresh and write:

$$D \sim \pi\_n(D) = \pi\_{n-1}\left(D|\mathcal{E}\_{n-1}\right) \tag{13}$$

and

$$
\mu \sim \pi\_n(\mu) = \ U(0, 1) \,, \tag{14}
$$

so that Equation (6) yields the following posterior distributions:

$$D \sim \pi\_n \left( D | \mathcal{E}\_n \right), \tag{15}$$

and

$$
u \sim \pi\_n \left( \mu | \mathcal{E}\_n \right). \tag{16}$$

While the above holds for the parameter analysis of DextranII and CCL19 throughout the experiment, in the analysis of DextranI for time step n, with 2 ≤ n ≤ 6, in order to overcome the issue of parameter identifiability, we assign the posterior distribution at time step n − 1 as the prior distribution at time step n for the parameter of advection, i.e., u ∼ π<sup>n</sup> (u) = πn−<sup>1</sup> (u|En−1). Then, for any subsequent time step n, with n ≥ 7, Equations (14) and (16) hold, as explained above.

Since the noise in the fluorescence images was not measured directly, the prior distribution πn(σ) at any subsequent time step n for the standard deviation σ is given by:

which gives rise to the following posterior distribution:

$$
\sigma \sim \pi\_n \left( \sigma \left| \mathcal{E}\_n \right. \right), \tag{18}
$$

$$\sigma \sim \pi\_n(\sigma), \text{with } \sigma = \left| \sigma' \right| \text{ and } \sigma' \sim N(0, 1), \tag{17}$$

where E<sup>n</sup> indicate the available experimental concentration data at time tn. At the first time step, as described above, the initial

conditions are extracted from the data. For any subsequent time step n ≥ 2, the initial conditions are updated using the values of the model parameters estimated through the sequential Bayesian approach which leads to a model-based concentration profile C (x, t = tn−1), at time t = tn<sup>−</sup> <sup>1</sup>.

#### Markov Chain Monte Carlo for Deriving the Posterior Distributions of the Model Parameters

The normalizing constant appearing in the denominator in Equation (5) is a multidimensional integral that can be cumbersome to determine analytically. Instead, simulationbased methods can be used for deriving the posterior distributions for each of the model parameters efficiently. In this study, we use a Markov Chain Monte Carlo (MCMC) algorithm (16) to efficiently generate samples from the posterior distribution which is considered the target distribution in our problem (17). We implement the widely-used random walk Metropolis-Hastings Algorithm (18, 19). The algorithms were implemented in the Python package PyMC which is intended for probabilistic machine learning and Bayesian stochastic modeling employing advanced Markov Chain Monte Carlo and variational fitting algorithms (20) using a Dell R720 with 2 x Intel(R) Xeon(R) E5-2665, 8-core processors and 512 Gb RAM.

The Metropolis-Hastings algorithm draws samples from the posterior distribution for each of the model parameters. Thus, we are able to summarize the posterior distribution and calculate the relevant statistical quantities of interest for each of the inferred parameters. These statistics include the mean, the median, the standard deviation and the Highest Posterior Density (HPD) intervals, which are the credible intervals in our Bayesian analysis.

At each time step n our prior knowledge for each transport parameter was updated through the posterior distribution at the previous time step n-1, as explained previously. However, the probability density functions of the posterior distributions resulting from the MCMC sampling are approximated well by a gamma distribution Ŵ (α, β), with the shape parameter α and the rate parameter β evaluated as follows (21):

$$E\left(\theta\right) = \frac{\alpha}{\beta} \tag{19}$$

and

$$\operatorname{Var}\left(\theta\right) = \frac{\alpha}{\beta^2}\ ,\tag{20}$$

with the mean E (θ) and the variance Var (θ) already known from the Bayesian statistical analysis for each transport parameter θ, with θ ∈ {D, u}.

#### RESULTS

The results of the Bayesian parameter analysis provide us with posterior distributions for each model parameter at each time point. For DextranI, representative posterior distributions at t = 600 s and t = 2,640 s are shown in **Figure 5A**, for DextranII representative posteriors at t = 480 s and t = 1,440 s are depicted in **Figure 8A** and for CCL19 representative posteriors at t = 60 s and t = 120 s are given in **Figure 11A**. These plots show that the hereby presented analysis provides us not only with a single point estimate (the median values of the distributions) for each model parameter at each time but also enables us to quantify the uncertainty connected with each one of them. In fact, at a single time point these plots can interpret graphically all the summary statistics for each one of the inferred parameters D, u, σ contained in the **Supplementary Material Tables 1–3** for DextranI, **Supplementary Material Tables 4**–**6** for DextranII and **Supplementary Material Tables 7**–**9** for CCL19. These summary statistics include measures of location (mean, median), measures of spread (standard deviation) as well as measures of confidence that the value of a parameter as estimated through its posterior distribution lies within a HPD (Highest Posterior Density) interval with 95% probability. **Supplementary Material Tables 1**–**9** show that the values of median and mean for the model parameters consistently lie within the 95% HPD intervals at every time step. The Bayesian parameter analysis performed in this study satisfies certain convergence criteria (see **Supplementary Material** for results related to convergence, mixing and autocorrelation) thus

allowing for efficient sampling of the posterior distribution for each model parameter at each time step.

In order to evaluate the predictability of the model and its ability to extract reliable values for the transport parameters, we use summary statistics of the posterior distributions of the estimated parameters as inputs into the mathematical model. Although following the analysis of the available datasets the median equals the mean of the posteriors for the vast majority of the time steps, we choose the median in order to account for the cases where the posterior distribution is skewed. The median

values for each of the parameter distributions are then substituted in the mathematical model to simulate the concentration profiles (red curves in **Figure 5B** for DextranI, **Figure 8B** for DextranII, and **Figure 11B** for CCL19) corresponding to each time point for which in vitro concentration profiles were extracted (blue curves in **Figure 5B** for DextranI, **Figure 8B** for DextranII, and **Figure 11B** for CCL19). **Figures 5B**, **8B**, **11B** show that at each time step the inferred transport parameters lead to a very good overall fit of the model consistently for all datasets. While for DextranI and CCL19 the fit is excellent at all time steps, some discrepancies between the data-based and the model-based concentration profiles are more clearly detected in DextranII at t = 720 s and t = 1,200 s (**Figure 8B**). The difference at these time points is a result of the poor polynomial fit to the boundary conditions at the corresponding time points (**Figure 4D**).

By fitting the model to experimental data at each time step we are also able to estimate the variation of the transport parameters over the course of the experiment (**Figure 6** for DextranI, **Figure 9** for DextranII and **Figure 12** for CCL19). The median values of diffusivity varied between 10−5mm<sup>2</sup> /s and 10−<sup>4</sup> mm<sup>2</sup> /s (**Figures 6A**, **9A**, **12A**). Based on the parameter estimation analysis, the advection across hydrogel varies over time (**Figures 6B**, **9B**, **12B**) due to limitations in the advection control in the microfluidic chamber.

different time points of DextranII.

Finally, we show that the probability density functions of the distributions are well approximated by a gamma distribution at each time step as explained in the Markov Chain section above. For all the datasets, **Figure 7** (DextranI), **Figure 10** (DextranII) and **Figure 13** (CCL19) show the evolution of the posterior distributions for the estimated transport parameters of diffusivity and advection over the duration of the experiments. The range of the distribution at later time steps changes, because knowledge about the estimated parameter at the previous time step is incorporated by informing the prior distribution for the next time step. These figures also provide a sound argument

to the above conclusion regarding the overall range of the diffusivity and advection over time guaranteeing that they are not distributed over multiple orders of magnitude.

# DISCUSSION

This study illustrates a robust parameter estimation approach that greatly facilitates the use of mathematical modeling in extracting quantitative information about key mechanisms from experimental data in chemokine biology. The inclusion of biologically relevant parameters, including the statistically sound evaluation of their experimental uncertainty and variability, is crucial in modeling efforts to describe chemokine transport phenomena. This truly enables the model equations to represent the functional mechanisms in a manner that will appropriately represent the in vivo reality.

every 30 *s* from 30 *s* to 120 *s*. (B) The estimated median values resulting from the posterior distribution for the advection *u* (*mm*/*s*) are plotted against time every 30 *s* from 30 *s* to 120 *s*. (C) The estimated median values resulting from the posterior distribution for the standard deviation σ (arbitrary units (a.u.) based on fluorescence intensity) at each time step are plotted against time every 30 *s* from 30 *s* to 120 *s*.

The example of parameter estimation shown here demonstrates an integrated pipeline for estimating key transport parameters from in vitro data using a mechanistic advectiondiffusion model. The Bayesian framework not only produces an overall good fit of the model to the experimental datasets but it also allows for diffusivity and advection to be estimated robustly. The resulting estimations of diffusivity for Dextran varied between 10−<sup>5</sup> and 10−<sup>4</sup> mm<sup>2</sup> /s and were close to the values of diffusivity predicted or measured in other ways. Indeed, AL-Barati et al. (22) and Takanori et al. (23) measured the diffusivity to range from 10−<sup>5</sup> and 10−<sup>4</sup> mm<sup>2</sup> /s depending on the experimental conditions such as temperature. These values are also close to the Stokes diffusivity. Regarding the estimation of diffusivity of CCL19, these values are coherent with the theoretical Stokes diffusivity of 1.3 x 10−<sup>4</sup> mm<sup>2</sup> /s

gamma distributions to the posterior distributions of advection *u* (*mm*/*s*) at the different time points of CCL19.

for A647-labeled CCL19 in water, calculated for an average molecular weight of 11.5 kDa for the fluorescently labeled chemokines (manufacturer batch documentation). The effective diffusivity in porous media is expected to be up to an order of magnitude lower than this estimated value. Similarly, the order of magnitude of the advection velocity is 10−<sup>4</sup> mm/s, i.e. a Péclet number lower than 1. This corresponds to the lower range of interstitial fluid velocities and is coherent with the fact that these data were obtained in devices intended for diffusive transport only. Because of the difficulty in balancing the system pressures, there was some variability in the advection velocity over time and this is captured by the parameter estimation algorithm. Diffusivity should not vary with time, so our estimates plateau out over time to the most representative value. The observation of advection variation over time is used in a feedback process for the refinement of the microfluidic chamber design. Its design aim is to enable precise and constant advection across the hydrogel, and the parameter estimations performed here help identify sources of error in the advection control strategy.

Fluorescence image noise is assumed to be independent for each time point, so it does not plateau. In addition, there were no data available about the fluorescence imaging experimental noise, which is quantifiable through the standard deviation σ (arbitrary units based on fluorescence intensity) as explained above and mathematically formulated in Equation (4). Although experimental noise is not known a priori (since we do not have multiple experimental repeats), our methodology enables us to estimate it. This is because our approach allows it to be treated as an extra parameter which can be inferred in tandem with both transport parameters successively throughout the duration of the experiment. The fact that our estimate for the noise was nominally about 1% of the fluorescence signal indicates that the data are of good quality.

This study also shows that Bayesian parameter analysis provides accurate posterior inference for all the estimated parameters at each time point during the course of the experiment. The framework provides point estimates of the three parameters of interest and assesses the uncertainty associated with each one by quantifying the corresponding statistical distribution. The resulting uncertainties in estimating diffusivity and advection are most likely a result of spatial variability due to hydrogel density variation and fluorescence imaging noise.

It is also worth noting that the initial and boundary conditions for the model simulations are extracted from the experimental data thus adding to the physical relevance of the estimated parameters of mathematical models and the reliability of the parameter inference approach itself. However, at certain time steps in one of the datasets (DextranII) the polynomial fit to the boundary condition fluorescence data was sufficiently poor to create disagreement with the model-based concentration profiles. Spline interpolation may be used as an alternative to address this issue.

The experimental set-up presented here is a prototype which only accounts for transport phenomena without incorporating binding kinetics. In future, the integrated pipeline for parameter estimation will be expanded to more complex experiments which also allow for binding kinetics, dynamic interactions between physical, biological, biochemical processes and cellular uptake. We will further perform experiments with different chemokines, as this could provide a broader understanding of chemokine

#### REFERENCES


gradient establishment and help stratify chemokines into relevant groups with respect to their gradient forming characteristics. This will also provide further support for the applicability and scalability of this integrated pipeline, since a quantitative understanding of a system with the complexity of chemokine transport dynamics requires not only a series of reductionist experimental approaches but also the ability to construct mathematical models with powerful prediction capabilities. The robust model parameter determination algorithm presented here provides the necessary foundation for this combined approach contributing to the emergence of a better knowledge base of the chemokine system and leukocyte trafficking. Thus, predictive modeling will provide invaluable insights into the potential therapeutic benefits of modulating immune response.

#### DATA AVAILABILITY

Data will be made available on request.

#### AUTHOR CONTRIBUTIONS

TK, BB, and JM designed the study. DK and MR developed the code and performed the simulations. WB, JF, and DW conducted the experiments and extracted the data. All authors contributed toward manuscript writing and revisions.

### FUNDING

This work was supported by the Sir Leon Bagrit Trust and Wellcome Trust Collaborative Award 206284/Z/17/Z.

#### ACKNOWLEDGMENTS

We would also like to thank R. J. Nibbs and his group for supporting the development of the chemokine model and the Imperial College FILM facility.

#### SUPPLEMENTARY MATERIAL

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fimmu. 2019.01986/full#supplementary-material

Curr Opin Biomed Eng. (2018) 5:90–5. doi: 10.1016/j.cobme.2018. 03.001


**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2019 Kalogiros, Russell, Bonneuil, Frattolin, Watson, Moore, Kypraios and Brook. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# Interleukin-15 Signaling in HIF-1α Regulation in Natural Killer Cells, Insights Through Mathematical Models

Anna Coulibaly 1†, Anja Bettendorf 2†, Ekaterina Kostina2,3, Ana Sofia Figueiredo1‡ , Sonia Y. Velásquez <sup>1</sup> , Hans-Georg Bock <sup>2</sup> , Manfred Thiel <sup>1</sup> , Holger A. Lindner <sup>1</sup> \* and Maria Vittoria Barbarossa<sup>3</sup> \*

#### Edited by:

Gennady Bocharov, Institute of Numerical Mathematics (RAS), Russia

#### Reviewed by:

Carmen Molina-Paris, University of Leeds, United Kingdom Marcus Rosenblatt, University of Freiburg, Germany

\*Correspondence:

Holger A. Lindner holgera.lindner@ medma.uni-heidelberg.de Maria Vittoria Barbarossa barbarossa@uni-heidelberg.de

†These authors have contributed equally to this work

#### ‡Present address:

Ana Sofia Figueiredo, Department of Modeling of Biological Processes, COS/BioQuant, Heidelberg University, Heidelberg, Germany

#### Specialty section:

This article was submitted to Molecular Innate Immunity, a section of the journal Frontiers in Immunology

Received: 30 November 2018 Accepted: 25 September 2019 Published: 16 October 2019

#### Citation:

Coulibaly A, Bettendorf A, Kostina E, Figueiredo AS, Velásquez SY, Bock H-G, Thiel M, Lindner HA and Barbarossa MV (2019) Interleukin-15 Signaling in HIF-1α Regulation in Natural Killer Cells, Insights Through Mathematical Models. Front. Immunol. 10:2401. doi: 10.3389/fimmu.2019.02401 <sup>1</sup> Department of Anesthesiology and Surgical Intensive Care Medicine, Medical Faculty Mannheim, University Medical Center Mannheim, Heidelberg University, Mannheim, Germany, <sup>2</sup> Interdisciplinary Center for Scientific Computing, Heidelberg University, Heidelberg, Germany, <sup>3</sup> Institute for Applied Mathematics, Heidelberg University, Heidelberg, Germany

Natural killer (NK) cells belong to the first line of host defense against infection and cancer. Cytokines, including interleukin-15 (IL-15), critically regulate NK cell activity, resulting in recognition and direct killing of transformed and infected target cells. NK cells have to adapt and respond in inflamed and often hypoxic areas. Cellular stabilization and accumulation of the transcription factor hypoxia-inducible factor-1α (HIF-1α) is a key mechanism of the cellular hypoxia response. At the same time, HIF-1α plays a critical role in both innate and adaptive immunity. While the HIF-1α hydroxylation and degradation pathway has been recently described with the help of mathematical methods, less is known concerning the mechanistic mathematical description of processes regulating the levels of HIF-1α mRNA and protein. In this work we combine mathematical modeling with experimental laboratory analysis and examine the dynamic relationship between HIF-1α mRNA, HIF-1α protein, and IL-15-mediated upstream signaling events in NK cells from human blood. We propose a system of non-linear ordinary differential equations with positive and negative feedback loops for describing the complex interplay of HIF-1α regulators. The experimental design is optimized with the help of mathematical methods, and numerical optimization techniques yield reliable parameter estimates. The mathematical model allows for the investigation and prediction of HIF-1α stabilization under different inflammatory conditions and provides a better understanding of mechanisms mediating cellular enrichment of HIF-1α. Thanks to the combination of in vitro experimental data and in silico predictions we identified the mammalian target of rapamycin (mTOR), the nuclear factor-κB (NF-κB), and the signal transducer and activator of transcription 3 (STAT3) as central regulators of HIF-1α accumulation. We hypothesize that the regulatory pathway proposed here for NK cells can be extended to other types of immune cells. Understanding the molecular mechanisms involved in the dynamic regulation of the HIF-1α pathway in immune cells is of central importance to the immune cell function and could be a promising strategy in the design of treatments for human inflammatory diseases and cancer.

Keywords: HIF-1α, IL-15, STAT3, NF-κB, mTOR, natural killer cells, parameter estimation, mathematical model

# 1. INTRODUCTION

As effector lymphocytes of innate immunity, natural killer (NK) cells are involved in the host defense against microbial infections and cancer (1). Sensing their environment, NK cells respond to cellular alterations including those caused by infections, cellular stress, and transformation (2).

Interleukin-15 (IL-15), produced by monocytes, macrophages and dendritic cells, critically regulates NK cell survival and activation (3, 4). While expression of IL-15 is low under homeostatic conditions, it is upregulated in inflammation (5). Upon receptor binding, IL-15 initiates Janus kinase/signal transducer and activator of transcription (JAK/STAT) signaling. This promotes growth of NK cells and enhances their ability to respond to activation. Activated NK cells infiltrate tissues containing pathogen-infected or malignant cells, resulting in their recognition and direct killing (6–8).

Sites of infection or cellular transformations are often characterized by inflammatory hypoxia. Thus, NK cells must adapt and respond under conditions of low oxygen tension. The critical cellular dependence of survival on oxygen led to the early evolution of adaptive cellular responses to hypoxia. Cellular adaptation to hypoxia is primarily orchestrated by the hypoxia inducible factor (HIF) family of transcription factors (9). To date, three HIF family members have been identified (HIF-1, HIF-2, and HIF-3) of which HIF-1 is the best characterized (10). Two subunits, HIF-1α and HIF-1β, form the transcriptionally active HIF-1 complex. The α-subunit is post-translationally hydroxylated by oxygen-sensitive prolyl hydroxylases (PHDs) which mark the protein for ubiquitination and continuous proteasomal degradation. A decrease in cellular oxygen availability stabilizes HIF-1α allowing its dimerization with HIF-1β. The dimer translocates to the nucleus, binds to hypoxia-response elements in promoters of adaptive genes, and activates their expression.

In immune cells, including T lymphocytes and myeloid cells, cellular activation of HIF-1α has also been reported to occur in an oxygen-independent manner during inflammation triggered by infection and cancer, and to involve transcriptional in addition to post-translational mechanisms (11). At sites of tissue damage and infection, both inflammation and decreased oxygen availability result in HIF-1α stabilization and its nuclear translocation. Recent insights from models of solid tumors in mice with an NK cell specific knockout of the HIF-1α gene and from chemical inhibition of HIF-1α in human NK cells (12, 13) suggest that HIF-1α limits NK cell anti-tumor activity.

In the past 15 years, several mathematical models for HIF-1α regulation based on systems of ordinary differential equations (ODEs) have been proposed (14–19). A review up to 2013 is given in (20). Nguyen and coauthors (19) have investigated the dynamics of the HIF-1α pathway, combining a mathematical mechanistic model and experimental analysis for human embryonic kidney 293 (HEK-293) cells. Their model studies accumulation of HIF-1α in hypoxia and its degradation in normoxia, considering hydroxylation of HIF-1α mediated through both prolyl hydroxylases and asparaginyl hydroxylase FIH (factor inhibiting HIF). Fábián et al. (10) have highlighted the importance of using system biology and mathematical modeling for understanding HIF signaling. Although lacking the comparison with experimental data, Fábián's models allowed to test different hypotheses on the HIF network, concluding that the negative feedback induced by PHDs plays a major role in triggering oscillations in the HIF-1α dynamics.

This work combines mathematical modeling and experimental analysis to understand processes regulating the levels of HIF-1α mRNA and protein in NK cells. The proposed mathematical model considers key features of HIF-1α regulation and is formulated as a system of non-linear ODEs with positive and negative feedbacks. In our in vitro studies, we isolated human peripheral NK cells and studied their behavior simulating hypoxic and inflammatory conditions, which were produced by the hypoxia-mimicking agent dimethyl-oxalyl glycine (DMOG) and the pro-inflammatory cytokine IL-15, respectively. Experimental trials were designed to collect time series data of HIF-1α protein expression and its upstream regulators in order to calibrate the mathematical model. Parameter estimation was performed by means of numerical methods based on a multiple shooting approach for dynamic systems and a generalized Gauss-Newton method for optimization. Our approach does not only explain experimental observations on HIF-1α dynamics but also allows to ask questions and test hypotheses with the help of in silico experiments. For example, we investigated how HIF-1α levels depend on the regulation of other upstream proteins, and identified the signal transducer and activator of transcription 3 (STAT3), the mammalian target of rapamycin (mTOR) and the nuclear factor-κB (NF-κB) as critical regulators. Further, we studied HIF-1α stabilization in dependence of DMOG-mediated PHD/FIH inhibition, determining a non-linear relation between HIF-1α levels and DMOG concentration. Our model provides new insights into the mechanisms mediating accumulation of HIF-1α in NK cells, by (i) highlighting the synergistic effects of IL-15 and chemical hypoxia, and (ii) suggesting that NF-κB and STAT3 are fundamental regulators of IL-15 induced HIF-1α enrichment.

# 2. MATERIALS AND METHODS

#### 2.1. NK Cell Purification and Cell Culture

The study was reviewed and approved by the Medical Ethics Commission II of the Medical Faculty Mannheim, Heidelberg University (2014-500N-MA). NK cells were isolated from buffy coats obtained through the local Red Cross Blood Donor Service (NK-Cell Isolation Kit, Miltenyi Biotec GmbH, Bergisch Gladbach, Germany). The purity of NK cells was determined by flow cytometry.

Freshly isolated NK cell preparations with a phenotype of ≥95% CD56+CD3<sup>−</sup> and ≤1% each CD3+, CD14+, CD15+, and CD19<sup>+</sup> were judged as pure and were further cultivated as previously described (21). In brief, cells were plated at a density of 10<sup>6</sup> cells/mL in RPMI 1640 medium (Sigma-Aldrich Chemie GmbH, Merck KGaA, Darmstadt, Germany) supplemented with 10% fetal bovine serum (FBS) and 2 mM L-glutamine and maintained in a standard tissue culture incubator (37◦C, 5% CO2, 21% O2, normoxia, standard condition). The cell permeable panhydroxylase inhibitor DMOG (Selleck Chemicals, Houston, TX, USA) was used to mimic hypoxia. The viability of the cells was determined by tryptan blue staining and was ≥95% (Countess, Invitrogen, ThermoFisher, Waltham, MA, USA).

#### 2.2. In vitro Treatments

Freshly isolated NK cells were maintained overnight under standard conditions and were stimulated with human recombinant IL-15 (45 ng/mL, PeproTech, NJ, USA), DMOG (20 µM, Selleck Chemicals), rapamycin (25 nM, Merck Chemicals GmbH, Darmstadt, Germany), STAT3 inhibitor (S3I-201, 200 µM, Merck Chemicals GmbH), or DMSO (Sigma-Aldrich Chemie GmbH) as control, on the next day for the indicated time periods. Protein concentrations in cell lysates were determined on a Direct Detect <sup>R</sup> infrared spectrometer (Merck Millipore) according to the manufacturer's instructions.

#### 2.3. Western Blotting

Total cell extracts were prepared by resuspending 3×10<sup>6</sup> NK cells in 100 µL NP-40 lysis buffer (50 mM Tris-HCl, pH 7.5, 120 mM NaCl, 20 mM NaF, 1 mM EDTA, 6 mM EGTA, 15 mM sodium pyrophosphate, 1 mM PMSF, 0.1% Nonident P-40). Fifteen minutes of cell lysis on ice was followed by centrifugation for 20 min at 14,000 × g. Cleared lysates were analyzed directly by SDS-PAGE and Western Blotting. Briefly, equal amounts of protein were separated by SDS-PAGE, transferred to nitrocellulose membranes (Thermo Fisher), blocked in 5% dry milk powder dissolved in 1×PBS-T, and then probed with primary antibody and HRP-conjugated secondary antibody (Santa Cruz Biotechnology, Dallas, TX, USA). Proteins were visualized using Enhanced Chemiluminescent solution (Thermo Fisher) and FUSION Vilber imager (Eberhardzell, Germany). The intensity of signals was quantified by densitometric analysis using the image analysis software ImageJ (Version 1.51j8). The value for HIF-1α was normalized to that for β-Actin. Anti-HIF-1α (# 2185) was obtained from Abcam (Cambridge, UK) and Anti-Actin (8H10D10) from Cell Signaling Technology (Frankfurt am Main, Germany). Representative experiments out of three performed are shown.

#### 2.4. MILLIPLEX Immunoassay

The MILLIPLEX MAP Multi-Pathway Signaling Phosphoprotein Kit 48-680MAG was used according to the manufacturer's protocol (Merck Millipore). Total NK cell extracts were diluted with MILLIPLEX MAP Assay Buffer to reach the protein concentration of 10 µg of total protein/well. Mixed magnetic beads were added to each well. To appropriate wells, 25 µL of Assay Buffer (background control), 25 µL of NK cell sample lysates and 25 µL of control cell lysates were added in duplicates. The plate was sealed and incubated overnight (20 h) at 4◦C on a plate shaker (750 rpm). After incubation and washing, 25 µL of Detection Antibody were added to each well. This incubation step was followed by addition of 25 µL Streptavidin-Phycoerythrin and incubation on the shaker. After resuspending the beads in 150 µL of Assay Buffer, the plate was read on a MAGPIX system (Luminex). Signals of phosphorylation of STAT3 and AKT were expressed as background-corrected median fluorescence intensities.

# 2.5. Modeling the Regulatory Network

The mathematical approach used in this study is based on a system of autonomous non-linear ODE, which can be in general written as

$$\begin{cases} \boldsymbol{\nu}'(t) &= \boldsymbol{f}(\boldsymbol{y}, \boldsymbol{u}, \boldsymbol{p}), \\ \boldsymbol{\nu}(t\_0) &= \boldsymbol{\nu}\_0(\boldsymbol{p}). \end{cases} \quad t \in [t\_0, t\_f] \subset \mathbb{R} \tag{1}$$

with states y, controls u and parameters p. The vector y(t) ∈ R ny indicates the "state of the system," that is, the concentration of the considered proteins, complexes and mRNA at time t ∈ [t0, t<sup>f</sup> ] ⊂ R, and y<sup>0</sup> ∈ R ny is the initial state. The parameter vector p ∈ R <sup>n</sup><sup>p</sup> contains non-negative constants describing the biochemical reaction rates (such as production, degradation, binding, etc.) in the system. The time-dependent experimental controls u(t) ∈ R <sup>n</sup><sup>u</sup> represent cell treatments, specifically with IL-15, DMOG, or other protein inhibitors. In this work we do not discuss basic theoretical properties of the solutions to system 1, such as existence and uniqueness of a global solution, or invariance of the positive cone of R ny . All these properties can be proven by applying elementary results and methods in ODE theory [see e.g., (22)], and we assume them to hold true in this manuscript. In the following we explain in detail the model assumptions and the resulting equations. Model variables and parameters are given in **Table S1**, and **Tables 1**, **2**, respectively. A diagram of the regulatory network is shown in **Figure 1**.

Recent results (24, 25) showed the connection between IL-15 and mTOR activity in NK cells, indicating that the AKTmTOR pathway is indispensable for efficient cell activity and immune functions of NK cells. We therefore focused on the latter signaling pathway, neglecting other cascades [such as Ras-Raf-MEK and JAK/STAT5 (26)] which are also known to be initiated by IL-15. Further, IL-15 stimulation in neutrophils and human peripheral blood lymphocytes has been shown to activate NF-κB and STAT3 (27–29). All in all, we assumed that IL-15 (y1) activates AKT (y2), NF-κB (y7), and STAT3 (y8). For a general formulation we further assumed that IL-15 enters the system at constant rate a<sup>1</sup> and decays at rate d1, that is, the IL-15 dynamics is given by

$$
\chi\_1'(t) = a\_1 - d\_1 \chi\_1. \tag{2}
$$

Activation of AKT is assumed to occur via IL-15 (30) (activation rate k1), other external mediators (basal activation rate a2), and also via STAT3 (maximal rate kS) (31). We further assumed that AKT constantly decays at rate d2, yielding

$$y\_2'(t) = a\_2 + k\_1 y\_1 + k\_8 \frac{y\_8^{n\_2}}{\xi\_{28}^{n\_2} + y\_8^{n\_2}} - d\_2 y\_2. \tag{3}$$

Following the literature (32) we assumed that AKT activates mTOR (y3) at rate k2. mTOR basal activation and decay rate are denoted by a<sup>3</sup> and d3, respectively. The inhibitory effect that hypoxia has on mTOR (33) is included in the model by means of TABLE 1 | Parameter description and values used for the mathematical model (2)–(11).


In the left column we report the parameter value used for the numerical simulations. In the right column we indicate whether the parameter value has been fixed because of previous literature, or pre-fitted and then fixed because of sensitivity analysis results (cf. section 2.7).

a negative feedback regulated by the HIF-1 complex (y6). Hence, for the mTOR dynamics we obtained

$$
\gamma\_3'(t) = \left(a\_3 + k\_2 \wp\_2\right) \frac{\alpha\_1}{\alpha\_2 + \wp\_6} - d\_3 \wp\_3. \tag{4}
$$

We denote phosphorylated STAT3 by y8. STAT3 basal activation and decay rates are a<sup>8</sup> and d8, respectively. Both IL-15 and mTOR are known to induce phosphorylation of STAT3 (29, 34), here assumed to occur at rate k<sup>6</sup> and k8, respectively. All in all, the differential equation for STAT3 reads

$$
\chi\_8'(t) = a s + k s \chi\_3 + k\_6 \chi\_1 - d s \chi\_8. \tag{5}
$$

The last protein upstream of HIF-1α that we considered in our model is NF-κB, for which we assumed basal activation (at rate a7) and decay (d7). NF-κB is further activated via IL-15 (27, 28) (activation rate k7), via mTOR (35) (k15) and via the HIF-1 complex (36, 37) (k14), yielding

$$y\_7'(t) = a\_7 + k\_7 y\_1 + k\_{14} y\_6 + k\_{15} y\_3 - d\_7 y\_7. \tag{6}$$

HIF-1α mRNA basal synthesis and degradation are defined to occur at rate a<sup>9</sup> and d9, respectively. Further, we assumed that HIF-1α mRNA is regulated by NF-κB (at rate k9) and STAT3 (at rate k3),

$$
\chi'\_9(t) = a\_9 + k\_9 \wp\_7 + k\_3 \wp\_8 - d\_9 \wp\_9. \tag{7}
$$

Following previous results (19), we assumed that asparaginyl hydroxylase FIH is at steady state (ϕ), whereas PHDs are TABLE 2 | Parameter description and values used for the mathematical model (2)–(11).


In the right column we report the estimated parameter value, indicating mean and standard deviation (s.d.).

upregulated by HIF-1 complex and we approximated their dynamics with quasi-steady state assumptions (see section S1 for detailed explanation). Further, we assumed that HIF-1α mRNA is translated at rate k<sup>α</sup> and HIF-1α protein decays at rate d4. We denote by KO<sup>2</sup> the oxygen-dependent binding force of FIH/PHD and HIF-1α (cf. section S1). In normoxia, HIF-1α is hydroxylated via FIH (assumed at maximal rate k10) and via PHD (maximal rate k13). The dynamics of HIF-1α protein (y4) is thus given by

$$y\_4'(t) = k\_{a}y\_9 - d\_4y\_4 - k\_{4}y\_4y\_5 + k\_5y\_6 - k\_{13}K\_{\mathcal{O}2}(\Delta y\_6 + a\_{11})\frac{y\_4}{\xi\_{44} + y\_4}$$

$$-k\_{10}K\_{\mathcal{O}2}\varphi \frac{y\_4}{\xi\_4 + y\_4} + k\_{11}y\_{10}.\tag{8}$$

In accordance with previous studies (19, 38), we assumed that asparaginyl-hydroxylated HIF-1α (HIF-1α-aOH, here denoted by y10) can be subsequently hydroxylated via PHD and then degraded, whereas prolyl-hydroxylated HIF-1α (HIF-1α-pOH) is quickly degraded. We henceforth neglected the dynamics of HIF-1α-pOH. Further, we assumed that there is some probability for HIF-1α-aOH dehydroxylation [cf. (19)]. The resulting dynamics of HIF-1α-aOH is given by

$$y\_{10}'(t) = k\_{10} K\_{\text{O}\_2} \varphi \frac{\wp\_4}{\xi\_4 + \wp\_4} - k\_{12} K\_{\text{O}\_2} (\Delta \wp\_6 + a\_{11}) \frac{\wp\_{10}}{\xi\_{10} + \wp\_{10}} \tag{9}$$

$$-k\_{11} \wp\_{10} - d\_{10} \wp\_{10}. \tag{9}$$

HIF-1β (y5) is constitutively expressed by the cells (synthesis rate a5), independently of the oxygen conditions (10). In hypoxia, HIF-1α accumulates and binds to HIF-1β (at rate k4) forming the transcriptional complex HIF-1, which can dissociate (rate k5). Hence, for HIF-1β and the HIF-1 complex we obtained the equations,

$$
\psi\_5'(t) = a\_5 - k\_4 \wp\_4 \wp\_5 + k\_5 \wp\_6 - d\_5 \wp\_5,\tag{10}
$$

and

$$
\chi\_6'(t) = k\_4 \chi\_4 \chi\_5 - k\_5 \chi\_6 - d\_6 \chi\_6,\tag{11}
$$

respectively. For model calibration and comparison with collected experimental data we extended the model (2)–(11) to include DMOG or other protein inhibitors. Details are given in the **Supplementary Material** (section S2).

FIGURE 1 | Diagram for HIF-1α regulatory network in NK cells, corresponding to model Equations (2)–(11). We study the interplay of HIF-1α, IL-15, mTOR, NF-κB, and STAT3 in normoxia and hypoxia. A signaling cascade starting with IL-15 activates NF-κB, STAT3, and AKT. This in turn activates mTOR, influencing HIF-1α mRNA and HIF-1α protein levels. In normoxia, HIF-1α is hydroxylated via FIH and PHD. Here, we consider only the FIH-mediated asparaginyl-hydroxylated HIF-1α (HIF-1α-aOH) and assume that a small fraction of HIF-1α-aOH can be dehydroxylated. In normoxia, hydroxylated HIF-1α is degraded via Von Hippel-Landau protein (not considered in the model). In hypoxia, HIF-1α accumulates and binds to HIF-1β, building the HIF-1 complex. The latter is responsible for both a positive and a negative feedback on HIF-1α, via activation of NF-κB and via inhibition of mTOR and upregulation of PHD. Black arrows indicate protein activation, translation or formation/dissociation of protein complexes. For simplicity, basal degradation of molecules is not depicted here. The blind-ended arrow indicates inhibition. Blue arrows indicate external regulation due to further stimuli not specifically considered in the mathematical model. Yellow arrows indicate HIF-1α hydroxylation via FIH or PHD.

# 2.6. Numerical Simulations

For the numerical integration of the non-linear ODE system (2)– (11) and numerical simulations shown in this manuscript we used the Runge-Kutta formula (4,5) and (3,2) in MATLAB <sup>R</sup> version 9.4 [routines ode45 and ode23s (39)], as well as a multistep Backward Differentiation Formula (BDF) method with variable step size and order control. The latter was implemented by mean of the solver DAESOL (40, 41) in VPLAN (42), a software for simulation, parameter estimation and optimum experimental design for non-linear processes described by differential equations.

We assumed that the initial load of IL-15 in primed cells is y1(0) = 1, whereas for unstimulated cells y1(0) = 0. Collected experimental data for HIF-1α, STAT3 and AKT were normalized with respect to measurements in untreated cells at the beginning of each experiment and used for model calibration. In setting the initial conditions, we normalized AKT, mTOR, NF-κB, STAT3, HIF-1α mRNA and HIF-1β with respect to the concentration at the beginning of each in silico experiment, meaning that yj(0) = 1, for j = 2, 3, 5, 7, 8, 9. The total HIF-1α level was normalized with respect to the initial measurement, corresponding to untreated cells in normoxia, hence we set y4(0) + y6(0) + y10(0) = 1. As cells were pre-cultivated in normoxia, we assumed that at the beginning of our observations (t = 0 h) most of HIF-1α is hydroxylated, hence, we set y4(0) = 0.05, y6(0) = 0.05 and y10(0) = 0.9.

# 2.7. Model Calibration

#### 2.7.1. Comparison With Experimental Data

For comparison with experimental data, the solution y(·) = y(·|u, p) of the mathematical model (1) is associated with an m-vector of observables,

#### g(t, y(t), u, p).

Typically it is not possible to observe all states, and in general g(·) is a non-linear function de-pending on the states and parameters. Given an experimental setting u ex(·), for each observable gi(t ex j , y ex(t ex j ), u ex , p), i = 1, . . . , m, at measurement times t ex <sup>j</sup> ∈ [t0, t<sup>f</sup> ], j = 1, . . . , k ex, experimental data η ex ij are collected in each experiment ex. Experimental measurements contain additive noise

$$
\eta\_{ij}^{\text{ex}} = \lg(t\_j^{\text{ex}}, \nu^{\text{ex}}(t\_j^{\text{ex}}), \mu^{\text{ex}}, p) + \varepsilon\_{ij}^{\text{ex}}, \tag{12}
$$

where the errors ε ex ij are assumed to be statistically independent and normally distributed with zero mean value and variances (σ ex ij ) 2 .

The experimental settings considered in this study present "perturbation" experiments, ex = 1, . . . , ne, which allow to investigate perturbations of the cellular processes from their equilibrium conditions. Each perturbation experiment corresponds to a different control u = u ex in the mathematical model (1), and the same quantities are measured in each experiment. It is biologically reasonable to assume that an unperturbed system is at the steady state. This corresponds in our case to unstimulated NK cells (u ex = u 0 ), hence to the initial condition y0. The steady state can be mathematically determined considering the dynamics of untreated cells and setting this in equilibrium (43, 44). Hence, the initial condition of the system y<sup>0</sup> and the model parameters must satisfy the steady state constraint,

$$0 = f(\wp\_0, \mu^0, p). \tag{13}$$

#### 2.7.2. Parameter Estimation Problem

In general, given a mathematical model (1) and experimental measurements, the goal of model calibration is to determine the model parameters in (1) from the collected data. This reduces to an optimization problem of minimizing the discrepancy between model observables and the experimental data using a particular metric (45). The weighted least squares functional is known to deliver a maximum likelihood estimate for the unknown parameters (46, 47). For the calibration of the parameters of the regulatory network (2)–(11) we used a weighted l2-norm of the measurement errors,

$$l\_2(\boldsymbol{\wp}, \boldsymbol{\wp}) := \frac{1}{2} \sum\_{i, \boldsymbol{i}, \boldsymbol{\epsilon} \boldsymbol{\chi}} \frac{(\eta\_{i\boldsymbol{j}}^{\boldsymbol{\varepsilon}\boldsymbol{\varepsilon}} - g\_i(t\_{\boldsymbol{j}}^{\boldsymbol{\varepsilon}\boldsymbol{\varepsilon}}, \boldsymbol{\wp}^{\boldsymbol{\varepsilon}\boldsymbol{\varepsilon}}(t\_{\boldsymbol{j}}^{\boldsymbol{\varepsilon}\boldsymbol{\varepsilon}}), \boldsymbol{\mu}^{\boldsymbol{\varepsilon}\boldsymbol{\varepsilon}}, \boldsymbol{\wp}))^2}{(\sigma\_{i\boldsymbol{j}}^{\boldsymbol{\varepsilon}\boldsymbol{\varepsilon}})^2},\tag{14}$$

and further included a priori information p<sup>0</sup> by adding a Tikhonov regularization term (48)

$$L(p, p^0, \lambda) := \frac{1}{2} \sum\_{\tilde{m} = 1}^{n\_p} \frac{(p\_{\tilde{m}} - p\_{\tilde{m}}^0)^2}{\lambda\_{\tilde{m}}^2},$$

where the vector λ ∈ R <sup>n</sup><sup>p</sup> controls the amount of regularization per parameter. Moreover we incorporated additional information about parameters and states (initial conditions, steady states, etc.) in the parameter estimation problem by formulating equality constraints (49, 50). We estimated parameters by solving the following multiple experiment parameter estimation (PEP) problem

$$\text{(PEP)} \quad \left\{ \begin{array}{l} \min\_{\mathbf{y}(\cdot),\mathbf{p}} \frac{1}{2} \sum\_{i,\mathbf{j},\mathbf{x}} \frac{(\eta^{\mathbf{x}\mathbf{x}}\_{i} - \eta\_{i}(t^{\mathbf{x}\mathbf{x}}\_{j},\mathbf{y}^{\mathbf{x}\mathbf{x}}(t^{\mathbf{x}\mathbf{x}}\_{j}),\boldsymbol{\mu}^{\mathbf{x}\mathbf{x}},\boldsymbol{\rho}))^{2}}{(\sigma^{\mathbf{x}\mathbf{x}}\_{\boldsymbol{\beta}})^{2}} + \frac{1}{2} \sum\_{\tilde{m}=1}^{n\_{P}} \frac{(\mathbb{p}\_{\tilde{m}} - \boldsymbol{p}^{0}\_{\tilde{m}})^{2}}{\lambda^{\mathbf{x}}\_{\tilde{m}}},\\ \text{s.t.} \quad \left(\boldsymbol{\mathcal{y}}^{\mathbf{x}\mathbf{x}}\right)'(\mathbf{t}) = f(\boldsymbol{\mathcal{y}}^{\mathbf{x}\mathbf{x}}, \boldsymbol{\mu}^{\mathbf{x}\mathbf{x}}, \boldsymbol{\rho}), \quad \mathbf{t} \in [t\_{0}, t\_{f}], \quad \boldsymbol{\mathcal{y}}^{\mathbf{x}\mathbf{x}}(t\_{0}) = \boldsymbol{\mathcal{y}}^{\mathbf{x}\mathbf{x}}\_{0}(\boldsymbol{\rho}),\\ \quad \quad r^{\mathbf{x}\mathbf{x}}(\boldsymbol{\mathcal{y}}^{\mathbf{x}\mathbf{x}}(t\_{1}^{\mathbf{x}\mathbf{x}}), \boldsymbol{\mu}^{\mathbf{x}\mathbf{x}}(t\_{2}^{\mathbf{x}\mathbf{x}}), \dots, \boldsymbol{\mathcal{y}}^{\mathbf{x}\mathbf{x}}(t\_{k}^{\mathbf{x}\mathbf{x}}), \boldsymbol{\rho}) = 0, \boldsymbol{\epsilon}\mathbf{x} = 1, \dots, n\_{\mathbf{c}}. \end{array} \} (10.10.1)$$

#### 2.7.3. Numerical Methods for Parameter Estimation

For least squares minimization, as those in (PEP), a frequently adopted approach is the derivative-based iterative Gauss-Newton method (45, 51). In this work we applied an "all-at-once" parameter estimation method based on a direct multiple shooting approach for dynamic systems (51) and a generalized Gauss-Newton method for optimization (50, 51). This is a boundary value problem approach, in which the system of differential Equations (1) is discretized including boundary conditions. The discretized system is treated as a non-linear constraint of the least squares objective function (52). For the multiple shooting approach, a suitable grid of multiple shooting nodes was chosen and, at each multiple shooting grid point, the values of the state variables were included as additional optimization variables. On each subinterval an additional initial value problem was solved. To maintain the continuity and feasibility of the solution, we included additional matching conditions (50, 52). The splitting of the integration interval leads to a numerically stable system (52). The resulting finite dimensional non-linear constrained least squares problem can be formally written as

$$\min\_{s,p} \frac{1}{2} ||F\_1(s,p)||\_2^2, \text{ s.t.} \quad F\_2(s,p) = 0,\tag{15}$$

where the constraints F2(s, p) = 0 include the multiple shooting parameterization of the dynamical model and s denotes the vector of states in the parametrized model. The problem (15) was solved by a generalized Gauss-Newton method. At each iteration of the Gauss Newton method

$$s^{k+1} = s^k + t^k \Delta s^k, \quad p^{k+1} = p^k + t^k \Delta p^k, \quad t^k \in [0, 1]$$

the increments 1s k , 1p k solve the linearized problem

$$\begin{cases} \min\_{\Delta s, \Delta p} & \frac{1}{2} ||F\_1(s^k, p^k) + \frac{\partial F\_1(s^k, p^k)}{\partial s} \Delta s + \frac{\partial F\_1(s^k, p^k)}{\partial p} \Delta p||\_2^2, \\ & \text{s.t.} \quad F\_2(s^k, p^k) + \frac{\partial F\_2(s^k, p^k)}{\partial s} \Delta s + \frac{\partial F\_2(s^k, p^k)}{\partial p} \Delta p = 0. \end{cases} \tag{16}$$

For the case considered in this work, the linearized problem (16) shows special structures due to multiple experiments and multiple shooting approaches. These structures are efficiently exploited in a tailored linear algebra method for the solution of (16). A numerical analysis of the well-posedness of the problem and an assessment of the error of the resulting parameter estimates were performed at the solution of the problem (PEP), based on the analysis of the corresponding sensitivity (or Jacobian) matrix J

$$J = J(s, p) = \begin{pmatrix} J\_1(s, p) \\ J\_2(s, p) \end{pmatrix} = \begin{pmatrix} \frac{\partial F\_1(s, p)}{\partial s} & \frac{\partial F\_1(s, p)}{\partial p} \\ \frac{\partial F\_2(s, p)}{\partial s} & \frac{\partial F\_2(s, p)}{\partial p} \end{pmatrix}. \tag{17}$$

Further details can be found in (49, 50). In particular, we computed the linear approximation of the variance-covariance matrix for the constrained parameter estimation problem (PEP)

$$\mathbf{Cov} = \mathbf{Cov}(\mathbf{s}, \mathbf{p}) = \boldsymbol{J}^{+T} \begin{pmatrix} \text{diag}\{\boldsymbol{\sigma}\_{ij}^{\text{ex}}\}\_{i, j, \text{ex}} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} \end{pmatrix} \boldsymbol{J}^{+},\tag{18}$$

and the standard deviations of states and parameters as square root of the corresponding diagonal elements of the matrix Cov. In (18) the matrix J <sup>+</sup> denotes the generalized inverse of the sensitivity matrix J(s, p), that is J <sup>+</sup>JJ<sup>+</sup> = J, and **0** denotes the zero matrices of the corresponding dimensions.

The above sketched methods are implemented in the software VPLAN (42), which includes the parameter estimation software PARFIT (53, 54). VPLAN was used for parameter estimation in this study.

#### 2.7.4. Initial Guesses

Before starting the actual optimization procedure (PEP), we determined initial guesses in a reasonable scale for the parameters that needed to be estimated. Good initial guesses are important for convergence of the parameter estimation methods used in this work. In certain cases prior information p 0 for initial guesses is found in the literature (cf. **Table 2**). When results published in previous studies did not help, we applied homotopy related methods (55) and pre-estimated the parameters using Wolfram Mathematica <sup>R</sup> version 11.3 with the Ndsolve and manipulate routines, as well as DAESOL in VPLAN.

#### 2.7.5. Sensitivity Analysis

In a second step, before starting parameter estimation, we performed a local sensitivity analysis at parameter values p 0 and corresponding solutions y ex(·|u ex , p 0 ), ex = 1, ..., n<sup>e</sup> of ODE systems. The goal of the local sensitivity analysis is to find those parameters, which can be estimated reliably with the model and the given measurements. The local sensitivity analysis is performed using the sensitivity matrix J <sup>0</sup> = J(s 0 , p 0 ). If the sensitivity matrix J <sup>0</sup> has almost linear dependent columns, then it is ill-conditioned and the parameter vector is badly identifiable (48, 56). We computed the singular value decomposition of the sensitivity matrix J 0 . The reciprocal of the minimal singular value yields the collinearity index, which, if it is large, indicates that the sensitivity matrix has almost linear dependent columns (48, 56, 57). In this work we set a minimal threshold 0.1 for rejecting small singular values and for determining the subset of parameters corresponding to almost linear independent columns of J 0 . This allowed to fix 13 parameters which correspond to almost linear dependent columns of J 0 . The remaining 25 parameters are reliably identifiable and were estimated by mean of (PEP).

#### 2.8. Model Robustness

We implemented extrinsic and intrinsic stochastic perturbations using a Monte Carlo analysis. For extrinsic perturbations we varied the input stimulation IL-15 (y1(t)) ±25% around its original value. We measured the effect of these perturbations on total HIF-1α in the model accounting for DMOG treatment and IL-15 stimulation at the steady-state level. For intrinsic perturbations we varied specific parameters (a2, a3) and measured the effect of these perturbations on total HIF-1α in the model accounting for DMOG treatment and IL-15 stimulation at the steady state level. We implemented the intrinsic and extrinsic stochastic perturbations by varying the specific elements 25% around their original value and sampling 1000 times from a uniform distribution. The Monte Carlo analysis was implemented in Wolfram Mathematica <sup>R</sup> version 11.2.

# 3. RESULTS

#### 3.1. IL-15-induced HIF-1α Protein Accumulation in Peripheral NK Cells

In cells exposed to hypoxia, the stabilization and activation of HIF-1α is well characterized (58). Instead of manipulating the oxygen tension to induce HIF-1α, pharmacological inhibitors of HIF-1α hydroxylation can be used as well. We used a cell-permeable pan-hydroxylase inhibitor, DMOG, to inhibit oxygen-sensitive hydroxylases that target HIF-1α for proteasomal degradation and its transcriptional inactivation. After preincubation under normoxia for 16 h, peripheral NK cells were stimulated with the pro-inflammatory cytokine IL-15 in the presence of DMOG for different time periods. Whole cell extracts were prepared and the response of NK cells to IL-15 stimulation and DMOG treatment was assessed by evaluating the expression of HIF-1α measured by Western Blot analysis (**Figure 2**). The expression of β-actin was monitored to confirm equal loading.

HIF-1α expression was barely detectable in the first 3 h of IL-15 and DMOG stimulation. However, after 4 h we detected accumulation of HIF-1α protein, which further increased over 8 h and was maintained to at least 27 h, although to a lesser extent than at 8 h (**Figure 2A**).

IL-15 signaling in NK cells through the kinase mTOR has previously been reported to be essential for their expansion in the bone marrow and sustained activation (24). Moreover, among other signaling pathways, the PI3K/mTOR pathway has been linked to the induction of HIF-1α protein expression in immune cells, including T lymphocytes (59). To study the role of mTOR in HIF-1α protein expression in NK cells, we stimulated the cells with IL-15 and DMOG in the presence or absence of pharmacological mTOR inhibitor rapamycin. As shown in **Figure 2B**, mTOR inhibition reduced HIF-1α levels. Nevertheless, HIF-1α signals remained detectable, pointing to other upstream regulators of HIF-1α protein accumulation in IL-15 stimulated NK cells.

#### 3.2. Model Parameters

The model (2)–(11) has been calibrated on time series for AKT, STAT3 and HIF-1α collected from NK cells under different experimental conditions (**Table S2**), namely under stimulation/treatment with: (i) IL-15; (ii) DMOG; (iii) IL-15 + DMOG + rapamycin; (iv) DMOG + IL-15 + S3I-201. We interpret (i)–(iv) as "perturbation experiments" from the initial (equilibrium) condition. Biological sense and previous literature on mathematical modeling of cellular dynamics (43, 44) suggest that it is important to assume that untreated cells are in the steady

state. Such an assumption yields 10 algebraic equations (steady state constrains) on the parameter, which together with the collected time series data of the four perturbation experiments (39 data points, cf. **Table S2**) were used to calibrate the model. **Table 2** reports the 25 estimated parameters and **Figure 3A** shows the model fit. Model parameters either estimated or fixed from previous literature are reported in **Tables 1**, **2**. With these parameter values we ran an in-silico experiment for NK cells stimulated with IL-15 and treated with DMOG. The obtained numerical simulations were used to validate our model, comparing the model predictions with collected experimental data for AKT, STAT3 and HIF-1α time series of NK cells stimulated with IL-15 and treated with DMOG (**Figure 3B**). To depict the statistical significance of the parameter estimates, **Table 2** also reports the standard deviation for each estimated parameter value.

#### 3.3. The Mathematical Model Explains the Dynamics of HIF-1α Accumulation

Besides the known hypoxia-induced HIF-1α stabilization and AKT-mTOR-mediated increase in protein translation, HIF-1α can also be induced through increased transcription involving activated transcription factors, among others STAT3 as shown in T lymphocytes (60) and B lymphocytes (61). To determine the role of AKT, mTOR and STAT3 as mediators of HIF-1α accumulation downstream of IL-15, we collected time series data (**Figure 3**, **Table S2**) for the phosphorylation status of AKT (Ser473) and STAT3 (Ser727), representing the activated forms of the proteins (62, 63). For optimal parameter estimation, we collected data from NK cells isolated from blood of the same donors, cultured in normoxia, chemical hypoxia (DMOG) and treated with a STAT3 or mTOR inhibitor.

The model is able to reproduce data collected for HIF-1α, STAT3 and AKT in different experimental settings (**Figure 3**). In particular, model predictions match time series data of HIF-1α protein expression and indicate that simultaneous exposure of NK cells to IL-15 and DMOG (**Figure 3B**) increases the levels of total HIF-1α, compared to HIF-1α levels in cells either stimulated with IL-15 or treated with DMOG (**Figure 3A**). Moreover, inhibition of mTOR or STAT3 leads to reduction of HIF-1α levels, suggesting that both proteins are involved in the regulation of IL-15 induced HIF-1α accumulation in DMOG treated cells. Model assumptions and calibration results (cf. **Tables 1**, **2**) indicate that the external regulation of IL-15, NF-κB, STAT3, and HIF-1αmRNA is negligible in order to explain collected time series in NK cells under the proposed experimental setting. Further, parameter estimation and model discrimination (results not shown here) suggest that DMOG reduces IL-15-mediated STAT3 activation (see section S2).

The collected data (cf. **Table S2**) further show that IL-15, DMOG or inhibition of either mTOR or STAT3 does not affect the levels of phosphorylated AKT (Ser473) in human NK cells. After preliminary steps in the parameter estimation procedure, we obtained k<sup>1</sup> ≈ 10−<sup>5</sup> , k<sup>S</sup> ≈ 10−<sup>4</sup> (cf. **Table 1**). The sensitivity analysis which followed indicated that the two values have small effects on the objective function in (PEP). This suggests that, the order of magnitude of k<sup>1</sup> and k<sup>S</sup> being much smaller than those of all other parameters, the two parameters could be set to zero without much affecting the simulation results, and the AKT dynamics in Equation (3) can be simplified and described by a linear equation,

$$
\wp\_2'(t) = a\_2 - d\_2 \wp\_2.
$$

With parameter values as indicated in **Tables 1**, **2** we ran numerical simulations of model (2)–(11) for NK cells under different experimental conditions. **Figures 4**, **5** show the simulation results for all<sup>1</sup> model components in normoxia without (N, O<sup>2</sup> = 21%) and with DMOG (20 µM, chemical hypoxia), treated with one (A panels) or two inhibitors at the same time (B panels). The level of total HIF-1α, which we define as the sum of HIF-1α protein, HIF-1 complex and HIF-1α-aOH, is significantly higher in DMOG treated cells than in untreated cells in normoxia. The major component

<sup>1</sup>The dynamics of IL-15 is trivial and neither affected by oxygen saturation nor other inhibitors (in the model assumptions there is no feedback on IL-15, cf. Equation 2), hence it is equivalent in all experimental conditions and omitted in **Figures 4**, **5**. The same holds for the AKT dynamics in view of the results of the sensitivity analysis.

FIGURE 3 | The mathematical model (2)–(11) explains collected time series for HIF-1α, STAT3, and AKT. Parameters used for numerical simulations are given in Tables 1, 2. (A) Model calibration results: comparison of numerical simulations (continuous curves) and collected experimental data (dots ± S.E.) of total HIF-1α (y<sup>4</sup> + y<sup>6</sup> + y10, red curves/dots), STAT3 (blue) and AKT (black). The model is fitted to data collected in different experimental settings: (upper left) IL-15-stimulated NK cells; (upper right) DMOG treated cells; (lower left) IL-15-stimulated NK cells treated with DMOG and STAT3 inhibitor (S3I-201); (lower right) IL-15-stimulated cells treated with DMOG and with mTOR inhibitor rapamycin (Rapa). (B) Model validation results: comparison of numerical simulations (continuous curves) and collected experimental data (dots ± S.E.) of total HIF-1α (y<sup>4</sup> + y<sup>6</sup> + y10, red curves/dots), STAT3 (blue), and AKT (black) for IL-15-stimulated cells treated with DMOG. Experimentally collected data points are reported in Table S2.

of the sum in DMOG treated cells is HIF-1α, whereas its hydroxylated form predominates in normoxic cells. As expected, we observe that HIF-1β is stable in normoxia. However, in the presence of available HIF-1α, HIF-1β is consumed for HIF-1 complex formation.

Consistent with the model assumptions, we observe mTOR inhibition by rapamycin. Moreover, the stabilization of HIF-1α in DMOG treated cells and the subsequent formation of HIF-1 results in a negative feedback on mTOR (**Figure 5**). NF-κB shows higher activity in DMOG treated cells compared to untreated cells and our simulations predict an essential role for NF-κB as a regulator of HIF-1α-mRNA and protein in DMOG treated cells. In contrast, the IL-15-induced STAT3 activity is higher in cells without DMOG and inhibition of STAT3 results in an important reduction of HIF-1α-mRNA and protein levels. Combined inhibition of both transcription factors abolishes HIF-1α enrichment in both, DMOG treated and untreated cells.

#### 3.4. Regulators of HIF-1α Enrichment

Parameter values used for data fitting (**Table 1**) in **Figure 3A** indicate that the external regulation rates of IL-15, mTOR, and STAT3 are small or negligible in the considered cell cultures. Nevertheless, such parameters could change from donor to donor, in particular if affected by inflammatory conditions or cancer (5, 64).

To investigate the influence of external regulators on the behavior of the total HIF-1α stabilization, we systematically perturbed the constant activation rate of different proteins in the network. First we studied the effect of external regulation of IL-15 on the stabilization of total HIF-1α in normoxia in the presence or absence of DMOG. For this we simulated ideal experiments in which the cells are exposed to continuous stimulation. We ran computer simulations varying the IL-15 external regulation parameter a<sup>1</sup> in the interval [0, 10] nM h−<sup>1</sup> and plotted the solution of total HIF-1α over time. All other parameter values as well as initial conditions were fixed as indicated in Materials and Methods. **Figure 6** shows the results for IL-15 (left), and for total HIF-1α in the absence (middle) or presence of DMOG (right), with dark blue curves corresponding to the lowest value (a<sup>1</sup> = 0 nM h−<sup>1</sup> ) and red curves to the highest value (a<sup>1</sup> = 10 nM h−<sup>1</sup> ). Simulations confirm the synergistic effect of IL-15 and DMOG treatment on HIF-1α. The stronger the continuous IL-15 stimulus, the higher are the total HIF-1α levels. Compared with untreated cells, HIF-1α levels reach two to three times higher steady state<sup>2</sup> values in the presence of DMOG.

Similarly, we investigated the dependence of total HIF-1α accumulation on the external regulation of mTOR and STAT3. We considered cells with or without DMOG and proceeded as above by varying the parameters a<sup>3</sup> (for mTOR) and a<sup>8</sup> (for STAT3) in the interval [0, 10] nM h−<sup>1</sup> . For investigations on the steady state of the systems (t = 100 h), it makes no difference if cells are initially stimulated with IL-15 or not, as the effect of initial stimulation has vanished at the steady state (cf. **Figures 4**, **5**). The results are shown in **Figure 7**, where dark blue curves correspond to the lowest value (a<sup>j</sup> = 0 nM h−<sup>1</sup> , j = 3, 8) and red curves to the highest value (a<sup>j</sup> = 10 nM h−<sup>1</sup> , j = 3, 8). Our computer simulations confirm that higher HIF-1α

<sup>2</sup>A complete analytical study of the steady states of system (2)–(11) and their stability was beyond the scope of this manuscript. In general, the non-linear system (2)–(11) might have, in certain parameter ranges, multiple biologically relevant steady states. However, the parameter values used for the numerical simulations in this work guarantee convergence to a unique non-negative steady state which is shifted to higher/lower values when different stimuli are considered (cf. simulations shown in **Figures 4**–**10**).

concentration in DMOG treated cells induces a negative feedback and downregulates mTOR (**Figure 7A**). As **Figure 7B** suggests, increasing STAT3 external regulation leads to higher HIF-1α levels, which are amplified by DMOG treatment, although STAT3 levels in DMOG treated cells are slightly lower than in cells without DMOG.

We further investigated the dependence of total HIF-1α enrichment on two signals at the same time. We started by varying the external activation rate of mTOR (a3) and STAT3 (a8) in the interval [0, 10] nM h−<sup>1</sup> , ran simulations up to t = 100 h and obtained numerical solutions of the mathematical model at the steady state. **Figure 8** shows the results for total

HIF-1α, STAT3, and mTOR in NK cells cultivated with or without DMOG. The same figure shows also the effect of simultaneous changes in the external activation rate of NF-κB (a<sup>7</sup> in the interval [0, 10] nM h−<sup>1</sup> ) and STAT3 (a<sup>8</sup> in the interval [0, 10] nM h−<sup>1</sup> ). Again, we observe a central role of STAT3 as regulator of HIF-1α enrichment, especially in synergy with NF-κB. An increase in STAT3 external regulation rate combined with an increase in the external regulation rate of NF-κB leads to a higher amplification of total HIF-1α compared to STAT3 combined with mTOR. Plots in **Figure 8** also reflect the negative feedback of induced HIF-1

FIGURE 6 | Effects of external IL-15 regulation on total HIF-1α in NK cells cultivated in the absence (N) or presence of DMOG. External IL-15 regulation rate (a1) is varied in [0, 10] nM h−<sup>1</sup> with regular steps. Other parameters and initial values are fixed as in Tables 1, 2. Curves with same color correspond to the same parameter value and follow the jet color map in MATLAB®, with dark blue corresponding to the lowest value (a<sup>1</sup> = 0 nM h−<sup>1</sup> ) and red to the highest value (a<sup>1</sup> = 10 nM h−<sup>1</sup> ). Left: IL-15 dynamics is equal in both, NK cells cultivated without and with DMOG as there is no feedback on IL-15 in the model (cf. Equation 2); middle: Total HIF-1α in untreated NK cells; right: Total HIF-1α in NK cells cultivated with DMOG.

FIGURE 7 | (A) Effects of external mTOR regulation on total HIF-1α in NK cells without (N) or with DMOG. External mTOR activation rate (a3) is varied in [0, 10] nM h−<sup>1</sup> with regular steps. (B) Effects of external STAT3 regulation on total HIF-1α. External STAT3 activation rate (a8) is varied in [0, 10] nM h−<sup>1</sup> with regular steps. All other parameters and initial values are fixed as in Tables 1, 2. Curves with same color correspond to the same parameter value and follow the jet color map in MATLAB®, with dark blue corresponding to the lowest value (a<sup>j</sup> = 0 nM h−<sup>1</sup> , j = 3, 8) and red to the highest value (a<sup>j</sup> = 10 nM h−<sup>1</sup> , j = 3, 8).

on mTOR: (i) in general, NK cells treated with DMOG have lower levels of activated mTOR than untreated cells and (ii) higher concentration of activated STAT3 (due to increasing a<sup>8</sup> rate) induces HIF-1α, resulting in higher levels of HIF-1 complex, which in turn is known to inhibit mTOR. Finally, **Figure 8** stresses the role of HIF-1 as activator of NF-κB. In normoxic cells, where HIF-1 levels are low, NF-κB activity is low, despite increasing of external regulation. In contrast, in DMOG treated cells, HIF-1 accumulation leads to upregulation of NF-κB. This is in accordance with data obtained in neutrophils demonstrating that NF-κB is an important downstream effector of the HIF-1αdependent response (37). **Figure 9** shows HIF-1α steady states in dependence on the external regulation rate of IL-15 (a<sup>1</sup> varying in [0, 10] nM h−<sup>1</sup> ) and activation of STAT3 (a<sup>8</sup> varying in the interval [0, 10] nM h−<sup>1</sup> ). The results confirm the synergistic effect of IL-15, STAT3 and DMOG in increasing HIF-1α levels.

Besides the above deterministic perturbations, we tested the network robustness with a stochastic approach. Robustness allows a system to maintain its function, regardless of external and internal perturbations (65). We perturbed specific elements of the system 25% around their originally estimated value (parameter values are otherwise fixed as in **Tables 1**, **2**). We applied stochastic perturbations using the Monte Carlo method (see section 2.8) and computed how external (in IL-15) and internal (in mTOR and AKT) changes affect the steady state of total HIF-1α in IL-15 stimulated cells treated with DMOG.

**Figure 10** shows the histograms for IL-15 (A), AKT (B), and mTOR (C) (green) and total HIF-1α (red). Our results show that the model is robust to internal and external stochastic perturbations, indicating that variations of ± 25% in IL-15, in the AKT activation rate a<sup>2</sup> ± 25% or in the mTOR activation rate a<sup>3</sup> ±25% result in minimal variations (< 10%) in the steady state of total HIF-1α.

With the help of numerical simulations we tested how HIF-1α stabilization is affected by increasing concentration of DMOG. Assuming that NK cells are treated with DMOG and stimulated with IL-15 at time t = 0 h, we changed the DMOG concentration from 0 to 100%, with 100% corresponding to 20 µM. **Figure 11A** shows the evolution of HIF-1α in time, with HIF-1α stabilization depending on the DMOG dosage. We computed the fold change of HIF-1α stabilization at the equilibrium (t = 100 h) and compared control cells (untreated) with cells treated with different DMOG concentrations (**Figure 11B**). The relation between HIF-1α stabilization and DMOG dosage is non-linear and doubling the DMOG dose does not lead to twice as high HIF-1α levels. Our results suggest an exponential trend in the relation between PHD/FIH inhibitor DMOG and HIF-1α stabilization, which is in accordance with what was previously observed for HEK cells (19).

## 3.5. Which Timing for Cell Treatment?

Model (2)–(11) can be used for a number of in silico experiments to test the validity of biological hypotheses or predict the outcome of laboratory tests. In this study we were particularly interested in the synergy of IL-15-stimulation and DMOG treatment in the stabilization of HIF-1α, already observed in the results described above. In all previous simulations, normoxic NK cells were stimulated at the beginning of the observation with IL-15 in the presence or absence of DMOG. To understand how the timing of the treatments affects HIF-1α stabilization in NK cells, we also simulated different possibilities for the timing of cell treatment combining chemical hypoxia and stimulation with IL-15 (**Figure 12**).

We compared the HIF-1α dynamics for the following in silico experiments: (gray) untreated NK cells (N) cultivated for 30 h; (dark blue) IL-15 stimulation at t = 0 h; (magenta) DMOG treatment for 30 h; (green) DMOG treatment for 30 h, with IL-15 stimulation at t = 0 h; (yellow) IL-15 stimulation at t = 6 h; (orange) DMOG treatment starting at t = 6 h; (red) IL-15

FIGURE 9 | Effects of external regulation of IL-15 (a1) and STAT3 (a8) in NK cells without DMOG (N, left) and DMOG treated cells (right). External regulation rates are varied in the intervals [0, 5] nM h−<sup>1</sup> for IL-15 and [0, 10] nM h−<sup>1</sup> for STAT3. Other parameters and initial values are fixed as in Tables 1, 2. Steady states (100 h) of the model solutions are computed for total HIF-1α (first row) and STAT3 (second row).

FIGURE 11 | Stabilization of HIF-1α in dependence on DMOG concentration after IL-15 stimulation. (A) Evolution of HIF-1α in time, depending on the DMOG dosage with dark blue corresponding to no DMOG (N) and red corresponding to 20 µM DMOG. In these simulations cells are initially stimulated with IL-15. (B) Fold change of HIF-1α stabilization at the equilibrium (t = 100 h). Simulation of NK cell treatment with increasing concentrations of DMOG. Parameter values for these simulations are chosen as in Tables 1, 2, with 100% corresponding to 20 µM DMOG.

t = 6 h; (light blue) DMOG treatment at t = 0 h and IL-15 stimulation at t = 6 h; (black) IL-15 stimulation at t = 0 h and DMOG treatment at t = 6 h.

stimulation at t = 6 h and DMOG treatment starting at t = 6 h; (light blue) DMOG treatment for 30 h with IL-15 stimulation at t = 6 h; (black) IL-15 stimulation at t = 0 h and DMOG treatment starting at t = 6 h.

**Figure 12A** shows the time evolution of HIF-1α stabilization over 30 h. We observe the impulses at t = 6 h due to changes in the experimental conditions. On the long term, the effect of IL-15 stimulation vanishes and HIF-1α levels converge to those reached in unstimulated cells. **Figure 12B** shows the fold change of HIF-1α at t =12 h. Values are normalized with respect to HIF-1α in untreated cells (N, gray bar). We observe that on the short time scale the timing of treatments importantly affects HIF-1α stabilization. In particular, treating the cells first with DMOG or first stimulating them with IL-15 is not equivalent (compare the black bar and the light blue bar). The highest HIF-1α levels after 12 h are reached when cells are first stimulated with IL-15 at t = 0 and treated with DMOG at t = 6 h (black bar). Cultivating cells in normoxia and treating them with IL-15 and DMOG at time t = 6 h (red) yields lower HIF-1α values than 12 h cultivation in the presence of DMOG after initial (t = 0 h) IL-15 stimulation (green).

#### 4. DISCUSSION

Being an essential mediator of cellular adaptation to hypoxia (66, 67), HIF-1α plays a critical role as regulator of inflammation and immune system response (36, 68). The understanding of its regulation is crucial in immunology.

While HIF-1α hydroxylation and degradation pathways have been recently described using mathematical methods (19, 20), less is known concerning the mechanistic description of processes regulating the levels of HIF-1α mRNA and protein (10). In this work we have presented a combined approach of experimental and mathematical analysis to understand HIF-1α regulation in human NK cells, in particular simulating hypoxic (DMOG) and inflammatory (IL-15) conditions. To the best of our knowledge, there is no previous interdisciplinary approach describing the interplay of hypoxia and IL-15 stimulation, and their effects on HIF-1α dynamics in immune cells. The proposed mathematical model (2)–(11) and the estimated parameter values (**Tables 1**, **2**) explain collected time series for HIF-1α, also catching the dynamics of other regulatory proteins (**Figure 3**). Our simulation results and in silico experiments highlight the synergy of IL-15 and hypoxia in HIF-1α stabilization, suggesting an important role for STAT3 and NF-κB as regulators of IL-15 induced HIF-1α enrichment in peripheral NK cells.

The mathematical model proposed in this work aimed at the qualitative mechanistic description of IL-15 induced biochemical processes regulating HIF-1α stabilization in NK cells. We made use of collected time series (HIF-1α, AKT, and STAT3) for quantitative investigation, data fitting and model predictions. A limitation of our results is that model predictions for quantities lacking experimental information (e.g., NF-κB, mTOR, and HIF-1α-mRNA) can be made only on a relative scale (43). While the calibration (**Figure 3A**) and validation results (**Figure 3B**) for HIF-1α are overall very satisfactory, the quantitative match for STAT3 in cells treated with DMOG and IL-15 (**Figure 3B**) could be improved. This might be achieved by refining the fit for STAT3 time series in IL-15-stimulated NK cells treated with DMOG and rapamycin (**Figure 3A**). In this study, we performed an all-at-once parameter estimation, applying a direct multiple shooting approach and a gradient-based (generalized Gauss-Newton) method (cf. section 2.7). The method is known to perform well and converge fast [cf. (45)] but, being a local optimization method, it might get stuck in a local optimal minimum. A possible method to overcome local minima is to perform many independent optimization runs starting from randomly selected starting points (69). Alternatively, one could adopt global optimization methods, which however can be computationally very costly (45, 69).

Concerning the statistical significance of the parameter estimates (**Table 2**) we have adopted here a first order approximation of non-linear confidence regions. Parameter estimation and identifiability could be further refined and investigated, e.g., performing a second-order analysis of the nonlinear confidence regions [cf. (70)] or exploiting the profile likelihood, as suggested by Raue et al. (69).

Our model captures essential features of HIF-1α regulation, making a number of simplifying assumptions. Several model extensions and refinements could be proposed. For example, we could include further steps in the degradation pathway of HIF-1α, as proposed by others (10, 19). Moreover, the dynamics of IL-15 is simply given by constant production and degradation rates [as it has previously been assumed by other authors, e.g., for IL-21 dynamics (71)], and sensitivity analysis indicates that the AKT dynamics is approximatively linear in the considered experimental setting (section 3.3). The reaction cascade downstream of IL-15 involves several components, including the IL-15Rβγ -subunits (4), which are known to be constitutively expressed on NK cells (72) but were neglected in the proposed mathematical model. Further, the IL-15-induced activation of AKT, NF-κB and STAT3 is modeled by means of linear terms. One possible model extension would include non-linear terms (Michaelis-Menten or higher order Hill functions) for the activation of IL-15 regulated proteins. Factors connected to IL-15 stimulations, such as IL-15 receptor binding and trafficking or other IL-15 induced signaling cascades (JAK/STAT5, Ras-Raf-MEK), might affect NK cell response to this cytokine and could also be taken into consideration. Further, the relation between mTOR and the HIF-1 complex could be investigated in detail. We have assumed that hypoxia downregulates mTOR (33) by means of a negative feedback of HIF-1 on mTOR. Nonetheless, the regulatory mechanism of mTOR is far more complicated, involving REDD1 and the Tsc1/Tsc2 complex (73). Our experimental data and modeling results show that HIF-1α accumulation in cells stimulated with IL-15 and treated with DMOG correlates with reduction of STAT3 activity. Our modeling approach suggests (cf. section S2) that the known negative feedback of HIF-1 on mTOR (33) is amplified by a direct inhibitory effect of DMOG on IL-15 induced STAT3 activation. This means that the observed STAT3 inhibition could not only be due to chemical hypoxia, stabilizing HIF-1α, but also due to additional effects of DMOG on NK cells. To further explore the role of DMOG on IL-15-induced STAT3 activation in NK cells, the experiments proposed in this study could be performed in cells cultivated in hypoxia (1% O2) instead of chemical hypoxia.

Spatial effects could also be taken into account for model refinement. In contrast to previous studies (19, 74), in our modeling approach we did not make any distinction between proteins in the cell cytoplasm and the nucleus, but simply consider total cellular concentrations. In general, increased model complexity necessarily calls for more detailed experimental data in order to achieve adequate model calibration and trustworthy predictions.

We hypothesize that the proposed regulatory network is appropriate for describing HIF-1α regulation not only in NK cells but also in other types of immune cells. Moreover, the model (2)–(11) can be applied to refine and extend mathematical models in which HIF-1α dynamics is involved, e.g., models of cell cycle regulation (75) or cell proliferation (76). When studying the effects of biochemical signaling at the cellular level, it might be convenient to adopt simpler regulatory networks than those proposed here [see for example (77) for a model for proliferation of IL-15 stimulated NK cells]. To this extent model reduction could be performed by mean of biological assumptions, e.g., via quasi-steady state approximations. Further, (global) sensitivity analysis results could be used to rank the relative influence of the model parameters on the model output, and could suggest how to simplify the regulatory network, identifying parameters that minimally impact model outputs [cf. (78, 79)].

Being involved in cytokine expression, myeloid cell migration and effector functions, HIF-1α regulates both innate and adaptive immunity (80). Understanding the molecular mechanisms involved in the regulation of the HIF-1α pathway, in particular in immune cells, is of central importance to the immune cell function and could be a promising strategy in the design of treatments for human inflammatory diseases and cancer. Our results indicate that NF-κB and STAT3 are important regulators of HIF-1α enrichment in IL-15 stimulated NK cells. It is tempting to speculate that a secondary effect of pharmacological STAT3 inhibition in cancer therapy may consist in a reduction of IL-15 dependent HIF-1α enrichment in NK cells, which may be expected to improve NK cell anti-tumor activity (12, 13).

#### DATA AVAILABILITY STATEMENT

All datasets generated for this study are included in the manuscript/**Supplementary Files**.

#### ETHICS STATEMENT

The study was reviewed and approved by the Medical Ethics Commission II of the Medical Faculty Mannheim, Heidelberg University (2014-500N-MA).

#### AUTHOR CONTRIBUTIONS

MB and HL: conceptualization. AC, MB, EK, and HL: methodology. SV: NK cell characterization. AB, MB, and AF: software. AF: robustness analysis. AC and MB: investigation and data curation. MB and AC: writing. AC, MB, SV, MT, and HL: review and editing (original manuscript). MB, AC, AB, AF, EK, and H-GB: review and editing (revision). All authors approved the final version of the manuscript.

#### REFERENCES


#### FUNDING

This research work was supported by funds to H-GB, EK, HL, and MT by the Klaus Tschira Foundation, Germany. MB was supported by the European Social Fund and by the Ministry of Science, Research and Arts Baden-Württemberg.

#### ACKNOWLEDGMENTS

We thank Jutta Schulte and Bianca S. Himmelhan (Department of Anesthesiology and Surgical Intensive Care Medicine, University Medical Center Mannheim, Medical Faculty Mannheim, Heidelberg University) for technical support. Further we acknowledge financial support by Deutsche Forschungsgemeinschaft within the funding program Open Access Publishing, by the Baden-Württemberg Ministry of Science, Research and the Arts and by Ruprecht-Karls-Universität Heidelberg.

The authors would like to thank the referees for their valuable critical comments which helped to improve the quality of this manuscript.

### SUPPLEMENTARY MATERIAL

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fimmu. 2019.02401/full#supplementary-material


**Conflict of Interest:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2019 Coulibaly, Bettendorf, Kostina, Figueiredo, Velásquez, Bock, Thiel, Lindner and Barbarossa. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# Immunological Paradigms, Mechanisms, and Models: Conceptual Understanding Is a Prerequisite to Effective Modeling

#### Zvi Grossman1,2 \*

*<sup>1</sup> Vaccine Research Center, National Institute of Allergy and Infectious Diseases, NIH, Bethesda, MD, United States, <sup>2</sup> Sackler Faculty of Medicine, Tel Aviv University, Tel Aviv, Israel*

Most mathematical models that describe the individual or collective actions of cells aim at creating faithful representations of limited sets of data in a self-consistent manner. Consistency with relevant physiological rules pertaining to the greater picture is rarely imposed. By themselves, such models have limited predictive or even explanatory value, contrary to standard claims. Here I try to show that a more critical examination of currently held paradigms is necessary and could potentially lead to models that pass the test of time. In considering the evolution of paradigms over the past decades I focus on the "smart surveillance" theory of how T cells can respond differentially, individually and collectively, to both self- and foreign antigens depending on various "contextual" parameters. The overall perspective is that physiological messages to cells are encoded not only in the biochemical connections of signaling molecules to the cellular machinery but also in the magnitude, kinetics, and in the time- and space-contingencies, of sets of stimuli. By rationalizing the feasibility of subthreshold interactions, the "dynamic tuning hypothesis," a central component of the theory, set the ground for further theoretical and experimental explorations of dynamically regulated immune tolerance, homeostasis and diversity, and of the notion that lymphocytes participate in nonclassical physiological functions. Some of these efforts are reviewed. Another focus of this review is the concomitant regulation of immune activation and homeostasis through the operation of a feedback mechanism controlling the balance between renewal and differentiation of activated cells. Different perspectives on the nature and regulation of chronic immune activation in HIV infection have led to conflicting models of HIV pathogenesis—a major area of research for theoretical immunologists over almost three decades—and can have profound impact on ongoing HIV cure strategies. Altogether, this critical review is intended to constructively influence the outlook of prospective model builders and of interested immunologists on the state of the art and to encourage conceptual work.

Keywords: smart surveillance, change detection, autoreactivity, adaptation, tuning, feedback control, selfrenewal, homeostasis

### INTRODUCTION

With some exceptions, the long-term impact of mathematical modeling on basic and clinical immunology has been modest (1–4), and sometimes counter-productive (5), despite claims to the contrary. The major reason is our incomplete understanding of the qualitative rules [or core principles (6)] that govern organized immune phenomena at the cellular and multicellular

#### Edited by:

*Andreas Meyerhans, Catalan Institute for Research and Advance Studies (ICREA), Spain*

#### Reviewed by:

*Alfred I. Tauber, Boston University, United States Irun R. Cohen, Weizmann Institute of Science, Israel Simon Wain-Hobson, Institut Pasteur, France*

> \*Correspondence: *Zvi Grossman grossmanz@niaid.nih.gov*

#### Specialty section:

*This article was submitted to T Cell Biology, a section of the journal Frontiers in Immunology*

Received: *27 March 2019* Accepted: *10 October 2019* Published: *05 November 2019*

#### Citation:

*Grossman Z (2019) Immunological Paradigms, Mechanisms, and Models: Conceptual Understanding Is a Prerequisite to Effective Modeling. Front. Immunol. 10:2522. doi: 10.3389/fimmu.2019.02522*

**259**

levels. Qualitative understanding is a prerequisite to a sensible quantitative analysis of empirical observations but rarely "emerges" from such analysis (1, 7)—again, contrary to claims. "Sensing which assumptions might be critical and which irrelevant to the question at hand is the art of modeling and, for this, there is no substitute for a deep understanding of the biology" (8). I shall get back to this repeatedly.

This contribution highlights these issues from a rather personal point of view. First, past and current paradigms pertaining to immune recognition and immune response are reviewed, focusing on a multipronged theory that I call here "smart surveillance" and related models. Second, a selective overview of past and present mainstream modeling efforts demonstrates the reality that mathematical models are typically adjusted to a change in paradigm, with a considerable delay, rather than producing such change. In this context, it is proposed that the omnipresent "ecological" models contributed little to theoretical immunology because they are too flexible; they were consistently used as data fitting tools while uncritically accommodating preconceived interpretations and fashionable trends. Third, I recount debates about cause-and-effect in HIV pathogenesis, arising after rudimentary mathematical descriptions of data coming from a handful of observations were over-interpreted (by the modelers) and over-evaluated (by others). These interpretations were rejected repeatedly by invoking basic immunological knowledge and were readily falsified when more data became available. The lesson is that models should be evaluated not according to their popular appeal but rather by whether the assumptions and arguments are biologically sound or not. I discuss how different sets of basic assumptions can lead to alternative HIV cure strategies. Standard low-dimensional mathematical models play a subsidiary heuristic role, at best, in making the choice. Fourth, common epistemic fallacies associated with mathematical modeling in immunology are briefly revisited. Those are too often compounded by lack of full openness, transparency, or even truthfulness in scientific reporting, which taints the scientific dealings among researchers and beyond. Finally, it is suggested that the advent of novel physiological perspectives should be considered essential part of the unavoidable iterative process that (ideally) transforms better understanding into increasingly accurate experimental and clinical predictions. Outstanding "big questions" need to be defined.

# THE EVOLUTION OF PARADIGMS AND THE IDENTIFICATION OF PROTOTYPICAL MECHANISMS

# Clonal Selection and the Self-Nonself Discrimination Paradigm

Macfarlane Burnet (9, 10) postulated that mature circulating lymphocytes responded specifically to foreign molecules, while those that were able to respond to the body's own tissues were depleted during a window of prenatal, actively acquired tolerance. Thus, two basic interrelated assumptions were taken for granted, then and in the following three decades. First, a lymphocyte that recognizes (i.e., binds specifically to) a foreign substance is normally activated, resulting in a stereotypic clonal response; elimination of pathogens is a consequence of their foreignness. Second, recognition of a self-antigen normally results in the lymphocyte's death or paralysis. (A major exception were self-antigens called idiotypes, in Jerne's idiotypic network theory; see below). Several two-signal theories were developed to provide a mechanistic explanation of how lymphocytes implemented this self-nonself discrimination paradigm. Depending also on the stage of development, the signals delivered to a lymphocyte at the time of antigen recognition served as a code, instructing a binary decision [Baxter and Hodgkin provided an excellent review (11)].

# The Pathogen-Nonpathogen Discrimination Paradigm

In 1989, Charles Janeway rejected the self-nonself paradigm, while retaining the two-signal concept. He contended that "the immune system evolved specifically to recognize and respond to infectious organisms, and that this involves recognition not only of specific antigenic determinants, but also of certain characteristics or patterns common on infectious organisms but absent from the host" (12). This explained the adjuvant effect of bacterial products that were usually added to antigens in the lab to raise antibody responses ("the immunologists' dirty little secret"). The work of Janeway and Medzhitov and of others led to the discovery of toll-like receptors on antigen-presenting cells (APC) that can bind bacterial and viral derivatives. Such binding often activates APC, enhancing antigen presentation (the specific "signal 1" for responding T cells) and the expression of other costimulatory molecules and inflammatory cytokines (the nonspecific "signal 2"). Accordingly, the signals no longer act as a code required for avoiding autoimmunity; rather, they act positively to initiate immunity to pathogens, while in the absence of pro-inflammatory signals, antigens—including self-antigens do not elicit a response (13).

Polly Matzinger's "danger theory" (14) was a variation on the theme (15, 16), postulating that pathogens are not recognized directly as such by APC but rather, primarily, via the tissue damage they have caused, which inevitably results in release of highly immunogenic byproducts. It also played strongly the self-nonself discrimination theme, however, proposing that autoreactive T cells coming out of the thymus are inactivated and rendered harmless as they typically first meet their cognate selfantigens in the absence of signal 2; this feature was necessary to prevent wide-spread autoimmunity in response to damage.

# Advent of the "Smart Surveillance" Paradigm

Some 27 years ago, my colleague William Paul and I proposed (17) what amounted to another change of paradigm (18) as to how the immune system, and T cells in particular, relates to self- and nonself antigens. Paul too believed that the immune system evolved to respond primarily to infectious agents, and that characteristics other than "foreignness" help trigger destructive responses. However, several observations suggested to us (a) that additional, "contextual" attributes of infection events were also important (1), besides the inherent proinflammatory properties of bacteria and viruses; and (b) that antigen-mediated signals delivered to lymphocytes in the absence of infection were capable of eliciting cellular responses that contributed to the lymphocytes' own functional integrity and to that of the cells with which they interacted.

The additional contextual attributes were not necessarily some other biochemical signals. For example, experimental allogeneic tumors could be immunologically rejected in the absence of apparent inflammation, but only if the initial number of transferred tumor cells were large enough; otherwise, the tumors could "sneak through" and grow despite immune surveillance (19–21). Conversely, chronic infection was often characterized by a quasi-stationary mode in which potentially responding lymphocytes were inactivated (22).

As for the second proposition, of important functional consequences of antigen recognition outside the setting of conventional immune responses—we and others had observed that while tissue self-antigens normally lack attributes required to trigger destructive responses, benign or controlled autoreactivity was too common to be dismissed as an aberration or epiphenomenon (1, 23–27). At the same time, unexpected complexity and plasticity of signaling networks and intercellular communication was being revealed, suggesting that cells of the immune system were required to deal adaptively with rich classificatory challenges in perception of their environment, richer than previously appreciated (1). It was reasoned that such advanced cognitive capabilities would be required in order to optimize response to pathogens and—given the preponderance of autoreactivity and other evidence—to perform a myriad of body maintenance functions [highlighted by the pioneering studies of Irun Cohen and colleague; see e.g. (28, 29)]. A foundation for a general theory of adaptive networks was laid down in Grossman (1) (see **Supplementary S1** for relevant extracts). In particular, it was proposed that the functional units are heterogeneous groups of interacting cells, assembled on ad hoc basis in response to infection or other forms of tissue perturbation; that lymphocytes are capable of tuning their responsiveness under the influence of recurring signals, antigenic and others; and that through such tuning and feedback from coresponding cells and from tissue cells, individual lymphocytes and the group as a whole "learn" (a) to identify recurring signal patterns as "meaningful," thus endowing the unit with appropriate discriminatory capacity (1); and (b) to adjust their response for better results. As discussed below, for lymphocytes, benign autoreactivity is key to maintaining relatively stable (but resilient) phenotypic profiles under stationary conditions and to selectively respond or not respond to perturbations.

#### Tuning, Change Detection, and Subthreshold Interactions

Given the broad range of qualitatively different challenges and responses, mapping a response to the challenge in each case by deciphering putative biochemical codes would be forbiddingly challenging. Fortunately, we identified a general organizing principle that reconciled the different and seemingly conflicting outcomes of immune recognition and allowed qualitative prediction. Encapsulated in a sentence, this organizing principle is that individual lymphocytes, as well as interacting lymphocytes and accessory cells collectively, sharply discriminate (in a threshold-dependent way) between small and large perturbations. Perturbation is generally defined as deviation of a system or process from its regular or normal state or path. The overall perspective is that physiological messages to cells are encoded not only in the biochemical connections of signaling molecules to the cellular machinery but also "in the magnitude, and in the time- and space-contingencies, of sets of stimuli" (30). Individual T cells respond differentially (adapt or become activated) to the rate of change in the level of stimulation, translated intracellularly into "state perturbations." The organization of the immune response at the cell-population level in space and time in turn is also conducive to discriminating "systemic perturbations," setting additional barriers to destructive immunity (17, 21, 30– 33). These propositions were later appropriated by others and "proposed again" (34–38).

Thus, the immune system was "designed" to respond in a characteristic explosive way mainly to episodes of acute or undulating infection and not to the continuous presence, or slow variation, of self- or foreign antigens. Inflammation is an important promoter of cellular perturbation and activation, but for a conventional immune response to occur, also required—and sometimes sufficient—are both high-affinity binding of a cognate antigen, against a background of weaker "tonic stimulation," and rapid convergence of the antigen/APC and the lymphocytes into dedicated sites within an inductive lymphoid tissue (17, 30, 39). These requirements enhance the selectivity of T-cell activation and conventional immune responses (32), but also define a wide range for subthreshold perturbations that can influence the viability, functional properties, and functional organization of T cells without overt activation [reviewed, (33)].

Autoreactivity is enforced during positive selection of T cells in thymus. We and others proposed that lymphocytes are selected to be moderately and variably autoreactive so that recognition of self-antigens in peripheral tissues can be used to actively and dynamically tune and shape their functional properties and regulate their numbers and diversity (17, 31, 32, 40–44) or to actually perform crucial tissue-maintenance functions (1, 7, 18, 28, 29, 31, 32, 45–53). The same subthreshold interactions dynamically tune the activation thresholds themselves, imposing a level of desensitization generally sufficient for preventing chance activation due to noisy ambient stimulation, and adapting that level to moderate variations in the local landscape. Similarly, activation-threshold tuning may impose tolerance to persisting foreign antigens—resident microorganism or pathogens in the context of chronic infection—averting immune pathogenesis (17, 33). In contrast, the tuned state of pathogen-specific lymphocytes that is associated with their physiological autoreactivity does not prevent them from responding vigorously to strong perturbations, which are typically associated with acute infection. When the increase in the level of stimulation—relative to the moving baseline—is too fast, the tuning apparatus fails to update the activation threshold quickly enough to avoid activation.

It was postulated that other cells are also endowed with similar adaptive properties, e.g., antigen-presenting cells, where bidirectional tuning of T cells and APC leads to the definition of homeostatic set points for T-cell clones, thus maximizing clonal diversity (32, 33). It was speculated also that twoway tuning of tumor-infiltrating lymphocytes and tumor cells (and/or stroma) might be involved in the induction of tumor dormancy, inhibiting local inflammation and tumor-cell growth (31). According to Philippe Kourilsky's normative-self model, which used and expanded our ideas on a self-oriented immune system (18), tuning applies to all the cells in the body, whether associated with immunity or not.

# Subthreshold Stimuli and Smart Surveillance

The smart surveillance paradigm combines the concepts of dynamic tuning and adaptive networks (1) (**Supplementary S1**). By rationalizing the feasibility of subthreshold interactions, the tuning hypothesis reconciled the physiological requirement of explosive immune responses to acute infections with the plethora of manifestations of intricate context discrimination and adaptive networking associated with homeostasis and functional preparedness, and set the ground for further theoretical and experimental explorations of generalized immunological functions.

The largely unexplored plasticity of interconnected molecular circuits is increasingly believed to conceal a potential for cellular "learning from experience" in real time—an extension of tuning—including detection of and selective response to recurring patterns, or "features," of external signals (1, 7, 17, 31, 49); concomitant generation of intracellular and intercellular associations, or "conditioning" (50); and even gradual reprogramming of the differentiation state ("adaptive differentiation") (17, 33). Adaptive responses of individual cells are coupled to the nonlinear dynamics of stimulatory and suppressive interactions operating at the cell-population level. Together, the nonlinear nature of such multilevel interactions provides rich opportunities for the selection of alternative forms of coordinated cellular activities, or responses, which may be guided by external feedback (e.g., stress signals from tissue cells). Such "feedback-reinforced learning" would facilitate (a) "quality control" and dynamic readjustment, including class selection, of ongoing responses to pathogens (1, 7, 31, 48); and (b) beneficial participation of lymphocytes and other immune cells in nonclassical immune functions such as wound healing and body maintenance (18, 29, 52, 53), including "immune surveillance without immunogenicity" (47), which is briefly discussed next.

While the widely accepted theory of immune surveillance against cancer was based on the premise that the immune system responds to antigenically modified cells in essentially the same way as it responds to invasive microorganisms, Ronald Herberman and I suggested that lymphoid cells also assist in regulating the differentiation of a variety of normal cells, and that they do so by recognizing self rather than foreign antigens. "By forcing and steering the turnover of tissue cells, lymphoid cells prevent the accumulation of small irregular phenotypic and karyotypic changes in the tissue" (47). Tumor escape from immune surveillance could accordingly be described as escape from homeostatic, immune-mediated differentiation pressures. Recently, some supportive evidence has been forthcoming. Studies showed that resident T cells and innate immune cells may indeed assist in tissue differentiation and development, and that disruption of such activities may result in tumorigenesis (54). The authors foresaw "increasing interest in immune cell functions that are outside the more canonical roles assigned to host defense, and that might be targeted with an aim toward improving human health" (54).

Irun Cohen's "cognitive paradigm" (51, 53) shares some elements with "smart surveillance." Cohen and colleagues pioneered the idea that the immune system is universally designed to pay particular attention to a preferred subset of tissue antigens, shared among different individuals and even cross-species, using recognition of these antigens to initiate and/or manage inflammatory responses that maintain or restore tissue integrity. In a series of elegant studies, the researchers demonstrated early selection of B and T cells recognizing overlapping subsets of self-antigens in different individuals, in different mice, and cross-species. Thus, healthy newborn humans manifest IgM autoantibody repertoires, produced in utero, that are highly correlated among unrelated babies, differ from the repertoires of their mothers, and target particular sets of self-antigens. A subset of T-cell receptor peptide-binding sites are also shared by individual healthy mice and cross-species by mice and humans. These public TCR repertoires manifest relatively large clone size and marked convergent recombination of different nucleic acid sequences into identical TCR amino acid sequences—evidence for strong repertoire selection. The public TCRs, at least in mice, are annotated for self-recognition. Cohen and Efroni proposed that a subset of the selecting selfantigens in the thymus collectively present a "wellness pattern," training the selected subset of lymphocytes to use those antigens as reference in performing tissue and organ surveillance tasks (53). The pattern is somehow imprinted into the phenotypic profiles of the selected cells. When a significantly different pattern is later encountered in a tissue, the difference is detected and elicits a response. Mechanistic description of these cognitive processes was not provided, but our dynamical tuning model can be invoked. In our model, the states of thymocytes become tuned to antigenic and other signals during thymic development; the tuned state is continuously updated in the periphery. Sufficiently strong perturbations of the tuned state elicit a hierarchy of responses. The collective responses of recruited cells can then be modified by quality-sensing feedback mechanisms (see above).

# Mechanistic Model of Signal Discrimination: Antagonistic Excitation-Deexcitation Processes

The postulated "organizing principle" was translated into a prototypic mechanistic model, which also became a paradigm of sorts. In this model, signal discrimination is based on a competition between "excitation" and "deexcitation" factors possessing different response kinetics (17). Notably, dynamic competition between stimulatory and regulatory forces was similarly invoked to account for growth-rate discrimination

at the cell-population level, as demonstrated in the "sneaking through" studies (further discussed in the next section). The single-cell level model was initially introduced in (17) and then applied successfully to interpret experimental results in T cells (32, 33, 55, 56), as briefly reviewed below. With some idiosyncratic modifications, it also basically accounts for kinetic discrimination and tuning in B cells and NK cells (57–59).

At various steps of the signal transduction pathway, local intracellular decision events depend on the balance between pertinent excitation- and deexcitation-inducing factors that are recruited by external signals. Excitation consists of biochemical changes that converge toward gene activation. Deexcitation consists of changes that reverse or negate the effects of excitation. Deexcitation arises either in response or in parallel to excitation, forming a feedback or a feedforward loop, respectively, in tandem with the excitation pathway. Signalinduced perturbations of intracellular modules such as the TCR complex translate into a kinetic competition between excitation and deexcitation. We assumed that the intracellular concentrations of excitation and deexcitation factors trace the changes in external stimulation with inherently different kinetics. A small increase in the level of external signals transiently perturbs an existing feedback-controlled homeostatic balance between excitation and deexcitation factors in each module. When the level of stimulation rises abruptly in the face of a low deexcitation factors' baseline (low tuning level), these factors may not be able to keep up with the rapid rise in excitation; once such imbalance exceeds a critical value, the unit's stability is lost, inducing additional events downstream, and so on. As usual, cooperativity between products of excitation is implicated in such stability switching; the combination of self-enhancing effects and a constitutive (homeostatic) feedback control gives rise to bistability. Note that high levels of tuning can protect a cell from potent stimulation, rendering it tolerant or "exhausted," though this state is reversible.

Tyrosine kinases and phosphatases were originally proposed as opposing factors at unspecified phases of the cellular activation pathway (17). This proposition, and the prediction that excitation involves a self-enhancing component, gained experimental support. Stefanova and Germain demonstrated the importance of a rapid rise allowing excitation signals to outcompete the negative signals (55).

In our model, the interactions initiated by TCR ligation consisted of rapid cycles of phosphorylation and dephosphorylation and of receptor binding and unbinding (32). The interactions are stochastic, but their population dynamics lead to robust outcomes. We reasoned that, when a small population of clustered TCRs collectively interacts, locally, with a small population of ligands, undergoing rapid engagement and disengagement, reengagement of the same TCR by the same ligand following disengagement is not necessary for a TCR to become progressively excited and then activated. Rather, we proposed that the buildup rate of excitation factors is linked to the cumulative binding time of the ligands to the clustered TCRs, which collectively act as the state-switching unit (32). Indeed, such micro-clusters were later demonstrated experimentally (60). The model explained why an increase in the number of ligands on an APC does not significantly compensate for weaker binding. This is due to the localized nature of the interactions; increase in ligand number would result mainly in a larger number of units rather than in a better signaling quality of individual units. In other words, individual TCR complexes gather signals from locally interacting signaling molecules and therefore measure mainly the quality of the ligand rather than its multiplicity.

A competing, influential model—based on McKeithan's kinetic proofreading hypothesis (61)—derived its appeal mainly from a simple explanation it provided for the apparent dominance of the TCR-ligand dissociation (off-) rate as a determinant of activation, with apparent insensitivity to the association rate. Our molecule-population dynamics model required additional ad-hoc assumptions to account for this bias (32). According to McKeithan's hypothesis, a single long occupancy of individual TCRs was required for activation. But more recent studies have shown that in the two-dimensional APC-T-cell interface, association and dissociation rates are much faster for agonists than what is measured in three-dimensional assays, and agonists tend to be characterized more by their high association rates than by the rates of dissociation. A long-lasting bond is not essential because "high bond formation frequency also accumulates a large fraction of engagement time" (62). Not surprisingly, the actual interplay of positive and negative factors observed experimentally is more complex than in our schematic models, but the concept that such an interplay plays a crucial role in signal discrimination has been established [reviewed, (33)].

Activation is a failure to adapt. Stimulation that does not reach the activation threshold results in "tuning," adaptive shifts in the size of the threshold and in that of additional parameters. Tuning reflects variation in the molecular residues of past subthreshold events. The traces of previous signaling events are gradually erased, actively and/or passively, in the absence of continued stimulation and are dynamically modified if stimulation continues but varies. Therefore, tuning mirrors the cell's stimulation experience, with more weight given to more recent signaling.

In the excitation-deexcitation model, activation-threshold tuning adjusts the levels of deexcitation factors to counter the ambient fluctuations in excitation. Following each relevant Tcell-APC encounter, excitation factors may initially rise more quickly than the associated deexcitation factors, as discussed, but the latter must outlive the former if a tuning state representing the cell's recent experience is to be sustained between encounters.

Under the cover of activation-threshold tuning, subthreshold interaction with self-antigens in the presence of other signals effect the tuning of other cellular properties. Such tuning can result in sensitization of signaling modules rather than desensitization. Thus, the ongoing integration of TCR-mediated signals and accessory signals in the interactive milieu could prepare lymphocytes to respond more efficiently, rather than less, upon activation by a potential pathogen (17, 32). The prediction that subthreshold interactions tune cellular characteristics in multiple ways, "positive" as well as "negative," went unnoticed at first. It is now supported by observations. [Pre-2015 evidence was reviewed in (33); see also references (63, 64) for recent reports.] At the cell-population level, diverse interactions create functional diversity through tuning. Moreover, by specifically tuning the viability and self-renewal of T-cell clones, subthreshold (or "tonic") interactions with self-antigens limit inter-clonal competition and sustain clonal diversity.

Our model originated from the identification of a general organizing principle linking together a range of phenomena (see above), including abundant autoreactivity and differential responses to perturbations. Dynamic tuning was inferred, inspired also by cell adaptation phenomena in other systems, especially in the nervous system. The point we wish to make is that general principles can be a powerful tool in modeling: Conceptual models can have considerable explanatory and predictive power and can, in turn, guide the formulation of quantitative mathematical models.

#### Related Mathematical Models

A hypothesis does not become more credible just because it is formulated in mathematical terms. Nevertheless, hypothesisdriven models can provide useful representations, or metaphors, of organized biological behavior and guide further study, e.g., by defining questions for experimentation or sensitive measures for comparing results. "Models can [also] corroborate a hypothesis by offering evidence to strengthen what may be already partly established through other means. . . . Thus, the primary value of models is heuristic" (65).

Several mathematical studies have integrated the "tunable activation threshold" into existing phenomenological models of immune regulation and autoimmunity [e.g., (66, 67)], or studied its generic pattern detection and pattern discrimination properties [e.g., (68, 69)]. Models that incorporate adaptive excitation-deexcitation processes ("push-pull"), which are fundamentally inherent in immunology, in mathematical representations of actual cellular or systemic data are scarce. Recently, Sontag and colleagues carried out an ambitious analysis of complex interactions involved in what appeared to be interference of the immune response to acute influenza infection in the lung with the (partial) immunologic control of a distal skin melanoma growth in the dermis (70). They described competing push-pull processes that they considered to be mechanistic instantiation of our conceptual signal discrimination model. The different activity levels of antagonistic excitation-deexcitation loops manifested themselves in phenotypically distinct outcomes.

As mentioned earlier, the immune system discriminates perturbations kinetically also at the systemic level. The classic example is the above-mentioned tumor escape from immune elimination via a sneaking through mechanism. Sneaking through was demonstrated in mice injected with different numbers of allogeneic tumor cells. Large numbers of dividing cells overwhelmed the host, as expected. Tumors arising from small numbers of cells, however, also grew and eventually killed the hosts, while intermediate numbers of cells were able to trigger effective responses. A generic mathematical model was proposed incorporating, for the first time, competing positive and negative processes with different kinetics (increasing concentrations of tumor antigens and, in parallel, of immunosuppressive molecules), demonstrating the rate-of-change discrimination capacity of such models (21). A rather simplistic model used at the time did not include physiologic controls on lymphocyte growth, for example regulatory T cells, other than the antigen itself. In that model, cooperativity at the effector-cell level was not required to produce sharp discrimination. When feedback control was later added, cooperativity in T-cell-APC interactions was also invoked (30, 33).

It was further proposed and demonstrated that acute immunogenic challenge could induce the immune system to eliminate the tumor completely, instead of reaching a predatorprey type equilibrium between the two. A built-in property of an explosive immune response to pathogens was proposed to play a role in reducing the likelihood that such equilibrium be established, namely, the fact that the response is not tightly geared to antigen concentration. Rather, its regulation (by the antigen itself and other feedback control mechanisms) involves time delays, allowing the effector cells to overshoot. Overshooting is a fundamental property of the immune response to acute challenge.

Generalizing, we proposed (21) that inherently slow infectious agents could also sneak through immune surveillance. This was demonstrated 24 years later by Gennady Bocharov and colleagues (71). Studying lymphocytic choriomeningitis virus (LCMV) infection in mice, they showed that slowly replicating viral strains induced weaker CTL responses than a more rapidly replicating strain and could thus persist in the host. Moreover, the clinical outcome of hepatitis C infection in humans was strongly associated with the rate of viral replication. A mathematical model reproduced the postulated overshooting versus adaptation modes to rapid and slow growing viruses, respectively. The authors invoked the analogy with "sneaking through" in the tumor context and interpreted their observations and analytical results in terms of our perturbation-dependency concept and prototypical models. [A later variation on this theme should be viewed as a "rediscovery" (34)].

It turns out that our simple mathematical statement of the sensitivity-to-change hypothesis (17) essentially defined an "incoherent loop," a ubiquitous motif in biological networks (72). Such a loop is characterized by the existence of two partly independent antagonistic pathways, from the input to the output, either direct or indirect (73). Sontag (73) explored the antigen discrimination properties of such motifs mathematically. Both his motif and our initial, simpler model possess the properties of log sensing, defined by the output approaching the logarithmic derivative of the stimulatory input. Such models serve heuristic purposes in attempts to conceptualize the actual operation of molecular circuits in cells or in tumor-host/viralhost interactions.

# ANTIGEN-DRIVEN AND FEEDBACK-REGULATED BALANCE OF GROWTH AND DIFFERENTIATION

Since the 1960s, mathematical models have been increasingly used in immunology. The first models were based on the hypothesis of a two-stage differentiation of cells participating in the antibody response (74), illustrated by the X –> Y –> Z scheme. Introduced to me by the late Richard Asofsky (75), this scheme was used by us as a starting point for a long series of elaborations and generalizations as our thinking and knowledge base developed. The following is a condensed summary:


Various simplifications or partial representations of this general scheme were translated into mathematical models. Such models were parameterized to simulate data numerically, e.g., the dependence of CD4 T-cell expansion on precursor number in experimental mice (log-linear relation between CD4 T-cell precursor number and factor of expansion, with a slope of ∼-0.5 over a range of 3–30,000 precursors) (83); used to qualitatively illustrate theoretical explanations of important observations, such as "sneaking through"; or qualitatively analyzed to demonstrate the soundness of theoretical arguments that apply to a broad range of observations, e.g., robustness of blood cell production in bone-marrow and its dynamic adaptation to external demand (76, 84). Such models were confirmed by the demonstration of agreement between observation and prediction, but confirmation is inherently partial.

Some of the most important consequences of the assumptions underlying this conceptual model did not require detailed mathematical analysis. An important corollary of these assumptions pertained to the concomitant regulation of immune activation and homeostasis. Thus, under recurrent clonal (or polyclonal) T-cell activation, the activated population must be in flux (77): extensively proliferating memoryphenotype T cells subject to feedback-mediated differentiation pressure are progressively pushed forward and out, along their preprogrammed developmental pathways, being replaced by the progeny of activated naïve cells. The number of naïve cells in turn is maintained dynamically via dynamically regulated incorporation of recent thymic emigrants (85, 86). Independent of the precise mechanisms of the feedback control, there is a sound physiologic rationale for a dynamic flux, in the context of recurring inflammation and activation. Constant cell replacement acts to reduce the accumulation of detrimental (e.g., tumorigenic) mutations associated with repeated episodes of extensive proliferation; and it also confers "functional resilience," flexibility in readjusting the composition of effector cells to varying physiologic needs (77).

Dynamic tuning of cellular properties during subthreshold interactions, including the activation thresholds, endows the system with additional levels of functional adaptability, resilience, and signal discrimination, as discussed. Dynamic, long-term interplay exists between the changing structure and size of the population in response to challenges or aging on the one hand, and adaptive changes in the function of the individual cells, as they patrol the tissues and perform "smart surveillance" functions, on the other hand. For recent, articulate and insightful overviews see (18, 53).

# WHILE PARADIGMS EVOLVED, MAINSTREAM MATHEMATICAL MODELING HAS BEEN SLOW TO CATCH UP

The mathematical models described so far were hypothesisdriven and heuristic in nature, supporting efforts to conceptualize data and predict trends. Modeling-experimental collaborations proved valuable also when using mathematics as an ancillary analytic tool. For example, mathematical analysis can accurately parameterize and depict complex data derived from monitoring the kinetics of cell populations in vivo using molecular markers and DNA-labeling agents or other reagents.

Other models, more ambitious, purportedly "captured" fundamental features of the immune system, in health or disease, professing an ability to explain and predict the consequences of experimental perturbations or clinical interventions through accurate mathematical analysis. But "capturing" is a misnomer. When Gunawardena writes about "accurate description of pathetic thinking" (8), a phrase originally coined by James Black in his 1988 Nobel Prize lecture, he refers to the fact that the assumptions that theoretical biologists accurately develop into mathematical models necessarily rely on phenomenology and guesswork rather than on the fundamental laws of nature. The higher the level of organization, the more "pathetic" (i.e., uncertain) are these assumptions. Indeed, many mathematical models in immunology usefully described local molecular events at the subcellular level, adding to our understanding of signal processing at this level. However, this has generally not been the case at the higher levels of organization.

#### The Idiotypic Network Impasse

In 1974 Jerne proposed that the immune system was regulated by a web of lymphocyte receptor-associated molecules (87). The receptor molecules on lymphocyte clones, created by random genetic mechanisms, differ from each other not only at the recognition sites, but also in related structures named "idiotypes," which serve as antigens for other clones. Each clone could activate some other clones, forming a network of interactions which encompassed the entire system. The functional consequence of such activation could be suppression, expansion, and/or induction of effector function in the clones involved, depending on the functional properties of the cells and perhaps also on the "direction" of the signal; no general rules were proposed.

As idiotypic interactions were demonstrated, their functional significance was under debate (1, 24, 88, 89). My own rejection of the network theory was based on the early recognition that self-antigens on tissue cells should be much more important than idiotypes in the shaping of immune responses. I argued that only pathogen-mediated activation of lymphocytes would be "acute" enough to elicit a meaningful, suprathreshold response, while responses further down the chain of anti-idiotypic clones would quickly dissipate, playing no significant role. Therefore, while a strongly coupled network would be uselessly "tied in a Gordian knot" (89), idiotype recognition might at best participate in regulation of the first line of responding lymphocytes. A more fundamental objection had to do with the "contextualist view" of immunity that I and several others were already advocating (1, 7, 27, 47, 51). Physiological autoreactivity, selforganization and integrated function of different kinds of cells are the hallmarks of the contextualist approach. In contrast, Jerne's idiotypic network did not recognize "self " in general (namely, self-antigens); it recognized only itself and foreign antigens. The immune system à la Jerne obeyed exclusive rules, unlike those governing other tissues and organs, which constituted a major obstacle to an integrative, physiologically sensible formulation of immune functions and to a genuine analogy with our other major cognitive system, the brain (90). In the end, the idiotypic network idea arguably had a considerable negative impact on progress.

For almost two decades mathematical immunologists uncritically adopted the network theory and analyzed in detail a variety of hypothetical realizations, until the theory went out of favor. The focus then shifted abruptly to AIDS research, where again most theoretical immunologists adopted the prevailing doctrines, teaming with clinical researchers and virologists and adding a "rigorous science" semblance to unfounded and simplistic interpretations.

# Inadequacy of the Ecological Metaphor

The pioneering work of Bell, Marchuk, Bruni, Mohler and others [reviewed, (91)] introduced population dynamics of organisms as a convenient metaphor for the dynamics of the cells of the immune system and the microorganisms with which they interact. This included a direct analogy with predator-prey interactions in ecology; comparing the spreading of pathogens in tissues to epidemics affecting human or other populations; and borrowing from the evolution of species to describe Darwinianlike mutation-selection processes affecting lymphocyte clones during development or in the course of immune responses or facilitating escape of pathogens from immune attack. This early work established the use of the mathematical language in immunology. 40 years later, mainstream mathematical immunology still adheres to the same ecological paradigm and variations on that theme [e.g., (92, 93)].

Interestingly, Burnet was not impressed by the "character of current research" in theoretical immunology in 1978, to which he referred as "disappointing" (94). In his words,

"T and B lymphocytes, with their myriad subpopulations, can be regarded almost as autonomous organisms, arising, interacting, and dying in a Darwinian evolutionary system at the cellular level. As in the orthodox evolutionary situation, we can identify genetic variation, proliferation and death (including something analogous to predation) among the lymphocytes. Intellectually, this provides an important road to understanding but little practical enlightenment."

In retrospect, it did not provide a road to understanding.

The myth that mathematical models made important conceptual contributions to basic and clinical immunology has been perpetuated in numerous reports and reviews, mostly in the biomathematical literature [e.g., (95)] but also in biological and general journals [e.g., (92, 96)]. In fact, to my knowledge no mathematical modeling-based studies in immunology at the cellular or systemic levels to date provided groundbreaking insights, or correct answers to key questions about causality. The reason for this predicament is that lymphocytes do not operate as "almost autonomous organisms." Rather, the immune system evolved along with other tissues and organs to operate interactively as a multifunctional, adaptive, dynamical and dynamically organized network (1, 18, 31, 33, 53).

Nowak and his colleagues were among the most prominent champions of the ecological approach (97–101). They were remarkably prolific, with mathematical models published in major journals. The way was paved by the "diversity threshold" model (98). The model "predicted" that, as antigenically different HIV mutants accumulate, a threshold number of strains is reached that the immune response can no longer contain, leading to a breakthrough of virus resulting in AIDS. In fact, the model was engineered to show this immunologically highly implausible restriction, by making the virus replication rate independent of target cell availability, and the lymphocyte activation rate independent of the lymphocyte concentration—implausible and unusual assumptions that give the virus an "unfair" advantage. Mathematically-innocent outsiders are unlikely to notice, and frank critiques from within are rare and subdued. Indeed, the specific predictions of the above model were disproven by data. It should be noted in this context that the observed evolution of viral strains under the pressure of anti-HIV immune responses does not necessarily imply a major role for such responses in the control of viral replication during the chronic infection phase in untreated individuals. Rather, it may only indicate progressive "selection of the fittest" in the presence of prolonged competition among strains, which would occur, given sufficient time, even if the selective pressure is modest and differences in fitness are otherwise inconsequential.

These and subsequent publications by Nowak and colleagues are summarized in a book where it is said that the work provided no less than "the basic principles for a quantitative approach to immunology, with practical implications for the design of therapy and vaccines" (97). Most immunologists would disagree. The central premise is that interactions among cells of the immune system, and with infectious agents, are conceptually very similar to classic predator–prey interactions. If the central premise does not hold, the mathematical edifice becomes rather irrelevant. Indeed, as Burnet seems to have observed 22 years earlier, the similarities of host-pathogen interactions to classic predator–prey dynamics are superficial and therefore not particularly enlightening. The Nowak–May model ignored crucial features of the immune response and of HIV dynamics, including the profound difference between the immune response to acute and to chronic infection, and the role of chronic immune activation in HIV pathogenesis. Importantly, "the immune system is not a well-stirred Erlenmeyer flask" (102); rather, antigenic activation of latently HIV-infected T cells occurs locally in lymphoid tissue, resulting in localized, transient, and highly structured proliferation, differentiation and death of infected cells and bystander cells and in proximal and local virus dissemination—arguably the only mode of viral replication that really matters (103–107). "Averaging out" such events in mathematical models by considering uniform distributions of T cells and virus particles, along with other "simplifying" assumptions, did not help advance our understanding of viral dynamics in vivo. On the contrary, the ecological paradigm, supporting a simplistic confrontational view of the interaction between a virus and the immune system, delayed serious consideration of alternative views, including the presently widely accepted view that chronic immune activation is the major force driving the progression of HIV disease.

The ecological paradigm continues to generate mathematical models of viral dynamics and anti-viral immunity and purports to predict the impact of antiretroviral treatment and of potential cure strategies; however, it is unlikely to provide novel insights. The importance of cell-to-cell transmission of HIV is starting to be appreciated (92) [with a 20-year delay (103, 105, 108–110)], but the "standard model of viral dynamics," which represented "the dominant and standard approach to analyze and quantify the spread of a viral infection within a host (92)," did not fundamentally change; depicting the concentrations of target cells, infected cells, and virions as piecewise uniform and piecewise aggregated, rather than fully uniform did not change the model's basic nature. Totally ignored is rich evidence supporting the existence of structured, proximal activation and transmission events, their transient ("burst-like") nature (see below), and the crucial role latentlyinfected cells arguably play in sparking these local events and in sustaining systemic infection during the chronic phase (102, 103, 105, 107, 111, 112).

A variant of "the standard model" "predicted" potential post-treatment control of HIV replication in patients treated very early post-infection to reduce the latently-infected cell reservoir (93). It was hypothesized that patients who exhibit post-treatment control (at least temporarily) generate earlier an adaptive immune response that is adequate to control infection after treatment interruption if the rate of generation of new productively infected cells is sufficiently small. The standard model, adapted to explicitly include the relevant entities, suggested a relationship among the latent reservoir size, the strength of the HIV-specific adaptive immune response, and post-treatment control, which the authors explored mathematically in detail using standard phase-space mapping methods. The explanatory power of the mathematical model is small, as it adds little to the verbal explanation of the simple hypothesis; a generic hypothesis was rephrased in mathematical terms with the help of the generic notion of bi-stability. In the Conway-Perelson model, the two species, CTLs and infected cells, interact in a stereotypical way. Following initial or recrudescent infection, infected cells grow in number rapidly while stimulating the growth of CTLs that in turn can kill infected cells. The predator-prey-like system is inherently bistable, but the kinetic parameters and the initial conditions are selected in such a way that chronic, full-blown infection usually results when people become infected, as observed. Blocking the infection process early enough through therapy can change the outcome, post-treatment, because the initial conditions are different—notably, the initial number of CTLs is larger. So is also the number of cells that can spark viral replication (latently infected cells), and therefore the outcome also depends on reservoir size.

The model is structured to be bi-stable in terms of the relevant variables, and therefore initial conditions that result in post-treatment control inevitably exist, as do near-control scenarios with delayed or slow viral rebound. The analysis and numerical simulations just illustrate these generic properties of the model. Whether the real host-virus system possesses these properties is not known. Note that delayed viral rebound is logically attributable to a small HIV reservoir independent of the operation of anti-HIV immune responses. Whether reducing the reservoir and concomitantly establishing strong antiviral immunity is a feasible strategy toward a functional cure of HIV infection remains an open question. As Burnet might have said – the mathematical model provided little practical enlightenment.

# CONTROVERSY OVER CAUSE AND EFFECT IN HIV PATHOGENESIS

The correlation between two hallmarks of untreated HIV infection, gradual CD4+ T-cell depletion and heightened immune activation and T-cell turnover, was the subject of a long debate. The dogma was at first that the death of infected cells gradually depleted the pool, while chronic immune activation was an epiphenomenon. Others reasoned that intact homeostatic mechanisms should have easily overcome the low rate of cell loss and invoked different mechanisms by which elevated immune activation and inflammation could be driving progressive CD4 depletion (113–116).

In 1995, two side-by-side publications in Nature presented results showing rapid decline in HIV concentration in blood of patients following initiation of antiretroviral treatment. Based on misinterpreting a parallel rapid increase in CD4 counts—which in fact reflected lymphocyte redistribution from tissues to blood following treatment initiation and reduced inflammation—the so-called "tap-and-drain" model was proposed (117–119) and received much attention. Accordingly, CD4 T cells are infected and killed by HIV at a very high rate, triggering a massive homeostatic response. To account for the very slow progression of CD4 T-cell depletion in most infected individuals in the face of such rapid killing, the rate of T-cell production was required to "almost" keep up with the rate of loss but always remain a little short of target—a highly implausible requirement, especially given the large inter-patient differences in viral load and other parameters.

Disease progression is more plausibly characterized as a steady state that is quasi-stable due to slow parametric variation. We and others identified the relevant parameter as the renewal capacity of uninfected CD4 T cells, mainly of the central memory subset, wearing down progressively as a result of recurrent activation and microenvironmental damage (104, 107, 112, 115, 120–129). We invoked basic immunology considerations in reasoning that even if productively infected cells are rapidly infected and killed by the virus or by CTLs, these infected cells would primarily be differentiated memory cells, which are inherently short-lived and/or lacking renewal capacity, that arise in the course of activation bursts and express high levels of the HIV coreceptor, CCR5. Such cells turn over rapidly and are physiologically "expendable," so that their infection is unlikely the cause of CD4 depletion (107, 112). Therefore, contradicting the view that ongoing CD4 depletion caused immune activation, promoted by proponents of the "tap-and-drain" model (and of its subsequent derivatives), we proposed just the opposite. This position prevailed and became broadly accepted.

One variant of "tap-and-drain" was the "source model," invoked to explain and simulate the results of in-vivo DNA labeling of activated T cells in SIV-infected rhesus macaques. The fraction of labeled memory cells dropped rapidly in both CD4 and CD8 T cells, with multiphasic kinetics. It was argued that this kinetics reflected rapid virusinduced killing of T cells and their steady-state replacement a homeostatic response—by uninfected cells coming from a "source" (130, 131). Why CD8 T cells, which are not targeted by HIV, showed similar decline post labeling was not satisfactorily explained. In our interpretation, activation was not a homeostatic response to virus-mediated killing. Rather, most of the labeled T cells had divided in response to stimulation by antigens, self and foreign, in which the otherwise weaker TCR-mediated signaling by self-peptides may have been enhanced by inflammation (106, 112). This kind of activation involves time-structured cell proliferation, differentiation and death. Untreated SIV/HIV infection continuously triggers asynchronous expansion-and-contraction episodes. When label is given over short periods, one is mainly tracking the rapid transient expansion (during the labeling period) and contraction (post labeling) of recently-activated cell populations, rather than the average turnover of the entire population. The death of activated CD4+ and CD8+ T cells counters their earlier accelerated proliferation, and therefore the observed decline in labeled cells post labeling had no bearing on the issue of CD4 depletion.

The key argument was that rapidly-dividing, short-lived differentiated cells were selectively labeled. To prove it, better characterization of the turnover of T cells in SIV-infected macaques was required. Tracking BrdU labeling and Ki67 expression simultaneously provided more information than BrdU (or deuterium) labeling alone. Picker and colleagues used 1–4 days BrdU labeling to tag dividing T cells in SIVinfected macaques and studied the kinetics of phenotypicallydistinct labeled cells in blood and tissues. Using the diminishing intensity of BrdU labeling of cells as a marker of continued division when BrdU was no-longer given, and that of Ki67 expression to estimate temporal proximity to last division (107), it was estimated that most of the cells that have divided in the previous day did it again once or twice during the following day, consistent with the concept of proliferation burst. A rapid decline of labeled effector memory T cells in blood was followed by a wave of these cells in mucosal tissue. Thus, as normally observed during isolated immune responses to pathogens, rapid successive division was the source of recirculating and tissueseeking cells. The lesson from this story is, again, that it is the soundness of the biological insights that matters, not the ability to simulate a given set of data using mathematical equations (8).

# Shock and Kill, or Rinse and Replace? Organizing Principles Matter

The ability of HIV to remain quiescent in a latent reservoir in long-lived CD4+ memory T cells is the main barrier to a cure. Unrealistic expectations of inducing the virus or CTLs to kill the latently infected cells by broadly reactivating the virus from latency ("shock and kill") in the presence of effective antiretroviral treatment (ART) were based in part on simplistic mathematical models. These models did not discern activation-associated death of infected cells from virus-mediated killing [(117, 119), see (107)]; instead of dying, latently-infected cells often proliferate when activated. My colleagues and I have proposed an alternative strategy, utilizing the natural homeostatic controls that govern the turnover and numbers of

T cells in order to boost the replacement of infected memory T cells with newly-generated, uninfected cells (132) 1 .

In essence, this strategy is aimed to imitate the one used by the body during untreated infection. Chronic immune activation is generally thought to result in a gradual loss of the immune system's regenerative capacity, but several observations show, when carefully analyzed, that in the shorter run the homeostatic response to immune activation delimits HIV-infected cell frequency. Integrated viral DNA is not efficiently detected by HIV-specific lymphocytes, but occasional activation of infected memory-phenotype CD4+ T cells in the lymphoid tissues generates sufficient inflammation to activate antigen-presenting cells and trigger bursts of immune activation in various locations. Bystander naïve and resting memory cells are selectively recruited into such localized bursts, based on cross-reactivity to selfantigens (and other common antigens) co-presented on the activated APC. Preexisting infected cells are progressively rinsed out and diluted by the influx of (initially) uninfected cells in the course of repeated expansion and contraction episodes, as the total number of nominally long-lived memory cells surviving these events is strictly controlled. This, in the absence of ART, results in accelerated turnover and reduced quasi steady-state level of proviral DNA, which in turn restricts viral replication and diversification (132) 1 .

To boost CD4+ T-cell turnover during ART, when residual immune activation alone can no longer drive a significant flux, sequential waves of polyclonal T-cell proliferation and differentiation can be deliberately triggered using a variety of tested agents over a protracted period of time<sup>1</sup> . ART will prevent infection of new cells. Adopting this strategy in practice would require a shift of paradigm and method, from "shock and kill" to "rinse and replace," although both strategies have provirus activation at core and latency-reversing agents could be incorporated in protocols. For the purposes of this commentary, we stress that our theory draws from previous observations and theoretical considerations regarding the concomitant regulation of immune activation and homeostasis, which suggested that memory T cells are subject to a structured dynamic replacement under conditions of recurrent activation (77, 85, 86). The model also relies on conceptual work that revealed the nature of chronic immune activation in untreated HIV infection (see above) and on the above-mentioned model of feedback-controlled balance of growth and differentiation (76). In the development of these concepts, and in their application, mathematical models were used for illustration purposes and to demonstrate consistency, not to validate assumptions or predict numerical results.

#### JUDGEMENT OF FACT AND JUDGEMENT OF VALUE

The inter-disciplinary differences—epistemic, methodological, and cultural—among immunologists, virologists and clinicians, and especially the gaps between all of them and the applied mathematicians who engage in the modeling of immune and related physiological phenomena, have often made it hard for the biologists to discriminate scientific progress from noise (102). As research in mathematical immunology shifts from low-dimensional models into more holistic, systemsbiology approaches, this problem will get worse. Theoretical immunologists often review their (collective) work, and they unanimously agree that the field has been coming of age into a gratifyingly prolific adulthood—a view that I and several colleagues do not entirely share. It is important to recognize the limitations and inadequacies of past and current modeling approaches and practices, because these will not necessarily disappear spontaneously with the availability of richer data and introduction of comprehensive mathematical and computational modeling methodologies. Listed below are archetypal shortcomings and fallacies—"the mathematical immunologists' dirty little secrets," to paraphrase Janeway—with occasional reference to earlier mentioned examples.


<sup>1</sup>Grossman Z, Singh NJ, Simonetti FR, Kawabe T, Bocharov G, Meier-Schellersheim M, et al. From shock and kill to rinse and replace: would boosting T-cell turnover reduce the HIV reservoir? Submitted, 2019.

(see above), providing new insights into "bystander activation"—mainstream mathematical immunologists insisted, and still do, on describing increased T-cell turnover during HIV infection simply as increased average rates of unstructured division and death. Ignoring the compelling evidence for the centrality of "T-cell activation bursts" and their dynamics led to problematic models such as the "tapand-drain" and the "source" models, to misinterpretations of in-vivo DNA-labeling results, and to the misconception that infected macrophages account for the "second phase" in viral-load decline following the initiation of antiretroviral treatment (leading some to believe even now that macrophages are an important reservoir for HIV, despite the absence of supportive evidence; R. Swanstrom, CROI 2019, Abstract 62). Sticking to obsolete or unfounded assumptions might be attributed in part to the striking publication success of this kind of work. Bill Paul, who knew something about immunology, and about AIDS research, found that success to be quite astonishing.


In view of such pitfalls and practices, mathematical modeling results presented in biomedical and general journals must be carefully evaluated. The above-listed problems plague the soundness of modeling results and belie the very notion that mainstream mathematical immunology makes substantial contributions to the science and is heading in the right direction.

# CONCLUSION

The myth that mathematical models have provided important insights into basic immunology, HIV pathogenesis, etc., has been consistently and successfully propagated by mainstream theoretical immunologists but has little foundation. Given that success, it has become fashionable for biologists to team with biomathematicians, believing this helps to make interpretations more "rigorous" and therefore more credible. But the obsolete "standard" ecological view of immunity and hostpathogen dynamics, which has subjugated the thinking of most professional modelers for decades, is as prevailing as ever.

In the general world of immunology, new paradigms gradually replace some of the older ones, including the classic "self-nonself discrimination" and the more recent "pathogennonpathogen discrimination" (134). Timely assimilation of these developments could have prevented the frequent failure of ambitious models, after they have received much attention, to stand the test of time. In this context, certain practices need to be avoided, e.g., grounding major propositions on insufficient data; considering related data sets in isolation, resulting in conflicting interpretations of the same phenomena by different groups; failing to acknowledge that a desired result was tacitly incorporated into a model's assumptions on ad-hoc basis; allowing personal dichotomies or mere opportunism to influence the scientific discourse (the latter is a rich topic that is not expounded here). Such practices impede progress.

For the most basic layer of immunologic research, outstanding "big questions" need to be defined. New challenges arise from the recognition that, in William Paul's words, "the behavior of immune cells is highly colored by the cellular/molecular environment in which they exist" (135). The new era of better technology and new methodology might allow interesting speculations to become subject to modeling that is falsifiable (8) (what is commonly called "testable"). By interesting I mean plausible enough, biologically, based on existing knowledge, to deserve the efforts involved in testing. My personal wish list would include, for example, the following issues:


of autoimmunity? Similarly, chronic HIV infection is associated with inflammation and ongoing polyclonal proliferation-and-differentiation bursts, but autoimmunity is not a major hallmark of the infection. Is it due to the low affinity of most activated cells for cognate self-antigens? It has been proposed that TCR-mediated destruction of tissue cells has a higher threshold than activation and differentiation (23, 24) but the issue has not been investigated. Cohen's rather abstract "wellness profile hypothesis" may provide a more fundamental perspective, which for the time being remains a "hopeful monster" (see next).


shaping of corrective responses? How is the choice—in-situ normalization (18) or induced turnover to rinse and replace (77) <sup>1</sup>—being managed? The feasibility of "deep learning" has been invoked by Cohen and Efroni (53), but it would imply extremely large number of ad-hoc phenotypic adjustments and unpredictable biochemical trajectory to the goal. These questions might be approached by using bioinformatics tools, such as co-expression network analysis and hierarchical clustering analysis of differentially expressed genes (142, 143), to characterize the evolution of gene-expression networks associated with different stages of the development and resolution of inflammatory processes. One might hope to identify robust local tissue signatures of "health," and of different kinds and levels of injury or malfunction, and in parallel features reflecting functional activity or adaptation of effector cells, especially those belonging to the trained subset. Once these key biomarkers are identified, controlscience tools could be utilized in adjusting therapeutic interventions to changes in the monitored biomarkers. "It may be more comfortable to ignore that our natural defense system permanently prevents and/or cures many infections, cancers, cardiovascular disorders, and so on. Nevertheless, understanding, then mastering better, these physiological dynamics, which maintain a stability slowly destroyed by physiological aging, will ultimately help improve our health" (18).

Current mathematical models of complex physio-pathological processes are inherently limited in their ability to provide new explanations for the observed phenomena and to predict the future course of events. The dimensionality of such models is necessarily limited, because the size of available biomedical data sets tends to be too small to allow for a reliable quantitative parameter estimation in models that involve many variables (144); there are too many ways to fit the data, including the (unknown) right way. The hope that a nominally high-dimensional biological system may effectively behave as a relatively low-dimensional dynamical system is not farfetched; several examples support the notion that there are simple organizing principles that allow a lower-dimensional representation. However, the variables which effectively represent such a minimal low-dimensional system cannot generally be expected to be simply a subset of the natural constituents of the biological system. Rather, they are likely to be some multivariate functions of these constituents. Similarly, the topological properties and the actual location of boundary surfaces between qualitatively different classes of behavior (e.g., surfaces defining domains of influence of attractors in the phase space) depend on the original parameters in an associative way. Full identification of a minimal set of dynamical equations that would enable robust mapping of different initial conditions to the asymptotic solutions may be feasible, using statistical inference and statistical pattern-recognition methods, but only if prior knowledge about the nature of the interactions underlying these dynamics exists (145). Conceptual understanding is a prerequisite.

In the future, reductionist research will remain as important as it has always been, but a systems biology outlook will become increasingly necessary for the integration of results. In a sense, systems biology extends the reductionist program, looking at complex motifs and circuits instead of at small sets of components one at a time. Even in its modern attire, using systems biology tools, reductionist research "does not shift our view of the immune system from a static schematic perception to a dynamic multi-level system" (see introductory comments to this article collection by the editors). Dynamical systems methods in the hands of applied mathematicians will be instrumental in proceeding to a full integrative approach. To participate effectively in the integration process, mathematical immunologists need to become more intimately engaged in the quest for the general rules—where a familiarity with the theory of dynamical systems and with model identification theory is helpful, but not sufficient—and to severely constrain models describing sets of observations by requiring confluence with such rules. The insights provided by sound hypotheses can aid in developing comprehensive yet appropriately simplified multiscale meta-models that circumvent "the curse of dimensionality" (144). The increasing interest in systems biology, and the development of powerful experimental and analytical tools, provide conditions whereby assumptions pertaining to cells, tissues and whole organisms can be efficiently assessed, and predictions can be tested experimentally and clinically. Where plenty of data and technology is available, it is important not to allow inadequate assumptions to be the weakest link.

# AUTHOR CONTRIBUTIONS

The author confirms being the sole contributor of this work and has approved it for publication.

# ACKNOWLEDGMENTS

My old colleague Dr. Sandy Livnat and my NIH colleagues Daniel Douek and Martin Meier-Schellersheim critically read through the manuscript and made many helpful suggestions. This review is dedicated to the memory of William E. Paul, my close colleague, mentor and friend, who was my partner in developing the smart surveillance concept delineated here. I know he found conceptual work important and personally fulfilling, dedicated as he was to conducting and leading empirical research in basic immunology, to which he made seminal and path-breaking contributions during his magnificent career.

# SUPPLEMENTARY MATERIAL

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fimmu. 2019.02522/full#supplementary-material

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**Conflict of Interest:** The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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