Frontiers in Applied Mathematics and Statistics | Systems Biology section | New and Recent Articles
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RSS Feed for Systems Biology section in the Frontiers in Applied Mathematics and Statistics journal | New and Recent Articlesen-usFrontiers Feed Generator,version:12019-11-15T08:43:21.1948843+00:0060https://www.frontiersin.org/articles/10.3389/fams.2019.00056
https://www.frontiersin.org/articles/10.3389/fams.2019.00056
Metastases Growth Patterns in vivo—A Unique Test Case of a Metastatic Colorectal Cancer Patient2019-11-12T00:00:00ZGili HochmanEinat Shacham-ShmueliTchia HeymannStephen RaskinSvetlana Bunimovich-MendrazitskyColorectal cancer (CRC) is one of the most common causes of cancer-related mortality worldwide. Most cases of deaths result from metastases, assumed to be shed, in many cases, before disease detection. Providing reliable predictions of the metastases' growth pattern may help planning treatment. Available mathematical tumor growth models rely mainly on primary tumor data, and rarely relate to metastases growth. The aim of this work was to explore CRC lung metastases growth patterns. We used data of a metastatic CRC patient, for whom 10 lung metastases were measured while untreated by seven serial computed tomography (CT) scans, during almost 3 years. Three mathematical growth models—Exponential, logistic, and Gompertzian—were fitted to the actual measurements. Goodness of fit of each of the models to actual growth was estimated using different scores. Factors affecting growth pattern were explored: size, location, and primary tumor resection. Exponential growth model demonstrated good fit to data of all metastases. Logistic and Gompertzian growth models, in most cases, were overfitted and hence unreliable. Metastases inception time, calculated by backwards extrapolation of the fitted growth models, was 8–19 years before primary tumor diagnosis date. Three out of ten metastases demonstrated enhanced growth rate shortly after primary tumor resection. Our unique data provide evidence that exponential growth of CRC lung metastases is a legitimate approximation, and encourage focusing research on short-term effects of surgery on metastases growth rate.Significance

Providing reliable predictions of the metastases' growth pattern using mathematical models may help determining the optimal treatment plan that fits a given patient best and maximizes the probability of cure.

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https://www.frontiersin.org/articles/10.3389/fams.2019.00055
MCMC Techniques for Parameter Estimation of ODE Based Models in Systems Biology2019-11-01T00:00:00ZGloria I. Valderrama-BahamóndezHolger FröhlichOrdinary differential equation systems (ODEs) are frequently used for dynamical system modeling in many science fields such as economics, physics, engineering, and systems biology. A special challenge in systems biology is that ODE systems typically contain kinetic rate parameters, which are unknown and have to be estimated from data. However, non-linearity of ODE systems together with noise in the data raise severe identifiability issues. Hence, Markov Chain Monte Carlo (MCMC) approaches have been frequently used to estimate posterior distributions of rate parameters. However, designing a good MCMC sampler for high dimensional and multi-modal parameter distributions remains a challenging task. Here we performed a systematic comparison of different MCMC techniques for this purpose using five public domain models. The comparison included Metropolis-Hastings, parallel tempering MCMC, adaptive MCMC, and parallel adaptive MCMC. In conclusion, we found specifically parallel adaptive MCMC to produce superior parameter estimates while benefitting from inclusion of our suggested informative Bayesian priors for rate parameters and noise variance.]]>https://www.frontiersin.org/articles/10.3389/fams.2019.00029
https://www.frontiersin.org/articles/10.3389/fams.2019.00029
Toward a Unification of System-Theoretical Principles in Biology and Ecology—The Stochastic Lyapunov Matrix Equation and Its Inverse Application2019-07-05T00:00:00ZWolfram WeckwerthSystem theory has its roots in mathematical formalisms developed by mathematicians and physicists, such as Leibniz, Euler, and Newton, and applied by congenial chemists and biologists such as Lotka and Bertalanffy. In these approaches, the dynamical system—may it be either single organisms or populations of organisms in their ecosystems—is defined and formally translated into an interaction matrix and first-order ordinary differential equations (ODEs) which are then solved. This provides the background for the quantitative analysis of any linear to non-linear system. In his inspiring article “Can a biologist fix a radio?,” Lazebnik made the differences very clear between a “guilt by association” hypothesis of a modern biologist vs. a Signal–Input–Output (SIO) model of an electrical engineer. The drawback of this “Gedankenexperiment” is that two rather different approaches are compared—a forward model predictive control approach in the case of the SIO model by an engineer and an inverse or reverse approach by the biologist or ecologist. Biological and ecological systems are much too complex to estimate all the underlying ODE's, parameter and input signals that generate a probability distribution. Thus, the combination of inverse data-driven modeling and stochastic simulation is a key process for understanding the control of a biological or ecological system. The challenge of the next decades of systems biology is to link these approaches more systematically. Over the last years, we have developed a hybrid modeling approach based on the stochastic Lyapunov matrix equation for the analysis of genome-scale molecular data. This workflow connects forward and inverse strategies such as the genome-scale-based metabolic reconstruction of an organism and the calculation of dynamics around a quasi-steady state using statistical features of large-scale multiomics data. Ultimately, this workflow is linked to physiology and phenotype (the output) to unambiguously define the genotype–environment–phenotype relationship. This system-theoretical formalism establishes the generic analysis of the genotype–environment–phenotype relationship to finally result in predictability of organismal function in the environmental context. The approach is based on fundamental mathematical control theory for the analysis of dynamical systems using eigenvalues and matrix algebra, stochastic differential equations (SDEs), and Langevin- and Fokker–Planck-type equations eventually leading to the continuous stochastic Lyapunov matrix equation. The stochastic Lyapunov matrix equation is also a fundamental approach for the analysis and control of artificial intelligence systems in model predictive control and thus opens up completely new perspectives for the integration of systems engineering and systems biology. Furthermore, similar mathematical formalisms—using a community matrix instead of a stoichiometric matrix of a metabolic network—were also conceptually developed and applied by ecologists such as Levins and May in the analysis of stability and complexity of model ecosystems. Thus, the generalization of this hybrid forward–inverse approach spans from biology to ecology and promises to be a systematic iterative process that finally leads to functional units able to explain living systems up to their interaction in complex ecosystems.]]>https://www.frontiersin.org/articles/10.3389/fams.2019.00032
https://www.frontiersin.org/articles/10.3389/fams.2019.00032
Improved Prediction of Periodicity Using Quartet Approximation in a Lattice Model of Intracellular Calcium Release2019-07-02T00:00:00ZRobert J. RovettiThis study uses a probabilistic cellular automata (PCA) to model the spatial and temporal dynamics of calcium release units (CRUs) within cardiac myocytes. The CRUs are subject to random activation, nearest-neighbor recruitment, and temporal refractoriness, and their interactions produce a physiologically-important condition called calcium alternans, a beat-to-beat oscillation in the amount of calcium released. In the PCA this manifests as a transition to period-2 behavior in the fraction of activated lattice sites. We investigate this phenomenon using PCA simulations and moment-closure approximation methods of zero order (mean-field), first order (pair), and second order (quartet). We show that only the quartet approximation (QA) accurately predicts the thresholds of the activation and recruitment probabilities for the onset of periodic behavior (alternans), as the lower-order approximations do not sufficiently account for important spatial correlations. The QA also accurately predicts the emergence of spatio-temporal clustering in the PCA, providing an analytical framework for investigating pattern formation dynamics in such models. Our analysis demonstrates a systematic approach to efficiently handling the increased combinatorial complexity of the QA, whose required computation time is non-trivially larger compared to the mean-field approximation but remains an order of magnitude lower than the numerical PCA simulations.]]>https://www.frontiersin.org/articles/10.3389/fams.2019.00017
https://www.frontiersin.org/articles/10.3389/fams.2019.00017
Constrained Covariance Matrices With a Biologically Realistic Structure: Comparison of Methods for Generating High-Dimensional Gaussian Graphical Models2019-04-12T00:00:00ZFrank Emmert-StreibShailesh TripathiMatthias DehmerHigh-dimensional data from molecular biology possess an intricate correlation structure that is imposed by the molecular interactions between genes and their products forming various different types of gene networks. This fact is particularly well-known for gene expression data, because there is a sufficient number of large-scale data sets available that are amenable for a sensible statistical analysis confirming this assertion. The purpose of this paper is two fold. First, we investigate three methods for generating constrained covariance matrices with a biologically realistic structure. Such covariance matrices are playing a pivotal role in designing novel statistical methods for high-dimensional biological data, because they allow to define Gaussian graphical models (GGM) for the simulation of realistic data; including their correlation structure. We study local and global characteristics of these covariance matrices, and derived concentration/partial correlation matrices. Second, we connect these results, obtained from a probabilistic perspective, to statistical results of studies aiming to estimate gene regulatory networks from biological data. This connection allows to shed light on the well-known heterogeneity of statistical estimation methods for inferring gene regulatory networks and provides an explanation for the difficulties inferring molecular interactions between highly connected genes.]]>https://www.frontiersin.org/articles/10.3389/fams.2019.00018
https://www.frontiersin.org/articles/10.3389/fams.2019.00018
Metabolic Games2019-04-12T00:00:00ZTaneli PusaMartin WannagatMarie-France SagotMetabolic networks have been used to successfully predict phenotypes based on optimization principles. However, a general framework that would extend to situations not governed by simple optimization, such as multispecies communities, is still lacking. Concepts from evolutionary game theory have been proposed to amend the situation. Alternative metabolic states can be seen as strategies in a “metabolic game,” and phenotypes can be predicted based on the equilibria of this game. In this survey, we review the literature on applying game theory to the study of metabolism, present the general idea of a metabolic game, and discuss open questions and future challenges.]]>https://www.frontiersin.org/articles/10.3389/fams.2018.00051
https://www.frontiersin.org/articles/10.3389/fams.2018.00051
Synchronization, Oscillator Death, and Frequency Modulation in a Class of Biologically Inspired Coupled Oscillators2018-11-08T00:00:00ZAlessio FranciMarco Arieli Herrera-ValdezMiguel Lara-AparicioPablo Padilla-LongoriaThe general purpose of this paper is to build up on our understanding of the basic mathematical principles that underlie the emergence of synchronous biological rhythms, in particular, the circadian clock. To do so, we study the role that the coupling strength, coupling type, and noise play in the synchronization of a system of coupled, non-linear oscillators. First, we study a deterministic model based on Van der Pol coupled oscillators, modeling a population of diffusively coupled cells, to find regions in the parameter space for which synchronous oscillations emerge and to provide conditions under which diffusive coupling kills the synchronous oscillation. Second, we study how noise and coupling interact and lead to synchronous oscillations in linearly coupled oscillators, modeling the interaction between various pacemaker populations, each having an endogenous circadian clock. To do so, we use the Fokker-Planck equation associated to the system. We show how coupling can tune the frequency of the emergent synchronous oscillation, which provides a general mechanism to make fast (ultradian) pacemakers slow (circadian) and synchronous via coupling. The basic mechanisms behind the generation of oscillations and the emergence of synchrony that we describe here can be used to guide further studies of coupled oscillations in biophysical non-linear models.]]>https://www.frontiersin.org/articles/10.3389/fams.2018.00028
https://www.frontiersin.org/articles/10.3389/fams.2018.00028
Spectral Functional-Digraph Theory, Stability, and Entropy for Gene Regulatory Networks2018-07-13T00:00:00ZDevin AkmanFüsun AkmanSpectral graph theory is an indispensable tool in the rich interdisciplinary field of network science, which includes as objects ordinary abstract graphs as well as directed graphs such as the Internet, semantic networks, electrical circuits, and gene regulatory networks (GRN). However, its contributions sometimes get lost in the code, and network theory occasionally becomes overwhelmed with problems specific to undirected graphs. In this paper, we will study functional digraphs, calculate the eigenvalues and eigenvectors of their adjacency matrices, describe how to compute their automorphism groups, and define a notion of entropy in terms of their symmetries. We will then introduce gene regulatory networks (GRNs) from scratch, and consider their phase spaces, which are functional digraphs describing the deterministic progression of the overall state of a GRN. Finally, we will redefine the stability of a GRN and assert that it is closely related to the entropy of its phase space.]]>https://www.frontiersin.org/articles/10.3389/fams.2018.00018
https://www.frontiersin.org/articles/10.3389/fams.2018.00018
Reduced Numerical Approximation of Reduced Fluid-Structure Interaction Problems With Applications in Hemodynamics2018-06-29T00:00:00ZClaudia M. ColciagoSimone DeparisThis paper deals with fast simulations of the hemodynamics in large arteries by considering a reduced model of the associated fluid-structure interaction problem, which in turn allows an additional reduction in terms of the numerical discretisation. The resulting method is both accurate and computationally cheap. This goal is achieved by means of two levels of reduction: first, we describe the model equations with a reduced mathematical formulation which allows to write the fluid-structure interaction problem as a Navier-Stokes system with non-standard boundary conditions; second, we employ numerical reduction techniques to further and drastically lower the computational costs. The non standard boundary condition is of a generalized Robin type, with a boundary mass and boundary stiffness terms accounting for the arterial wall compliance. The numerical reduction is obtained coupling two well-known techniques: the proper orthogonal decomposition and the reduced basis method, in particular the greedy algorithm. We start by reducing the numerical dimension of the problem at hand with a proper orthogonal decomposition and we measure the system energy with specific norms; this allows to take into account the different orders of magnitude of the state variables, the velocity and the pressure. Then, we introduce a strategy based on a greedy procedure which aims at enriching the reduced discretization space with low offline computational costs. As application, we consider a realistic hemodynamics problem with a perturbation in the boundary conditions and we show the good performances of the reduction techniques presented in the paper. The results obtained with the numerical reduction algorithm are compared with the one obtained by a standard finite element method. The gains obtained in term of CPU time are of three orders of magnitude.]]>