- 1Medical School, Swansea University, Swansea, United Kingdom
- 2Flow Cytometry Core Facility and Innovation, Methodology and Application Research Theme, Biosciences Institute, Faculty of Medical Sciences, Newcastle University, Newcastle upon Tyne, United Kingdom
- 3College of Engineering, Swansea University, Swansea, United Kingdom
Many chemotherapeutic drugs target cell processes in specific cell cycle phases. Determining the specific phases targeted is key to understanding drug mechanism of action and efficacy against specific cancer types. Flow cytometry experiments, combined with cell cycle phase and division round specific staining, can be used to quantify the current cell cycle phase and number of mitotic events of each cell within a population. However, quantification of cell interphase times and the efficacy of cytotoxic drugs targeting specific cell cycle phases cannot be determined directly. We present a data driven computational cell population model for interpreting experimental results, where in-silico populations are initialized to match observable results from experimental populations. A two-stage approach is used to determine the efficacy of cytotoxic drugs in blocking cell-cycle phase transitions. In the first stage, our model is fitted to experimental multi-parameter flow cytometry results from untreated cell populations to identify parameters defining probability density functions for phase transitions. In the second stage, we introduce a blocking routine to the model which blocks a percentage of attempted transitions between cell-cycle phases due to therapeutic treatment. The resulting model closely matches the percentage of cells from experiment in each cell-cycle phase and division round. From untreated cell populations, interphase and intermitotic times can be inferred. We then identify the specific cell-cycle phases that cytotoxic compounds target and quantify the percentages of cell transitions that are blocked compared with the untreated population, which will lead to improved understanding of drug efficacy and mechanism of action.
Introduction
The use of flow cytometry and more recently, imaging flow cytometry are well established as methods for investigating single cell properties and proliferation in large cell populations (Vermeulen et al., 2003; Darzynkiewicz et al., 2004; Filby et al., 2011). These multi-parameter flow cytometry techniques are particularly useful in assessing the effectiveness of therapeutics which inhibit progress through the cell cycle (Vermeulen et al., 2003). Staining asynchronously dividing cells using CellTraceViolet (CTV), Propidium Iodide (PI) and phospho-histone H3 (pH3) staining (Filby et al., 2011) allows the determination of cell numbers in G1, S, G2 and M phases and the number of division rounds each cell has undergone. When a therapeutic agent is introduced in-vitro, the change cell population percentages in each phase at a given time point, can indicate which phase transitions are inhibited. However, quantifying the effectiveness of therapeutics is complicated by the complex dynamics and heterogeneity within a cell population. Certainly, the asynchronous nature of a proliferating cell line means that while two cells are in the same cell cycle phase, they may exist in different rounds of division and thus are temporally distinct form one another. This can have a significant impact on interpreting data from cell cycle inhibitory compounds as both cell cycle phase and cell division history, may determine therapeutic response (Filby et al., 2016).
A range of mathematical and computational models have been developed to predict the dynamics of the cell cycle, both for wild type populations and populations treated with therapeutics. Models fall into two categories; explanatory models designed around specific experiments and used to interpret the results or predictive models formulated based on theoretical cell cycle hypothesis (Fuß et al., 2005).
Both types of model rely on an underlying hypothesis and mathematical formulations. Two commonly used mathematical schemes are continuous systems of ODE’s (Sible and Tyson, 2007; Novak et al., 2009) and discrete Boolean models (Davidich and Bornholdt, 2008; Davidich and Bornholdt, 2013; Fauré et al., 2006). While these models are effective in capturing wildtype behavior of specific cell populations and the effect of modulation of modeled regulatory systems, they are often found to be too simplistic to capture the intrinsic processes of the cell cycle (Chaffey et al., 2014; Fuentes-Garí et al., 2015).
Population balance models (PBM) based on systems of first order PDE’s, Faraday et al. (2001) and Basse and Ubezio (2007) are common in the literature. The population of cells in each phase are modeled by continuous population density functions. The number of cell cycle phases modeled varies dependent on the application (Abroudi et al., 2015). The rates of cells entering and exiting each phase are governed by transition functions, which are dependent on cell properties, such as cell DNA content (Liu et al., 2007), mass (Mantzaris et al., 1999), age (Liou et al., 1997; Chaffey et al., 2014), cell size (Liu et al., 2007; Chapman et al., 2008) and cyclin content (García Münzer et al., 2014), either individually or in combination. From a modeling perspective, age-structured models are advantageous as without defining a maximum phase time, cells may remain within a phase indefinitely (Chaffey et al., 2014). However, they can be difficult to verify as cell age is difficult to measure experimentally, compared with more easily observable parameters such as DNA content (Liu et al., 2007).
In contrast to the continuous population distributions utilized in PBMs, cell ensemble models (CEM) model the population as an ensemble of individual cells (Henson, 2005). Simulating cells discretely allows heterogeneity of key cellular parameters across the population to be captured. Altinok et al. (2011) develop a stochastic CEM to study the desynchronization of cell populations and entrainment into the circadian clock. The effects of changing mean interphase times and variability are analyzed. While similar in structure to the model presented herein, Altinok et al. (2011) do not compare directly to experimental data.
Cadart et al. (2018) develop a mathematical frame work to investigate the relative contributions of growth rate and cell cycle duration modulation to mammalian cell size homeostasis. Linear regression is used to fit the growth rate and duration modulation coefficients to experimental data in a balance equation. Designing the model and experiment in tandem is clearly advantageous to confirm model accuracy and identifying key parameters pertaining to cell population growth.
Many models attempt to quantify the efficacy of cancer therapeutics and determine optimal treatment plans. Fuentes-Garí et al. (2015) compare three different cell cycle models to evaluate their suitability for modeling the effect of chemotherapy treatments on leukemia cell populations, demonstrating the difficulty of selecting models that accurately replicates experimental results while producing meaningful biological quantities.
The mode of action of therapeutics often modifies processes only during certain phases of the cell cycle e.g., DNA synthesis (Schwartz and Dickson, 2009). Therefore, to assess the efficacy of the intervention it is essential to determine at what points in the cell cycle the cells are affected. Given the number cells present in all phases in an asynchronous cell population and the heterogeneity of the timings of cell cycle transition processes it is difficult to determine and quantify which phases have been affected.
In previous work we have shown the effectiveness of explanatory stochastic data driven models to assess the distribution of quantum dots markers in asynchronously dividing cell populations, where more traditional statistical analysis has been unsuccessful (Brown et al., 2010; Errington et al., 2010). Virtual in-silico cell populations, containing large numbers of cells, are initialized and optimized to match experimentally determined phase populations. These models capture the variability and heterogeneity of individual cell cycles and events, giving greater insight into population dynamics than population statistics approaches.
In this work we present an explanatory virtual in-silico cell population cell cycle model. The model is used as an analysis tool to extract information from imaging flow cytometry experimental data, extracting inter-phase and inter-mitotic times and quantifying the efficacy of therapeutics in inducing arrest in cell cycle phases.
The model consists of two main steps. The first step requires experimental data from untreated cell populations to determine cell-cycle parameters defining distributions governing the time that cells reside in each cell-cycle phase. The parameters values are obtained by fitting the measured cell precursor frequency (cpF) of the experimental data to the cpF of the virtual population. The second step uses experimental data from the same population of cells plus the addition of a therapeutic after a set number of hours after at the initiation of the experiment. The parameters governing inter-phase time from the untreated populations are used in the model for the treated population. To account for therapeutic induced cell-cycle-phase arrest a blocking probability is assigned to each cell cycle phase, which gives the probability that any attempted transition to the next cell cycle phase is blocked. The values of the blocking parameter are determined by fitting the cpF of the treated in-silico population to the treated experimental population.
We demonstrate the effectiveness of this approach using 4 different cytotoxic agents known to target specific cell cycle phases. The model identifies the cell cycle phases targeted by the agents and quantifies the percentage of transitions blocked in each phase compared to the untreated population.
Methods
Experiment
To generate the experimental dataset, we perform imaging flow-cytometry measurements on populations of T-lymphocyte cells, from the Jurkat cell-line. Cells were stained using CellTraceViolet (CTV), Propidium Iodide (PI) and phospho-histone H3 (pH3) to determine the number of cells in G1, S, G2, and M phases. Using a gating strategy previous defined (Begum et al., 2013) we determine the cumulative precursory frequency (cpF), defined as the percentage of the total cell population in each cell cycle phase for each division round. Successive rounds are normalized by a factor of 2n where n is the division round, n = 0 corresponding the initial undivided population. For further experimental details see supplementary information. Figure 1 shows experimental cpF’s for an untreated Jurkat cell population measured at times T = 0, 32, 48 and 72 h, generated by the methods discussed above and in our previous work (Barteneva et al., 2016). It is evident that cells become increasingly dispersed throughout different phases and division rounds as measurement T increases.
FIGURE 1. Experimental and model cpF for untreated sample. Light blue bars show cpF from experimental population and dark blue bars show cpF from in-silico population at (A) 0 h, (B) 32 h, (C) 48 h, and (D) 72 h (note at 0 h cpF of in-silico population is matched to experimental population). Error bars on experimental data show the largest percentage error for any population observed in the control data. Error bars on model data show standard error from fits of 100 in-silico cell populations.
Using the same methodology, further experiments were performed where a cytotoxic drug is introduced to cell populations after 32 h. The cpF is measured at the time the drug is added and again at 48 h. Four different therapeutic treatments were used; Nocodazole, Demecolcine, 5-Fluorouracil, and Etoposide.
All experiments were carried out in triplicate and the mean and standard deviation of the proportion of cells in each phase calculated. As with previous work (Begum et al., 2013; Filby et al., 2011) we observed very small errors, especially with the shorter cell cycle phases, therefore we choose to take a pessimistic approach and assume an error for each phase equal to the largest percentage error observed which was 14% (Figure 1). However, we note that the least squares cost function for fitting the model is robust to the inclusion of Gaussian errors (especially small and uncorrelated errors observed in these experiments) which makes this approach easily applicable.
Model
To determine cell phases targeted by each therapeutic, we have developed a transient cell cycle model utilizing a discrete in-silico cell population. The model was developed using MATLAB (Mathworks Inc., 2016). Each cell in an in-silico is represented by three parameters; i) its current cell-cycle phase (G1, G2, S or M), ii) the time it has resided in the current cell cycle phase, t, and iii) the number of division rounds the cell has undergone, N (undivided cells are labeled as division round 0). The transition of cells to their next cell-cycle phase is a stochastic process, governed by time dependent probability distribution functions (PDF),
Initialization
For each run of the model, an in-silico population of 20,000 individual cells is initialized with cpF matched to experimental data at time
Transition Through Cell Cycle
Starting at
where
By using GEV PDF’s with
Fitting to Experimental Data
Model Applied to Untreated Population
To obtain optimal values of
Initially, we used separate values of
Given the similarities of the fit parameters across different division rounds, the number of fit parameters was reduced by using the same values of
In the experimental cpF plots it is evident that a small percentage of cells are arrested in division round 0 for the duration of the experiment (see Figure 1, panels C and D). These arrested cells are considered within the model, by randomly selecting a subset of cells from the initial population that cannot transition from their original phase or division round. The cpF of the total population of in-silico cells arrested is matched to the experimental data.
Model Applied to Treated Population
To quantify the effect of therapeutics on cell cycle progression, we fitted the cpF of in-silico cell populations to the treated experimental populations, using the DE algorithm (Storn and Price, 1997). The values of
In the experiments, therapeutics were added at 32 h. Correspondingly, for T < 32 h, we set
Results
Untreated Population
Figures 1A-D show cpF's of model and experimental populations at 0, 32, 48 and 72 hours respectively. There is good correspondence between the cpF’s of the model and experimental data for all cell cycle phases and division rounds. The final values of
TABLE 1. Mean vaues of transition PDF location, µ, and scale parameters, σ, for final untreated model, averaged over 100 individual fits.
The time of transition through division round 0 is expected to be shorter than for other division rounds, as cells in the experimental are not synchronized and already part way through their current cell cycle. For G1 phase,
Treated Population
The cpF of in-silico populations fitted to experimental data after 48 h for the four therapeutics, Nocodazole, Demecolcine, 5-Fluorouracil, and Etoposide is shown in Figure 2. There is clear correspondence between model and experimental cpF’s. The assumption that the phase transition PDF’s of the untreated population can be used for the treated populations is shown to be reasonable given the quality of the fits of the in-silico populations cpF to experiment (Figure 2). The blocking factors for each phase gives a measure of the probability of a cell attempting to transition into its next phase being blocked. If a transition is unsuccessful the phase time t is not reset and it is still able to attempt a transition in a later time step. This approach is designed to capture delay in transitions due to the therapeutic, rather than explicitly capturing cell cycle arrest. However, due to the use of GEV PDF’s (with
FIGURE 2. Experimental and model cpF results for cell populations treated with different inhibitor drugs. Experimental (light blue bars) and model (dark blue bars) cpFs after 48 h, after introducing therapeutics; (A) Nocodazole (B) Demecolcine (C) 5-Fluorouracil (D) Etoposide at 32 h. Error bars on experimental data show the largest percentage error for any population observed in the control data. Error bars on model data show standard error from fits of 100 in-silico cell populations.
The values of
FIGURE 3. Percentage of transitions blocked from each cell phase of treated cell populations compared with the untreated population. Box plots show median (red horizontal line), interquartile range (blue boxes), range of data points within a factor of 1 of the interquartile range (whiskers) and outliers (red crosses) of the blocking factors for (A) Nocodazole (B) Demecolcine (C) 5-Fluorouracil (D) Etoposide blocking drugs.
Figure 3B shows the percentage of transitions blocked for Demecolcine treatment. Demecolcine depolymerizes microtubules and limits microtubule formation arresting cells in the M phase. This depolymerization effect can also cause the breakdown cytoskeletal structures resulting in arrest in other phases (Mukhtar et al., 2014). This behavior is reflected in the blocking percentages of each phase with 63.2% blocked on the M phase and smaller but high percentages blocked in the other three phases (see Supplementary Table S5).
Figure 3C the shows percentage of transitions blocked for 5-Fluorouracil treatment. 5-Fluorouracil is principally a thymidylate synthase inhibitor, blocking synthesis of pyrimidine thymidine and in consequence disrupting DNA replication (Miura et al., 2010). Cells are predominately arrested in the S phase. This is reflected by the model with 58.7% of transitions from the S phase blocked, while the percentage of transitions blocked in the other phases are negligible.
Figure 3D shows the percentage of transitions blocked for Etoposide treatment. Etoposide inhibits DNA re-litigation (Montecucco et al., 2015), causing apoptosis of cells in the S phase. Errors in the DNA due to a sub-optimal re-litigation process can result in arrest in other phases, particularly at the G2 phase DNA damage checkpoint (Nam et al., 2010). The model shows that 61.3% of transitions from the S phase are blocked, 99.7% of transitions from the G2 phase are blocked. This indicates that even if re-litigation does occur the damage caused to DNA by Etoposide is too significant to pass the DNA checkpoint for all effected cells.
The phases targeted by each inhibitor drug predicted by the model correspond well with expected behavior, indicating that the model is successfully capturing the dynamics of the cell population and the effect of blocking drugs. The efficacy of inhibitor drugs is quantified using a simple and easily interpretable parameter
Discussion
Extraction of population wide statistical quantities from cell proliferation experiments is difficult due to heterogeneity and limitations of measurement techniques. Explanatory models can be used to extract information from experimental data sets which cannot be obtained directly from measurements.
In this work we present an explanatory modeling approach which is used as a data analysis tool to quantify the efficacy of therapeutics which cause arrest in specific cell-cycle phases. We validate the modeling approach using four different therapeutics on cpF data from imaging flow cytometry experiments on the Jurkat cell line. For both treated and untreated populations, the predicted cpF’s of the in-silico population closely match experimental population at the measurement time points. The model accurately identifies the cell cycle phases targeted by each therapeutic. In addition, the efficacy of the treatment in arresting cells in each cell-cycle phase is quantified by the percentage of phase transitions blocked compared with the untreated population.
For compounds with unknown mechanisms of action or drug combination therapies, this information provides far greater insight into how cell-cycle phases are targeted when compared with simply assessing the difference in accumulation of cells in different cell-cycle phases in treated and untreated samples. Assessing targets based on accumulation alone may ignore subtle effects where a high percentage of cells being arrested in one phase masks arrest events in other phases. This could be particularly important for setting optimal does in combination therapies where multiple phases are targeted by the different treatments.
Data Availability Statement
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding authors.
Author Contributions
PR and MB contributed to conception and design of the study. DJ performed modeling work. MB assisted with modeling. AF provided the experimental dataset which formed the basis of the model. LF helped respond to referees comments and provided expertise in redrafting the manuscript. HS secured funding for this project. All authors contributed to manuscript revision, read, and approved thesubmitted version.
Funding
This work was funded by Engineering and Physical Science research council Grant EP/N013506/1. Part of this grant was used to fund DJ’s salary during this project.
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.
Supplementary Material
The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fbinf.2021.662210/full#supplementary-material.
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Keywords: cell-cycle, cancer, therapeutic, model, differential-evolution, flow-cytometory
Citation: James DW, Filby A, Brown MR, Summers HD, Francis LW and Rees P (2021) Data Driven Cell Cycle Model to Quantify the Efficacy of Cancer Therapeutics Targeting Specific Cell-Cycle Phases From Flow Cytometry Results. Front. Bioinform. 1:662210. doi: 10.3389/fbinf.2021.662210
Received: 31 January 2021; Accepted: 01 April 2021;
Published: 27 April 2021.
Edited by:
Luis Fernando Saraiva Macedo Timmers, Universidade do Vale do Taquari - Univates, BrazilReviewed by:
Bibo Tan, Fourth Hospital of Hebei Medical University, ChinaTim Lezon, University of Pittsburgh, United States
Copyright © 2021 James, Filby, Brown, Summers, Francis and Rees. 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.
*Correspondence: David W. James, ZC53LmphbWVzQHN3YW5zZWEuYWMudWs=; Paul Rees, cC5yZWVzQHN3YW5zZWEuYWMudWs=