A general mathematical framework to model generation structure in a population of asynchronously dividing cells

Counting generations in birth and death processes with competing Erlang and exponential waiting times

Lymphocyte populations, stimulated in vitro or in vivo, grow as cells divide. Stochastic models are appropriate because some cells undergo multiple rounds of division, some die, and others of the same type in the same conditions do not divide at all. If individual cells behave independently, then each cell can be imagined as sampling from a probability density of times to division and death. The exponential density is the most mathematically and computationally convenient choice. It has the advantage of satisfying the memoryless property, consistent with a Markov process, but it overestimates the probability of short division times. With the aim of preserving the advantages of a Markovian framework while improving the representation of experimentally-observed division times, we consider a multi-stage model of cellular division and death. We use Erlang-distributed (or, more generally, phase-type distributed) times to division, and exponentially distributed times to death. We classify cells into generations, using the rule that the daughters of cells in generation n are in generation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n+1$$\end{document}n+1. In some circumstances, our representation is equivalent to established models of lymphocyte dynamics. We find the growth rate of the cell population by calculating the proportions of cells by stage and generation. The exponent describing the late-time cell population growth, and the criterion for extinction of the population, differs from what would be expected if N steps with rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}λ were equivalent to a single step of rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda /N$$\end{document}λ/N. We link with a published experimental dataset, where cell counts were reported after T cells were transferred to lymphopenic mice, using Approximate Bayesian Computation. In the comparison, the death rate is assumed to be proportional to the generation and the Erlang time to division for generation 0 is allowed to differ from that of subsequent generations. The multi-stage representation is preferred to a simple exponential in posterior distributions, and the mean time to first division is estimated to be longer than the mean time to subsequent divisions.

FGF2 modulates simultaneously the mode, the rate of division and the growth fraction in cultures of radial glia

Radial glial progenitors in the mammalian developing neocortex have been shown to follow a deterministic differentiation program restricted to an asymmetric-only mode of division. This feature seems incompatible with their well-known ability to increase in number when cultured in vitro, driven by fibroblast growth factor 2 and other mitogenic signals. The changes in their differentiation dynamics that allow this transition from in vivo asymmetric-only division mode to an in vitro self-renewing culture have not been fully characterized. Here, we combine experiments of radial glia cultures with numerical models and a branching process theoretical formalism to show that fibroblast growth factor 2 has a triple effect by simultaneously increasing the growth fraction, promoting symmetric divisions and shortening the length of the cell cycle. These combined effects partner to establish and sustain a pool of rapidly proliferating radial glial progenitors in vitro. We also show that, in conditions of variable proliferation dynamics, the branching process tool outperforms other commonly used methods based on thymidine analogs, such as BrdU and EdU, in terms of accuracy and reliability.

Highlighted Article: When mode and/or rate of division are changing, a branching process, rather than a thymidine analog method, provides temporal resolution, it is more robust and does not interfere with cell homeostasis.

Modeling cell population dynamics

 Quantitative modeling is quickly becoming an integral part of biology, due to the ability of mathematical models and computer simulations to generate insights and predict the behavior of living systems. Single-cell models can be incapable or misleading for inferring population dynamics, as they do not consider the interactions between cells via metabolites or physical contact, nor do they consider competition for limited resources such as nutrients or space. Here we examine methods that are commonly used to model and simulate cell populations. First, we cover simple models where analytic solutions are available, and then move on to more complex scenarios where computational methods are required. Overall, we present a summary of mathematical models used to describe cell population dynamics, which may aid future model development and highlights the importance of population modeling in biology.

Modeling cell population dynamics

 Quantitative modeling is quickly becoming an integral part of biology, due to the ability of mathematical models and computer simulations to generate insights and predict the behavior of living systems. Single-cell models can be incapable or misleading for inferring population dynamics, as they do not consider the interactions between cells via metabolites or physical contact, nor do they consider competition for limited resources such as nutrients or space. Here we examine methods that are commonly used to model and simulate cell populations. First, we cover simple models where analytic solutions are available, and then move on to more complex scenarios where computational methods are required. Overall, we present a summary of mathematical models used to describe cell population dynamics, which may aid future model development and highlights the importance of population modeling in biology.

Converse Smith-Martin cell cycle kinetics by transformed B lymphocytes

Recent studies using direct live cell imaging have reported that individual B lymphocytes have correlated transit times between their G1 and S/G2/M phases. This finding is in contradiction with the influential model of Smith and Martin that assumed the bulk of the total cell cycle time variation arises in the G1 phase of the cell cycle with little contributed by the S/G2/M phase. Here we extend these studies to examine the relation between cell cycle phase lengths in two B lymphoma cell lines. We report that transformed B lymphoma cells undergo a short G1 period that displays little correlation with the time taken for the subsequent S/G2/M phase. Consequently, the bulk of the variation noted for total division times within a population is found in the S/G2/M phases and not the G1 phase. Models that reverse the expected source of variation and assume a single deterministic time in G1 followed by a lag + exponential distribution for S/G2/M fit the data well. These models can be improved further by adopting two sequential distributions or by using the stretched lognormal model developed for primary lymphocytes. We propose that shortening of G1 transit times and uncoupling from other cell cycle phases may be a hallmark of lymphocyte transformation that could serve as an observable phenotypic marker of cancer evolution.

Itraconazole targets cell cycle heterogeneity in colorectal cancer

Cellular dormancy and heterogeneity in cell cycle length provide important explanations for treatment failure after adjuvant therapy with S-phase cytotoxics in colorectal cancer (CRC), yet the molecular control of the dormant versus cycling state remains unknown. We sought to understand the molecular features of dormant CRC cells to facilitate rationale identification of compounds to target both dormant and cycling tumor cells. Unexpectedly, we demonstrate that dormant CRC cells are differentiated, yet retain clonogenic capacity. Mouse organoid drug screening identifies that itraconazole generates spheroid collapse and loss of dormancy. Human CRC cell dormancy and tumor growth can also be perturbed by itraconazole, which is found to inhibit Wnt signaling through noncanonical hedgehog signaling. Preclinical validation shows itraconazole to be effective in multiple assays through Wnt inhibition, causing both cycling and dormant cells to switch to global senescence. These data provide preclinical evidence to support an early phase trial of itraconazole in CRC.

Stochastic system identification without an a priori chosen kinetic model-exploring feasible cell regulation with piecewise linear functions

Kinetic models are at the heart of system identification. A priori chosen rate functions may, however, be unfitting or too restrictive for complex or previously unanticipated regulation. We applied general purpose piecewise linear functions for stochastic system identification in one dimension using published flow cytometry data on E.coli and report on identification results for equilibrium state and dynamic time series. In metabolic labelling experiments during yeast osmotic stress response, we find mRNA production and degradation to be strongly co-regulated. In addition, mRNA degradation appears overall uncorrelated with mRNA level. Comparison of different system identification approaches using semi-empirical synthetic data revealed the superiority of single-cell tracking for parameter identification. Generally, we find that even within restrictive error bounds for deviation from experimental data, the number of viable regulation types may be large. Indeed, distinct regulation can lead to similar expression behaviour over time. Our results demonstrate that molecule production and degradation rates may often differ from classical constant, linear or Michaelis–Menten type kinetics.

Classical cell-regulation models are often imperfectly fitting or even inconsistent with experimental data suggesting inappropriate model assumptions. Martin Hoffmann from Fraunhofer ITEM Regensburg and Jörg Galle from IZBI Leipzig analysed different protein and gene expression data using general purpose piecewise linear functions for system identification. They assessed data corresponding to various experimental techniques for their potential to determine the parameters of their models. Single-cell recordings of expression values over time were most effective for parameter identification. Generally, different and often non-classical cell-regulation models were consistent with the experimental data, even for restrictive error bounds. The authors used virtual treatment experiments to demonstrate that precise knowledge of cell regulation is important for assessing therapy effects. Their findings clearly argue in favour of system identification being performed without an a priori chosen kinetic model.

A drift-diffusion checkpoint model predicts a highly variable and growth-factor-sensitive portion of the cell cycle G1 phase

Even among isogenic cells, the time to progress through the cell cycle, or the intermitotic time (IMT), is highly variable. This variability has been a topic of research for several decades and numerous mathematical models have been proposed to explain it. Previously, we developed a top-down, stochastic drift-diffusion+threshold (DDT) model of a cell cycle checkpoint and showed that it can accurately describe experimentally-derived IMT distributions [Leander R, Allen EJ, Garbett SP, Tyson DR, Quaranta V. Derivation and experimental comparison of cell-division probability densities. J. Theor. Biol. 2014;358:129-135]. Here, we use the DDT modeling approach for both descriptive and predictive data analysis. We develop a custom numerical method for the reliable maximum likelihood estimation of model parameters in the absence of a priori knowledge about the number of detectable checkpoints. We employ this method to fit different variants of the DDT model (with one, two, and three checkpoints) to IMT data from multiple cell lines under different growth conditions and drug treatments. We find that a two-checkpoint model best describes the data, consistent with the notion that the cell cycle can be broadly separated into two steps: the commitment to divide and the process of cell division. The model predicts one part of the cell cycle to be highly variable and growth factor sensitive while the other is less variable and relatively refractory to growth factor signaling. Using experimental data that separates IMT into G1 vs. S, G2, and M phases, we show that the model-predicted growth-factor-sensitive part of the cell cycle corresponds to a portion of G1, consistent with previous studies suggesting that the commitment step is the primary source of IMT variability. These results demonstrate that a simple stochastic model, with just a handful of parameters, can provide fundamental insights into the biological underpinnings of cell cycle progression.

A Multi-stage Representation of Cell Proliferation as a Markov Process

The stochastic simulation algorithm commonly known as Gillespie’s algorithm (originally derived for modelling well-mixed systems of chemical reactions) is now used ubiquitously in the modelling of biological processes in which stochastic effects play an important role. In well-mixed scenarios at the sub-cellular level it is often reasonable to assume that times between successive reaction/interaction events are exponentially distributed and can be appropriately modelled as a Markov process and hence simulated by the Gillespie algorithm. However, Gillespie’s algorithm is routinely applied to model biological systems for which it was never intended. In particular, processes in which cell proliferation is important (e.g. embryonic development, cancer formation) should not be simulated naively using the Gillespie algorithm since the history-dependent nature of the cell cycle breaks the Markov process. The variance in experimentally measured cell cycle times is far less than in an exponential cell cycle time distribution with the same mean.

Here we suggest a method of modelling the cell cycle that restores the memoryless property to the system and is therefore consistent with simulation via the Gillespie algorithm. By breaking the cell cycle into a number of independent exponentially distributed stages, we can restore the Markov property at the same time as more accurately approximating the appropriate cell cycle time distributions. The consequences of our revised mathematical model are explored analytically as far as possible. We demonstrate the importance of employing the correct cell cycle time distribution by recapitulating the results from two models incorporating cellular proliferation (one spatial and one non-spatial) and demonstrating that changing the cell cycle time distribution makes quantitative and qualitative differences to the outcome of the models. Our adaptation will allow modellers and experimentalists alike to appropriately represent cellular proliferation—vital to the accurate modelling of many biological processes—whilst still being able to take advantage of the power and efficiency of the popular Gillespie algorithm.

Applications and implications of the exponentially modified gamma distribution as a model for time variabilities related to cell proliferation and gene expression

A panel of published distributions of cell interdivision times (IDT) comprising 77 datasets related to 16 cell types, some studied under different conditions, was used to evaluate their conformance to the exponentially modified gamma distribution (EMGD) in comparison with distributions suggested for IDT data earlier. Lognormal, gamma, inverse Gaussian, and shifted Weibull and gamma distributions were found to be generally inferior to EMGD. Exponentially modified Gaussian (EMG) performed equally well. Although EMGD or EMG may be worse than some other distributions in specific cases, the reason that IDT distributions must be generated by a common mechanism of the cell cycle makes it unlikely that they differ essentially in different cell types. Therefore, exponentially modified peak functions, such as EMGD or EMG, are most appropriate if the use of a single distribution for IDT data is reasonable. EMGD is also shown to be the best descriptive tool for published data on the distribution of times between the bursts of mRNA synthesis at defined genes in single cells. EMG is inadequate to such data because its Gaussian component markedly extends to the negative time domain. The applicability of EMGD to comparable features of cells and genes behaviors are discussed to support the validity of the transition probability model and to relate the exponential component of EMGD to the times of cell dwelling in the restriction point of the cell cycle.

Stochastic Measurement Models for Quantifying Lymphocyte Responses Using Flow Cytometry

Adaptive immune responses are complex dynamic processes whereby B and T cells undergo division and differentiation triggered by pathogenic stimuli. Deregulation of the response can lead to severe consequences for the host organism ranging from immune deficiencies to autoimmunity. Tracking cell division and differentiation by flow cytometry using fluorescent probes is a major method for measuring progression of lymphocyte responses, both in vitro and in vivo. In turn, mathematical modeling of cell numbers derived from such measurements has led to significant biological discoveries, and plays an increasingly important role in lymphocyte research. Fitting an appropriate parameterized model to such data is the goal of these studies but significant challenges are presented by the variability in measurements. This variation results from the sum of experimental noise and intrinsic probabilistic differences in cells and is difficult to characterize analytically. Current model fitting methods adopt different simplifying assumptions to describe the distribution of such measurements and these assumptions have not been tested directly. To help inform the choice and application of appropriate methods of model fitting to such data we studied the errors associated with flow cytometry measurements from a wide variety of experiments. We found that the mean and variance of the noise were related by a power law with an exponent between 1.3 and 1.8 for different datasets. This violated the assumptions inherent to commonly used least squares, linear variance scaling and log-transformation based methods. As a result of these findings we propose a new measurement model that we justify both theoretically, from the maximum entropy standpoint, and empirically using collected data. Our evaluation suggests that the new model can be reliably used for model fitting across a variety of conditions. Our work provides a foundation for modeling measurements in flow cytometry experiments thus facilitating progress in quantitative studies of lymphocyte responses.

Quantifying the length and variance of the eukaryotic cell cycle phases by a stochastic model and dual nucleoside pulse labelling

A fundamental property of cell populations is their growth rate as well as the time needed for cell division and its variance. The eukaryotic cell cycle progresses in an ordered sequence through the phases and and is regulated by environmental cues and by intracellular checkpoints. Reflecting this regulatory complexity, the length of each phase varies considerably in different kinds of cells but also among genetically and morphologically indistinguishable cells. This article addresses the question of how to describe and quantify the mean and variance of the cell cycle phase lengths. A phase-resolved cell cycle model is introduced assuming that phase completion times are distributed as delayed exponential functions, capturing the observations that each realization of a cycle phase is variable in length and requires a minimal time. In this model, the total cell cycle length is distributed as a delayed hypoexponential function that closely reproduces empirical distributions. Analytic solutions are derived for the proportions of cells in each cycle phase in a population growing under balanced growth and under specific non-stationary conditions. These solutions are then adapted to describe conventional cell cycle kinetic assays based on pulse labelling with nucleoside analogs. The model fits well to data obtained with two distinct proliferating cell lines labelled with a single bromodeoxiuridine pulse. However, whereas mean lengths are precisely estimated for all phases, the respective variances remain uncertain. To overcome this limitation, a redesigned experimental protocol is derived and validated in silico. The novelty is the timing of two consecutive pulses with distinct nucleosides that enables accurate and precise estimation of both the mean and the variance of the length of all phases. The proposed methodology to quantify the phase length distributions gives results potentially equivalent to those obtained with modern phase-specific biosensor-based fluorescent imaging.

Among the important characteristics of dividing cell populations is the time necessary for cells to complete each of the cell cycle phases, that is, to increase the cell's mass, to duplicate and repair its genome, to properly segregate its chromosomes, and to make decisions whether to continue dividing or enter a quiescent state. The cycle phase times also determine the maximal rate at which a dividing cell population can grow in size. Cell cycle phase completion times largely differ between cell types, cellular environments as well as metabolic stages, and can thus be considered as part of the phenotype of a given cell. Our article advances the methods to quantitatively characterize this phenotype. We introduce a novel phase-resolved cell cycle progression model and use it to estimate the mean and variance of the cycle phase completion times based on nucleoside analog pulse labelling experiments. This classic workhorse of cell cycle kinetic studies is revamped by our approach to potentially rival in accuracy and precision with modern phase-specific biosensor-based fluorescent imaging, while superseding the latter in its application scope.

Stretched cell cycle model for proliferating lymphocytes

Stochastic variation in cell cycle time is a consistent feature of otherwise similar cells within a growing population. Classic studies concluded that the bulk of the variation occurs in the G1 phase, and many mathematical models assume a constant time for traversing the S/G2/M phases. By direct observation of transgenic fluorescent fusion proteins that report the onset of S phase, we establish that dividing B and T lymphocytes spend a near-fixed proportion of total division time in S/G2/M phases, and this proportion is correlated between sibling cells. This result is inconsistent with models that assume independent times for consecutive phases. Instead, we propose a stretching model for dividing lymphocytes where all parts of the cell cycle are proportional to total division time. Data fitting based on a stretched cell cycle model can significantly improve estimates of cell cycle parameters drawn from DNA labeling data used to monitor immune cell dynamics.

Quantal and graded stimulation of B lymphocytes as alternative strategies for regulating adaptive immune responses

Lymphocytes undergo a typical response pattern following stimulation in vivo: they proliferate, differentiate to effector cells, cease dividing and predominantly die, leaving a small proportion of long-lived memory and effector cells. This pattern results from cell-intrinsic processes following activation and the influence of external regulation. Here we apply quantitative methods to study B-cell responses in vitro. Our results reveal that B cells stimulated through two Toll-like receptors (TLRs) require minimal external direction to undergo the basic pattern typical of immunity. Altering the stimulus strength regulates the outcome in a quantal manner by varying the number of cells that participate in the response. In contrast, the T-cell-dependent CD40 activation signal induces a response where division times and differentiation rates vary in relation to stimulus strength. These studies offer insight into how the adaptive antibody response may have evolved from simple autonomous response patterns to the highly regulable state that is now observed in mammals.

Mechanisms Underlying CD4+ Treg Immune Regulation in the Adult: From Experiments to Models

To maintain immunological balance the organism has to be tolerant to self while remaining competent to mount an effective immune response against third-party antigens. An important mechanism of this immune regulation involves the action of regulatory T-cell (Tregs). In this mini-review, we discuss some of the known and proposed mechanisms by which Tregs exert their influence in the context of immune regulation, and the contribution of mathematical modeling for these mechanistic studies. These models explore the mechanisms of action of regulatory T cells, and include hypotheses of multiple signals, delivered through simultaneous antigen-presenting cell (APC) conjugation; interaction of feedback loops between APC, Tregs, and effector cells; or production of specific cytokines that act on effector cells. As the field matures, and competing models are winnowed out, it is likely that we will be able to quantify how tolerance-inducing strategies, such as CD4-blockade, affect T-cell dynamics and what mechanisms explain the observed behavior of T-cell based tolerance.

Quantifying T lymphocyte turnover

Peripheral T cell populations are maintained by production of naive T cells in the thymus, clonal expansion of activated cells, cellular self-renewal (or homeostatic proliferation), and density dependent cell life spans. A variety of experimental techniques have been employed to quantify the relative contributions of these processes. In modern studies lymphocytes are typically labeled with 5-bromo-2'-deoxyuridine (BrdU), deuterium, or the fluorescent dye carboxy-fluorescein diacetate succinimidyl ester (CFSE), their division history has been studied by monitoring telomere shortening and the dilution of T cell receptor excision circles (TRECs) or the dye CFSE, and clonal expansion has been documented by recording changes in the population densities of antigen specific cells. Proper interpretation of such data in terms of the underlying rates of T cell production, division, and death has proven to be notoriously difficult and involves mathematical modeling. We review the various models that have been developed for each of these techniques, discuss which models seem most appropriate for what type of data, reveal open problems that require better models, and pinpoint how the assumptions underlying a mathematical model may influence the interpretation of data. Elaborating various successful cases where modeling has delivered new insights in T cell population dynamics, this review provides quantitative estimates of several processes involved in the maintenance of naive and memory, CD4(+) and CD8(+) T cell pools in mice and men.

Analysis and simulation of division- and label-structured population models : a new tool to analyze proliferation assays

In most biological studies and processes, cell proliferation and population dynamics play an essential role. Due to this ubiquity, a multitude of mathematical models has been developed to describe these processes. While the simplest models only consider the size of the overall populations, others take division numbers and labeling of the cells into account. In this work, we present a modeling and computational framework for proliferating cell populations undergoing symmetric cell division, which incorporates both the discrete division number and continuous label dynamics. Thus, it allows for the consideration of division number-dependent parameters as well as the direct comparison of the model prediction with labeling experiments, e.g., performed with Carboxyfluorescein succinimidyl ester (CFSE), and can be shown to be a generalization of most existing models used to describe these data. We prove that under mild assumptions the resulting system of coupled partial differential equations (PDEs) can be decomposed into a system of ordinary differential equations (ODEs) and a set of decoupled PDEs, which drastically reduces the computational effort for simulating the model. Furthermore, the PDEs are solved analytically and the ODE system is truncated, which allows for the prediction of the label distribution of complex systems using a low-dimensional system of ODEs. In addition to modeling the label dynamics, we link the label-induced fluorescence to the measure fluorescence which includes autofluorescence. Furthermore, we provide an analytical approximation for the resulting numerically challenging convolution integral. This is illustrated by modeling and simulating a proliferating population with division number-dependent proliferation rate.

A division-dependent compartmental model for computing cell numbers in CFSE-based lymphocyte proliferation assays

Some key features of a mathematical description of an immune response are an estimate of the number of responding cells and the manner in which those cells divide, differentiate, and die. The intracellular dye CFSE is a powerful experimental tool for the analysis of a population of dividing cells, and numerous mathematical treatments have been aimed at using CFSE data to describe an immune response [30,31,32,37,38,42,48,49]. Recently, partial differential equation structured population models, with measured CFSE fluorescence intensity as the structure variable, have been shown to accurately fit histogram data obtained from CFSE flow cytometry experiments [18,19,52,54]. In this report, the population of cells is mathematically organized into compartments, with all cells in a single compartment having undergone the same number of divisions. A system of structured partial differential equations is derived which can be fit directly to CFSE histogram data. From such a model, cell counts (in terms of the number of divisions undergone) can be directly computed and thus key biological parameters such as population doubling time and precursor viability can be determined. Mathematical aspects of this compartmental model are discussed, and the model is fit to a data set. As in [18,19], we find temporal and division dependence in the rates of proliferation and death to be essential features of a structured population model for CFSE data. Variability in cellular autofluorescence is found to play a significant role in the data, as well. Finally, the compartmental model is compared to previous work, and statistical aspects of the experimental data are discussed.

Modeling the role of IL2 in the interplay between CD4+ helper and regulatory T cells: studying the impact of IL2 modulation therapies

Several reports in the literature have drawn a complex picture of the effect of treatments aiming to modulate IL2 activity in vivo. They seem to promote indistinctly immunity or tolerance, probably depending on the specific context, dose and timing of their application. Such complexity might derives from the dual role of IL2 on T-cell dynamics. To theoretically address the latter possibility, we develop a mathematical model for helper, regulatory and memory T-cells dynamics, which account for most well-known facts relative to their relationship with IL2. We simulate the effect of three types of therapies: IL2 injections, IL2 depletion using anti-IL2 antibodies and IL2/anti-IL2 immune complexes injection. We focus in the qualitative and quantitative conditions of dose and timing for these treatments which allow them to potentate either immunity or tolerance. Our results provide reasonable explanations for the existent pre-clinical and clinical data and further provide interesting practical guidelines to optimize the future application of these types of treatments. Particularly, our results predict that: (i) Immune complexes IL2/anti-IL2 mAbs, using mAbs which block the interaction of IL2 and CD25 (the alpha chain of IL2 receptor), is the best option to potentate immunity alone or in combination with vaccines. These complexes are optimal when a 1:2 molar ratio of mAb:IL2 is used and the mAbs have the largest possible affinity; (ii) Immune complexes IL2/anti-IL2 mAbs, using mAbs which block the interaction of IL2 and CD122 (the beta chain of IL2 receptor), are the best option to reinforce preexistent natural tolerance, for instance to prevent allograft rejection. These complexes are optimal when a 1:2 molar ratio of mAb:IL2 is used and the mAbs have intermediate affinities; (iii) mAbs anti-IL2 can be successfully used alone to treat an ongoing autoimmune disorder, promoting the re-induction of tolerance. The best strategy in this therapy is to start treatment with an initially high dose of the mAbs (one capable to induce some immune suppression) and then scales down slowly the dose of mAb in subsequent applications.

A new model for the estimation of cell proliferation dynamics using CFSE data

CFSE analysis of a proliferating cell population is a popular tool for the study of cell division and divisionlinked changes in cell behavior. Recently Banks et al. (2011), Luzyanina et al. (2009), Luzyanina et al. (2007), a partial differential equation (PDE) model to describe lymphocyte dynamics in a CFSE proliferation assay was proposed. We present a significant revision of this model which improves the physiological understanding of several parameters. Namely, the parameter used previously as a heuristic explanation for the dilution of CFSE dye by cell division is replaced with a more physical component, cellular autofluorescence. The rate at which label decays is also quantified using a Gompertz decay process. We then demonstrate a revised method of fitting the model to the commonly used histogram representation of the data. It is shown that these improvements result in a model with a strong physiological basis which is fully capable of replicating the behavior observed in the data.

Evaluation of multitype mathematical models for CFSE-labeling experiment data

Carboxy-fluorescein diacetate succinimidyl ester (CFSE) labeling is an important experimental tool for measuring cell responses to extracellular signals in biomedical research. However, changes of the cell cycle (e.g., time to division) corresponding to different stimulations cannot be directly characterized from data collected in CFSE-labeling experiments. A number of independent studies have developed mathematical models as well as parameter estimation methods to better understand cell cycle kinetics based on CFSE data. However, when applying different models to the same data set, notable discrepancies in parameter estimates based on different models has become an issue of great concern. It is therefore important to compare existing models and make recommendations for practical use. For this purpose, we derived the analytic form of an age-dependent multitype branching process model. We then compared the performance of different models, namely branching process, cyton, Smith-Martin, and a linear birth-death ordinary differential equation (ODE) model via simulation studies. For fairness of model comparison, simulated data sets were generated using an agent-based simulation tool which is independent of the four models that are compared. The simulation study results suggest that the branching process model significantly outperforms the other three models over a wide range of parameter values. This model was then employed to understand the proliferation pattern of CD4+ and CD8+ T cells under polyclonal stimulation.

Stochastic models of lymphocyte proliferation and death

Quantitative understanding of the kinetics of lymphocyte proliferation and death upon activation with an antigen is crucial for elucidating factors determining the magnitude, duration and efficiency of the immune response. Recent advances in quantitative experimental techniques, in particular intracellular labeling and multi-channel flow cytometry, allow one to measure the population structure of proliferating and dying lymphocytes for several generations with high precision. These new experimental techniques require novel quantitative methods of analysis. We review several recent mathematical approaches used to describe and analyze cell proliferation data. Using a rigorous mathematical framework, we show that two commonly used models that are based on the theories of age-structured cell populations and of branching processes, are mathematically identical. We provide several simple analytical solutions for a model in which the distribution of inter-division times follows a gamma distribution and show that this model can fit both simulated and experimental data. We also show that the estimates of some critical kinetic parameters, such as the average inter-division time, obtained by fitting models to data may depend on the assumed distribution of inter-division times, highlighting the challenges in quantitative understanding of cell kinetics.

Estimation of cell proliferation dynamics using CFSE data

Advances in fluorescent labeling of cells as measured by flow cytometry have allowed for quantitative studies of proliferating populations of cells. The investigations (Luzyanina et al. in J. Math. Biol. 54:57-89, 2007; J. Math. Biol., 2009; Theor. Biol. Med. Model. 4:1-26, 2007) contain a mathematical model with fluorescence intensity as a structure variable to describe the evolution in time of proliferating cells labeled by carboxyfluorescein succinimidyl ester (CFSE). Here, this model and several extensions/modifications are discussed. Suggestions for improvements are presented and analyzed with respect to statistical significance for better agreement between model solutions and experimental data. These investigations suggest that the new decay/label loss and time dependent effective proliferation and death rates do indeed provide improved fits of the model to data. Statistical models for the observed variability/noise in the data are discussed with implications for uncertainty quantification. The resulting new cell dynamics model should prove useful in proliferation assay tracking and modeling, with numerous applications in the biomedical sciences.

A minimum of two distinct heritable factors are required to explain correlation structures in proliferating lymphocytes

During the adaptive immune response, lymphocyte populations undergo a characteristic three-phase process: expansion through a series of cell divisions; cessation of expansion; and, finally, most of the accumulated lymphocytes die by apoptosis. The data used, thus far, to inform understanding of these processes, both in vitro and in vivo, are taken from flow cytometry experiments. One significant drawback of flow cytometry is that individual cells cannot be tracked, so that it is not possible to investigate interdependencies in the fate of cells within a family tree. This deficit in experimental information has recently been overcome by Hawkins et al. (Hawkins et al. 2009 Proc. Natl Acad. Sci. USA106, 13 457–13 462 (doi:10.1073/pnas.0905629106)), who reported on time-lapse microscopy experiments in which B-cells were stimulated through the TLR-9 receptor. Cells stimulated in this way do not aggregate, so that data regarding family trees can be recorded. In this article, we further investigate the Hawkins et al. data. Our conclusions are striking: in order to explain the familial correlation structure in division times, death times and propensity to divide, a minimum of two distinct heritable factors are necessary. As the data show that two distinct factors are necessary, we develop a stochastic model that has two heritable factors and demonstrate that it can reproduce the key features of the data. This model shows that two heritable factors are sufficient. These deductions have a clear impact upon biological understanding of the adaptive immune response. They also necessitate changes to the fundamental premises behind the tools developed by statisticians to draw deductions from flow cytometry data. Finally, they affect the mathematical modelling paradigms that are used to study these systems, as these are widely developed based on assumptions of cellular independence that are not accurate.

Modeling the role of IL-2 in the interplay between CD4+ helper and regulatory T cells: assessing general dynamical properties

Mathematical models accounting for well-known evidences relating to the dynamics of interleukin 2, helper and regulatory T cells are presented. These models extend an existent model (the so-called cross-regulation model of immunity), by assuming IL-2 as the growth factor produced by helper cells, but used by both helper and regulatory cells to proliferate and survive. Two model variants, motivated by current literature, are explored. The first variant assumes that regulatory cells suppress helper cells by limiting IL-2 production and consuming the available IL-2; i.e. they just trigger competition for IL-2. The second model variant adds to the latter competitive mechanism the direct inhibition of helper cells activation by regulatory cells. The extended models retain key dynamical features of the cross-regulation model. But such reasonable behavior depends on parameter constraints, which happen to be realistic and lead to interesting biological discussions. Furthermore, the introduction of IL-2 in these models breaks the local/specific character of interactions, providing new properties to them. In the extended models, but not in the cross-regulation model, the response triggered by an antigen affects the response to other antigens in the same lymph node. The first model variant predicts an unrealistic coupling of the immune reactions to all the antigens in the lymph node. In contrast, the second model variant allows the coexistent of concomitant tolerant and immune responses to different antigens. The IL-2 derived from an ongoing immune reaction reinforces tolerance to other antigens in the same lymph node. Overall the models introduced here are useful extensions of the cross-regulation formalism. In particular, they might allow future studies of the effect of different IL-2 modulation therapies on CD4+ T cell dynamics.

Interpreting CFSE obtained division histories of B cells in vitro with Smith-Martin and cyton type models

The fluorescent dye carboxyfluorescin diacetate succinimidyl ester (CFSE) classifies proliferating cell populations into groups according to the number of divisions each cell has undergone (i.e., its division class). The pulse labeling of cells with radioactive thymidine provides a means to determine the distribution of times of entry into the first cell division. We derive in analytic form the number of cells in each division class as a function of time using the cyton approach that utilizes independent stochastic distributions for the time to divide and the time to die. We confirm that our analytic form for the number of cells in each division class is consistent with the numerical solution of a set of delay differential equations representing the generalized Smith-Martin model with cell death rates depending on the division class. Choosing the distribution of time to the first division to fit thymidine labeling data for B cells stimulated in vitro with lipopolysaccharide (LPS) and either with or without interleukin-4 (IL-4), we fit CFSE data to determine the dependence of B cell kinetic parameters on the presence of IL-4. We find when IL-4 is present, a greater proportion of cells are recruited into division with a longer average time to first division. The most profound effect of the presence of IL-4 was decreased death rates for smaller division classes, which supports a role of IL-4 in the protection of B cells from apoptosis.

Distributed parameter identification for a label-structured cell population dynamics model using CFSE histogram time-series data

In this work we address the problem of the robust identification of unknown parameters of a cell population dynamics model from experimental data on the kinetics of cells labelled with a fluorescence marker defining the division age of the cell. The model is formulated by a first order hyperbolic PDE for the distribution of cells with respect to the structure variable x (or z) being the intensity level (or the log(10)-transformed intensity level) of the marker. The parameters of the model are the rate functions of cell division, death, label decay and the label dilution factor. We develop a computational approach to the identification of the model parameters with a particular focus on the cell birth rate alpha(z) as a function of the marker intensity, assuming the other model parameters are scalars to be estimated. To solve the inverse problem numerically, we parameterize alpha(z) and apply a maximum likelihood approach. The parametrization is based on cubic Hermite splines defined on a coarse mesh with either equally spaced a priori fixed nodes or nodes to be determined in the parameter estimation procedure. Ill-posedness of the inverse problem is indicated by multiple minima. To treat the ill-posed problem, we apply Tikhonov regularization with the regularization parameter determined by the discrepancy principle. We show that the solution of the regularized parameter estimation problem is consistent with the data set with an accuracy within the noise level in the measurements.

Noise-driven stem cell and progenitor population dynamics

BACKGROUND

The balance between maintenance of the stem cell state and terminal differentiation is influenced by the cellular environment. The switching between these states has long been understood as a transition between attractor states of a molecular network. Herein, stochastic fluctuations are either suppressed or can trigger the transition, but they do not actually determine the attractor states.

METHODOLOGY/PRINCIPAL FINDINGS

We present a novel mathematical concept in which stem cell and progenitor population dynamics are described as a probabilistic process that arises from cell proliferation and small fluctuations in the state of differentiation. These state fluctuations reflect random transitions between different activation patterns of the underlying regulatory network. Importantly, the associated noise amplitudes are state-dependent and set by the environment. Their variability determines the attractor states, and thus actually governs population dynamics. This model quantitatively reproduces the observed dynamics of differentiation and dedifferentiation in promyelocytic precursor cells.

CONCLUSIONS/SIGNIFICANCE

Consequently, state-specific noise modulation by external signals can be instrumental in controlling stem cell and progenitor population dynamics. We propose follow-up experiments for quantifying the imprinting influence of the environment on cellular noise regulation.

Quantitative proliferation dynamics and random chromosome segregation of hair follicle stem cells

Regulation of stem cell (SC) proliferation is central to tissue homoeostasis, injury repair, and cancer development. Accumulation of replication errors in SCs is limited by either infrequent division and/or by chromosome sorting to retain preferentially the oldest 'immortal' DNA strand. The frequency of SC divisions and the chromosome-sorting phenomenon are difficult to examine accurately with existing methods. To address this question, we developed a strategy to count divisions of hair follicle (HF) SCs over time, and provide the first quantitative proliferation history of a tissue SC during its normal homoeostasis. We uncovered an unexpectedly high cellular turnover in the SC compartment in one round of activation. Our study provides quantitative data in support of the long-standing infrequent SC division model, and shows that HF SCs do not retain the older DNA strands or sort their chromosome. This new ability to count divisions in vivo has relevance for obtaining basic knowledge of tissue kinetics.

Quantitative analysis of T cell homeostatic proliferation

T cell homeostatic proliferation occurs on transfer of T cells into lymphopenic recipients; transferred cells undergo several rounds of division in the absence of specific antigen stimulation. For a quantitative analysis of this phenomenon, we applied a mathematical method to describe proliferating T cells to match peak distributions from actual CFSE dilution data. For in vitro stimulation of T cells with anti-CD3/anti-CD28, our simulation confirmed a high proportion of cells entering cell cycle with a low proportion undergoing apoptosis. When applied to homeostatic proliferation, it described striking differences in CD4 and CD8 T cell proliferation rates, and accurately predicted that successive divisions were accompanied by higher rates of apoptosis, limiting the accumulation of proliferating cells. Thus, the presence of multiple CFSE dilution peaks cannot be considered equivalent to lymphocyte expansion. Finally, genetic effects were identified that may help explain links between homeostatic proliferation and autoimmunity.

Analysis of cell differentiation by division tracking cytometry

We propose a quantitative method to characterize growth and differentiation dynamics of multipotent cells from time series carboxyfluorescein diacetate, succinimidyl ester (CFDA-SE) division tracking data. The dynamics of cell proliferation and differentiation was measured by combining (CFDA-SE) division tracking with phenotypic analysis. We define division tracking population statistics such as precursor cell frequency, generation time and renewal rate that characterize growth of various phenotypes in a heterogeneous culture system. This method is illustrated by study of the divisional recruitment of cord blood CD34(+) cells by hematopoietic growth factors. The technical issue of assigning the correct generation number to cells was addressed by employing high-resolution division tracking methodology and daily histogram analysis. We also quantified division-tracking artifacts such as CFDA-SE degeneration and cellular auto-fluorescence. Mitotic activation of cord blood CD34(+) cells by cytokines commenced after 2 days of cytokine stimulation. Mean generation number increased linearly thereafter, and it was conclusively shown that CD34(+) cells cycle slower than CD34(-) cells. Generation times for CD34(+) and CD34(-) cells were 24.7 +/- 0.8 h and 15.1 +/- 0.9 h (+/-SD, n = 5), respectively. The 20-fold increase in CD34(+) cell numbers at Day 6 could be attributed to a high CD34(+) cell renewal rate (91% +/- 2% per division). Although cultures were initiated with highly purified CD34(+) cells (approximately 96%), CD34(-) numbers had expanded rapidly by Day 6. This rapid expansion could be explained by their short generation time as well as a small fraction of CD34(+) cells (approximately 5%) that differentiated into CD34(-) cells. Multitype division tracking provides a detailed analysis of multipotent cell differentiation dynamics.

Determining the expected variability of immune responses using the cyton model

During an adaptive immune response, lymphocytes proliferate for 5-20 cell divisions, then stop and die over a period of weeks. The cyton model for regulation of lymphocyte proliferation and survival was introduced by Hawkins et al. (Proc. Natl. Acad. Sci. USA 104, 5032-5037, 2007) to provide a framework for understanding this response and its regulation. The model assumes stochastic values for division and survival times for each cell in a responding population. Experimental evidence indicates that the choice of times is drawn from a skewed distribution such as the lognormal, with the fate of individual cells being potentially highly variable. For this reason we calculate the higher moments of the model so that the expected variability can be determined. To do this we formulate a new analytic framework for the cyton model by introducing a generalization to the Bellman-Harris branching process. We use this framework to introduce two distinct approaches to predicting variability in the immune response to a mitogenic signal. The first method enables explicit calculations for certain distributions and qualitatively exhibits the full range of observed immune responses. The second approach does not facilitate analytic solutions, but allows simple numerical schemes for distributions for which there is little prospect of analytic formulae. We compare the predictions derived from the second method to experimentally observed lymphocyte population sizes from in vivo and in vitro experiments. The model predictions for both data sets are remarkably accurate. The important biological conclusion is that there is limited variation around the expected value of the population size irrespective of whether the response is mediated by small numbers of cells undergoing many divisions or for many cells pursuing a small number of divisions. Therefore, we conclude the immune response is robust and predictable despite the potential for great variability in the experience of each individual cell.

TCR affinity promotes CD8+ T cell expansion by regulating survival

Ligation with high affinity ligands are known to induce T lymphocytes to become fully activated effector cells while ligation with low affinity ligands (or partial agonists) may result in a delayed or incomplete response. We have examined the quantitative features of CD8(+) T cell proliferation induced by peptides of different TCR affinities at a range of concentrations in the mouse OT-I model. Both the frequency of cells responding and the average time taken for cells to reach their first division are affected by peptide concentration and affinity. Consecutive division times, however, remained largely unaffected by these variables. Importantly, we identified affinity to be the sole regulator of cell death in subsequent division. These results suggest a mechanism whereby TCR affinity detection can modulate the subsequent rate of T cell growth and ensure the dominance of higher affinity clones over time.

IL-2 regulates expansion of CD4+ T cell populations by affecting cell death: insights from modeling CFSE data

It is generally accepted that IL-2 influences the dynamics of populations of T cells in vitro and in vivo. However, which parameters for cell division and/or death are affected by IL-2 is not well understood. To get better insights into the potential ways of how IL-2 may influence the population dynamics of T cells, we analyze data on the dynamics of CFSE-labeled polyclonal CD4(+) T lymphocytes in vitro after anti-CD3 stimulation at different concentrations of exogenous IL-2. Inferring cell division and death rates from CFSE-delabeling experiments is not straightforward and requires the use of mathematical models. We find that to adequately describe the dynamics of T cells at low concentrations of exogenous IL-2, the death rate of divided cells has to increase with the number of divisions cells have undergone. IL-2 hardly affects the average interdivision time. At low IL-2 concentrations 1) fewer cells are recruited into the response and successfully complete their first division; 2) the stochasticity of cell division is increased; and 3) the rate, at which the death rate increases with the division number, increases. Summarizing, our mathematical reinterpretation suggests that the main effect of IL-2 on the in vitro dynamics of naive CD4(+) T cells occurs by affecting the rate of cell death and not by changing the rate of cell division.

Modeling T cell proliferation and death in vitro based on labeling data: generalizations of the Smith-Martin cell cycle model

The fluorescent dye carboxyfluorescein diacetate succinimidyl ester (CFSE) classifies proliferating cell populations into groups according to the number of divisions each cell has undergone (i.e., its division class). The pulse labeling of cells with radioactive thymidine provides a means to determine the distribution of times of entry into the first cell division. We derive in analytic form the number of cells in each division class as a function of time based on the distribution of times to the first division. Choosing the distribution of time to the first division to fit thymidine labeling data for T cells stimulated in vitro under different concentrations of IL-2, we fit CFSE data to determine the dependence of T cell kinetic parameters on the concentration of IL-2. As the concentration of IL-2 increases, the average cell cycle time is shortened, the death rate of cells is decreased, and a higher fraction of cells is recruited into division. We also find that if the average cell cycle time increases with division class then the qualify of our fit to the data improves.

Numerical modelling of label-structured cell population growth using CFSE distribution data

Background

The flow cytometry analysis of CFSE-labelled cells is currently one of the most informative experimental techniques for studying cell proliferation in immunology. The quantitative interpretation and understanding of such heterogenous cell population data requires the development of distributed parameter mathematical models and computational techniques for data assimilation.

Methods and Results

The mathematical modelling of label-structured cell population dynamics leads to a hyperbolic partial differential equation in one space variable. The model contains fundamental parameters of cell turnover and label dilution that need to be estimated from the flow cytometry data on the kinetics of the CFSE label distribution. To this end a maximum likelihood approach is used. The Lax-Wendroff method is used to solve the corresponding initial-boundary value problem for the model equation. By fitting two original experimental data sets with the model we show its biological consistency and potential for quantitative characterization of the cell division and death rates, treated as continuous functions of the CFSE expression level.

Conclusion

Once the initial distribution of the proliferating cell population with respect to the CFSE intensity is given, the distributed parameter modelling allows one to work directly with the histograms of the CFSE fluorescence without the need to specify the marker ranges. The label-structured model and the elaborated computational approach establish a quantitative basis for more informative interpretation of the flow cytometry CFSE systems.

Reconstruction of cell population dynamics using CFSE

Background

Quantifying cell division and death is central to many studies in the biological sciences. The fluorescent dye CFSE allows the tracking of cell division in vitro and in vivo and provides a rich source of information with which to test models of cell kinetics. Cell division and death have a stochastic component at the single-cell level, and the probabilities of these occurring in any given time interval may also undergo systematic variation at a population level. This gives rise to heterogeneity in proliferating cell populations. Branching processes provide a natural means of describing this behaviour.

Results

We present a likelihood-based method for estimating the parameters of branching process models of cell kinetics using CFSE-labeling experiments, and demonstrate its validity using synthetic and experimental datasets. Performing inference and model comparison with real CFSE data presents some statistical problems and we suggest methods of dealing with them.

Conclusion

The approach we describe here can be used to recover the (potentially variable) division and death rates of any cell population for which division tracking information is available.

A model of immune regulation as a consequence of randomized lymphocyte division and death times

The magnitude of an adaptive immune response is controlled by the interplay of lymphocyte quiescence, proliferation, and apoptosis. How lymphocytes integrate receptor-mediated signals influencing these cell fates is a fundamental question for understanding this complex system. We examined how lymphocytes interleave times to divide and die to develop a mathematical model of lymphocyte growth regulation. This model provides a powerful method for fitting and analyzing fluorescent division tracking data and reveals how summing receptor-mediated kinetic changes can modify the immune response progressively from rapid tolerance induction to strong immunity. An important consequence of our results is that intrinsic variability in otherwise identical cells, usually dismissed as noise, may have evolved to be an essential feature of immune regulation.

Immunological characteristics of subjects with asymptomatic skin sensitization to birch and grass pollen

BACKGROUND

Asymptomatic skin sensitization (AS) has been shown to be a risk factor for respiratory allergic disease.

OBJECTIVE

We investigated allergen and recall antigen-driven T cell proliferation, cytokine production and T cell expression of the chemokine receptor CCR4, in cultures derived from symptomatic atopics (SA), subjects with AS and healthy controls (HC). Numbers of allergen-specific precursor T cells in all three groups were also estimated.

METHODS

Peripheral blood mononuclear cells from the three groups were isolated and stimulated with allergen and tetanus toxoid. Proliferation, cytokine production and CCR4 expression were measured by flow cytometry.

RESULTS

A significantly increased proportion of CD4(+) memory T cells proliferated in response to allergen in SA as compared with subjects with AS (P<0.001) and HC (P<0.001). Only in SA was expansion of CD4(+)CCR4(+) T cells, after allergen stimulation observed. SA had higher frequencies of allergen-specific T cells than subjects with AS and HC (P=0.02, for both). With regard to allergen-induced production of T-helper type 1 (Th1) and Th2 cytokines, subjects with AS and HC resembled each other, while differing significantly from SA.

CONCLUSION

We conclude, that subjects with AS, although clearly IgE sensitized, have significant diminished numbers of allergen-specific T cells as well as decreased allergen-induced CD4(+) memory T cell proliferation as compared with SA. To a large extent, our findings are capable of explaining the immunological characteristics associated with AS. Our findings may serve as better prognostic markers for subsequent allergic progression, than previously described clinical and paraclinical characteristics.

Quantifying lymphocyte kinetics in vivo using carboxyfluorescein diacetate succinimidyl ester (CFSE)

The cytoplasmic dye carboxyfluorescein diacetate succinimidyl ester (CFSE) is used to quantify cell kinetics. It is particularly important in studies of lymphocyte homeostasis where its labelling of cells irrespective of their stage in the cell cycle makes it preferable to deuterated glucose and BrdU, which only label dividing cells and thus produce unrepresentative results. In the past, experiments have been limited by the need to obtain a clear separation of CFSE peaks forcing scientists to adopt a strategy of in vitro labelling of cells followed by their injection into the host. Here we develop a framework for analysis of in vivo CFSE labelling data. This enables us to estimate the rate of proliferation and death of lymphocytes in situ, and thus represents a considerable advance over current procedures. We illustrate this approach using in vivo CFSE labelling of B lymphocytes in sheep.

Estimating Lymphocyte Division and Death Rates from CFSE Data

The division tracking dye, carboxyfluorescin diacetate succinimidyl ester (CFSE) is currently the most informative labeling technique for characterizing the division history of cells in the immune system. Gett and Hodgkin [Nat. Immunol. 1:239-244, 2000] have pioneered the quantitative analysis of CFSE data. We confirm and extend their data analysis approach using simple mathematical models. We employ the extended Gett and Hodgkin [Nat. Immunol. 1:239-244, 2000] method to estimate the time to first division, the fraction of cells recruited into division, the cell cycle time, and the average death rate from CFSE data on T cells stimulated under different concentrations of IL-2. The same data is also fitted with a simple mathematical model that we derived by reformulating the numerical model of Deenick et al. [J. Immunol. 170:4963-4972, 2003]. By a non-linear fitting procedure we estimate parameter values and confidence intervals to identify the parameters that are influenced by the IL-2 concentration. We obtain a significantly better fit to the data when we assume that the T cell death rate depends on the number of divisions cells have completed. We provide an outlook on future work that involves extending the Deenick et al. [J. Immunol. 170:4963-4972, 2003] model into the classical smith-martin model, and into a model with arbitrary probability distributions for death and division through subsequent divisions.