Stochastic models of lymphocyte proliferation and death

Description of chemical systems by means of response functions

In this paper we introduce a formalism that allows to describe the response of a part of a biochemical system in terms of renewal equations. In particular, we examine under which conditions the interactions between the different parts of a chemical system, described by means of linear ODEs, can be represented in terms of renewal equations. We show also how to apply the formalism developed in this paper to some particular types of linear and non-linear ODEs, modelling some biochemical systems of interest in biology (for instance, some time-dependent versions of the classical Hopfield model of kinetic proofreading). We also analyse some of the properties of the renewal equations that we are interested in, as the long-time behaviour of their solution. Furthermore, we prove that the kernels characterising the renewal equations derived by biochemical system with reactions that satisfy the detail balance condition belong to the class of completely monotone functions.

High-resolution mapping of cell cycle dynamics during steady-state T cell development and regeneration in vivo

Control of cell proliferation is critical for the lymphocyte life cycle. However, little is known about how stage-specific alterations in cell cycle behavior drive proliferation dynamics during T cell development. Here, we employed in vivo dual-nucleoside pulse labeling combined with the determination of DNA replication over time as well as fluorescent ubiquitination-based cell cycle indicator mice to establish a quantitative high-resolution map of cell cycle kinetics of thymocytes. We developed an agent-based mathematical model of T cell developmental dynamics. To generate the capacity for proliferative bursts, cell cycle acceleration followed a "stretch model" characterized by the simultaneous and proportional contraction of both G1 and S phases. Analysis of cell cycle phase dynamics during regeneration showed tailored adjustments of cell cycle phase dynamics. Taken together, our results highlight intrathymic cell cycle regulation as an adjustable system to maintain physiologic tissue homeostasis and foster our understanding of dysregulation of the T cell developmental program.

A comprehensive review of computational cell cycle models in guiding cancer treatment strategies

This article reviews the current knowledge and recent advancements in computational modeling of the cell cycle. It offers a comparative analysis of various modeling paradigms, highlighting their unique strengths, limitations, and applications. Specifically, the article compares deterministic and stochastic models, single-cell versus population models, and mechanistic versus abstract models. This detailed analysis helps determine the most suitable modeling framework for various research needs. Additionally, the discussion extends to the utilization of these computational models to illuminate cell cycle dynamics, with a particular focus on cell cycle viability, crosstalk with signaling pathways, tumor microenvironment, DNA replication, and repair mechanisms, underscoring their critical roles in tumor progression and the optimization of cancer therapies. By applying these models to crucial aspects of cancer therapy planning for better outcomes, including drug efficacy quantification, drug discovery, drug resistance analysis, and dose optimization, the review highlights the significant potential of computational insights in enhancing the precision and effectiveness of cancer treatments. This emphasis on the intricate relationship between computational modeling and therapeutic strategy development underscores the pivotal role of advanced modeling techniques in navigating the complexities of cell cycle dynamics and their implications for cancer therapy.

Integration of quantitative methods and mathematical approaches for the modeling of cancer cell proliferation dynamics

Physiological processes rely on the control of cell proliferation, and the dysregulation of these processes underlies various pathological conditions, including cancer. Mathematical modeling can provide new insights into the complex regulation of cell proliferation dynamics. In this review, we first examine quantitative experimental approaches for measuring cell proliferation dynamics in vitro and compare the various types of data that can be obtained in these settings. We then explore the toolbox of common mathematical modeling frameworks that can describe cell behavior, dynamics, and interactions of proliferation. We discuss how these wet-laboratory studies may be integrated with different mathematical modeling approaches to aid the interpretation of the results and to enable the prediction of cell behaviors, specifically in the context of cancer.

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 [Formula: see text]. 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 [Formula: see text] were equivalent to a single step of rate [Formula: see text]. 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.

Cyton2: A Model of Immune Cell Population Dynamics That Includes Familial Instructional Inheritance

Lymphocytes are the central actors in adaptive immune responses. When challenged with antigen, a small number of B and T cells have a cognate receptor capable of recognising and responding to the insult. These cells proliferate, building an exponentially growing, differentiating clone army to fight off the threat, before ceasing to divide and dying over a period of weeks, leaving in their wake memory cells that are primed to rapidly respond to any repeated infection. Due to the non-linearity of lymphocyte population dynamics, mathematical models are needed to interrogate data from experimental studies. Due to lack of evidence to the contrary and appealing to arguments based on Occam's Razor, in these models newly born progeny are typically assumed to behave independently of their predecessors. Recent experimental studies, however, challenge that assumption, making clear that there is substantial inheritance of timed fate changes from each cell by its offspring, calling for a revision to the existing mathematical modelling paradigms used for information extraction. By assessing long-term live-cell imaging of stimulated murine B and T cells in vitro, we distilled the key phenomena of these within-family inheritances and used them to develop a new mathematical model, Cyton2, that encapsulates them. We establish the model's consistency with these newly observed fine-grained features. Two natural concerns for any model that includes familial correlations would be that it is overparameterised or computationally inefficient in data fitting, but neither is the case for Cyton2. We demonstrate Cyton2's utility by challenging it with high-throughput flow cytometry data, which confirms the robustness of its parameter estimation as well as its ability to extract biological meaning from complex mixed stimulation experiments. Cyton2, therefore, offers an alternate mathematical model, one that is, more aligned to experimental observation, for drawing inferences on lymphocyte population dynamics.

Modeling the Dynamics of T-Cell Development in the Thymus

The thymus hosts the development of a specific type of adaptive immune cells called T cells. T cells orchestrate the adaptive immune response through recognition of antigen by the highly variable T-cell receptor (TCR). T-cell development is a tightly coordinated process comprising lineage commitment, somatic recombination of Tcr gene loci and selection for functional, but non-self-reactive TCRs, all interspersed with massive proliferation and cell death. Thus, the thymus produces a pool of T cells throughout life capable of responding to virtually any exogenous attack while preserving the body through self-tolerance. The thymus has been of considerable interest to both immunologists and theoretical biologists due to its multi-scale quantitative properties, bridging molecular binding, population dynamics and polyclonal repertoire specificity. Here, we review experimental strategies aimed at revealing quantitative and dynamic properties of T-cell development and how they have been implemented in mathematical modeling strategies that were reported to help understand the flexible dynamics of the highly dividing and dying thymic cell populations. Furthermore, we summarize the current challenges to estimating in vivo cellular dynamics and to reaching a next-generation multi-scale picture of T-cell development.

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.

What Will B Will B: Identifying Molecular Determinants of Diverse B-Cell Fate Decisions Through Systems Biology

B-cells are the poster child for cellular diversity and heterogeneity. The diverse repertoire of B lymphocytes, each expressing unique antigen receptors, provides broad protection against pathogens. However, B-cell diversity goes beyond unique antigen receptors. Side-stepping B-cell receptor (BCR) diversity through BCR-independent stimuli or engineered organisms with monoclonal BCRs still results in seemingly identical B-cells reaching a wide variety of fates in response to the same challenge. Identifying to what extent the molecular state of a B-cell determines its fate is key to gaining a predictive understanding of B-cells and consequently the ability to control them with targeted therapies. Signals received by B-cells through transmembrane receptors converge on intracellular molecular signaling networks, which control whether each B-cell divides, dies, or differentiates into a number of antibody-secreting distinct B-cell subtypes. The signaling networks that interpret these signals are well known to be susceptible to molecular variability and noise, providing a potential source of diversity in cell fate decisions. Iterative mathematical modeling and experimental studies have provided quantitative insight into how B-cells achieve distinct fates in response to pathogenic stimuli. Here, we review how systems biology modeling of B-cells, and the molecular signaling networks controlling their fates, is revealing the key determinants of cell-to-cell variability in B-cell destiny.

A mathematical solution to Peto's paradox using Polya's urn model: implications for the aetiology of cancer in general

Ageing is the leading risk factor for the emergence of cancer in humans. Accumulation of pro-carcinogenic events throughout life is believed to explain this observation; however, the lack of direct correlation between the number of cells in an organism and cancer incidence, known as Peto's Paradox, is at odds with this assumption. Finding the events responsible for this discrepancy can unveil mechanisms with potential uses in prevention and treatment of cancer in humans. On the other hand, the immune system is important in preventing the development of clinically relevant tumours by maintaining a fine equilibrium between reactive and suppressive lymphocyte clones. It is suggested here that the loss of this equilibrium is what ultimately leads to increased risk of cancer and to propose a mechanism for the changes in clonal proportions based on decreased proliferative capacity of lymphocyte clones as a natural phenomenon of ageing. This mechanism, being a function of the number of cells, provides an explanation for Peto's Paradox.

FAMoS: A Flexible and dynamic Algorithm for Model Selection to analyse complex systems dynamics

Most biological systems are difficult to analyse due to a multitude of interacting components and the concomitant lack of information about the essential dynamics. Finding appropriate models that provide a systematic description of such biological systems and that help to identify their relevant factors and processes can be challenging given the sheer number of possibilities. Model selection algorithms that evaluate the performance of a multitude of different models against experimental data provide a useful tool to identify appropriate model structures. However, many algorithms addressing the analysis of complex dynamical systems, as they are often used in biology, compare a preselected number of models or rely on exhaustive searches of the total model space which might be unfeasible dependent on the number of possibilities. Therefore, we developed an algorithm that is able to perform model selection on complex systems and searches large model spaces in a dynamical way. Our algorithm includes local and newly developed non-local search methods that can prevent the algorithm from ending up in local minima of the model space by accounting for structurally similar processes. We tested and validated the algorithm based on simulated data and showed its flexibility for handling different model structures. We also used the algorithm to analyse experimental data on the cell proliferation dynamics of CD4⁺ and CD8⁺ T cells that were cultured under different conditions. Our analyses indicated dynamical changes within the proliferation potential of cells that was reduced within tissue-like 3D ex vivo cultures compared to suspension. Due to the flexibility in handling various model structures, the algorithm is applicable to a large variety of different biological problems and represents a useful tool for the data-oriented evaluation of complex model spaces.

Identifying the systematic interactions of multiple components within a complex biological system can be challenging due to the number of potential processes and the concomitant lack of information about the essential dynamics. Selection algorithms that allow an automated evaluation of a large number of different models provide a useful tool in identifying the systematic relationships between experimental data. However, many of the existing model selection algorithms are not able to address complex model structures, such as systems of differential equations, and partly rely on local or exhaustive search methods which are inappropriate for the analysis of various biological systems. Therefore, we developed a flexible model selection algorithm that performs a robust and dynamical search of large model spaces to identify complex systems dynamics and applied it to the analysis of T cell proliferation dynamics within different culture conditions. The algorithm, which is available as an R-package, provides an advanced tool for the analysis of complex systems behaviour and, due to its flexible structure, can be applied to a large variety of biological problems.

Mathematical modelling reveals unexpected inheritance and variability patterns of cell cycle parameters in mammalian cells

The cell cycle is the fundamental process of cell populations, it is regulated by environmental cues and by intracellular checkpoints. Cell cycle variability in clonal cell population is caused by stochastic processes such as random partitioning of cellular components to progeny cells at division and random interactions among biomolecules in cells. One of the important biological questions is how the dynamics at the cell cycle scale, which is related to family dependencies between the cell and its descendants, affects cell population behavior in the long-run. We address this question using a “mechanistic” model, built based on observations of single cells over several cell generations, and then extrapolated in time. We used cell pedigree observations of NIH 3T3 cells including FUCCI markers, to determine patterns of inheritance of cell-cycle phase durations and single-cell protein dynamics. Based on that information we developed a hybrid mathematical model, involving bifurcating autoregression to describe stochasticity of partitioning and inheritance of cell-cycle-phase times, and an ordinary differential equation system to capture single-cell protein dynamics. Long-term simulations, concordant with in vitro experiments, demonstrated the model reproduced the main features of our data and had homeostatic properties. Moreover, heterogeneity of cell cycle may have important consequences during population development. We discovered an effect similar to genetic drift, amplified by family relationships among cells. In consequence, the progeny of a single cell with a short cell cycle time had a high probability of eventually dominating the population, due to the heritability of cell-cycle phases. Patterns of epigenetic heritability in proliferating cells are important for understanding long-term trends of cell populations which are either required to provide the influx of maturing cells (such as hematopoietic stem cells) or which started proliferating uncontrollably (such as cancer cells).

All cells in multicellular organisms obey orchestrated sequences of signals to ensure developmental and homeostatic fitness under a variety of external stimuli. However, there also exist self-perpetuating stem-cell populations, the function of which is to provide a steady supply of differentiated progenitors that in turn ensure persistence of organism functions. This “cell production engine” is an important element of biological homeostasis. A similar process, albeit distorted in many respects, plays a major role in cancer development; here the robustness of homeostasis contributes to difficulty in eradication of malignancy. An important role in homeostasis seems to be played by generation of heterogeneity among cell phenotypes, which then can be shaped by selection and other genetic forces. In the present paper, we present a model of a cultured cell population, which factors in relationships among related cells and the dynamics of cell growth and important proteins regulating cell division. We find that the model not only maintains homeostasis, but that it also responds to perturbations in a manner that is similar to that exhibited by the Wright-Fisher model of population genetics. The model-cell population can become dominated by the progeny of the fittest individuals, without invoking advantageous mutations. If confirmed, this may provide an alternative mode of evolution of cell populations.

Stochastic Inheritance of Division and Death Times Determines the Size and Phenotype of CD8+ T Cell Families

After antigen stimulation cognate naïve CD8⁺ T cells undergo rapid proliferation and ultimately their progeny differentiates into short-lived effectors and longer-lived memory T cells. Although the expansion of individual cells is very heterogeneous, the kinetics are reproducible at the level of the total population of cognate cells. After the expansion phase, the population contracts, and if antigen is cleared, a population of memory T cells remains behind. Different markers like CD62L, CD27, and KLRG1 have been used to define several T cell subsets (or cell fates) developing from individual naïve CD8⁺ T cells during the expansion phase. Growing evidence from high-throughput experiments, like single cell RNA sequencing, epigenetic profiling, and lineage tracing, highlights the need to model this differentiation process at the level of single cells. We model CD8⁺ T cell proliferation and differentiation as a competitive process between the division and death probabilities of individual cells (like in the Cyton model). We use an extended form of the Cyton model in which daughter cells inherit the division and death times from their mother cell in a stochastic manner (using lognormal distributions). We show that this stochastic model reproduces the dynamics of CD8⁺ T cells both at the population and at the single cell level. Modeling the expression of the CD62L, CD27, and KLRG1 markers of each individual cell, we find agreement with the changing phenotypic distributions of these markers in single cell RNA sequencing data. Retrospectively re-defining conventional T-cell subsets by “gating” on these markers, we find agreement with published population data, without having to assume that these subsets have different properties, i.e., correspond to different fates.

Selection for synchronized cell division in simple multicellular organisms

The evolution of multicellularity was a major transition in the history of life on earth. Conditions under which multicellularity is favored have been studied theoretically and experimentally. But since the construction of a multicellular organism requires multiple rounds of cell division, a natural question is whether these cell divisions should be synchronous or not. We study a population model in which there compete simple multicellular organisms that grow by either synchronous or asynchronous cell divisions. We demonstrate that natural selection can act differently on synchronous and asynchronous cell division, and we offer intuition for why these phenotypes are generally not neutral variants of each other.

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.

Modeling Cell-to-Cell Communication Networks Using Response-Time Distributions

Cell-to-cell communication networks have critical roles in coordinating diverse organismal processes, such as tissue development or immune cell response. However, compared with intracellular signal transduction networks, the function and engineering principles of cell-to-cell communication networks are far less understood. Major complications include: cells are themselves regulated by complex intracellular signaling networks; individual cells are heterogeneous; and output of any one cell can recursively become an additional input signal to other cells. Here, we make use of a framework that treats intracellular signal transduction networks as "black boxes" with characterized input-to-output response relationships. We study simple cell-to-cell communication circuit motifs and find conditions that generate bimodal responses in time, as well as mechanisms for independently controlling synchronization and delay of cell-population responses. We apply our modeling approach to explain otherwise puzzling data on cytokine secretion onset times in T cells. Our approach can be used to predict communication network structure using experimentally accessible input-to-output measurements and without detailed knowledge of intermediate steps.

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.

Quantitative analysis of cell proliferation by a dye dilution assay: Application to cell lines and cocultures

Cell proliferation represents one of the most fundamental processes in biological systems, thus the quantitative analysis of cell proliferation is important in many biological applications such as drug screening, production of biologics, and assessment of cytotoxicity. Conventional proliferation assays mainly quantify cell number based on a calibration curve of a homogeneous cell population, and therefore are not applicable for the analysis of cocultured cells. Moreover, these assays measure cell proliferation indirectly, based on cellular metabolic activity or DNA content. To overcome these shortcomings, a dye dilution assay employing fluorescent cell tracking dyes that are retained within cells was applied and was diluted proportionally by subsequent cell divisions. Here, it was demonstrated that this assay could be implemented to quantitatively analyze the cell proliferation of different types of cell lines, and to concurrently analyze the proliferation of two types of cell lines in coculture by utilizing cell tracking dyes with different spectral characteristics. The mean division time estimated by the dye dilution assay is compared with the population doubling time obtained from conventional methods and values from literature. Additionally, dye transfer between cocultured cells was investigated and it was found that it is a characteristic of the cells rather than a characteristic of the dye. It was suggested that this method can be easily combined with other flow cytometric analyses of cellular properties, providing valuable information on cell status under diverse conditions. © 2017 International Society for Advancement of Cytometry.

Effects of cell-cycle-dependent expression on random fluctuations in protein levels

Expression of many genes varies as a cell transitions through different cell-cycle stages. How coupling between stochastic expression and cell cycle impacts cell-to-cell variability (noise) in the level of protein is not well understood. We analyse a model where a stable protein is synthesized in random bursts, and the frequency with which bursts occur varies within the cell cycle. Formulae quantifying the extent of fluctuations in the protein copy number are derived and decomposed into components arising from the cell cycle and stochastic processes. The latter stochastic component represents contributions from bursty expression and errors incurred during partitioning of molecules between daughter cells. These formulae reveal an interesting trade-off: cell-cycle dependencies that amplify the noise contribution from bursty expression also attenuate the contribution from partitioning errors. We investigate the existence of optimum strategies for coupling expression to the cell cycle that minimize the stochastic component. Intriguingly, results show that a zero production rate throughout the cell cycle, with expression only occurring just before cell division, minimizes noise from bursty expression for a fixed mean protein level. By contrast, the optimal strategy in the case of partitioning errors is to make the protein just after cell division. We provide examples of regulatory proteins that are expressed only towards the end of the cell cycle, and argue that such strategies enhance robustness of cell-cycle decisions to the intrinsic stochasticity of gene expression.

Intercellular Variability in Protein Levels from Stochastic Expression and Noisy Cell Cycle Processes

Inside individual cells, expression of genes is inherently stochastic and manifests as cell-to-cell variability or noise in protein copy numbers. Since proteins half-lives can be comparable to the cell-cycle length, randomness in cell-division times generates additional intercellular variability in protein levels. Moreover, as many mRNA/protein species are expressed at low-copy numbers, errors incurred in partitioning of molecules between two daughter cells are significant. We derive analytical formulas for the total noise in protein levels when the cell-cycle duration follows a general class of probability distributions. Using a novel hybrid approach the total noise is decomposed into components arising from i) stochastic expression; ii) partitioning errors at the time of cell division and iii) random cell-division events. These formulas reveal that random cell-division times not only generate additional extrinsic noise, but also critically affect the mean protein copy numbers and intrinsic noise components. Counter intuitively, in some parameter regimes, noise in protein levels can decrease as cell-division times become more stochastic. Computations are extended to consider genome duplication, where transcription rate is increased at a random point in the cell cycle. We systematically investigate how the timing of genome duplication influences different protein noise components. Intriguingly, results show that noise contribution from stochastic expression is minimized at an optimal genome-duplication time. Our theoretical results motivate new experimental methods for decomposing protein noise levels from synchronized and asynchronized single-cell expression data. Characterizing the contributions of individual noise mechanisms will lead to precise estimates of gene expression parameters and techniques for altering stochasticity to change phenotype of individual cells.

Inside individual cells, gene products often occur at low molecular counts and are subject to considerable stochastic fluctuations (noise) in copy numbers over time. An important consequence of noisy expression is that the level of a protein can vary considerably even among genetically identical cells exposed to the same environment. Such non-genetic phenotypic heterogeneity is physiologically relevant and critically influences diverse cellular processes. In addition to noise sources inherent in gene product synthesis, recent experimental studies have uncovered additional noise mechanisms that critically effect expression. For example, the time within the cell cycle when a gene duplicates, and the time taken to complete cell cycle are governed by random processes. The key contribution of this work is development of novel mathematical results quantifying how cell cycle-related noise sources combine with stochastic expression to drive intercellular variability in protein molecular counts. Derived formulas lead to many counterintuitive results, such as increasing randomness in the timing of cell division can lower noise in the level of a protein. Finally, these results inform experimental strategies to systematically dissect the contributions of different noise sources in the expression of a gene of interest.

A Hierarchical Kinetic Theory of Birth, Death and Fission in Age-Structured Interacting Populations

We develop mathematical models describing the evolution of stochastic age-structured populations. After reviewing existing approaches, we formulate a complete kinetic framework for age-structured interacting populations undergoing birth, death and fission processes in spatially dependent environments. We define the full probability density for the population-size age chart and find results under specific conditions. Connections with more classical models are also explicitly derived. In particular, we show that factorial moments for non-interacting processes are described by a natural generalization of the McKendrick-von Foerster equation, which describes mean-field deterministic behavior. Our approach utilizes mixed-type, multidimensional probability distributions similar to those employed in the study of gas kinetics and with terms that satisfy BBGKY-like equation hierarchies.

Mathematics in modern immunology

Mathematical and statistical methods enable multidisciplinary approaches that catalyse discovery. Together with experimental methods, they identify key hypotheses, define measurable observables and reconcile disparate results. We collect a representative sample of studies in T-cell biology that illustrate the benefits of modelling–experimental collaborations and that have proven valuable or even groundbreaking. We conclude that it is possible to find excellent examples of synergy between mathematical modelling and experiment in immunology, which have brought significant insight that would not be available without these collaborations, but that much remains to be discovered.

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.

Division time-based amplifiers for stochastic gene expression

While cell-to-cell variability is a phenotypic consequence of gene expression noise, sources of this noise may be complex – apart from intrinsic sources such as the random birth/death of mRNA and stochastic switching between promoter states, there are also extrinsic sources of noise such as cell division where division times are either constant or random.

Quantifying gene expression variability arising from randomness in cell division times

The level of a given mRNA or protein exhibits significant variations from cell-to-cell across a homogeneous population of living cells. Much work has focused on understanding the different sources of noise in the gene-expression process that drive this stochastic variability in gene-expression. Recent experiments tracking growth and division of individual cells reveal that cell division times have considerable inter-cellular heterogeneity. Here we investigate how randomness in the cell division times can create variability in population counts. We consider a model by which mRNA/protein levels in a given cell evolve according to a linear differential equation and cell divisions occur at times spaced by independent and identically distributed random intervals. Whenever the cell divides the levels of mRNA and protein are halved. For this model, we provide a method for computing any statistical moment (mean, variance, skewness, etcetera) of the mRNA and protein levels. The key to our approach is to establish that the time evolution of the mRNA and protein statistical moments is described by an upper triangular system of Volterra equations. Computation of the statistical moments for physiologically relevant parameter values shows that randomness in the cell division process can be a major factor in driving difference in protein levels across a population of cells.

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.

Labour-efficient in vitro lymphocyte population tracking and fate prediction using automation and manual review

Interest in cell heterogeneity and differentiation has recently led to increased use of time-lapse microscopy. Previous studies have shown that cell fate may be determined well in advance of the event. We used a mixture of automation and manual review of time-lapse live cell imaging to track the positions, contours, divisions, deaths and lineage of 44 B-lymphocyte founders and their 631 progeny in vitro over a period of 108 hours. Using this data to train a Support Vector Machine classifier, we were retrospectively able to predict the fates of individual lymphocytes with more than 90% accuracy, using only time-lapse imaging captured prior to mitosis or death of 90% of all cells. The motivation for this paper is to explore the impact of labour-efficient assistive software tools that allow larger and more ambitious live-cell time-lapse microscopy studies. After training on this data, we show that machine learning methods can be used for realtime prediction of individual cell fates. These techniques could lead to realtime cell culture segregation for purposes such as phenotype screening. We were able to produce a large volume of data with less effort than previously reported, due to the image processing, computer vision, tracking and human-computer interaction tools used. We describe the workflow of the software-assisted experiments and the graphical interfaces that were needed. To validate our results we used our methods to reproduce a variety of published data about lymphocyte populations and behaviour. We also make all our data publicly available, including a large quantity of lymphocyte spatio-temporal dynamics and related lineage information.

Fluctuation analysis: can estimates be trusted?

The estimation of mutation rates and relative fitnesses in fluctuation analysis is based on the unrealistic hypothesis that the single-cell times to division are exponentially distributed. Using the classical Luria-Delbrück distribution outside its modelling hypotheses induces an important bias on the estimation of the relative fitness. The model is extended here to any division time distribution. Mutant counts follow a generalization of the Luria-Delbrück distribution, which depends on the mean number of mutations, the relative fitness of normal cells compared to mutants, and the division time distribution of mutant cells. Empirical probability generating function techniques yield precise estimates both of the mean number of mutations and the relative fitness of normal cells compared to mutants. In the case where no information is available on the division time distribution, it is shown that the estimation procedure using constant division times yields more reliable results. Numerical results both on observed and simulated data are reported.

Asymmetry of Cell Division in CFSE-Based Lymphocyte Proliferation Analysis

Flow cytometry-based analysis of lymphocyte division using carboxyfluorescein succinimidyl ester (CFSE) dye dilution permits acquisition of data describing cellular proliferation and differentiation. For example, CFSE histogram data enable quantitative insight into cellular turnover rates by applying mathematical models and parameter estimation techniques. Several mathematical models have been developed using different types of deterministic or stochastic approaches. However, analysis of CFSE proliferation assays is based on the premise that the label is halved in the two daughter cells. Importantly, asymmetry of protein distribution in lymphocyte division is a basic biological feature of cell division with the degree of the asymmetry depending on various factors. Here, we review the recent literature on asymmetric lymphocyte division and CFSE-based lymphocyte proliferation analysis. We suggest that division- and label-structured mathematical models describing CFSE-based cell proliferation should take into account asymmetry and time-lag in cell proliferation. Utilization of improved modeling algorithms will permit straightforward quantification of essential parameters describing the performance of activated lymphocytes.

Modeling sequence evolution in HIV-1 infection with recombination

Previously we proposed two simplified models of early HIV-1 evolution. Both showed that under a model of neutral evolution and exponential growth, the mean Hamming distance (HD) between genetic sequences grows linearly with time. In this paper we describe a more realistic continuous-time, age-dependent mathematical model of infection and viral replication, and show through simulations that even in this more complex description, the mean Hamming distance grows linearly with time. This remains unchanged when we introduce recombination, though the confidence intervals of the mean HD obtained ignoring recombination are overly conservative.

FlowMax: A Computational Tool for Maximum Likelihood Deconvolution of CFSE Time Courses

The immune response is a concerted dynamic multi-cellular process. Upon infection, the dynamics of lymphocyte populations are an aggregate of molecular processes that determine the activation, division, and longevity of individual cells. The timing of these single-cell processes is remarkably widely distributed with some cells undergoing their third division while others undergo their first. High cell-to-cell variability and technical noise pose challenges for interpreting popular dye-dilution experiments objectively. It remains an unresolved challenge to avoid under- or over-interpretation of such data when phenotyping gene-targeted mouse models or patient samples. Here we develop and characterize a computational methodology to parameterize a cell population model in the context of noisy dye-dilution data. To enable objective interpretation of model fits, our method estimates fit sensitivity and redundancy by stochastically sampling the solution landscape, calculating parameter sensitivities, and clustering to determine the maximum-likelihood solution ranges. Our methodology accounts for both technical and biological variability by using a cell fluorescence model as an adaptor during population model fitting, resulting in improved fit accuracy without the need for ad hoc objective functions. We have incorporated our methodology into an integrated phenotyping tool, FlowMax, and used it to analyze B cells from two NFκB knockout mice with distinct phenotypes; we not only confirm previously published findings at a fraction of the expended effort and cost, but reveal a novel phenotype of nfkb1/p105/50 in limiting the proliferative capacity of B cells following B-cell receptor stimulation. In addition to complementing experimental work, FlowMax is suitable for high throughput analysis of dye dilution studies within clinical and pharmacological screens with objective and quantitative conclusions.

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.

Intracellular competition for fates in the immune system

During an adaptive immune response, lymphocytes proliferate for five to 20 generations, differentiating to take on effector functions, before cessation and cell death become dominant. Recent experimental methodologies enable direct observation of individual lymphocytes and the times at which they adopt fates. Data from these experiments reveal diversity in fate selection, heterogeneity and involved correlation structures in times to fate, as well as considerable familial correlations. Despite the significant complexity, these data are consistent with the simple hypothesis that each cell possesses autonomous processes, subject to temporal competition, leading to each fate. This article addresses the evidence for this hypothesis, its hallmarks, and, should it be an appropriate description of a cell system, its ramifications for manipulation.

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.

Functional role of kallikrein 6 in regulating immune cell survival

BACKGROUND

Kallikrein 6 (KLK6) is a newly identified member of the kallikrein family of secreted serine proteases that prior studies indicate is elevated at sites of central nervous system (CNS) inflammation and which shows regulated expression with T cell activation. Notably, KLK6 is also elevated in the serum of multiple sclerosis (MS) patients however its potential roles in immune function are unknown. Herein we specifically examine whether KLK6 alters immune cell survival and the possible mechanism by which this may occur.

METHODOLOGY/PRINCIPAL FINDINGS

Using murine whole splenocyte preparations and the human Jurkat T cell line we demonstrate that KLK6 robustly supports cell survival across a range of cell death paradigms. Recombinant KLK6 was shown to significantly reduce cell death under resting conditions and in response to camptothecin, dexamethasone, staurosporine and Fas-ligand. Moreover, KLK6-over expression in Jurkat T cells was shown to generate parallel pro-survival effects. In mixed splenocyte populations the vigorous immune cell survival promoting effects of KLK6 were shown to include both T and B lymphocytes, to occur with as little as 5 minutes of treatment, and to involve up regulation of the pro-survival protein B-cell lymphoma-extra large (Bcl-XL), and inhibition of the pro-apoptotic protein Bcl-2-interacting mediator of cell death (Bim). The ability of KLK6 to promote survival of splenic T cells was also shown to be absent in cell preparations derived from PAR1 deficient mice.

CONCLUSION/SIGNIFICANCE

KLK6 promotes lymphocyte survival by a mechanism that depends in part on activation of PAR1. These findings point to a novel molecular mechanism regulating lymphocyte survival that is likely to have relevance to a range of immunological responses that depend on apoptosis for immune clearance and maintenance of homeostasis.