Desynchronization rate in cell populations: mathematical modeling and experimental data

Evolutionary dynamics in non-Markovian models of microbial populations

In the past decade, great strides have been made to quantify the dynamics of single-cell growth and division in microbes. In order to make sense of the evolutionary history of these organisms, we must understand how features of single-cell growth and division influence evolutionary dynamics. This requires us to connect processes on the single-cell scale to population dynamics. Here, we consider a model of microbial growth in finite populations which explicitly incorporates the single-cell dynamics. We study the behavior of a mutant population in such a model and ask: can the evolutionary dynamics be coarse-grained so that the forces of natural selection and genetic drift can be expressed in terms of the long-term fitness? We show that it is in fact not possible, as there is no way to define a single fitness parameter (or reproductive rate) that defines the fate of an organism even in a constant environment. This is due to fluctuations in the population averaged division rate. As a result, various details of the single-cell dynamics affect the fate of a new mutant independently from how they affect the long-term growth rate of the mutant population. In particular, we show that in the case of neutral mutations, variability in generation times increases the rate of genetic drift, and in the case of beneficial mutations, variability decreases its fixation probability. Furthermore, we explain the source of the persistent division rate fluctuations and provide analytic solutions for the fixation probability as a multispecies generalization of the Euler-Lotka equation.

Generalization of Powell's results to population out of steady state

Since the seminal work of Powell, the relationships between the population growth rate, the probability distributions of generation time, and the distribution of cell age have been known for the bacterial population in a steady state of exponential growth. Here we generalize these relationships to include an unsteady (transient) state for both the batch culture and the mother machine experiment. In particular, we derive a time-dependent Euler-Lotka equation (relating the generation-time distributions to the population growth rate) and a generalization of the inequality between the mean generation time and the population doubling time. To do this, we use a model proposed by Lebowitz and Rubinow, in which each cell is described by its age and generation time. We show that our results remain valid for a class of more complex models that use other state variables in addition to cell age and generation time, as long as the integration of these additional variables reduces the model to Lebowitz-Rubinow form. As an application of this formalism, we calculate the fitness landscapes for phenotypic traits (cell age, generation time) in a population that is not growing exponentially. We clarify that the known fitness landscape formula for the cell age as a phenotypic trait is an approximation to the exact time-dependent formula.

Impact of variability in cell cycle periodicity on cell population dynamics

The cell cycle consists of a series of orchestrated events controlled by molecular sensing and feedback networks that ultimately drive the duplication of total DNA and the subsequent division of a single parent cell into two daughter cells. The ability to block the cell cycle and synchronize cells within the same phase has helped understand factors that control cell cycle progression and the properties of each individual phase. Intriguingly, when cells are released from a synchronized state, they do not maintain synchronized cell division and rapidly become asynchronous. The rate and factors that control cellular desynchronization remain largely unknown. In this study, using a combination of experiments and simulations, we investigate the desynchronization properties in cervical cancer cells (HeLa) starting from the G1/S boundary following double-thymidine block. Propidium iodide (PI) DNA staining was used to perform flow cytometry cell cycle analysis at regular 8 hour intervals, and a custom auto-similarity function to assess the desynchronization and quantify the convergence to an asynchronous state. In parallel, we developed a single-cell phenomenological model the returns the DNA amount across the cell cycle stages and fitted the parameters using experimental data. Simulations of population of cells reveal that the cell cycle desynchronization rate is primarily sensitive to the variability of cell cycle duration within a population. To validate the model prediction, we introduced lipopolysaccharide (LPS) to increase cell cycle noise. Indeed, we observed an increase in cell cycle variability under LPS stimulation in HeLa cells, accompanied with an enhanced rate of cell cycle desynchronization. Our results show that the desynchronization rate of artificially synchronized in-phase cell populations can be used a proxy of the degree of variance in cell cycle periodicity, an underexplored axis in cell cycle research.

Regulation of Cell Cycle Entry and Exit: A Single Cell Perspective

The transition between proliferating and quiescent states must be carefully regulated to ensure that cells divide to create the cells an organism needs only at the appropriate time and place. Cyclin-dependent kinases (CDKs) are critical for both transitioning cells from one cell cycle state to the next, and for regulating whether cells are proliferating or quiescent. CDKs are regulated by association with cognate cyclins, activating and inhibitory phosphorylation events, and proteins that bind to them and inhibit their activity. The substrates of these kinases, including the retinoblastoma protein, enforce the changes in cell cycle status. Single cell analysis has clarified that competition among factors that activate and inhibit CDK activity leads to the cell's decision to enter the cell cycle, a decision the cell makes before S phase. Signaling pathways that control the activity of CDKs regulate the transition between quiescence and proliferation in stem cells, including stem cells that generate muscle and neurons. © 2020 American Physiological Society. Compr Physiol 10:317-344, 2020.

Accurate delineation of cell cycle phase transitions in living cells with PIP-FUCCI

Cell cycle phase transitions are tightly orchestrated to ensure efficient cell cycle progression and genome stability. Interrogating these transitions is important for understanding both normal and pathological cell proliferation. By quantifying the dynamics of the popular FUCCI reporters relative to the transitions into and out of S phase, we found that their dynamics are substantially and variably offset from true S phase boundaries. To enhance detection of phase transitions, we generated a new reporter whose oscillations are directly coupled to DNA replication and combined it with the FUCCI APC/C reporter to create "PIP-FUCCI". The PIP degron fusion protein precisely marks the G1/S and S/G2 transitions; shows a rapid decrease in signal in response to large doses of DNA damage only during G1; and distinguishes cell type-specific and DNA damage source-dependent arrest phenotypes. We provide guidance to investigators in selecting appropriate fluorescent cell cycle reporters and new analysis strategies for delineating cell cycle transitions.

A checkpoint-oriented cell cycle simulation model

Modeling and in silico simulations are of major conceptual and applicative interest in studying the cell cycle and proliferation in eukaryotic cells. In this paper, we present a cell cycle checkpoint-oriented simulator that uses agent-based simulation modeling to reproduce the dynamics of a cancer cell population in exponential growth. Our in silico simulations were successfully validated by experimental in vitro supporting data obtained with HCT116 colon cancer cells. We demonstrated that this model can simulate cell confluence and the associated elongation of the G1 phase. Using nocodazole to synchronize cancer cells at mitosis, we confirmed the model predictivity and provided evidence of an additional and unexpected effect of nocodazole on the overall cell cycle progression. We anticipate that this cell cycle simulator will be a potential source of new insights and research perspectives.

Cell cycle time series gene expression data encoded as cyclic attractors in Hopfield systems

Modern time series gene expression and other omics data sets have enabled unprecedented resolution of the dynamics of cellular processes such as cell cycle and response to pharmaceutical compounds. In anticipation of the proliferation of time series data sets in the near future, we use the Hopfield model, a recurrent neural network based on spin glasses, to model the dynamics of cell cycle in HeLa (human cervical cancer) and S. cerevisiae cells. We study some of the rich dynamical properties of these cyclic Hopfield systems, including the ability of populations of simulated cells to recreate experimental expression data and the effects of noise on the dynamics. Next, we use a genetic algorithm to identify sets of genes which, when selectively inhibited by local external fields representing gene silencing compounds such as kinase inhibitors, disrupt the encoded cell cycle. We find, for example, that inhibiting the set of four kinases AURKB, NEK1, TTK, and WEE1 causes simulated HeLa cells to accumulate in the M phase. Finally, we suggest possible improvements and extensions to our model.

Cell cycle—the process in which a parent cell replicates its DNA and divides into two daughter cells—is an upregulated process in many forms of cancer. Identifying gene inhibition targets to regulate cell cycle is important to the development of effective therapies. Although modern high throughput techniques offer unprecedented resolution of the molecular details of biological processes like cell cycle, analyzing the vast quantities of the resulting experimental data and extracting actionable information remains a formidable task. Here, we create a dynamical model of the process of cell cycle using the Hopfield model (a type of recurrent neural network) and gene expression data from human cervical cancer cells and yeast cells. We find that the model recreates the oscillations observed in experimental data. Tuning the level of noise (representing the inherent randomness in gene expression and regulation) to the “edge of chaos” is crucial for the proper behavior of the system. We then use this model to identify potential gene targets for disrupting the process of cell cycle. This method could be applied to other time series data sets and used to predict the effects of untested targeted perturbations.

Are Tumor Cell Lineages Solely Shaped by Mechanical Forces?

This paper investigates cell proliferation dynamics in small tumor cell aggregates using an individual-based model (IBM). The simulation model is designed to study the morphology of the cell population and of the cell lineages as well as the impact of the orientation of the division plane on this morphology. Our IBM model is based on the hypothesis that cells are incompressible objects that grow in size and divide once a threshold size is reached, and that newly born cell adhere to the existing cell cluster. We performed comparisons between the simulation model and experimental data by using several statistical indicators. The results suggest that the emergence of particular morphologies can be explained by simple mechanical interactions.

Cell cycle proliferation decisions: the impact of single cell analyses

Cell proliferation is a fundamental requirement for organismal development and homeostasis. The mammalian cell division cycle is tightly controlled to ensure complete and precise genome duplication and segregation of replicated chromosomes to daughter cells. The onset of DNA replication marks an irreversible commitment to cell division, and the accumulated efforts of many decades of molecular and cellular studies have probed this cellular decision, commonly called the restriction point. Despite a long-standing conceptual framework of the restriction point for progression through G1 phase into S phase or exit from G1 phase to quiescence (G0), recent technical advances in quantitative single cell analysis of mammalian cells have provided new insights. Significant intercellular heterogeneity revealed by single cell studies and the discovery of discrete subpopulations in proliferating cultures suggests the need for an even more nuanced understanding of cell proliferation decisions. In this review, we describe some of the recent developments in the cell cycle field made possible by quantitative single cell experimental approaches.

Mathematical determination of cell population doubling times for multiple cell lines

Cell cycle times are vital parameters in cancer research, and short cell cycle times are often related to poor survival of cancer patients. A method for experimental estimation of cell cycle times, or doubling times of cultured cancer cell populations, based on addition of paclitaxel (an inhibitor of cell division) has been proposed in literature. We use a mathematical model to investigate relationships between essential parameters of the cell division cycle following inhibition of cell division. The reduction in the number of cells engaged in DNA replication reaches a plateau as the concentration of paclitaxel is increased; this can be determined experimentally. From our model we have derived a plateau log reduction formula for proliferating cells and established that there are linear relationships between the plateau log reduction values and the reciprocal of doubling times (i.e. growth rates of the populations). We have therefore provided theoretical justification of an important experimental technique to determine cell doubling times. Furthermore, we have applied Monte Carlo experiments to justify the suggested linear relationships used to estimate doubling time from 5-day cell culture assays. We show that our results are applicable to cancer cell populations with cell loss present.

The dynamics of Turing patterns for morphogen-regulated growing domains with cellular response delays

Since its conception in 1952, the Turing paradigm for pattern formation has been the subject of numerous theoretical investigations. Experimentally, this mechanism was first demonstrated in chemical reactions over 20 years ago and, more recently, several examples of biological self-organisation have also been implicated as Turing systems. One criticism of the Turing model is its lack of robustness, not only with respect to fluctuations in the initial conditions, but also with respect to the inclusion of delays in critical feedback processes such as gene expression. In this work we investigate the possibilities for Turing patterns on growing domains where the morphogens additionally regulate domain growth, incorporating delays in the feedback between signalling and domain growth, as well as gene expression. We present results for the proto-typical Schnakenberg and Gierer-Meinhardt systems: exploring the dynamics of these systems suggests a reconsideration of the basic Turing mechanism for pattern formation on morphogen-regulated growing domains as well as highlighting when feedback delays on domain growth are important for pattern formation.

Can telomere shortening explain sigmoidal growth curves?

A general branching process model is proposed to describe the shortening of telomeres in eukaryotic chromosomes. The model is flexible and incorporates many special cases to be found in the literature. In particular, we show how telomere shortening can give rise to sigmoidal growth curves, an idea first expressed by Portugal et al. [A computational model for telomere-dependent cell-replicative aging, BioSystems 91 (2008), pp. 262-267]. We also demonstrate how other types of growth curves arise if telomere shortening is mitigated by other cellular processes. We compare our results with published data sets from the biological literature.

Synchrony in reaction-diffusion models of morphogenesis: applications to curvature-dependent proliferation and zero-diffusion front waves

The paper presents the classical age-dependent approach of the morphogenesis in the framework of the von Foerster equation, in which we introduce a new constraint and study a new feature: (i) the new constraint concerns cell proliferation along the contour lines of the cell density, depending on the local curvature such as it favours the amplification of the concavities (like in the gastrulation process) and (ii) the new feature consists of considering, on the cell density surface, a remarkable line (the null mean Gaussian curvature line), on which the normal diffusion vanishes, favouring local coexistence of diffusing morphogens, metabolites or cells, and hence the auto-assemblages of these entities. Two applications to biological multi-agents systems are studied, gastrulation and feather morphogenesis.

A branching process model for flow cytometry and budding index measurements in cell synchrony experiments

We present a flexible branching process model for cell population dynamics in synchrony/time-series experiments used to study important cellular processes. Its formulation is constructive, based on an accounting of the unique cohorts in the population as they arise and evolve over time, allowing it to be written in closed form. The model can attribute effects to subsets of the population, providing flexibility not available using the models historically applied to these populations. It provides a tool for in silico synchronization of the population and can be used to deconvolve population-level experimental measurements, such as temporal expression profiles. It also allows for the direct comparison of assay measurements made from multiple experiments. The model can be fit either to budding index or DNA content measurements, or both, and is easily adaptable to new forms of data. The ability to use DNA content data makes the model applicable to almost any organism. We describe the model and illustrate its utility and flexibility in a study of cell cycle progression in the yeast Saccharomyces cerevisiae.

A stochastic model of cell cycle desynchronization

A general branching process model is suggested to describe cell cycle desynchronization. Cell cycle phase times are modeled as random variables and a formula for the expected fraction of cells in S phase as a function of time is established. The model is compared to data from the literature and is also compared to previously suggested deterministic and stochastic models.

A stochastic model of a cell population with quiescence

A cell population in which cells are allowed to enter a quiescent (nonproliferating) phase is analyzed using a stochastic approach. A general branching process is used to model the population which, under very mild conditions, exhibits balanced exponential growth. A formula is given for the asymptotic fraction of quiescent cells, and a numerical example illustrates how convergence toward the asymptotic fraction exhibits a typical oscillatory pattern. The model is compared with deterministic models based on semigroup analysis of systems of differential equations.

A quantitative study of growth variability of tumour cell clones in vitro

OBJECTIVES

In this study, we quantify growth variability of tumour cell clones from a human leukaemia cell line.

MATERIALS AND METHODS

We have used microplate spectrophotometry to measure growth kinetics of hundreds of individual cell clones from the Molt3 cell line. Growth rate of each clonal population has been estimated by fitting experimental data with the logistic equation.

RESULTS

Growth rates were observed to vary between different clones. Up to six clones with growth rates above or below mean growth rate of the parent population were further cloned and growth rates of their offspring were measured. Distribution of growth rates of the subclones did not significantly differ from that of the parent population, thus suggesting that growth variability has an epigenetic origin. To explain observed distributions of clonal growth rates, we have developed a probabilistic model, assuming that fluctuation in the number of mitochondria through successive cell cycles is the leading cause of growth variability. For fitting purposes, we have estimated experimentally by flow cytometry the average maximum number of mitochondria in Molt3 cells. The model fits nicely observed distributions in growth rates; however, cells in which mitochondria were rendered non-functional (rho(0) cells) showed only 30% reduction in clonal growth variability with respect to normal cells.

CONCLUSIONS

A tumour cell population is a dynamic ensemble of clones with highly variable growth rates. At least part of this variability is due to fluctuations in the initial number of mitochondria in daughter cells.

Identification of age-structured models: cell cycle phase transitions

A methodology is developed that determines age-specific transition rates between cell cycle phases during balanced growth by utilizing age-structured population balance equations. Age-distributed models are the simplest way to account for varied behavior of individual cells. However, this simplicity is offset by difficulties in making observations of age distributions, so age-distributed models are difficult to fit to experimental data. Herein, the proposed methodology is implemented to identify an age-structured model for human leukemia cells (Jurkat) based only on measurements of the total number density after the addition of bromodeoxyuridine partitions the total cell population into two subpopulations. Each of the subpopulations will temporarily undergo a period of unbalanced growth, which provides sufficient information to extract age-dependent transition rates, while the total cell population remains in balanced growth. The stipulation of initial balanced growth permits the derivation of age densities based on only age-dependent transition rates. In fitting the experimental data, a flexible transition rate representation, utilizing a series of cubic spline nodes, finds a bimodal G(0)/G(1) transition age probability distribution best fits the experimental data. This resolution may be unnecessary as convex combinations of more restricted transition rates derived from normalized Gaussian, lognormal, or skewed lognormal transition-age probability distributions corroborate the spline predictions, but require fewer parameters. The fit of data with a single log normal distribution is somewhat inferior suggesting the bimodal result as more likely. Regardless of the choice of basis functions, this methodology can identify age distributions, age-specific transition rates, and transition-age distributions during balanced growth conditions.

Ab initio phenomenological simulation of the growth of large tumor cell populations

In a previous paper we have introduced a phenomenological model of cell metabolism and of the cell cycle to simulate the behavior of large tumor cell populations (Chignola and Milotti 2005 Phys. Biol. 2 8). Here we describe a refined and extended version of the model that includes some of the complex interactions between cells and their surrounding environment. The present version takes into consideration several additional energy-consuming biochemical pathways such as protein and DNA synthesis, the tuning of extracellular pH and of the cell membrane potential. The control of the cell cycle, which was previously modeled by means of ad hoc thresholds, has been directly addressed here by considering checkpoints from proteins that act as targets for phosphorylation on multiple sites. As simulated cells grow, they can now modify the chemical composition of the surrounding environment which in turn acts as a feedback mechanism to tune cell metabolism and hence cell proliferation: in this way we obtain growth curves that match quite well those observed in vitro with human leukemia cell lines. The model is strongly constrained and returns results that can be directly compared with actual experiments, because it uses parameter values in narrow ranges estimated from experimental data, and in perspective we hope to utilize it to develop in silico studies of the growth of very large tumor cell populations (10(6) cells or more) and to support experimental research. In particular, the program is used here to make predictions on the behavior of cells grown in a glucose-poor medium: these predictions are confirmed by experimental observation.

A phenomenological approach to the simulation of metabolism and proliferation dynamics of large tumour cell populations

A major goal of modern computational biology is to simulate the collective behaviour of large cell populations starting from the intricate web of molecular interactions occurring at the microscopic level. In this paper we describe a simplified model of cell metabolism, growth and proliferation, suitable for inclusion in a multicell simulator, now under development (Chignola R and Milotti E 2004 Physica A 338 261-6). Nutrients regulate the proliferation dynamics of tumour cells which adapt their behaviour to respond to changes in the biochemical composition of the environment. This modelling of nutrient metabolism and cell cycle at a mesoscopic scale level leads to a continuous flow of information between the two disparate spatiotemporal scales of molecular and cellular dynamics that can be simulated with modern computers and tested experimentally.

Interpreting cell cycle effects of drugs: the case of melphalan

Multiple effects usually occur in the cell cycle, during and after the exposure to a drug, while treated cells flowing through the cycle encounter G1, S and G2M checkpoints. We developed a simulation tool connecting the microscopic level of the cellular response in G1, S and G2M with the experimental data of growth inhibition and flow cytometry. We found that multiple-often not intuitive-combinations of cytostatic and cytotoxic effects can be in keeping with the observations. This multiplicity of interpretation can be strongly reduced by considering together data with different methods, ideally reaching a reconstruction of the underlying cell cycle perturbations. Here, we propose an experimental plan including a time course of DNA flow cytometry and absolute cell count measurements with several drug concentrations and a limited number of flow cytometric DNA-Bromodeoxyuridine and TUNEL analyses, coupled with computer simulation. We showed its use in the attempt to define the complete time course of the effects of melphalan on three cancer cell lines. After drug treatment, each subset of cells experienced blocks and lethality in all phases of the cell cycle, but the dynamics was different, the differences being strongly dose-dependent. Our approach allows a better appreciation of the complexity of the cell cycle phenomena associated with drug treatment. It is expected that such level of understanding of the time- and dose-dependence of the cytostatic and cytotoxic effects of a drug might support rational therapeutic design.

The application of mathematical modelling to aspects of adjuvant chemotherapy scheduling

In this paper simple models for tumour growth incorporating age-structured cell cycle dynamics are considered in the presence of two non-cross-resistant S-phase specific chemotherapeutic drugs. According to the seminal work of Goldie and Coldman, if one cannot deliver two cell cycle phase non-specific, non-cross-resistant drugs simultaneously, for example due to toxicity, and both drugs are identical apart from resistance, one should alternate their delivery as rapidly as possible. However consider S-phase specific drugs. One might speculate that, for example, alternating the two drugs at intervals of T, where T is the mean cell cycle time, is better than alternating the drugs at intervals of T/2, as the latter strategy allows the possibility of a cell cycle sanctuary. Such speculation implicitly requires a sufficiently low variance of the cell cycle time, and hence it is not clear if such reasoning prevents a generalisation of the results of Goldie and Coldman. This question is addressed in this paper via a detailed modelling investigation, as motivated by suggestions for future colorectal adjuvant chemotherapy trials and developments in hepatic arterial infusion technology. It is shown that the cell cycle distribution of the resistant cell populations is strongly influenced by the chemotherapy schedule. The consequences of this can be dramatic, and can lead to chemotherapy failure at resonant chemotherapy timings, especially for a small standard deviation of the cell cycle time. The novel aspects of this observation are highlighted compared to other models in the literature exhibiting resonant behaviour in the timing of a periodic chemotherapy protocol. The above investigation also results in the principal prediction of this paper that reducing the drug alternation time to approximately a few hours, if possible, can result in substantial improvements in predicted chemotherapy outcomes. Critically, such improvements are not predicted by the Goldie Coldman model or other chemotherapy scheduling models in the literature.

[Deterministic differential equations in cytokinetics]

Carcinogenesis is generally viewed as the result of a disrupted equilibrium between the processes of proliferation, differentiation, migration and loss. In cell kinetics, perturbation of fundamental kinetic parameters such as the mitotic index or the growth fraction may reflect a carcinogenic process. In this paper, we present a system of deterministic differential equations describing the dynamical evolution of a cell population. We show that this model can account for asynchronicity of cells and simulate the fraction of labelled mitoses experiment as well.

Variability in the timing of G(1)/S transition

Cell cycle duration and phase transition times are not fixed, even within homogeneous cell populations growing under optimal environmental conditions. We investigate G(1) phase variability from the molecular point of view and propose a mathematical approach to model the protein interactions regulating the transition from the G(1) phase to the phase of DNA synthesis. The mathematical model has some connections with flow cytometry experimental data.