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.

Stochastic Gene Expression in Proliferating Cells: Differing Noise Intensity in Single-Cell and Population Perspectives

Random fluctuations (noise) in gene expression can be studied from two complementary perspectives: following expression in a single cell over time or comparing expression between cells in a proliferating population at a given time. Here, we systematically investigated scenarios where both perspectives lead to different levels of noise in a given gene product. We first consider a stable protein, whose concentration is diluted by cellular growth, and the protein inhibits growth at high concentrations, establishing a positive feedback loop. For a stochastic model with molecular bursting of gene products, we analytically predict and contrast the steady-state distributions of protein concentration in both frameworks. Although positive feedback amplifies the noise in expression, this amplification is much higher in the population framework compared to following a single cell over time. We also study other processes that lead to different noise levels even in the absence of such dilution-based feedback. When considering randomness in the partitioning of molecules between daughters during mitosis, we find that in the single-cell perspective, the noise in protein concentration is independent of noise in the cell cycle duration. In contrast, partitioning noise is amplified in the population perspective by increasing randomness in cell-cycle time. Overall, our results show that the commonly used single-cell framework that does not account for proliferating cells can, in some cases, underestimate the noise in gene product levels. These results have important implications for studying the inter-cellular variation of different stress-related expression programs across cell types that are known to inhibit cellular growth.

Cell size homeostasis is tightly controlled throughout the cell cycle

To achieve a stable size distribution over multiple generations, proliferating cells require a means of counteracting stochastic noise in the rate of growth, the time spent in various phases of the cell cycle, and the imprecision in the placement of the plane of cell division. In the most widely accepted model, cell size is thought to be regulated at the G1/S transition, such that cells smaller than a critical size pause at the end of G1 phase until they have accumulated mass to a predetermined size threshold, at which point the cells proceed through the rest of the cell cycle. However, a model, based solely on a specific size checkpoint at G1/S, cannot readily explain why cells with deficient G1/S control mechanisms are still able to maintain a very stable cell size distribution. Furthermore, such a model would not easily account for stochastic variation in cell size during the subsequent phases of the cell cycle, which cannot be anticipated at G1/S. To address such questions, we applied computationally enhanced quantitative phase microscopy (ceQPM) to populations of cultured human cell lines, which enables highly accurate measurement of cell dry mass of individual cells throughout the cell cycle. From these measurements, we have evaluated the factors that contribute to maintaining cell mass homeostasis at any point in the cell cycle. Our findings reveal that cell mass homeostasis is accurately maintained, despite disruptions to the normal G1/S machinery or perturbations in the rate of cell growth. Control of cell mass is generally not confined to regulation of the G1 length. Instead mass homeostasis is imposed throughout the cell cycle. In the cell lines examined, we find that the coefficient of variation (CV) in dry mass of cells in the population begins to decline well before the G1/S transition and continues to decline throughout S and G2 phases. Among the different cell types tested, the detailed response of cell growth rate to cell mass differs. However, in general, when it falls below that for exponential growth, the natural increase in the CV of cell mass is effectively constrained. We find that both mass-dependent cell cycle regulation and mass-dependent growth rate modulation contribute to reducing cell mass variation within the population. Through the interplay and coordination of these 2 processes, accurate cell mass homeostasis emerges. Such findings reveal previously unappreciated and very general principles of cell size control in proliferating cells. These same regulatory processes might also be operative in terminally differentiated cells. Further quantitative dynamical studies should lead to a better understanding of the underlying molecular mechanisms of cell size control.

First-passage-time statistics of growing microbial populations carry an imprint of initial conditions

In exponential population growth, variability in the timing of individual division events and environmental factors (including stochastic inoculation) compound to produce variable growth trajectories. In several stochastic models of exponential growth we show power-law relationships that relate variability in the time required to reach a threshold population size to growth rate and inoculum size. Population-growth experiments in E. coli and S. aureus with inoculum sizes ranging between 1 and 100 are consistent with these relationships. We quantify how noise accumulates over time, finding that it encodes-and can be used to deduce-information about the early growth rate of a population.

Myrcene: A Natural Compound Showing Anticancer Activity in HeLa Cells

γ-terpinene, α-terpinene, p-cymene, and myrcene are monoterpenes found in many essential oils extracted from a variety of plants and spices. Myrcene also occurs naturally in plants such as hops, cannabis, lemongrass, and verbena and is used as a flavoring agent in food and beverage manufacturing. In this research, the biological efficacy of γ-terpinene, α-terpinene, p-cymene, and myrcene was studied in human cell lines (HeLa, SH-SY5Y, and HDFa). Cytotoxicity, cell proliferation, cell migration, and morphology assays were performed to obtain detailed information on the anticancer properties. Our results show that myrcene has potential biological activity, especially in HeLa cells. In this cell line, it leads to an arrest of proliferation, a decrease in motility and morphological changes with loss of sphericity and thickness, and DNA damage. In addition, the interaction of γ-terpinene, α-terpinene, p-terpinene, and myrcene with calf thymus DNA (ct-DNA) was studied by UV-visible spectrophotometry. DNA binding experiments show that only myrcene can interact with DNA with an apparent dissociation constant (Kd) of 29 × 10-6 M.

Bacterial cell-size changes resulting from altering the relative expression of Min proteins

The timing of cell division, and thus cell size in bacteria, is determined in part by the accumulation dynamics of the protein FtsZ, which forms the septal ring. FtsZ localization depends on membrane-associated Min proteins, which inhibit FtsZ binding to the cell pole membrane. Changes in the relative concentrations of Min proteins can disrupt FtsZ binding to the membrane, which in turn can delay cell division until a certain cell size is reached, in which the dynamics of Min proteins frees the cell membrane long enough to allow FtsZ ring formation. Here, we study the effect of Min proteins relative expression on the dynamics of FtsZ ring formation and cell size in individual Escherichia coli bacteria. Upon inducing overexpression of minE, cell size increases gradually to a new steady-state value. Concurrently, the time required to initiate FtsZ ring formation grows as the size approaches the new steady-state, at which point the ring formation initiates as early as before induction. These results highlight the contribution of Min proteins to cell size control, which may be partially responsible for the size fluctuations observed in bacterial populations, and may clarify how the size difference acquired during asymmetric cell division is offset.

Multigenerational memory in bacterial size control

Cells maintain a stable size as they grow and divide. Inspired by the available experimental data, most proposed models for size homeostasis assume size-control mechanisms that act on a timescale of one generation. Such mechanisms lead to short-lived autocorrelations in size fluctuations that decay within less than two generations. However, recent evidence from comparing sister lineages suggests that correlations in size fluctuations can persist for many generations. Here we develop a minimal model that explains these seemingly contradictory results. Our model proposes that different environments result in different control parameters, leading to distinct inheritance patterns. Multigenerational memory is revealed in constant environments but obscured when averaging over many different environments. Inferring the parameters of our model from Escherichia coli size data in microfluidic experiments, we recapitulate the observed statistics. Our paper elucidates the impact of the environment on cell homeostasis and growth and division dynamics.

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.

PyEcoLib: a python library for simulating stochastic cell size dynamics

Recently, there has been an increasing need for tools to simulate cell size regulation due to important applications in cell proliferation and gene expression. However, implementing the simulation usually presents some difficulties, as the division has a cycle-dependent occurrence rate. In this article, we gather a recent theoretical framework inPyEcoLib, a python-based library to simulate the stochastic dynamics of the size of bacterial cells. This library can simulate cell size trajectories with an arbitrarily small sampling period. In addition, this simulator can include stochastic variables, such as the cell size at the beginning of the experiment, the cycle duration timing, the growth rate, and the splitting position. Furthermore, from a population perspective, the user can choose between tracking a single lineage or all cells in a colony. They can also simulate the most common division strategies (adder, timer, and sizer) using the division rate formalism and numerical methods. As an example of PyecoLib applications, we explain how to couple size dynamics with gene expression predicting, from simulations, how the noise in protein levels increases by increasing the noise in division timing, the noise in growth rate and the noise in cell splitting position. The simplicity of this library and its transparency about the underlying theoretical framework yield the inclusion of cell size stochasticity in complex models of gene expression.

Control of Cell Size by c-di-GMP Requires a Two-Component Signaling System in the Cyanobacterium Anabaena sp. Strain PCC 7120

Each bacterial species possesses a specific cell size and morphology, which constitute important and recognizable physical traits. How bacteria maintain their particular cell size and morphology remains an essential question in microbiology. Cyanobacteria are oxygen-evolving photosynthetic prokaryotes. Although monophyletic, these organisms are highly diverse in their cell morphology and cell size. How these physical traits of cyanobacteria are controlled is poorly understood. Here, we report the identification of a two-component signaling system, composed of a histidine kinase CdgK and a response regulator CdgS, involved in cell size regulation in the filamentous, heterocyst-forming cyanobacterium Anabaena sp. PCC 7120. Inactivation of cdgK or cdgS led to reduction of cell length and width with little effect on cell growth capacity. CdgS has a GGDEF domain responsible for the synthesis of the second messenger c-di-GMP. Based on genetic and biochemical studies, we proposed a signaling pathway initiated by CdgK, leading to the phosphorylation of CdgS, and thereby an enhanced enzymatic activity for c-di-GMP synthesis of the latter. The GGDEF domain of CdgS was essential in cell size control, and the reduction of cell size observed in various mutants could be rescued by the expression of a c-di-GMP synthetase from E. coli. These results provided evidence that a minimal threshold of c-di-GMP level was required for maintaining cell size in Anabaena. IMPORTANCE Cyanobacteria are considered the first organisms to produce oxygen on Earth, and their activities shaped the evolution of our ecosystems. Cell size is an important trait fixed early in evolution, with the diversification of micro- and macrocyanobacterial species during the Great Oxidation Event. However, the genetic basis underlying cell size control in cyanobacteria was not understood. Our studies demonstrated that the CdgK-CdgS signaling pathway participates in the control of cell size, and their absence did not affect cell growth. CdgK has multiple domains susceptible to signal input, which are necessary for cell size regulation. This observation suggests that cell size in Anabaena could respond to environmental signals. These studies paved the way for genetic dissection of cell size regulation in cyanobacteria.

Patterns of interdivision time correlations reveal hidden cell cycle factors

The time taken for cells to complete a round of cell division is a stochastic process controlled, in part, by intracellular factors. These factors can be inherited across cellular generations which gives rise to, often non-intuitive, correlation patterns in cell cycle timing between cells of different family relationships on lineage trees. Here, we formulate a framework of hidden inherited factors affecting the cell cycle that unifies known cell cycle control models and reveals three distinct interdivision time correlation patterns: aperiodic, alternator, and oscillator. We use Bayesian inference with single-cell datasets of cell division in bacteria, mammalian and cancer cells, to identify the inheritance motifs that underlie these datasets. From our inference, we find that interdivision time correlation patterns do not identify a single cell cycle model but generally admit a broad posterior distribution of possible mechanisms. Despite this unidentifiability, we observe that the inferred patterns reveal interpretable inheritance dynamics and hidden rhythmicity of cell cycle factors. This reveals that cell cycle factors are commonly driven by circadian rhythms, but their period may differ in cancer. Our quantitative analysis thus reveals that correlation patterns are an emergent phenomenon that impact cell proliferation and these patterns may be altered in disease.

High-throughput determination of dry mass of single bacterial cells by ultrathin membrane resonators

How bacteria are able to maintain their size remains an open question. Techniques that can measure the biomass (dry mass) of single cells with high precision and high-throughput are demanded to elucidate this question. Here, we present a technological approach that combines the transport, guiding and focusing of individual bacteria from solution to the surface of an ultrathin silicon nitride membrane resonator in vacuum. The resonance frequencies of the membrane undergo abrupt variations at the instants where single cells land on the membrane surface. The resonator design displays a quasi-symmetric rectangular shape with an extraordinary capture area of 0.14 mm², while maintaining a high mass resolution of 0.7 fg (1 fg = 10⁻¹⁵g) to precisely resolve the dry mass of single cells. The small rectangularity of the membrane provides unprecedented frequency density of vibration modes that enables to retrieve the mass of individual cells with high accuracy by specially developed inverse problem theory. We apply this approach for profiling the dry mass distribution in Staphylococcus epidermidis and Escherichia coli cells. The technique allows the determination of the dry mass of single bacterial cells with an accuracy of about 1% at an unparalleled throughput of 20 cells/min. Finally, we revisit Koch & Schaechter model developed during 60 s to assess the intrinsic sources of stochasticity that originate cell size heterogeneity in steady-state populations. The results reveal the importance of mass resolution to correctly describe these mechanisms.

A technological approach combines transport, guiding and focusing of individual bacteria from solution to ultrathin membrane resonators for dry mass determination of single cells with accuracy within 1% and throughput of 20 cells/min.

Eukaryotic Cell Size Control and Its Relation to Biosynthesis and Senescence

The most fundamental feature of cellular form is size, which sets the scale of all cell biological processes. Growth, form, and function are all necessarily linked in cell biology, but we often do not understand the underlying molecular mechanisms nor their specific functions. Here, we review progress toward determining the molecular mechanisms that regulate cell size in yeast, animals, and plants, as well as progress toward understanding the function of cell size regulation. It has become increasingly clear that the mechanism of cell size regulation is deeply intertwined with basic mechanisms of biosynthesis, and how biosynthesis can be scaled (or not) in proportion to cell size. Finally, we highlight recent findings causally linking aberrant cell size regulation to cellular senescence and their implications for cancer therapies.

Scaling of biosynthesis and metabolism with cell size

Cells adopt a size that is optimal for their function, and pushing them beyond this limit can cause cell aging and death by senescence or reduce proliferative potential. However, by increasing their genome copy number (ploidy), cells can increase their size dramatically and homeostatically maintain physiological properties such as biosynthesis rate. Recent studies investigating the relationship between cell size and rates of biosynthesis and metabolism under normal, polyploid, and pathological conditions are revealing new insights into how cells attain the best function or fitness for their size by tuning processes including transcription, translation, and mitochondrial respiration. A new frontier is to connect single-cell scaling relationships with tissue and whole-organism physiology, which promises to reveal molecular and evolutionary principles underlying the astonishing diversity of size observed across the tree of life.

Single-cell monitoring of dry mass and dry mass density reveals exocytosis of cellular dry contents in mitosis

Cell mass and composition change with cell cycle progression. Our previous work characterized buoyant mass dynamics in mitosis (Miettinen et al., 2019), but how dry mass and cell composition change in mitosis has remained unclear. To better understand mitotic cell growth and compositional changes, we develop a single-cell approach for monitoring dry mass and the density of that dry mass every ~75 s with 1.3% and 0.3% measurement precision, respectively. We find that suspension grown mammalian cells lose dry mass and increase dry mass density following mitotic entry. These changes display large, non-genetic cell-to-cell variability, and the changes are reversed at metaphase-anaphase transition, after which dry mass continues accumulating. The change in dry mass density causes buoyant and dry mass to differ specifically in early mitosis, thus reconciling existing literature on mitotic cell growth. Mechanistically, cells in early mitosis increase lysosomal exocytosis, and inhibition of lysosomal exocytosis decreases the dry mass loss and dry mass density increase in mitosis. Overall, our work provides a new approach for monitoring single-cell dry mass and dry mass density, and reveals that mitosis is coupled to extensive exocytosis-mediated secretion of cellular contents.

Coupling between DNA replication, segregation, and the onset of constriction in Escherichia coli

Escherichia coli cell cycle features two critical cell-cycle checkpoints: initiation of replication and the onset of constriction. While the initiation of DNA replication has been extensively studied, it is less clear what triggers the onset of constriction and when exactly it occurs during the cell cycle. Here, using high-throughput fluorescence microscopy in microfluidic devices, we determine the timing for the onset of constriction relative to the replication cycle in different growth rates. Our single-cell data and modeling indicate that the initiation of constriction is coupled to replication-related processes in slow growth conditions. Furthermore, our data suggest that this coupling involves the mid-cell chromosome blocking the onset of constriction via some form of nucleoid occlusion occurring independently of SlmA and the Ter linkage proteins. This work highlights the coupling between replication and division cycles and brings up a new nucleoid mediated control mechanism in E. coli.

Using high-throughput microscopy, Tiruvadi-Krishnan et al. determine timings for critical cell-cycle checkpoints related to division and replication in Escherichia coli. The data, combined with cell-cycle modeling, show that the onset of constriction is blocked by the mid-cell nucleoid. In slow-growth conditions, the blockage is limiting for cell division.

ppGpp is a bacterial cell size regulator

Growth and division are central to cell size. Bacteria achieve size homeostasis by dividing when growth has added a constant size since birth, termed the adder principle, by unknown mechanisms.^1,2 Growth is well known to be regulated by guanosine tetraphosphate (ppGpp), which controls diverse processes from ribosome production to metabolic enzyme activity and replication initiation and whose absence or excess can induce stress, filamentation, and small growth-arrested cells.^3-6 These observations raise unresolved questions about the relation between ppGpp and size homeostasis mechanisms during normal exponential growth. Here, to untangle effects of ppGpp and nutrients, we gained control of cellular ppGpp by inducing the synthesis and hydrolysis enzymes RelA and Mesh1. We found that ppGpp not only exerts control over the growth rate but also over cell division and thus the steady state cell size. In response to changes in ppGpp level, the added size already establishes its new constant value while the growth rate still adjusts, aided by accelerated or delayed divisions. Moreover, the magnitude of the added size and resulting steady-state birth size correlate consistently with the ppGpp level, rather than with the growth rate, which results in cells of different size that grow equally fast. Our findings suggest that ppGpp serves as a key regulator that coordinates cell size and growth control.

Distinguishing different modes of growth using single-cell data

Collection of high-throughput data has become prevalent in biology. Large datasets allow the use of statistical constructs such as binning and linear regression to quantify relationships between variables and hypothesize underlying biological mechanisms based on it. We discuss several such examples in relation to single-cell data and cellular growth. In particular, we show instances where what appears to be ordinary use of these statistical methods leads to incorrect conclusions such as growth being non-exponential as opposed to exponential and vice versa. We propose that the data analysis and its interpretation should be done in the context of a generative model, if possible. In this way, the statistical methods can be validated either analytically or against synthetic data generated via the use of the model, leading to a consistent method for inferring biological mechanisms from data. On applying the validated methods of data analysis to infer cellular growth on our experimental data, we find the growth of length in E. coli to be non-exponential. Our analysis shows that in the later stages of the cell cycle the growth rate is faster than exponential.

Continuous rate modeling of bacterial stochastic size dynamics

Bacterial division is an inherently stochastic process with effects on fluctuations of protein concentration and phenotype variability. Current modeling tools for the stochastic short-term cell-size dynamics are scarce and mainly phenomenological. Here we present a general theoretical approach based on the Chapman-Kolmogorov equation incorporating continuous growth and division events as jump processes. This approach allows us to include different division strategies, noisy growth, and noisy cell splitting. Considering bacteria synchronized from their last division, we predict oscillations in both the central moments of the size distribution and its autocorrelation function. These oscillations, barely discussed in past studies, can arise as a consequence of the discrete time displacement invariance of the system with a period of one doubling time, and they do not disappear when including stochasticity on either division times or size heterogeneity on the starting population but only after inclusion of noise in either growth rate or septum position. This result illustrates the usefulness of having a solid mathematical description that explicitly incorporates the inherent stochasticity in various biological processes, both to understand the process in detail and to evaluate the effect of various sources of variability when creating simplified descriptions.

Cell size controlled in plants using DNA content as an internal scale

How eukaryotic cells assess and maintain sizes specific for their species and cell type remains unclear. We show that in the Arabidopsis shoot stem cell niche, cell size variability caused by asymmetric divisions is corrected by adjusting the growth period before DNA synthesis. KIP-related protein 4 (KRP4) inhibits progression to DNA synthesis and associates with mitotic chromosomes. The F BOX-LIKE 17 (FBL17) protein removes excess KRP4. Consequently, daughter cells are born with comparable amounts of KRP4. Inhibitor dilution models predicted that KRP4 inherited through chromatin would robustly regulate size, whereas inheritance of excess free KRP4 would disrupt size homeostasis, as confirmed by mutant analyses. We propose that a cell cycle regulator, stabilized by association with mitotic chromosomes, reads DNA content as a cell size-independent scale.

Exponential trajectories, cell size fluctuations, and the adder property in bacteria follow from simple chemical dynamics and division control

Experiments on steady-state bacterial cultures have uncovered several quantitative regularities at the system level. These include, first, the exponential growth of cell size with time and the balanced growth of intracellular chemicals between cell birth and division, which are puzzling given the nonlinear and decentralized chemical dynamics in the cell. We model a cell as a set of chemical populations undergoing nonlinear mass action kinetics in a container whose volume is a linear function of the chemical populations. This turns out to be a special class of dynamical systems that generically has attractors in which all populations grow exponentially with time at the same rate. This explains exponential balanced growth of bacterial cells without invoking any regulatory mechanisms and suggests that this could be a robust property of protocells as well. Second, we consider the hypothesis that cells commit themselves to division when a certain internal chemical population reaches a threshold of N molecules. We show that this hypothesis leads to a simple explanation of some of the variability observed across cells in a bacterial culture. In particular, it reproduces the adder property of cell size fluctuations observed recently in E. coli; the observed correlations among interdivision time, birth volume, and added volume in a generation; and the observed scale of the fluctuations (CV ≈ 10-30%) when N is between 10 and 100. Third, upon including a suitable regulatory mechanism that optimizes the growth rate of the cell, the model reproduces the observed bacterial growth laws including the dependence of the growth rate and ribosomal protein fraction on the medium. Thus, the models provide a framework for unifying diverse aspects of bacterial growth physiology under one roof. They also suggest new questions for experimental and theoretical enquiry.

Cell size: a key determinant of meristematic potential in plant protoplasts

Metabolic pathway reconstruction and gene edits for native natural product synthesis in single plant cells are considered to be less complicated when compared to the production of non-native metabolites. Being an efficient eukaryotic system, plants encompass suitable post-translational modifications. However, slow cell division rate and heterogeneous nature is an impediment for consistent product retrieval from plant cells. Plant cell synchrony can be attained in cultures developed in vitro. Isolated plant protoplasts capable of division, can potentially enhance the unimpaired yield of target bioactives, similar to microbes and unicellular eukaryotes. Evidence from yeast experiments suggests that 'critical cell size' and division rates for enhancement machinery, primarily depend on culture conditions and nutrient availability. The cell size control mechanisms in Arabidopsis shoot apical meristem is analogous to yeast notably, fission yeast. If protoplasts isolated from plants are subjected to cell size studies and cell cycle progression in culture, it will answer the underlying molecular mechanisms such as, unicellular to multicellular transition states, longevity, senescence, 'cell-size resetting' during organogenesis, and adaptation to external cues.

Non-genetic inheritance restraint of cell-to-cell variation

Heterogeneity in physical and functional characteristics of cells (e.g. size, cycle time, growth rate, protein concentration) proliferates within an isogenic population due to stochasticity in intracellular biochemical processes and in the distribution of resources during divisions. Conversely, it is limited in part by the inheritance of cellular components between consecutive generations. Here we introduce a new experimental method for measuring proliferation of heterogeneity in bacterial cell characteristics, based on measuring how two sister cells become different from each other over time. Our measurements provide the inheritance dynamics of different cellular properties, and the 'inertia' of cells to maintain these properties along time. We find that inheritance dynamics are property specific and can exhibit long-term memory (∼10 generations) that works to restrain variation among cells. Our results can reveal mechanisms of non-genetic inheritance in bacteria and help understand how cells control their properties and heterogeneity within isogenic cell populations.

From single-cell variability to population growth

Single-cell experiments have revealed cell-to-cell variability in generation times and growth rates for genetically identical cells. Theoretical models relating the fluctuating generation times of single cells to the population growth rate are usually based on the assumption that the generation times of mother and daughter cells are uncorrelated. This assumption, however, is inconsistent with the exponential growth of cell volume in time observed for many cell types. Here we develop a more general and biologically relevant model in which cells grow exponentially and generation times are correlated in a manner which controls cell size. In addition to the fluctuating generation times, we also allow the single-cell growth rates to fluctuate and account for their correlations across the lineage tree. Surprisingly, we find that the population growth rate only depends on the distribution of single-cell growth rates and their correlations.

Contributions to the 'noise floor' in gene expression in a population of dividing cells

Experiments with cells reveal the existence of a lower bound for protein noise, the noise floor, in highly expressed genes. Its origins are still debated. We propose a minimal model of gene expression in a proliferating bacterial cell population. The model predicts the existence of a noise floor and it semi-quantitatively reproduces the curved shape of the experimental noise vs. mean protein concentration plots. When the cell volume increases in a different manner than does the mean protein copy number, the noise floor level is determined by the cell population’s age structure and by the dependence of the mean protein concentration on cell age. Additionally, the noise floor level may depend on a biological limit for the mean number of bursts in the cell cycle. In that case, the noise floor level depends on the burst size distribution width but it is insensitive to the mean burst size. Our model quantifies the contributions of each of these mechanisms to gene expression noise.

Large Deviation Principle Linking Lineage Statistics to Fitness in Microbial Populations

In exponentially proliferating populations of microbes, the population doubles at a rate less than the average doubling time of a single-cell due to variability at the single-cell level. It is known that the distribution of generation times obtained from a single lineage is, in general, insufficient to determine a population's growth rate. Is there an explicit relationship between observables obtained from a single lineage and the population growth rate? We show that a population's growth rate can be represented in terms of averages over isolated lineages. This lineage representation is related to a large deviation principle that is a generic feature of exponentially proliferating populations. Due to the large deviation structure of growing populations, the number of lineages needed to obtain an accurate estimate of the growth rate depends exponentially on the duration of the lineages, leading to a nonmonotonic convergence of the estimate, which we verify in both synthetic and experimental data sets.

Cell-size regulation in budding yeast does not depend on linear accumulation of Whi5

Cells must couple cell-cycle progress to their growth rate to restrict the spread of cell sizes present throughout a population. Linear, rather than exponential, accumulation of Whi5, was proposed to provide this coordination by causing a higher Whi5 concentration in cells born at a smaller size. We tested this model using the inducible GAL1 promoter to make the Whi5 concentration independent of cell size. At an expression level that equalizes the mean cell size with that of wild-type cells, the size distributions of cells with galactose-induced Whi5 expression and wild-type cells are indistinguishable. Fluorescence microscopy confirms that the endogenous and GAL1 promoters produce different relationships between Whi5 concentration and cell volume without diminishing size control in the G1 phase. We also expressed Cln3 from the GAL1 promoter, finding that the spread in cell sizes for an asynchronous population is unaffected by this perturbation. Our findings indicate that size control in budding yeast does not fundamentally originate from the linear accumulation of Whi5, contradicting a previous claim and demonstrating the need for further models of cell-cycle regulation to explain how cell size controls passage through Start.

A Mechanistic Model of the Regulation of Division Timing by the Circadian Clock in Cyanobacteria

The cyanobacterium Synechococcus elongatus possesses a circadian clock in the form of a group of proteins whose concentrations and phosphorylation states oscillate with daily periodicity under constant conditions. The circadian clock regulates the cell cycle such that the timing of the cell divisions is biased toward certain times during the circadian period, but the mechanism underlying this phenomenon remains unclear. Here, we propose a mechanism in which a protein limiting for division accumulates at a rate proportional to the cell volume growth and is modulated by the clock. This "modulated rate" model, in which the clock signal is integrated over time to affect division timing, differs fundamentally from the previously proposed "gating" concept, in which the clock is assumed to suppress divisions during a specific time window. We found that although both models can capture the single-cell statistics of division timing in S. elongatus, only the modulated rate model robustly places divisions away from darkness during changes in the environment. Moreover, within the framework of the modulated rate model, existing experiments on S. elongatus are consistent with the simple mechanism that division timing is regulated by the accumulation of a division limiting protein in a phase with genes whose activity peaks at dusk.

Correlation between protein concentration and bacterial cell size can reveal mechanisms of gene expression

Classically, gene expression is modeled as a chemical process with reaction rates dependent on the concentration of the reactants (typically, DNA loci, plasmids, RNA, enzymes, etc). Other variables like cell size are in general ignored. Size dynamics can become an important variable due to the low number of many of these reactants, imperfectly symmetric cell partitioning and molecule segregation. In this work we measure the correlation between size and protein concentration by observing the gene expression of the RpOD gene from a low-copy plasmid in Escherichia coli during balanced growth in different media. A positive correlation was found, and we used it to examine possible models of cell size dynamics and plasmid replication. We implemented a previously developed model describing the full gene expression process including transcription, translation, loci replication, cell division and molecule segregation. By comparing with the observed correlation, we determine that the transcription rate must be proportional to the size times the number of plasmids. We discuss how fluctuations in plasmid segregation, due to the low copy number, can impose limits in this correlation.

The interplay of phenotypic variability and fitness in finite microbial populations

In isogenic microbial populations, phenotypic variability is generated by a combination of stochastic mechanisms, such as gene expression, and deterministic factors, such as asymmetric segregation of cell volume. Here we address the question: how does phenotypic variability of a microbial population affect its fitness? While this question has previously been studied for exponentially growing populations, the situation when the population size is kept fixed has received much less attention, despite its relevance to many natural scenarios. We show that the outcome of competition between multiple microbial species can be determined from the distribution of phenotypes in the culture using a generalization of the well-known Euler-Lotka equation, which relates the steady-state distribution of phenotypes to the population growth rate. We derive a generalization of the Euler-Lotka equation for finite cultures, which relates the distribution of phenotypes among cells in the culture to the exponential growth rate. Our analysis reveals that in order to predict fitness from phenotypes, it is important to understand how distributions of phenotypes obtained from different subsets of the genealogical history of a population are related. To this end, we derive a mapping between the various ways of sampling phenotypes in a finite population and show how to obtain the equivalent distributions from an exponentially growing culture. Finally, we use this mapping to show that species with higher growth rates in exponential growth conditions will have a competitive advantage in the finite culture.

Hidden long-range memories of growth and cycle speed correlate cell cycles in lineage trees

Cell heterogeneity may be caused by stochastic or deterministic effects. The inheritance of regulators through cell division is a key deterministic force, but identifying inheritance effects in a systematic manner has been challenging. Here, we measure and analyze cell cycles in deep lineage trees of human cancer cells and mouse embryonic stem cells and develop a statistical framework to infer underlying rules of inheritance. The observed long-range intra-generational correlations in cell-cycle duration, up to second cousins, seem paradoxical because ancestral correlations decay rapidly. However, this correlation pattern is naturally explained by the inheritance of both cell size and cell-cycle speed over several generations, provided that cell growth and division are coupled through a minimum-size checkpoint. This model correctly predicts the effects of inhibiting cell growth or cycle progression. In sum, we show how fluctuations of cell cycles across lineage trees help in understanding the coordination of cell growth and division.

Efficient computation of stochastic cell-size transient dynamics

BACKGROUND

How small, fast-growing bacteria ensure tight cell-size distributions remains elusive. High-throughput measurement techniques have propelled efforts to build modeling tools that help to shed light on the relationships between cell size, growth and cycle progression. Most proposed models describe cell division as a discrete map between size at birth and size at division with stochastic fluctuations assumed. However, such models underestimate the role of cell size transient dynamics by excluding them.

RESULTS

We propose an efficient approach for estimation of cell size transient dynamics. Our technique approximates the transient size distribution and statistical moment dynamics of exponential growing cells following an adder strategy with arbitrary precision.

CONCLUSIONS

We approximate, up to arbitrary precision, the distribution of division times and size across time for the adder strategy in rod-shaped bacteria cells. Our approach is able to compute statistical moments like mean size and its variance from such distributions efficiently, showing close match with numerical simulations. Additionally, we observed that these distributions have periodic properties. Our approach further might shed light on the mechanisms behind gene product homeostasis.

Cell Size Regulation Induces Sustained Oscillations in the Population Growth Rate

We study the effect of correlations in generation times on the dynamics of population growth of microorganisms. We show that any nonzero correlation that is due to cell-size regulation, no matter how small, induces long-term oscillations in the population growth rate. The population only reaches its steady state when we include the often-neglected variability in the growth rates of individual cells. We discover that the relaxation timescale of the population to its steady state is determined by the distribution of single-cell growth rates and is surprisingly independent of details of the division process such as the noise in the timing of division and the mechanism of cell-size regulation. We validate the predictions of our model using existing experimental data and propose an experimental method to measure single-cell growth variability by observing how long it takes for the population to reach its steady state or balanced growth.

Optimal Segregation of Proteins: Phase Transitions and Symmetry Breaking

Asymmetric segregation of key proteins at cell division-be it a beneficial or deleterious protein-is ubiquitous in unicellular organisms and often considered as an evolved trait to increase fitness in a stressed environment. Here, we provide a general framework to describe the evolutionary origin of this asymmetric segregation. We compute the population fitness as a function of the protein segregation asymmetry a, and show that the value of a which optimizes the population growth manifests a phase transition between symmetric and asymmetric partitioning phases. Surprisingly, the nature of phase transition is different for the case of beneficial proteins as opposed to deleterious proteins: a smooth (second order) transition from purely symmetric to asymmetric segregation is found in the former, while a sharp transition occurs in the latter. Our study elucidates the optimization problem faced by evolution in the context of protein segregation, and motivates further investigation of asymmetric protein segregation in biological systems.

Homeostasis of protein and mRNA concentrations in growing cells

Many experiments show that the numbers of mRNA and protein are proportional to the cell volume in growing cells. However, models of stochastic gene expression often assume constant transcription rate per gene and constant translation rate per mRNA, which are incompatible with these experiments. Here, we construct a minimal gene expression model to fill this gap. Assuming ribosomes and RNA polymerases are limiting in gene expression, we show that the numbers of proteins and mRNAs both grow exponentially during the cell cycle and that the concentrations of all mRNAs and proteins achieve cellular homeostasis; the competition between genes for the RNA polymerases makes the transcription rate independent of the genome number. Furthermore, by extending the model to situations in which DNA (mRNA) can be saturated by RNA polymerases (ribosomes) and becomes limiting, we predict a transition from exponential to linear growth of cell volume as the protein-to-DNA ratio increases.

The Empirical Fluctuation Pattern of E. coli Division Control

In physics, it is customary to represent the fluctuations of a stochastic system at steady state in terms of linear response to small random perturbations. Previous work has shown that the same framework describes effectively the trade-off between cell-to-cell variability and correction in the control of cell division of single E. coli cells. However, previous analyses were motivated by specific models and limited to a subset of the measured variables. For example, most analyses neglected the role of growth rate variability. Here, we take a comprehensive approach and consider several sets of available data from both microcolonies and microfluidic devices in different growth conditions. We evaluate all the coupling coefficients between the three main measured variables (interdivision times, cell sizes and individual-cell growth rates). The linear-response framework correctly predicts consistency relations between a priori independent experimental measurements, which confirms its validity. Additionally, the couplings between the cell-specific growth rate and the other variables are typically non zero. Finally, we use the framework to detect signatures of mechanisms in experimental data involving growth rate fluctuations, finding that (1) noise-generating coupling between size and growth rate is a consequence of inter-generation growth rate correlations and (2) the correlation patterns agree with a near-adder model where the added size has a dependence on the single-cell growth rate. Our findings define relevant constraints that any theoretical description should reproduce, and will help future studies aiming to falsify some of the competing models of the cell cycle existing today in the literature.

Coordination of Growth, Chromosome Replication/Segregation, and Cell Division in E. coli

Bacterial cells growing in steady state maintain a 1:1:1 relationship between an appropriate mass increase, a round of DNA replication plus sister chromosome segregation, and cell division. This is accomplished without the cell cycle engine found in eukaryotic cells. We propose here a formal logic, and an accompanying mechanism, for how such coordination could be provided in E. coli. Completion of chromosomal and divisome-related events would lead, interactively, to a “progression control complex” (PCC) which provides integrated physical coupling between sister terminus regions and the nascent septum. When a cell has both (i) achieved a sufficient mass increase, and (ii) the PCC has developed, a conformational change in the PCC occurs. This change results in “progression permission,” which triggers both onset of cell division and release of terminus regions. Release of the terminus region, in turn, directly enables a next round of replication initiation via physical changes transmitted through the nucleoid. Division and initiation are then implemented, each at its own rate and timing, according to conditions present. Importantly: (i) the limiting step for progression permission may be either completion of the growth requirement or the chromosome/divisome processes required for assembly of the PCC; and, (ii) the outcome of the proposed process is granting of permission to progress, not determination of the absolute or relative timings of downstream events. This basic logic, and the accompanying mechanism, can explain coordination of events in both slow and fast growth conditions; can accommodate diverse variations and perturbations of cellular events; and is compatible with existing mathematical descriptions of the E. coli cell cycle. Also, while our proposition is specifically designed to provide 1:1:1 coordination among basic events on a “per-cell cycle” basis, it is a small step to further envision permission progression is also the target of basic growth rate control. In such a case, the rate of mass accumulation (or its equivalent) would determine the length of the interval between successive permission events and, thus, successive cell divisions and successive replication initiations.