Unification of cell division control strategies through continuous rate models

A Generalized Adder mechanism for Cell Size Homeostasis: Implications for Stochastic Dynamics of Clonal Proliferation

Measurements of cell size dynamics have revealed phenomenological principles by which individual cells control their size across diverse organisms. One of the emerging paradigms of cell size homeostasis is the adder, where the cell cycle duration is established such that the cell size increase from birth to division is independent of the newborn cell size. We provide a mechanistic formulation of the adder considering that cell size follows any arbitrary non-exponential growth law. Our results show that the main requirement to obtain an adder regardless of the growth law (the time derivative of cell size) is that cell cycle regulators are produced at a rate proportional to the growth law and cell division is triggered when these molecules reach a prescribed threshold level. Among the implications of this generalized adder, we investigate fluctuations in the proliferation of single-cell derived colonies. Considering exponential cell size growth, random fluctuations in clonal size show a transient increase and then eventually decay to zero over time (i.e., clonal populations become asymptotically more similar). In contrast, several forms of non-exponential cell size dynamics (with adder-based cell size control) yield qualitatively different results: clonal size fluctuations monotonically increase over time reaching a non-zero value. These results characterize the interplay between cell size homeostasis at the single-cell level and clonal proliferation at the population level, explaining the broad fluctuations in clonal sizes seen in barcoded human cell lines.

Mechanisms of cell size regulation in slow-growing Escherichia coli cells: discriminating models beyond the adder

Under ideal conditions, Escherichia coli cells divide after adding a fixed cell size, a strategy known as the adder. This concept applies to various microbes and is often explained as the division that occurs after a certain number of stages, associated with the accumulation of precursor proteins at a rate proportional to cell size. However, under poor media conditions, E. coli cells exhibit a different size regulation. They are smaller and follow a sizer-like division strategy where the added size is inversely proportional to the size at birth. We explore three potential causes for this deviation: degradation of the precursor protein and two models where the propensity for accumulation depends on the cell size: a nonlinear accumulation rate, and accumulation starting at a threshold size termed the commitment size. These models fit the mean trends but predict different distributions given the birth size. To quantify the precision of the models to explain the data, we used the Akaike information criterion and compared them to open datasets of slow-growing E. coli cells in different media. We found that none of the models alone can consistently explain the data. However, the degradation model better explains the division strategy when cells are larger, whereas size-related models (power-law and commitment size) account for smaller cells. Our methodology proposes a data-based method in which different mechanisms can be tested systematically.

Synergistic and antagonistic effects of deterministic and stochastic cell-cell variations

By diversifying, cells in a clonal population can together overcome the limits of individuals. Diversity in single-cell growth rates allows the population to survive environmental stresses, such as antibiotics, and grow faster than the undiversified population. These functional cell-cell variations can arise stochastically, from noise in biochemical reactions, or deterministically, by asymmetrically distributing damaged components. While each of the mechanisms is well understood, the effect of the combined mechanisms is unclear. To evaluate the contribution of the deterministic component we developed a mathematical model by mapping the growing population to the Ising model. To analyze the combined effects of stochastic and deterministic contributions we introduced the analytical results of the Ising-mapping into an Euler-Lotka framework. Model results, confirmed by simulations and experimental data, show that deterministic cell-cell variations increase near-linearly with stress. As a consequence, we predict that the gain in population doubling time from cell-cell variations is primarily stochastic at low stress but may cross over to deterministic at higher stresses. Furthermore, we find that while the deterministic component minimizes population damage, stochastic variations antagonize this effect. Together our results may help identifying stress-tolerant pathogenic cells and thus inspire novel antibiotic strategies.

Effects of Molecular Noise on Cell Size Control

Cells employ control strategies to maintain a stable size. Dividing at a target size (the "sizer" strategy) is thought to produce the tightest size distribution. However, this result follows from phenomenological models that ignore the molecular mechanisms required to implement the strategy. Here we investigate a simple mechanistic model for exponentially growing cells whose division is triggered at a molecular abundance threshold. We find that size noise inherits the molecular noise and is consequently minimized not by the sizer but by the "adder" strategy, where a cell divides after adding a target amount to its birth size. We derive a lower bound on size noise that agrees with publicly available data from six microfluidic studies on Escherichia coli bacteria.

Coupling Cell Size Regulation and Proliferation Dynamics of C. glutamicum Reveals Cell Division Based on Surface Area

Single cells actively coordinate growth and division to regulate their size, yet how this size homeostasis at the single-cell level propagates over multiple generations to impact clonal expansion remains fundamentally unexplored. Classical timer models for cell proliferation (where the duration of the cell cycle is an independent variable) predict that the stochastic variation in colony size will increase monotonically over time. In stark contrast, implementing size control according to adder strategy (where on average a fixed size added from cell birth to division) leads to colony size variations that eventually decay to zero. While these results assume a fixed size of the colony-initiating progenitor cell, further analysis reveals that the magnitude of the intercolony variation in population number is sensitive to heterogeneity in the initial cell size. We validate these predictions by tracking the growth of isogenic microcolonies of Corynebacterium glutamicum in microfluidic chambers. Approximating their cell shape to a capsule, we observe that the degree of random variability in cell size is different depending on whether the cell size is quantified as per length, surface area, or volume, but size control remains an adder regardless of these size metrics. A comparison of the observed variability in the colony population with the predictions suggests that proliferation matches better with a cell division based on the cell surface. In summary, our integrated mathematical-experimental approach bridges the paradigms of single-cell size regulation and clonal expansion at the population levels. This innovative approach provides elucidation of the mechanisms of size homeostasis from the stochastic dynamics of colony size for rod-shaped microbes.

Mechanisms of Cell Size Regulation in Slow-Growing Escherichia coli Cells: Discriminating Models Beyond the Adder

Under ideal conditions, Escherichia coli cells divide after adding a fixed cell size, a strategy known as the adder. This concept applies to various microbes and is often explained as the division that occurs after a certain number of stages, associated with the accumulation of precursor proteins at a rate proportional to cell size. However, under poor media conditions, E. coli cells exhibit a different size regulation. They are smaller and follow a sizer-like division strategy where the added size is inversely proportional to the size at birth. We explore three potential causes for this deviation: precursor protein degradation, nonlinear accumulation rate, and a threshold size termed the commitment size. These models fit mean trends but predict different distributions given the birth size. To validate these models, we used the Akaike information criterion and compared them to open datasets of slow-growing E. coli cells in different media. the degradation model could explain the division strategy for media where cells are larger, while the commitment size model could account for smaller cells. The power-law model, finally, better fits the data at intermediate regimes.

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.

Characterization of proteome-size scaling by integrative omics reveals mechanisms of proliferation control in cancer

Almost all living cells maintain size uniformity through successive divisions. Proteins that over and underscale with size can act as rheostats, which regulate cell cycle progression. Using a multiomic strategy, we leveraged the heterogeneity of melanoma cell lines to identify peptides, transcripts, and phosphorylation events that differentially scale with cell size. Subscaling proteins are enriched in regulators of the DNA damage response and cell cycle progression, whereas super-scaling proteins included regulators of the cytoskeleton, extracellular matrix, and inflammatory response. Mathematical modeling suggested that decoupling growth and proliferative signaling may facilitate cell cycle entry over senescence in large cells when mitogenic signaling is decreased. Regression analysis reveals that up-regulation of TP53 or CDKN1A/p21CIP1 is characteristic of proliferative cancer cells with senescent-like sizes/proteomes. This study provides one of the first demonstrations of size-scaling phenomena in cancer and how morphology influences the chemistry of the cell.

Analytical cell size distribution: lineage-population bias and parameter inference

We derive analytical steady-state cell size distributions for size-controlled cells in single-lineage experiments, such as the mother machine, which are fundamentally different from batch cultures where populations of cells grow freely. For exponential single-cell growth, characterizing most bacteria, the lineage-population bias is obtained explicitly. In addition, if volume is evenly split between the daughter cells at division, we show that cells are on average smaller in populations than in lineages. For more general power-law growth rates and deterministic volume partitioning, both symmetric and asymmetric, we derive the exact lineage distribution. This solution is in good agreement with Escherichia coli mother machine data and can be used to infer cell cycle parameters such as the strength of the size control and the asymmetry of the division. When introducing stochastic volume partitioning, we derive the large-size and small-size tails of the lineage distribution and show that the lineage-population bias only depends on the single-cell growth rate. These asymptotic behaviours are extended to the adder model of cell size control. When considering noisy single-cell growth rate, we derive the large-size lineage and population distributions. Finally, we show that introducing noise, either on the volume partitioning or on the single-cell growth rate, can cancel the lineage-population bias.

Microfluidics for long-term single-cell time-lapse microscopy: Advances and applications

Cells are inherently dynamic, whether they are responding to environmental conditions or simply at equilibrium, with biomolecules constantly being made and destroyed. Due to their small volumes, the chemical reactions inside cells are stochastic, such that genetically identical cells display heterogeneous behaviors and gene expression profiles. Studying these dynamic processes is challenging, but the development of microfluidic methods enabling the tracking of individual prokaryotic cells with microscopy over long time periods under controlled growth conditions has led to many discoveries. This review focuses on the recent developments of one such microfluidic device nicknamed the mother machine. We overview the original device design, experimental setup, and challenges associated with this platform. We then describe recent methods for analyzing experiments using automated image segmentation and tracking. We further discuss modifications to the experimental setup that allow for time-varying environmental control, replicating batch culture conditions, cell screening based on their dynamic behaviors, and to accommodate a variety of microbial species. Finally, this review highlights the discoveries enabled by this technology in diverse fields, such as cell-size control, genetic mutations, cellular aging, and synthetic biology.

Concentration fluctuations in growing and dividing cells: Insights into the emergence of concentration homeostasis

Intracellular reaction rates depend on concentrations and hence their levels are often regulated. However classical models of stochastic gene expression lack a cell size description and cannot be used to predict noise in concentrations. Here, we construct a model of gene product dynamics that includes a description of cell growth, cell division, size-dependent gene expression, gene dosage compensation, and size control mechanisms that can vary with the cell cycle phase. We obtain expressions for the approximate distributions and power spectra of concentration fluctuations which lead to insight into the emergence of concentration homeostasis. We find that (i) the conditions necessary to suppress cell division-induced concentration oscillations are difficult to achieve; (ii) mRNA concentration and number distributions can have different number of modes; (iii) two-layer size control strategies such as sizer-timer or adder-timer are ideal because they maintain constant mean concentrations whilst minimising concentration noise; (iv) accurate concentration homeostasis requires a fine tuning of dosage compensation, replication timing, and size-dependent gene expression; (v) deviations from perfect concentration homeostasis show up as deviations of the concentration distribution from a gamma distribution. Some of these predictions are confirmed using data for E. coli, fission yeast, and budding yeast.

Characterizing non-exponential growth and bimodal cell size distributions in fission yeast: An analytical approach

Unlike many single-celled organisms, the growth of fission yeast cells within a cell cycle is not exponential. It is rather characterized by three distinct phases (elongation, septation, and reshaping), each with a different growth rate. Experiments also showed that the distribution of cell size in a lineage can be bimodal, unlike the unimodal distributions measured for the bacterium Escherichia coli. Here we construct a detailed stochastic model of cell size dynamics in fission yeast. The theory leads to analytic expressions for the cell size and the birth size distributions, and explains the origin of bimodality seen in experiments. In particular, our theory shows that the left peak in the bimodal distribution is associated with cells in the elongation phase, while the right peak is due to cells in the septation and reshaping phases. We show that the size control strategy, the variability in the added size during a cell cycle, and the fraction of time spent in each of the three cell growth phases have a strong bearing on the shape of the cell size distribution. Furthermore, we infer all the parameters of our model by matching the theoretical cell size and birth size distributions to those from experimental single-cell time-course data for seven different growth conditions. Our method provides a much more accurate means of determining the size control strategy (timer, adder or sizer) than the standard method based on the slope of the best linear fit between the birth and division sizes. We also show that the variability in added size and the strength of size control in fission yeast depend weakly on the temperature but strongly on the culture medium. More importantly, we find that stronger size homeostasis and larger added size variability are required for fission yeast to adapt to unfavorable environmental conditions.

Advances in microscopy enable us to follow single cells over long timescales from which we can understand how their size varies with time and the nature of innate strategies developed to control cell size. These data show that in many cell types, growth is exponential and the distribution of cell size has one peak, namely there is a single characteristic cell size. However data for fission yeast show remarkable differences: growth is non-exponential and the distribution of cell sizes has two peaks, corresponding to different growth phases. Here we construct a detailed stochastic mathematical model of this organism; by solving the model analytically, we show that it is able to predict the two peaked distributions of cell size seen in data and provide an explanation for each peak in terms of various growth phases of the single-celled organism. Furthermore, by fitting the model to the data, we infer values for the rates of all microscopic processes in our model. This method is shown to provide a much more reliable inference than current methods and shed light on how the strategy used by fission yeast cells to control their size varies with external conditions.

Quantitative Examination of Five Stochastic Cell-Cycle and Cell-Size Control Models for Escherichia coli and Bacillus subtilis

We examine five quantitative models of the cell-cycle and cell-size control in Escherichia coli and Bacillus subtilis that have been proposed over the last decade to explain single-cell experimental data generated with high-throughput methods. After presenting the statistical properties of these models, we test their predictions against experimental data. Based on simple calculations of the defining correlations in each model, we first dismiss the stochastic Helmstetter-Cooper model and the Initiation Adder model, and show that both the Replication Double Adder (RDA) and the Independent Double Adder (IDA) model are more consistent with the data than the other models. We then apply a recently proposed statistical analysis method and obtain that the IDA model is the most likely model of the cell cycle. By showing that the RDA model is fundamentally inconsistent with size convergence by the adder principle, we conclude that the IDA model is most consistent with the data and the biology of bacterial cell-cycle and cell-size control. Mechanistically, the Independent Adder Model is equivalent to two biological principles: (i) balanced biosynthesis of the cell-cycle proteins, and (ii) their accumulation to a respective threshold number to trigger initiation and division.

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 distribution of lineage data: analytic results and parameter inference

Recent advances in single-cell technologies have enabled time-resolved measurements of the cell size over several cell cycles. These data encode information on how cells correct size aberrations so that they do not grow abnormally large or small. Here, we formulate a piecewise deterministic Markov model describing the evolution of the cell size over many generations, for all three cell size homeostasis strategies (timer, sizer, and adder). The model is solved to obtain an analytical expression for the non-Gaussian cell size distribution in a cell lineage; the theory is used to understand how the shape of the distribution is influenced by the parameters controlling the dynamics of the cell cycle and by the choice of cell tracking protocol. The theoretical cell size distribution is found to provide an excellent match to the experimental cell size distribution of E. coli lineage data collected under various growth conditions.

The Ethanologenic Bacterium Zymomonas mobilis Divides Asymmetrically and Exhibits Heterogeneity in DNA Content

The alphaproteobacterium Zymomonas mobilis exhibits extreme ethanologenic physiology, making this species a promising biofuel producer. Numerous studies have investigated its biology relevant to industrial applications and mostly at the population level. However, the organization of single cells in this industrially important polyploid species has been largely uncharacterized. In the present study, we characterized basic cellular behavior of Z. mobilis strain Zm6 under anaerobic conditions at the single-cell level. We observed that growing Z. mobilis cells often divided at a nonmidcell position, which contributed to variant cell size at birth. However, the cell size variance was regulated by a modulation of cell cycle span, mediated by a correlation of bacterial tubulin homologue FtsZ ring accumulation with cell growth. The Z. mobilis culture also exhibited heterogeneous cellular DNA content among individual cells, which might have been caused by asynchronous replication of chromosome that was not coordinated with cell growth. Furthermore, slightly angled divisions might have resulted in temporary curvatures of attached Z. mobilis cells. Overall, the present study uncovers a novel bacterial cell organization in Z. mobilis IMPORTANCE With increasing environmental concerns about the use of fossil fuels, development of a sustainable biofuel production platform has been attracting significant public attention. Ethanologenic Z. mobilis species are endowed with an efficient ethanol fermentation capacity that surpasses, in several respects, that of baker's yeast (Saccharomyces cerevisiae), the most-used microorganism for ethanol production. For development of a Z. mobilis culture-based biorefinery, an investigation of its uncharacterized cell biology is important, because bacterial cellular organization and metabolism are closely associated with each other in a single cell compartment. In addition, the current work demonstrates that the polyploid bacterium Z. mobilis exhibits a distinctive mode of bacterial cell organization, likely reflecting its unique metabolism that does not prioritize incorporation of nutrients for cell growth. Thus, another significant result of this work is to advance our general understanding in the diversity of bacterial cell architecture.

Modeling homeostasis mechanisms that set the target cell size

How organisms maintain cell size homeostasis is a long-standing problem that remains unresolved, especially in multicellular organisms. Recent experiments in diverse animal cell types demonstrate that within a cell population, cellular proliferation is low for small and large cells, but high at intermediate sizes. Here we use mathematical models to explore size-control strategies that drive such a non-monotonic profile resulting in the proliferation capacity being maximized at a target cell size. Our analysis reveals that most models of size control yield proliferation capacities that vary monotonically with cell size, and non-monotonicity requires two key mechanisms: (1) the growth rate decreases with increasing size for excessively large cells; and (2) cell division occurs as per the Adder model (division is triggered upon adding a fixed size from birth), or a Sizer-Adder combination. Consistent with theory, Jurkat T cell growth rates increase with size for small cells, but decrease with size for large cells. In summary, our models show that regulation of both growth and cell-division timing is necessary for size control in animal cells, and this joint mechanism leads to a target cell size where cellular proliferation capacity is maximized.

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.

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.