Concerted control of Escherichia coli cell division

Evolutionary Advantage of Cell Size Control

We analyze the advantage of cell size control strategies in growing populations under mortality constraints and show that growth-dependent mortality can select for accurate size control. We determine how mortality, noise, and nongenetic heritability of cell size impact long-term population growth. We derive an analytical expression for the optimal cell size. We demonstrate that size heritability enables selection to act on the distribution of cell sizes in a population to avoid viability thresholds and adapt to size- and growth-dependent mortality landscapes.

Microscopic origin of the spatial and temporal precision in biological systems

All living systems display remarkable spatial and temporal precision, despite operating in intrinsically fluctuating environments. It is even more surprising given that biological phenomena are regulated by multiple chemical reactions that are also random. Although the underlying molecular mechanisms of surprisingly high precision in biology remain not well understood, a novel theoretical picture that relies on the coupling of relevant stochastic processes has recently been proposed and applied to explain different phenomena. To illustrate this approach, in this review, we discuss two systems that exhibit precision control: spatial regulation in bacterial cell size and temporal regulation in the timing of cell lysis by λ bacteriophage. In cell-size regulation, it is argued that a balance between stochastic cell growth and cell division processes leads to a narrow distribution of cell sizes. In cell lysis, it is shown that precise timing is due to the coupling of holin protein accumulation and the breakage of the cellular membrane. The stochastic coupling framework also allows us to explicitly evaluate dynamic properties for both biological systems, eliminating the need to utilize the phenomenological concept of thresholds. Excellent agreement with experimental observations is observed, supporting the proposed theoretical ideas. These observations also suggest that the stochastic coupling method captures the important aspects of molecular mechanisms of precise cellular regulation, providing a powerful new tool for more advanced investigations of complex biological phenomena.

From noisy cell size control to population growth: When variability can be beneficial

Single-cell experiments revealed substantial variability in generation times, growth rates, but also in birth and division sizes between genetically identical cells. Understanding how these fluctuations determine the fitness of the population, i.e., its growth rate, is necessary in any quantitative theory of evolution. Here, we develop a biologically relevant model which accounts for the stochasticity in single-cell growth rates, birth sizes, and division sizes. We derive expressions for the population growth rate and mean birth size in the population in terms of single-cell fluctuations. Allowing division sizes to fluctuate reveals how the mechanism of cell size control (timer, sizer, and adder) influences population growth. Surprisingly, we find that fluctuations in single-cell growth rates can be beneficial for population growth when slow-growing cells tend to divide at smaller sizes than fast-growing cells. Our framework is not limited to exponentially growing cells like Escherichia coli, and we derive similar expressions for cells with linear and bilinear growth laws, such as Mycobacterium tuberculosis and fission yeast Schizosaccharomyces pombe, respectively.

Intracellular competition shapes plasmid population dynamics

Conflicts between levels of biological organization are central to evolution, from populations of multicellular organisms to selfish genetic elements in microbes. Plasmids are extrachromosomal, self-replicating genetic elements that underlie much of the evolutionary flexibility of bacteria. Evolving plasmids face selective pressures on their hosts, but also compete within the cell for replication, making them an ideal system for studying the joint dynamics of multilevel selection. While theory indicates that within-cell selection should matter for plasmid evolution, experimental measurement of within-cell plasmid fitness and its consequences has remained elusive. Here we measure the within-cell fitness of competing plasmids and characterize drift and selective dynamics. We achieve this by the controlled splitting of synthetic plasmid dimers to create balanced competition experiments. We find that incompatible plasmids co-occur for longer than expected due to methylation-based plasmid eclipsing. During this period of co-occurrence, less transcriptionally active plasmids display a within-cell selective advantage over their competing plasmids, leading to preferential fixation of silent plasmids. When the transcribed gene is beneficial to the cell, for example an antibiotic resistance gene, there is a cell-plasmid fitness tradeoff mediated by the dominance of the beneficial trait. Surprisingly, more dominant plasmid-encoded traits are less likely to fix but more likely to initially invade than less dominant traits. Taken together, our results show that plasmid evolution is driven by dynamics at two levels, with a transient, but critical, contribution of within-cell fitness.

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.

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.

The roles of nucleoid-associated proteins and topoisomerases in chromosome structure, strand segregation, and the generation of phenotypic heterogeneity in bacteria

How to adapt to a changing environment is a fundamental, recurrent problem confronting cells. One solution is for cells to organize their constituents into a limited number of spatially extended, functionally relevant, macromolecular assemblies or hyperstructures, and then to segregate these hyperstructures asymmetrically into daughter cells. This asymmetric segregation becomes a particularly powerful way of generating a coherent phenotypic diversity when the segregation of certain hyperstructures is with only one of the parental DNA strands and when this pattern of segregation continues over successive generations. Candidate hyperstructures for such asymmetric segregation in prokaryotes include those containing the nucleoid-associated proteins (NAPs) and the topoisomerases. Another solution to the problem of creating a coherent phenotypic diversity is by creating a growth-environment-dependent gradient of supercoiling generated along the replication origin-to-terminus axis of the bacterial chromosome. This gradient is modulated by transcription, NAPs, and topoisomerases. Here, we focus primarily on two topoisomerases, TopoIV and DNA gyrase in Escherichia coli, on three of its NAPs (H-NS, HU, and IHF), and on the single-stranded binding protein, SSB. We propose that the combination of supercoiling-gradient-dependent and strand-segregation-dependent topoisomerase activities result in significant differences in the supercoiling of daughter chromosomes, and hence in the phenotypes of daughter cells.

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.

Implications of differential size-scaling of cell-cycle regulators on cell size homeostasis

Accurate timing of division and size homeostasis is crucial for cells. A potential mechanism for cells to decide the timing of division is the differential scaling of regulatory protein copy numbers with cell size. However, it remains unclear whether such a mechanism can lead to robust growth and division, and how the scaling behaviors of regulatory proteins influence the cell size distribution. Here we study a mathematical model combining gene expression and cell growth, in which the cell-cycle activators scale superlinearly with cell size while the inhibitors scale sublinearly. The cell divides once the ratio of their concentrations reaches a threshold value. We find that the cell can robustly grow and divide within a finite range of the threshold value with the cell size proportional to the ploidy. In a stochastic version of the model, the cell size at division is uncorrelated with that at birth. Also, the more differential the cell-size scaling of the cell-cycle regulators is, the narrower the cell-size distribution is. Intriguingly, our model with multiple regulators rationalizes the observation that after the deletion of a single regulator, the coefficient of variation of cell size remains roughly the same though the average cell size changes significantly. Our work reveals that the differential scaling of cell-cycle regulators provides a robust mechanism of cell size control.

Dynamic proteome trade-offs regulate bacterial cell size and growth in fluctuating nutrient environments

Bacteria dynamically regulate cell size and growth to thrive in changing environments. While previous studies have characterized bacterial growth physiology at steady-state, a quantitative understanding of bacterial physiology in time-varying environments is lacking. Here we develop a quantitative theory connecting bacterial growth and division rates to proteome allocation in time-varying nutrient environments. In such environments, cell size and growth are regulated by trade-offs between prioritization of biomass accumulation or division, resulting in decoupling of single-cell growth rate from population growth rate. Specifically, bacteria transiently prioritize biomass accumulation over production of division machinery during nutrient upshifts, while prioritizing division over growth during downshifts. When subjected to pulsatile nutrient concentration, we find that bacteria exhibit a transient memory of previous metabolic states due to the slow dynamics of proteome reallocation. This allows for faster adaptation to previously seen environments and results in division control which is dependent on the time-profile of fluctuations.

Evolutionary scaling of maximum growth rate with organism size

Data from nearly 1000 species reveal the upper bound to rates of biomass production achievable by natural selection across the Tree of Life. For heterotrophs, maximum growth rates scale positively with organism size in bacteria but negatively in eukaryotes, whereas for phototrophs, the scaling is negligible for cyanobacteria and weakly negative for eukaryotes. These results have significant implications for understanding the bioenergetic consequences of the transition from prokaryotes to eukaryotes, and of the expansion of some groups of the latter into multicellularity. The magnitudes of the scaling coefficients for eukaryotes are significantly lower than expected under any proposed physical-constraint model. Supported by genomic, bioenergetic, and population-genetic data and theory, an alternative hypothesis for the observed negative scaling in eukaryotes postulates that growth-diminishing mutations with small effects passively accumulate with increasing organism size as a consequence of associated increases in the power of random genetic drift. In contrast, conditional on the structural and functional features of ribosomes, natural selection has been able to promote bacteria with the fastest possible growth rates, implying minimal conflicts with both bioenergetic constraints and random genetic drift. If this extension of the drift-barrier hypothesis is correct, the interpretations of comparative studies of biological traits that have traditionally ignored differences in population-genetic environments will require revisiting.

Cellular resource allocation strategies for cell size and shape control in bacteria

Bacteria are highly adaptive microorganisms that thrive in a wide range of growth conditions via changes in cell morphologies and macromolecular composition. How bacterial morphologies are regulated in diverse environmental conditions is a long-standing question. Regulation of cell size and shape implies control mechanisms that couple the growth and division of bacteria to their cellular environment and macromolecular composition. In the past decade, simple quantitative laws have emerged that connect cell growth to proteomic composition and the nutrient availability. However, the relationships between cell size, shape, and growth physiology remain challenging to disentangle and unifying models are lacking. In this review, we focus on regulatory models of cell size control that reveal the connections between bacterial cell morphology and growth physiology. In particular, we discuss how changes in nutrient conditions and translational perturbations regulate the cell size, growth rate, and proteome composition. Integrating quantitative models with experimental data, we identify the physiological principles of bacterial size regulation, and discuss the optimization strategies of cellular resource allocation for size control.

Cell growth and the cell cycle: New insights about persistent questions

Before a cell divides into two daughter cells, it typically doubles not only its DNA, but also its mass. Numerous studies in cells ranging from yeast to mammals have shown that cellular growth, stimulated by nutrients and/or growth factor signaling, is a prerequisite for cell cycle progression in most types of cells. The textbook view of growth-regulated cell cycles is that growth signaling activates the transcription of G1 Cyclin genes to induce cell proliferation, and also stimulates anabolic metabolism and cell growth in parallel. However, genetic knockout tests in model organisms indicate that this is not the whole story, and new studies show that additional, "smarter" mechanisms help to coordinate the cell cycle with growth itself. Here we summarize recent advances in this field, and discuss current models in which growth signaling regulates cell proliferation by targeting core cell cycle regulators via non-transcriptional mechanisms.

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.

The possible modes of microbial reproduction are fundamentally restricted by distribution of mass between parent and offspring

Cells and simple cell colonies reproduce by fragmenting their bodies into pieces. Produced newborns need to grow before they can reproduce again. How big a cell or a cell colony should grow? How many offspring should be produced? Should they be of equal size or diverse? We show that the simple fact that the immediate mass of offspring cannot exceed the mass of parents restricts possible answers to these questions. For example, our theory states that, when mass is conserved in the course of fragmentation, the evolutionarily optimal reproduction mode is fragmentation into exactly two, typically equal, parts. Our theory also shows conditions which promote evolution of asymmetric division or fragmentation into multiple pieces.

Multiple modes of asexual reproduction are observed among microbial organisms in natural populations. These modes are not only subject to evolution, but may drive evolutionary competition directly through their impact on population growth rates. The most prominent transition between two such modes is the one from unicellularity to multicellularity. We present a model of the evolution of reproduction modes, where a parent organism fragments into smaller parts. While the size of an organism at fragmentation, the number of offspring, and their sizes may vary a lot, the combined mass of fragments is limited by the mass of the parent organism. We found that mass conservation can fundamentally limit the number of possible reproduction modes. This has important direct implications for microbial life: For unicellular species, the interplay between cell shape and kinetics of the cell growth implies that the largest and the smallest possible cells should be rod shaped rather than spherical. For primitive multicellular species, these considerations can explain why rosette cell colonies evolved a mechanistically complex binary split reproduction. Finally, we show that the loss of organism mass during sporulation can explain the macroscopic sizes of the formally unicellular microorganism Myxomycetes plasmodium. Our findings demonstrate that a number of seemingly unconnected phenomena observed in unrelated species may be different manifestations of the same underlying process.

Multiple timescales in bacterial growth homeostasis

In balanced exponential growth, bacteria maintain many properties statistically stable for a long time: cell size, cell cycle time, and more. As these are strongly coupled variables, it is not a-priori obvious which are directly regulated and which are stabilized through interactions. Here, we address this problem by separating timescales in bacterial single-cell dynamics. Disentangling homeostatic set points from fluctuations around them reveals that some variables, such as growth-rate, cell size and cycle time, are "sloppy" with highly volatile set points. Quantifying the relative contribution of environmental and internal sources, we find that sloppiness is primarily driven by the environment. Other variables such as fold-change define "stiff" combinations of coupled variables with robust set points. These results are manifested geometrically as a control manifold in the space of variables: set points span a wide range of values within the manifold, whereas out-of-manifold deviations are constrained. Our work offers a generalizable data-driven approach for identifying control variables in a multidimensional system.

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.

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.

Analytics and visualization tools to characterize single-cell stochasticity using bacterial single-cell movie cytometry data

Background

Time-lapse microscopy live-cell imaging is essential for studying the evolution of bacterial communities at single-cell resolution. It allows capturing detailed information about the morphology, gene expression, and spatial characteristics of individual cells at every time instance of the imaging experiment. The image analysis of bacterial "single-cell movies" (videos) generates big data in the form of multidimensional time series of measured bacterial attributes. If properly analyzed, these datasets can help us decipher the bacterial communities' growth dynamics and identify the sources and potential functional role of intra- and inter-subpopulation heterogeneity. Recent research has highlighted the importance of investigating the role of biological "noise" in gene regulation, cell growth, cell division, etc. Single-cell analytics of complex single-cell movie datasets, capturing the interaction of multiple micro-colonies with thousands of cells, can shed light on essential phenomena for human health, such as the competition of pathogens and benign microbiome cells, the emergence of dormant cells (“persisters”), the formation of biofilms under different stress conditions, etc. However, highly accurate and automated bacterial bioimage analysis and single-cell analytics methods remain elusive, even though they are required before we can routinely exploit the plethora of data that single-cell movies generate.

Results

We present visualization and single-cell analytics using R (ViSCAR), a set of methods and corresponding functions, to visually explore and correlate single-cell attributes generated from the image processing of complex bacterial single-cell movies. They can be used to model and visualize the spatiotemporal evolution of attributes at different levels of the microbial community organization (i.e., cell population, colony, generation, etc.), to discover possible epigenetic information transfer across cell generations, infer mathematical and statistical models describing various stochastic phenomena (e.g., cell growth, cell division), and even identify and auto-correct errors introduced unavoidably during the bioimage analysis of a dense movie with thousands of overcrowded cells in the microscope's field of view.

Conclusions

ViSCAR empowers researchers to capture and characterize the stochasticity, uncover the mechanisms leading to cellular phenotypes of interest, and decipher a large heterogeneous microbial communities' dynamic behavior. ViSCAR source code is available from GitLab at https://gitlab.com/ManolakosLab/viscar.

Supplementary Information

The online version contains supplementary material available at 10.1186/s12859-021-04409-9.

Cell Growth Model with Stochastic Gene Expression Helps Understand the Growth Advantage of Metabolic Exchange and Auxotrophy

During cooperative growth, microbes often experience higher fitness by sharing resources via metabolite exchange. How competitive species evolve to cooperate is, however, not known. Moreover, existing models (based on optimization of steady-state resources or fluxes) are often unable to explain the growth advantage for the cooperating species, even for simple reciprocally cross-feeding auxotrophic pairs. We present here an abstract model of cell growth that considers the stochastic burst-like gene expression of biosynthetic pathways of limiting biomass precursor metabolites and directly connect the amount of metabolite produced to cell growth and division, using a "metabolic sizer/adder" rule. Our model recapitulates Monod's law and yields the experimentally observed right-skewed long-tailed distribution of cell doubling times. The model further predicts the growth effect of secretion and uptake of metabolites by linking it to changes in the internal metabolite levels. The model also explains why auxotrophs may grow faster when supplied with the metabolite they cannot produce and why two reciprocally cross-feeding auxotrophs can grow faster than prototrophs. Overall, our framework allows us to predict the growth effect of metabolic interactions in independent microbes and microbial communities, setting up the stage to study the evolution of these interactions. IMPORTANCE Cooperative behaviors are highly prevalent in the wild, but their evolution is not understood. Metabolic flux models can demonstrate the viability of metabolic exchange as cooperative interactions, but steady-state growth models cannot explain why cooperators grow faster. We present a stochastic model that connects growth to the cell's internal metabolite levels and quantifies the growth effect of metabolite exchange and auxotrophy. We show that a reduction in gene expression noise can explain why cells that import metabolites or become auxotrophs can grow faster and why reciprocal cross-feeding of metabolites between complementary auxotrophs allows them to grow faster. Furthermore, our framework can simulate the growth of interacting cells, which will enable us to understand the possible trajectories of the evolution of cooperation in silico.

Limits and Constraints on Mechanisms of Cell-Cycle Regulation Imposed by Cell Size-Homeostasis Measurements

High-throughput imaging has led to an explosion of observations about cell-size homeostasis across the kingdoms of life. Among bacteria, "adder" behavior-in which a constant size increment appears to be added during each cell cycle-is ubiquitous, while various eukaryotes show other size-homeostasis behaviors. Since interactions between cell-cycle progression and growth ultimately determine such behaviors, we developed a general model of cell-cycle regulation. Our analyses reveal a range of scenarios that are plausible but fail to regulate cell size, indicating that mechanisms of cell-cycle regulation are stringently limited by size-control requirements, and possibly why certain cell-cycle features are strongly conserved. Cell-cycle features can play unintuitive roles in altering size-homeostasis behaviors: noisy regulator production can enhance adder behavior, while Whi5-like inhibitor dilutors respond sensitively to perturbations to G2/M control and noisy G1/S checkpoints. Our model thus provides holistic insights into the mechanistic implications of size-homeostasis experimental measurements.

Coordination of gene expression noise with cell size: analytical results for agent-based models of growing cell populations

The chemical master equation and the Gillespie algorithm are widely used to model the reaction kinetics inside living cells. It is thereby assumed that cell growth and division can be modelled through effective dilution reactions and extrinsic noise sources. We here re-examine these paradigms through developing an analytical agent-based framework of growing and dividing cells accompanied by an exact simulation algorithm, which allows us to quantify the dynamics of virtually any intracellular reaction network affected by stochastic cell size control and division noise. We find that the solution of the chemical master equation-including static extrinsic noise-exactly agrees with the agent-based formulation when the network under study exhibits stochastic concentration homeostasis, a novel condition that generalizes concentration homeostasis in deterministic systems to higher order moments and distributions. We illustrate stochastic concentration homeostasis for a range of common gene expression networks. When this condition is not met, we demonstrate by extending the linear noise approximation to agent-based models that the dependence of gene expression noise on cell size can qualitatively deviate from the chemical master equation. Surprisingly, the total noise of the agent-based approach can still be well approximated by extrinsic noise models.

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.

Threshold accumulation of a constitutive protein explains E. coli cell-division behavior in nutrient upshifts

Despite a boost of recent progress in dynamic single-cell measurements and analyses in Escherichia coli, we still lack a mechanistic understanding of the determinants of the decision to divide. Specifically, the debate is open regarding the processes linking growth and chromosome replication to division and on the molecular origin of the observed "adder correlations," whereby cells divide, adding roughly a constant volume independent of their initial volume. In order to gain insight into these questions, we interrogate dynamic size-growth behavior of single cells across nutrient upshifts with a high-precision microfluidic device. We find that the division rate changes quickly after nutrients change, much before growth rate goes to a steady state, and in a way that adder correlations are robustly conserved. Comparison of these data to simple mathematical models falsifies proposed mechanisms, where replication-segregation or septum completions are the limiting step for cell division. Instead, we show that the accumulation of a putative constitutively expressed "P-sector divisor" protein explains the behavior during the shift.

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.

Bacterial Growth Control Mechanisms Inferred from Multivariate Statistical Analysis of Single-Cell Measurements

Analysis of single-cell measurements of bacterial growth and division often relied on testing preconceived models of cell size control mechanisms. Such an approach could limit the scope of data analysis and prevent us from uncovering new information. Here, we take an "agnostic" approach by applying regression methods to multiple simultaneously measured cellular variables, which allow us to infer dependencies among those variables from their apparent correlations. Besides previously observed correlations attributed to particular cell size control mechanisms, we identify dependencies that point to potentially new mechanisms. In particular, cells born smaller than their sisters tend to grow faster and make up for the size difference acquired during division. We also find that sister cells are correlated beyond what single-cell, size-control models predict. These trends are consistently found in repeat experiments, although the dependencies vary quantitatively. Such variation highlights the sensitivity of cell growth to environmental variations and the limitation of currently used experimental setups.

Bacterial cell proliferation: from molecules to cells

Bacterial cell proliferation is highly efficient, both because bacteria grow fast and multiply with a low failure rate. This efficiency is underpinned by the robustness of the cell cycle and its synchronization with cell growth and cytokinesis. Recent advances in bacterial cell biology brought about by single-cell physiology in microfluidic chambers suggest a series of simple phenomenological models at the cellular scale, coupling cell size and growth with the cell cycle. We contrast the apparent simplicity of these mechanisms based on the addition of a constant size between cell cycle events (e.g. two consecutive initiation of DNA replication or cell division) with the complexity of the underlying regulatory networks. Beyond the paradigm of cell cycle checkpoints, the coordination between the DNA and division cycles and cell growth is largely mediated by a wealth of other mechanisms. We propose our perspective on these mechanisms, through the prism of the known crosstalk between DNA replication and segregation, cell division and cell growth or size. We argue that the precise knowledge of these molecular mechanisms is critical to integrate the diverse layers of controls at different time and space scales into synthetic and verifiable models.

Cell Size Is Coordinated with Cell Cycle by Regulating Initiator Protein DnaA in E. coli

Sixty years ago, bacterial cell size was found to be an exponential function of growth rate. Fifty years ago, a more general relationship was proposed, in which cell mass was equal to the initiation mass multiplied by 2 to the power of the ratio of the total time of C and D periods to the doubling time. This relationship has recently been experimentally confirmed by perturbing doubling time, C period, D period, or initiation mass. However, the underlying molecular mechanism remains unclear. Here, we developed a theoretical model for initiator protein DnaA mediating DNA replication initiation in Escherichia coli. We introduced an initiation probability function for competitive binding of DnaA-ATP and DnaA-ADP at oriC. We established a kinetic description of regulatory processes (e.g., expression regulation, titration, inactivation, and reactivation) of DnaA. Cell size as a spatial constraint also participates in the regulation of DnaA. By simulating DnaA kinetics, we obtained a regular DnaA oscillation coordinated with cell cycle and a converged cell size that matches replication initiation frequency to the growth rate. The relationship between the simulated cell size and growth rate, C period, D period, or initiation mass reproduces experimental results. The model also predicts how DnaA number and initiation mass vary with perturbation parameters, comparable with experimental data. The results suggest that 1) when growth rate, C period, or D period changes, the regulation of DnaA determines the invariance of initiation mass; 2) ppGpp inhibition of replication initiation may be important for the growth rate independence of initiation mass because three possible mechanisms therein produce different DnaA dynamics, which is experimentally verifiable; and 3) perturbation of some DnaA regulatory process causes a changing initiation mass or even an abnormal cell cycle. This study may provide clues for concerted control of cell size and cell cycle in synthetic biology.

Microbial metabolic noise

From the time a cell was first placed under the microscope, it became apparent that identifying two clonal cells that "look" identical is extremely challenging. Since then, cell-to-cell differences in shape, size, and protein content have been carefully examined, informing us of the ultimate limits that hinder two cells from occupying an identical phenotypic state. Here, we present recent experimental and computational evidence that similar limits emerge also in cellular metabolism. These limits pertain to stochastic metabolic dynamics and, thus, cell-to-cell metabolic variability, including the resulting adapting benefits. We review these phenomena with a focus on microbial metabolism and conclude with a brief outlook on the potential relationship between metabolic noise and adaptive evolution. This article is categorized under: Metabolic Diseases > Computational Models Metabolic Diseases > Biomedical Engineering.

Artificial modulation of cell width significantly affects the division time of Escherichia coli

Bacterial cells have characteristic spatial and temporal scales. For instance, Escherichia coli, the typical rod-shaped bacteria, always maintains a relatively constant cell width and cell division time. However, whether the external physical perturbation of cell width has an impact on cell division time remains largely unexplored. In this work, we developed two microchannel chips, namely straight channels and ‘necked’ channels, to precisely regulate the width of E. coli cells and to investigate the correlation between cell width and division time of the cells. Our results show that, in the straight channels, the wide cells divide much slower than narrow cells. In the ‘necked’ channels, the cell division is remarkably promoted compared to that in straight channels with the same width. Besides, fluorescence time-lapse microscopy imaging of FtsZ dynamics shows that the cell pre-constriction time is more sensitive to cell width perturbation than cell constriction time. Finally, we revealed a significant anticorrelation between the death rate and the division rate of cell populations with different widths. Our work provides new insights into the correlation between the geometrical property and division time of E. coli cells and sheds new light on the future study of spatial–temporal correlation in cell physiology.

Stochastic Mechanisms of Cell-Size Regulation in Bacteria

How bacteria are able to maintain their sizes remains an open question. It is believed that cells have narrow distributions of sizes as a consequence of a homeostasis that allows bacteria to function at the optimal conditions. Several phenomenological approaches to explain these observations have been presented, but the microscopic origins of the cell-size regulation are still not understood. Here, we propose a new stochastic approach to investigate the molecular mechanisms of maintaining the cell sizes in bacteria. It is argued that the cell-size regulation is a result of coupling of two stochastic processes, cell growth and division, which eliminates the need for introducing the thresholds. Dynamic properties of the system are explicitly evaluated, and it is shown that the model is consistent with the experimentally supported adder principle of the cell-size regulation. In addition, theoretical predictions agree with experimental observations on E. coli bacteria. Theoretical analysis clarifies some important features of bacterial cell growth.

Experimental Realization of Diffusion with Stochastic Resetting

Stochastic resetting is prevalent in natural and man-made systems, giving rise to a long series of nonequilibrium phenomena. Diffusion with stochastic resetting serves as a paradigmatic model to study these phenomena, but the lack of a well-controlled platform by which this process can be studied experimentally has been a major impediment to research in the field. Here, we report the experimental realization of colloidal particle diffusion and resetting via holographic optical tweezers. We provide the first experimental corroboration of central theoretical results and go on to measure the energetic cost of resetting in steady-state and first-passage scenarios. In both cases, we show that this cost cannot be made arbitrarily small because of fundamental constraints on realistic resetting protocols. The methods developed herein open the door to future experimental study of resetting phenomena beyond diffusion.

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.

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.

Fluctuation relations and fitness landscapes of growing cell populations

We construct a pathwise formulation of a growing population of cells, based on two different samplings of lineages within the population, namely the forward and backward samplings. We show that a general symmetry relation, called fluctuation relation relates these two samplings, independently of the model used to generate divisions and growth in the cell population. These relations lead to estimators of the population growth rate, which can be very efficient as we demonstrate by an analysis of a set of mother machine data. These fluctuation relations lead to general and important inequalities between the mean number of divisions and the doubling time of the population. We also study the fitness landscape, a concept based on the two samplings mentioned above, which quantifies the correlations between a phenotypic trait of interest and the number of divisions. We obtain explicit results when the trait is the age or the size, for age and size-controlled models.

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.

Second-order phase transition in phytoplankton trait dynamics

Key traits of unicellular species, such as cell size, often follow scale-free or self-similar distributions, hinting at the possibility of an underlying critical process. However, linking such empirical scaling laws to the critical regime of realistic individual-based model classes is difficult. Here, we reveal new empirical scaling evidence associated with a transition in the population and the chlorophyll dynamics of phytoplankton. We offer a possible explanation for these observations by deriving scaling laws in the vicinity of the critical point of a new universality class of non-local cell growth and division models. This "criticality hypothesis" can be tested through new scaling predictions derived for our model class, for the response of chlorophyll distributions to perturbations. The derived scaling laws may also be generalized to other cellular traits and environmental drivers relevant to phytoplankton ecology.

On the Molecular Mechanisms Regulating Animal Cell Size Homeostasis

Cell size is fundamental to cell physiology because it sets the scale of intracellular geometry, organelles, and biosynthetic processes. In animal cells, size homeostasis is controlled through two phenomenologically distinct mechanisms. First, size-dependent cell cycle progression ensures that smaller cells delay cell cycle progression to accumulate more biomass than larger cells prior to cell division. Second, size-dependent cell growth ensures that larger and smaller cells grow slower per unit mass than more optimally sized cells. This decade has seen dramatic progress in single-cell technologies establishing the diverse phenomena of cell size control in animal cells. Here, we review this recent progress and suggest pathways forward to determine the underlying molecular mechanisms.

Loss of phenotypic inheritance associated with ydcI mutation leads to increased frequency of small, slow persisters in Escherichia coli

Whenever a genetically homogenous population of bacterial cells is exposed to antibiotics, a tiny fraction of cells survives the treatment, the phenomenon known as bacterial persistence [G.L. Hobby et al., Exp. Biol. Med. 50, 281-285 (1942); J. Bigger, The Lancet 244, 497-500 (1944)]. Despite its biomedical relevance, the origin of the phenomenon is still unknown, and as a rare, phenotypically resistant subpopulation, persisters are notoriously hard to study and define. Using computerized tracking we show that persisters are small at birth and slowly replicating. We also determine that the high-persister mutant strain of Escherichia coli, HipQ, is associated with the phenotype of reduced phenotypic inheritance (RPI). We identify the gene responsible for RPI, ydcI, which encodes a transcription factor, and propose a mechanism whereby loss of phenotypic inheritance causes increased frequency of persisters. These results provide insight into the generation and maintenance of phenotypic variation and provide potential targets for the development of therapeutic strategies that tackle persistence in bacterial infections.

Models of protein production along the cell cycle: An investigation of possible sources of noise

In this article, we quantitatively study, through stochastic models, the effects of several intracellular phenomena, such as cell volume growth, cell division, gene replication as well as fluctuations of available RNA polymerases and ribosomes. These phenomena are indeed rarely considered in classic models of protein production and no relative quantitative comparison among them has been performed. The parameters for a large and representative class of proteins are determined using experimental measures. The main important and surprising conclusion of our study is to show that despite the significant fluctuations of free RNA polymerases and free ribosomes, they bring little variability to protein production contrary to what has been previously proposed in the literature. After verifying the robustness of this quite counter-intuitive result, we discuss its possible origin from a theoretical view, and interpret it as the result of a mean-field effect.

Comparison of Escherichia coli surface attachment methods for single-cell microscopy

For in vivo, single-cell imaging bacterial cells are commonly immobilised via physical confinement or surface attachment. Different surface attachment methods have been used both for atomic force and optical microscopy (including super resolution), and some have been reported to affect bacterial physiology. However, a systematic comparison of the effects these attachment methods have on the bacterial physiology is lacking. Here we present such a comparison for bacterium Escherichia coli, and assess the growth rate, size and intracellular pH of cells growing attached to different, commonly used, surfaces. We demonstrate that E. coli grow at the same rate, length and internal pH on all the tested surfaces when in the same growth medium. The result suggests that tested attachment methods can be used interchangeably when studying E. coli physiology.

Organ-wide and ploidy-dependent regulation both contribute to cell-size determination: evidence from a computational model of tomato fruit

The development of a new organ is the result of coordinated events of cell division and expansion, in strong interaction with each other. This study presents a dynamic model of tomato fruit development that includes cell division, endoreduplication, and expansion processes. The model is used to investigate the potential interactions among these developmental processes within the context of the neo-cellular theory. In particular, different control schemes (either cell-autonomous or organ-controlled) are tested and compared to experimental data from two contrasting genotypes. The model shows that a pure cell-autonomous control fails to reproduce the observed cell-size distribution, and that an organ-wide control is required in order to get realistic cell-size variations. The model also supports the role of endoreduplication as an important determinant of the final cell size and suggests that a direct effect of endoreduplication on cell expansion is needed in order to obtain a significant correlation between size and ploidy, as observed in real data.

Microfluidic time-lapse analysis and reevaluation of the Bacillus subtilis cell cycle

Recent studies taking advantage of automated single-cell time-lapse analysis have reignited interest in the bacterial cell cycle. Several studies have highlighted alternative models, such as Sizer and Adder, which differ essentially in relation to whether cells can measure their present size or their amount of growth since birth. Most of the recent work has been done with Escherichia coli. We set out to study the well-characterized Gram-positive bacterium, Bacillus subtilis, at the single-cell level, using an accurate fluorescent marker for division as well as a marker for completion of chromosome replication. Our results are consistent with the Adder model in both fast and slow growth conditions tested, and with Sizer but only at the slower growth rate. We also find that cell size variation arises not only from the expected variation in size at division but also that division site offset from mid-cell contributes to a significant degree. Finally, although traditional cell cycle models imply a strong connection between the termination of a round of replication and subsequent division, we find that at the single-cell level these events are largely disconnected.