Stochasticity and homeostasis in the E. coli replication and division cycle

Live tracking of replisomes reveals nutrient-dependent regulation of replication elongation rates in Caulobacter crescentus

In bacteria, commitment to genome replication (initiation) is intricately linked to nutrient availability. Whether growth conditions affect other stages of replication beyond initiation remains to be systematically studied. To address this, we assess the replication dynamics of Caulobacter crescentus, a bacterium that undergoes only a single round of replication per cell cycle, by tracking the replisome across various growth phases and nutrient conditions. We find that the replication elongation rates slow down as cells transition from exponential (high-nutrient) to stationary (low-nutrient) phase, and this contributes significantly to the overall cell-cycle delay. Although elongation rates are correlated with growth rates, both properties are differentially influenced by nutrient status. This slowdown in replication progression is reversed via supplementation with dNTPs and is not associated with increased mutagenesis or upregulation of the DNA damage responses. We conclude that growth conditions not only dictate the commitment to replication but also the rates of genome duplication. Such regulation appears to be distinct from stress-induced replication slowdown and likely serves as an adaptive mechanism to cope with fluctuations in nutrient availability in the environment.

Direct single-cell observation of a key Escherichia coli cell-cycle oscillator

Initiation of DNA replication in Escherichia coli is coupled to cell size via the DnaA protein, whose activity is dependent on its nucleotide-bound state. However, the oscillations in DnaA activity have never been observed at the single-cell level. By measuring the volume-specific production rate of a reporter protein under control of a DnaA-regulated promoter, we could distinguish two distinct cell-cycle oscillators. The first, driven by both DnaA activity and SeqA repression, shows a causal relationship with cell size and divisions, similarly to initiation events. The second one, a reporter of DnaA activity alone, loses the synchrony and causality properties. Our results show that transient inhibition of gene expression by SeqA keeps the oscillation of volume-sensing DnaA activity in phase with the subsequent division event and suggest that DnaA activity peaks do not correspond directly to initiation events.

Apparent simplicity and emergent robustness in the control of the Escherichia coli cell cycle

To examine how bacteria achieve robust cell proliferation across diverse conditions, we developed a method that quantifies 77 cell morphological, cell cycle, and growth phenotypes of a fluorescently labeled Escherichia coli strain and >800 gene deletion derivatives under multiple nutrient conditions. This approach revealed extensive phenotypic plasticity and deviating mutant phenotypes were often nutrient dependent. From this broad phenotypic landscape emerged simple and robust unifying rules (laws) that connect DNA replication initiation, nucleoid segregation, FtsZ ring formation, and cell constriction to specific aspects of cell size (volume, length, or added length) at the population level. Furthermore, completion of cell division followed the initiation of cell constriction after a constant time delay across strains and nutrient conditions, identifying cell constriction as a key control point for cell size determination. Our work provides a population-level description of the governing principles by which E. coli integrates cell cycle processes and growth rate with cell size to achieve its robust proliferative capability. A record of this paper's transparent peer review process is included in the supplemental information.

The Quantification of Bacterial Cell Size: Discrepancies Arise from Varied Quantification Methods

The robust regulation of the cell cycle is critical for the survival and proliferation of bacteria. To gain a comprehensive understanding of the mechanisms regulating the bacterial cell cycle, it is essential to accurately quantify cell-cycle-related parameters and to uncover quantitative relationships. In this paper, we demonstrate that the quantification of cell size parameters using microscopic images can be influenced by software and by the parameter settings used. Remarkably, even if the consistent use of a particular software and specific parameter settings is maintained throughout a study, the type of software and the parameter settings can significantly impact the validation of quantitative relationships, such as the constant-initiation-mass hypothesis. Given these inherent characteristics of microscopic image-based quantification methods, it is recommended that conclusions be cross-validated using independent methods, especially when the conclusions are associated with cell size parameters that were obtained under different conditions. To this end, we presented a flexible workflow for simultaneously quantifying multiple bacterial cell-cycle-related parameters using microscope-independent methods.

Beyond the average: An updated framework for understanding the relationship between cell growth, DNA replication, and division in a bacterial system

Our understanding of the bacterial cell cycle is framed largely by population-based experiments that focus on the behavior of idealized average cells. Most famously, the contributions of Cooper and Helmstetter help to contextualize the phenomenon of overlapping replication cycles observed in rapidly growing bacteria. Despite the undeniable value of these approaches, their necessary reliance on the behavior of idealized average cells masks the stochasticity inherent in single-cell growth and physiology and limits their mechanistic value. To bridge this gap, we propose an updated and agnostic framework, informed by extant single-cell data, that quantitatively accounts for stochastic variations in single-cell dynamics and the impact of medium composition on cell growth and cell cycle progression. In this framework, stochastic timers sensitive to medium composition impact the relationship between cell cycle events, accounting for observed differences in the relationship between cell cycle events in slow- and fast-growing cells. We conclude with a roadmap for potential application of this framework to longstanding open questions in the bacterial cell cycle field.

Robust replication initiation from coupled homeostatic mechanisms

The bacterium Escherichia coli initiates replication once per cell cycle at a precise volume per origin and adds an on average constant volume between successive initiation events, independent of the initiation size. Yet, a molecular model that can explain these observations has been lacking. Experiments indicate that E. coli controls replication initiation via titration and activation of the initiator protein DnaA. Here, we study by mathematical modelling how these two mechanisms interact to generate robust replication-initiation cycles. We first show that a mechanism solely based on titration generates stable replication cycles at low growth rates, but inevitably causes premature reinitiation events at higher growth rates. In this regime, the DnaA activation switch becomes essential for stable replication initiation. Conversely, while the activation switch alone yields robust rhythms at high growth rates, titration can strongly enhance the stability of the switch at low growth rates. Our analysis thus predicts that both mechanisms together drive robust replication cycles at all growth rates. In addition, it reveals how an origin-density sensor yields adder correlations.

Scaling of biosynthesis and metabolism with cell size

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

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.

Two different cell-cycle processes determine the timing of cell division in Escherichia coli

Cells must control the cell cycle to ensure that key processes are brought to completion. In Escherichia coli, it is controversial whether cell division is tied to chromosome replication or to a replication-independent inter-division process. A recent model suggests instead that both processes may limit cell division with comparable odds in single cells. Here, we tested this possibility experimentally by monitoring single-cell division and replication over multiple generations at slow growth. We then perturbed cell width, causing an increase of the time between replication termination and division. As a consequence, replication became decreasingly limiting for cell division, while correlations between birth and division and between subsequent replication-initiation events were maintained. Our experiments support the hypothesis that both chromosome replication and a replication-independent inter-division process can limit cell division: the two processes have balanced contributions in non-perturbed cells, while our width perturbations increase the odds of the replication-independent process being limiting.

Interconnecting solvent quality, transcription, and chromosome folding in Escherichia coli

All cells fold their genomes, including bacterial cells, where the chromosome is compacted into a domain-organized meshwork called the nucleoid. How compaction and domain organization arise is not fully understood. Here, we describe a method to estimate the average mesh size of the nucleoid in Escherichia coli. Using nucleoid mesh size and DNA concentration estimates, we find that the cytoplasm behaves as a poor solvent for the chromosome when the cell is considered as a simple semidilute polymer solution. Monte Carlo simulations suggest that a poor solvent leads to chromosome compaction and DNA density heterogeneity (i.e., domain formation) at physiological DNA concentration. Fluorescence microscopy reveals that the heterogeneous DNA density negatively correlates with ribosome density within the nucleoid, consistent with cryoelectron tomography data. Drug experiments, together with past observations, suggest the hypothesis that RNAs contribute to the poor solvent effects, connecting chromosome compaction and domain formation to transcription and intracellular organization.

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.

Nucleoid Size Scaling and Intracellular Organization of Translation across Bacteria

The scaling of organelles with cell size is thought to be exclusive to eukaryotes. Here, we demonstrate that similar scaling relationships hold for the bacterial nucleoid. Despite the absence of a nuclear membrane, nucleoid size strongly correlates with cell size, independent of changes in DNA amount and across various nutrient conditions. This correlation is observed in diverse bacteria, revealing a near-constant ratio between nucleoid and cell size for a given species. As in eukaryotes, the nucleocytoplasmic ratio in bacteria varies greatly among species. This spectrum of nucleocytoplasmic ratios is independent of genome size, and instead it appears linked to the average population cell size. Bacteria with different nucleocytoplasmic ratios have a cytoplasm with different biophysical properties, impacting ribosome mobility and localization. Together, our findings identify new organizational principles and biophysical features of bacterial cells, implicating the nucleocytoplasmic ratio and cell size as determinants of the intracellular organization of translation.

Initiation of chromosome replication controls both division and replication cycles in E. coli through a double-adder mechanism

Living cells proliferate by completing and coordinating two cycles, a division cycle controlling cell size and a DNA replication cycle controlling the number of chromosomal copies. It remains unclear how bacteria such as Escherichia coli tightly coordinate those two cycles across a wide range of growth conditions. Here, we used time-lapse microscopy in combination with microfluidics to measure growth, division and replication in single E. coli cells in both slow and fast growth conditions. To compare different phenomenological cell cycle models, we introduce a statistical framework assessing their ability to capture the correlation structure observed in the data. In combination with stochastic simulations, our data indicate that the cell cycle is driven from one initiation event to the next rather than from birth to division and is controlled by two adder mechanisms: the added volume since the last initiation event determines the timing of both the next division and replication initiation events.

Deciphering Metabolic Heterogeneity by Single-Cell Analysis

Single-cell analysis provides insights into cellular heterogeneity and dynamics of individual cells. This Feature highlights recent developments in key analytical techniques suited for single-cell metabolic analysis with a special focus on mass spectrometry-based analytical platforms and RNA-seq as well as imaging techniques that reveal stochasticity in metabolism.

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.

Mechanistic Origin of Cell-Size Control and Homeostasis in Bacteria

Evolutionarily divergent bacteria share a common phenomenological strategy for cell-size homeostasis under steady-state conditions. In the presence of inherent physiological stochasticity, cells following this "adder" principle gradually return to their steady-state size by adding a constant volume between birth and division, regardless of their size at birth. However, the mechanism of the adder has been unknown despite intense efforts. In this work, we show that the adder is a direct consequence of two general processes in biology: (1) threshold-accumulation of initiators and precursors required for cell division to a respective fixed number-and (2) balanced biosynthesis-maintenance of their production proportional to volume growth. This mechanism is naturally robust to static growth inhibition but also allows us to "reprogram" cell-size homeostasis in a quantitatively predictive manner in both Gram-negative Escherichia coli and Gram-positive Bacillus subtilis. By generating dynamic oscillations in the concentration of the division protein FtsZ, we were able to oscillate cell size at division and systematically break the adder. In contrast, periodic induction of replication initiator protein DnaA caused oscillations in cell size at initiation but did not alter division size or the adder. Finally, we were able to restore the adder phenotype in slow-growing E. coli, the only known steady-state growth condition wherein E. coli significantly deviates from the adder, by repressing active degradation of division proteins. Together, these results show that cell division and replication initiation are independently controlled at the gene-expression level and that division processes exclusively drive cell-size homeostasis in bacteria. VIDEO ABSTRACT.

Concurrent processes set E. coli cell division

A cell can divide only upon completion of chromosome segregation; otherwise, its daughters would lose genetic material. However, we do not know whether the partitioning of chromosomes is the key event for the decision to divide. We show how key trends in single-cell data reject the classic idea of replication-segregation as the rate-limiting process for cell division. Instead, the data agree with a model where two concurrent processes (setting replication initiation and interdivision time) set cell division on competing time scales. During each cell cycle, division is set by the slowest process (an "AND" gate). The concept of transitions between cell cycle stages as decisional processes integrating multiple inputs instead of cascading from orchestrated steps can affect the way we think of the cell cycle in general.

Size control in mammalian cells involves modulation of both growth rate and cell cycle duration

Despite decades of research, how mammalian cell size is controlled remains unclear because of the difficulty of directly measuring growth at the single-cell level. Here we report direct measurements of single-cell volumes over entire cell cycles on various mammalian cell lines and primary human cells. We find that, in a majority of cell types, the volume added across the cell cycle shows little or no correlation to cell birth size, a homeostatic behavior called "adder". This behavior involves modulation of G1 or S-G2 duration and modulation of growth rate. The precise combination of these mechanisms depends on the cell type and the growth condition. We have developed a mathematical framework to compare size homeostasis in datasets ranging from bacteria to mammalian cells. This reveals that a near-adder behavior is the most common type of size control and highlights the importance of growth rate modulation to size control in mammalian cells.

The Empirical Fluctuation Pattern of E. coli Division Control

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

A Markovian Approach towards Bacterial Size Control and Homeostasis in Anomalous Growth Processes

Regardless of the progress achieved during recent years, the mechanisms coupling growth and division to attain cell size homeostasis in bacterial populations are still not well understood. In particular, there is a gap of knowledge about the mechanisms controlling anomalous growth events that are ubiquitous even in wild-type phenotypes. Thus, when cells exceed the doubling size the divisome dynamics sets a characteristic length scale that suggests a sizer property. Yet, it has been recently shown that the size at birth and the size increment still satisfy an adder-like correlation. Herein we propose a Markov chain model, that we complement with computational and experimental approaches, to clarify this issue. In this context, we show that classifying cells as a function of the characteristic size set by the divisome dynamics provides a compelling framework to understand size convergence, growth, and division at the large length scale, including the adaptation to, and rescue from, filamentation processes. Our results reveal the independence of size homeostasis on the division pattern of long cells and help to reconcile sizer concepts at the single cell level with an adder-like behavior at a population level.

Surface Area to Volume Ratio: A Natural Variable for Bacterial Morphogenesis

An immediately observable feature of bacteria is that cell size and shape are remarkably constant and characteristic for a given species in a particular condition, but vary quantitatively with physiological parameters such as growth rate, indicating both genetic and environmental regulation. However, despite decades of research, the molecular mechanisms underlying bacterial morphogenesis have remained incompletely characterized. We recently demonstrated that a wide range of bacterial species exhibit a robust surface area to volume ratio (SA/V) homeostasis. Because cell size, shape, and SA/V are mathematically interconnected, if SA/V is indeed the natural variable that cells actively monitor, this finding has critical implications for our understanding of bacterial morphogenesis, placing fundamental constraints on the sizes and shapes that cells can adopt. In this Opinion article we discuss the broad implications that this novel perspective has for the field of bacterial growth and morphogenesis.

Modeling Cell Size Regulation: From Single-Cell-Level Statistics to Molecular Mechanisms and Population-Level Effects

Most microorganisms regulate their cell size. In this article, we review some of the mathematical formulations of the problem of cell size regulation. We focus on coarse-grained stochastic models and the statistics that they generate. We review the biologically relevant insights obtained from these models. We then describe cell cycle regulation and its molecular implementations, protein number regulation, and population growth, all in relation to size regulation. Finally, we discuss several future directions for developing understanding beyond phenomenological models of cell size regulation.

Analysis of Factors Limiting Bacterial Growth in PDMS Mother Machine Devices

The microfluidic mother machine platform has attracted much interest for its potential in studies of bacterial physiology, cellular organization, and cell mechanics. Despite numerous experiments and development of dedicated analysis software, differences in bacterial growth and morphology in narrow mother machine channels compared to typical liquid media conditions have not been systematically characterized. Here we determine changes in E. coli growth rates and cell dimensions in different sized dead-end microfluidic channels using high resolution optical microscopy. We find that E. coli adapt to the confined channel environment by becoming narrower and longer compared to the same strain grown in liquid culture. Cell dimensions decrease as the channel length increases and width decreases. These changes are accompanied by increases in doubling times in agreement with the universal growth law. In channels 100 μm and longer, cell doublings can completely stop as a result of frictional forces that oppose cell elongation. Before complete cessation of elongation, mechanical stresses lead to substantial deformation of cells and changes in their morphology. Our work shows that mechanical forces rather than nutrient limitation are the main growth limiting factor for bacterial growth in long and narrow channels.

Fundamental principles in bacterial physiology-history, recent progress, and the future with focus on cell size control: a review

Bacterial physiology is a branch of biology that aims to understand overarching principles of cellular reproduction. Many important issues in bacterial physiology are inherently quantitative, and major contributors to the field have often brought together tools and ways of thinking from multiple disciplines. This article presents a comprehensive overview of major ideas and approaches developed since the early 20th century for anyone who is interested in the fundamental problems in bacterial physiology. This article is divided into two parts. In the first part (sections 1-3), we review the first 'golden era' of bacterial physiology from the 1940s to early 1970s and provide a complete list of major references from that period. In the second part (sections 4-7), we explain how the pioneering work from the first golden era has influenced various rediscoveries of general quantitative principles and significant further development in modern bacterial physiology. Specifically, section 4 presents the history and current progress of the 'adder' principle of cell size homeostasis. Section 5 discusses the implications of coarse-graining the cellular protein composition, and how the coarse-grained proteome 'sectors' re-balance under different growth conditions. Section 6 focuses on physiological invariants, and explains how they are the key to understanding the coordination between growth and the cell cycle underlying cell size control in steady-state growth. Section 7 overviews how the temporal organization of all the internal processes enables balanced growth. In the final section 8, we conclude by discussing the remaining challenges for the future in the field.

Cell size control and gene expression homeostasis in single-cells

Growth of a cell and its subsequent division into daughters is a fundamental aspect of all cellular living systems. During these processes, how do individual cells correct size aberrations so that they do not grow abnormally large or small? How do cells ensure that the concentration of essential gene products are maintained at desired levels, in spite of dynamic/stochastic changes in cell size during growth and division? Both these questions have fascinated researchers for over a century. We review how advances in singe-cell technologies and measurements are providing unique insights into these questions across organisms from prokaryotes to human cells. More specifically, diverse strategies based on timing of cell-cycle events, regulating growth, and number of daughters are employed to maintain cell size homeostasis. Interestingly, size homeostasis often results in size optimality - proliferation of individual cells in a population is maximized at an optimal cell size. We further discuss how size-dependent expression or gene-replication timing can buffer concentration of a gene product from cell-to-cell size variations within a population. Finally, we speculate on an intriguing hypothesis that specific size control strategies may have evolved as a consequence of gene-product concentration homeostasis.

Stable Regulation of Cell Cycle Events in Mycobacteria: Insights From Inherently Heterogeneous Bacterial Populations

Model bacteria, such as E. coli and B. subtilis, tightly regulate cell cycle progression to achieve consistent cell size distributions and replication dynamics. Many of the hallmark features of these model bacteria, including lateral cell wall elongation and symmetric growth and division, do not occur in mycobacteria. Instead, mycobacterial growth is characterized by asymmetric polar growth and division. This innate asymmetry creates unequal birth sizes and growth rates for daughter cells with each division, generating a phenotypically heterogeneous population. Although the asymmetric growth patterns of mycobacteria lead to a larger variation in birth size than typically seen in model bacterial populations, the cell size distribution is stable over time. Here, we review the cellular mechanisms of growth, division, and cell cycle progression in mycobacteria in the face of asymmetry and inherent heterogeneity. These processes coalesce to control cell size. Although Mycobacterium smegmatis and Mycobacterium bovis Bacillus Calmette-Guérin (BCG) utilize a novel model of cell size control, they are similar to previously studied bacteria in that initiation of DNA replication is a key checkpoint for cell division. We compare the regulation of DNA replication initiation and strategies used for cell size homeostasis in mycobacteria and model bacteria. Finally, we review the importance of cellular organization and chromosome segregation relating to the physiology of mycobacteria and consider how new frameworks could be applied across the wide spectrum of bacterial diversity.

Cell-to-cell variability and robustness in S-phase duration from genome replication kinetics

Genome replication, a key process for a cell, relies on stochastic initiation by replication origins, causing a variability of replication timing from cell to cell. While stochastic models of eukaryotic replication are widely available, the link between the key parameters and overall replication timing has not been addressed systematically. We use a combined analytical and computational approach to calculate how positions and strength of many origins lead to a given cell-to-cell variability of total duration of the replication of a large region, a chromosome or the entire genome. Specifically, the total replication timing can be framed as an extreme-value problem, since it is due to the last region that replicates in each cell. Our calculations identify two regimes based on the spread between characteristic completion times of all inter-origin regions of a genome. For widely different completion times, timing is set by the single specific region that is typically the last to replicate in all cells. Conversely, when the completion time of all regions are comparable, an extreme-value estimate shows that the cell-to-cell variability of genome replication timing has universal properties. Comparison with available data shows that the replication program of three yeast species falls in this extreme-value regime.

Stochastic timing in gene expression for simple regulatory strategies

Timing is essential for many cellular processes, from cellular responses to external stimuli to the cell cycle and circadian clocks. Many of these processes are based on gene expression. For example, an activated gene may be required to reach in a precise time a threshold level of expression that triggers a specific downstream process. However, gene expression is subject to stochastic fluctuations, naturally inducing an uncertainty in this threshold-crossing time with potential consequences on biological functions and phenotypes. Here, we consider such ‘timing fluctuations’ and we ask how they can be controlled. Our analytical estimates and simulations show that, for an induced gene, timing variability is minimal if the threshold level of expression is approximately half of the steady-state level. Timing fluctuations can be reduced by increasing the transcription rate, while they are insensitive to the translation rate. In presence of self-regulatory strategies, we show that self-repression reduces timing noise for threshold levels that have to be reached quickly, while self-activation is optimal at long times. These results lay a framework for understanding stochasticity of endogenous systems such as the cell cycle, as well as for the design of synthetic trigger circuits.

The Adder Phenomenon Emerges from Independent Control of Pre- and Post-Start Phases of the Budding Yeast Cell Cycle

Although it has long been clear that cells actively regulate their size, the molecular mechanisms underlying this regulation have remained poorly understood. In budding yeast, cell size primarily modulates the duration of the cell-division cycle by controlling the G1/S transition known as Start. We have recently shown that the rate of progression through Start increases with cell size, because cell growth dilutes the cell-cycle inhibitor Whi5 in G1. Recent phenomenological studies in yeast and bacteria have shown that these cells add an approximately constant volume during each complete cell cycle, independent of their size at birth. These results seem to be in conflict, as the phenomenological studies suggest that cells measure the amount they grow, rather than their size, and that size control acts over the whole cell cycle, rather than specifically in G1. Here, we propose an integrated model that unifies the adder phenomenology with the molecular mechanism of G1/S cell-size control. We use single-cell microscopy to parameterize a full cell-cycle model based on independent control of pre- and post-Start cell-cycle periods. We find that our model predicts the size-independent amount of cell growth during the full cell cycle. This suggests that the adder phenomenon is an emergent property of the independent regulation of pre- and post-Start cell-cycle periods rather than the consequence of an underlying molecular mechanism measuring a fixed amount of growth.

A simple molecular mechanism explains multiple patterns of cell-size regulation

Increasingly accurate and massive data have recently shed light on the fundamental question of how cells maintain a stable size trajectory as they progress through the cell cycle. Microbes seem to use strategies ranging from a pure sizer, where the end of a given phase is triggered when the cell reaches a critical size, to pure adder, where the cell adds a constant size during a phase. Yet the biological origins of the observed spectrum of behavior remain elusive. We analyze a molecular size-control mechanism, based on experimental data from the yeast S. cerevisiae, that gives rise to behaviors smoothly interpolating between adder and sizer. The size-control is obtained from the accumulation of an activator protein that titrates an inhibitor protein. Strikingly, the size-control is composed of two different regimes: for small initial cell size, the size-control is a sizer, whereas for larger initial cell size, it is an imperfect adder, in agreement with recent experiments. Our model thus indicates that the adder and critical size behaviors may just be different dynamical regimes of a single simple biophysical mechanism.

Careful accounting of extrinsic noise in protein expression reveals correlations among its sources

In order to grow and replicate, living cells must express a diverse array of proteins, but the process by which proteins are made includes a great deal of inherent randomness. Understanding this randomness-whether it arises from the discrete stochastic nature of chemical reactivity ("intrinsic" noise), or from cell-to-cell variability in the concentrations of molecules involved in gene expression, or from the timings of important cell-cycle events like DNA replication and cell division ("extrinsic" noise)-remains a challenge. In this article we analyze a model of gene expression that accounts for several extrinsic sources of noise, including those associated with chromosomal replication, cell division, and variability in the numbers of RNA polymerase, ribonuclease E, and ribosomes. We then attempt to fit our model to a large proteomics and transcriptomics data set and find that only through the introduction of a few key correlations among the extrinsic noise sources can we accurately recapitulate the experimental data. These include significant correlations between the rate of mRNA degradation (mediated by ribonuclease E) and the rates of both transcription (RNA polymerase) and translation (ribosomes) and, strikingly, an anticorrelation between the transcription and the translation rates themselves.

Robustness of synthetic oscillators in growing and dividing cells

Synthetic biology sets out to implement new functions in cells, and to develop a deeper understanding of biological design principles. Elowitz and Leibler [Nature (London) 403, 335 (2000)NATUAS0028-083610.1038/35002125] showed that by rational design of the reaction network, and using existing biological components, they could create a network that exhibits periodic gene expression, dubbed the repressilator. More recently, Stricker et al. [Nature (London) 456, 516 (2008)NATUAS0028-083610.1038/nature07389] presented another synthetic oscillator, called the dual-feedback oscillator, which is more stable. Detailed studies have been carried out to determine how the stability of these oscillators is affected by the intrinsic noise of the interactions between the components and the stochastic expression of their genes. However, as all biological oscillators reside in growing and dividing cells, an important question is how these oscillators are perturbed by the cell cycle. In previous work we showed that the periodic doubling of the gene copy numbers due to DNA replication can couple not only natural, circadian oscillators to the cell cycle [Paijmans et al., Proc. Natl. Acad. Sci. (USA) 113, 4063 (2016)PNASA60027-842410.1073/pnas.1507291113], but also these synthetic oscillators. Here we expand this study. We find that the strength of the locking between oscillators depends not only on the positions of the genes on the chromosome, but also on the noise in the timing of gene replication: noise tends to weaken the coupling. Yet, even in the limit of high levels of noise in the replication times of the genes, both synthetic oscillators show clear signatures of locking to the cell cycle. This work enhances our understanding of the design of robust biological oscillators inside growing and diving cells.

Is cell size a spandrel?

All organisms control the size of their cells. We focus here on the question of size regulation in bacteria, and suggest that the quantitative laws governing cell size and its dependence on growth rate may arise as byproducts of a regulatory mechanism which evolved to support multiple DNA replication forks. In particular, we show that the increase of bacterial cell size during Lenski’s long-term evolution experiments is a natural outcome of this proposal. This suggests that, in the context of evolution, cell size may be a 'spandrel'

DOI:http://dx.doi.org/10.7554/eLife.22186.001

Interrogating the Escherichia coli cell cycle by cell dimension perturbations

Bacteria tightly regulate and coordinate the various events in their cell cycles to duplicate themselves accurately and to control their cell sizes. Growth of Escherichia coli, in particular, follows a relation known as Schaechter's growth law. This law says that the average cell volume scales exponentially with growth rate, with a scaling exponent equal to the time from initiation of a round of DNA replication to the cell division at which the corresponding sister chromosomes segregate. Here, we sought to test the robustness of the growth law to systematic perturbations in cell dimensions achieved by varying the expression levels of mreB and ftsZ We found that decreasing the mreB level resulted in increased cell width, with little change in cell length, whereas decreasing the ftsZ level resulted in increased cell length. Furthermore, the time from replication termination to cell division increased with the perturbed dimension in both cases. Moreover, the growth law remained valid over a range of growth conditions and dimension perturbations. The growth law can be quantitatively interpreted as a consequence of a tight coupling of cell division to replication initiation. Thus, its robustness to perturbations in cell dimensions strongly supports models in which the timing of replication initiation governs that of cell division, and cell volume is the key phenomenological variable governing the timing of replication initiation. These conclusions are discussed in the context of our recently proposed "adder-per-origin" model, in which cells add a constant volume per origin between initiations and divide a constant time after initiation.

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

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

Chromosomal Replication Complexity: A Novel DNA Metrics and Genome Instability Factor

As the ratio of the copy number of the most replicated to the unreplicated regions in the same chromosome, the definition of chromosomal replication complexity (CRC) appears to leave little room for variation, being either two during S-phase or one otherwise. However, bacteria dividing faster than they replicate their chromosome spike CRC to four and even eight. A recent experimental inquiry about the limits of CRC in Escherichia coli revealed two major reasons to avoid elevating it further: (i) increased chromosomal fragmentation and (ii) complications with subsequent double-strand break repair. Remarkably, examples of stable elevated CRC in eukaryotic chromosomes are well known under various terms like "differential replication," "underreplication," "DNA puffs," "onion-skin replication," or "re-replication" and highlight the phenomenon of static replication fork (sRF). To accurately describe the resulting "amplification by overinitiation," I propose a new term: "replification" (subchromosomal overreplication). In both prokaryotes and eukaryotes, replification, via sRF processing, causes double-strand DNA breaks and, with their repair elevating chromosomal rearrangements, represents a novel genome instability factor. I suggest how static replication bubbles could be stabilized and speculate that some tandem duplications represent such persistent static bubbles. Moreover, I propose how static replication bubbles could be transformed into tandem duplications, double minutes, or inverted triplications. Possible experimental tests of these models are discussed.

Ribosome biogenesis in replicating cells: Integration of experiment and theory

Ribosomes-the primary macromolecular machines responsible for translating the genetic code into proteins-are complexes of precisely folded RNA and proteins. The ways in which their production and assembly are managed by the living cell is of deep biological importance. Here we extend a recent spatially resolved whole-cell model of ribosome biogenesis in a fixed volume [Earnest et al., Biophys J 2015, 109, 1117-1135] to include the effects of growth, DNA replication, and cell division. All biological processes are described in terms of reaction-diffusion master equations and solved stochastically using the Lattice Microbes simulation software. In order to determine the replication parameters, we construct and analyze a series of Escherichia coli strains with fluorescently labeled genes distributed evenly throughout their chromosomes. By measuring these cells' lengths and number of gene copies at the single-cell level, we could fit a statistical model of the initiation and duration of chromosome replication. We found that for our slow-growing (120 min doubling time) E. coli cells, replication was initiated 42 min into the cell cycle and completed after an additional 42 min. While simulations of the biogenesis model produce the correct ribosome and mRNA counts over the cell cycle, the kinetic parameters for transcription and degradation are lower than anticipated from a recent analytical time dependent model of in vivo mRNA production. Describing expression in terms of a simple chemical master equation, we show that the discrepancies are due to the lack of nonribosomal genes in the extended biogenesis model which effects the competition of mRNA for ribosome binding, and suggest corrections to parameters to be used in the whole-cell model when modeling expression of the entire transcriptome. © 2016 Wiley Periodicals, Inc. Biopolymers 105: 735-751, 2016.

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

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

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

A mechanistic stochastic framework for regulating bacterial cell division

How exponentially growing cells maintain size homeostasis is an important fundamental problem. Recent single-cell studies in prokaryotes have uncovered the adder principle, where cells add a fixed size (volume) from birth to division, irrespective of their size at birth. To mechanistically explain the adder principle, we consider a timekeeper protein that begins to get stochastically expressed after cell birth at a rate proportional to the volume. Cell-division time is formulated as the first-passage time for protein copy numbers to hit a fixed threshold. Consistent with data, the model predicts that the noise in division timing increases with size at birth. Intriguingly, our results show that the distribution of the volume added between successive cell-division events is independent of the newborn cell size. This was dramatically seen in experimental studies, where histograms of the added volume corresponding to different newborn sizes collapsed on top of each other. The model provides further insights consistent with experimental observations: the distribution of the added volume when scaled by its mean becomes invariant of the growth rate. In summary, our simple yet elegant model explains key experimental findings and suggests a mechanism for regulating both the mean and fluctuations in cell-division timing for controlling size.

Stochastic modeling of cell growth with symmetric or asymmetric division

We consider a class of biologically motivated stochastic processes in which a unicellular organism divides its resources (volume or damaged proteins, in particular) symmetrically or asymmetrically between its progeny. Assuming the final amount of the resource is controlled by a growth policy and subject to additive and multiplicative noise, we derive the recursive integral equation describing the evolution of the resource distribution over subsequent generations and use it to study the properties of stable resource distributions. We find conditions under which a unique stable resource distribution exists and calculate its moments for the class of affine linear growth policies. Moreover, we apply an asymptotic analysis to elucidate the conditions under which the stable distribution (when it exists) has a power-law tail. Finally, we use the results of this asymptotic analysis along with the moment equations to draw a stability phase diagram for the system that reveals the counterintuitive result that asymmetry serves to increase stability while at the same time widening the stable distribution. We also briefly discuss how cells can divide damaged proteins asymmetrically between their progeny as a form of damage control. In the appendixes, motivated by the asymmetric division of cell volume in Saccharomyces cerevisiae, we extend our results to the case wherein mother and daughter cells follow different growth policies.