Scaling laws governing stochastic growth and division of single bacterial cells

Stochastic maps, continuous approximation, and stable distribution

A continuous approximation framework for general nonlinear stochastic as well as deterministic discrete maps is developed. For the stochastic map with uncorelated Gaussian noise, by successively applying the Itô lemma, we obtain a Langevin type of equation. Specifically, we show how nonlinear maps give rise to a Langevin description that involves multiplicative noise. The multiplicative nature of the noise induces an additional effective force, not present in the absence of noise. We further exploit the continuum description and provide an explicit formula for the stable distribution of the stochastic map and conditions for its existence. Our results are in good agreement with numerical simulations of several maps.

Analysis of Noise Mechanisms in Cell-Size Control

At the single-cell level, noise arises from multiple sources, such as inherent stochasticity of biomolecular processes, random partitioning of resources at division, and fluctuations in cellular growth rates. How these diverse noise mechanisms combine to drive variations in cell size within an isoclonal population is not well understood. Here, we investigate the contributions of different noise sources in well-known paradigms of cell-size control, such as adder (division occurs after adding a fixed size from birth), sizer (division occurs after reaching a size threshold), and timer (division occurs after a fixed time from birth). Analysis reveals that variation in cell size is most sensitive to errors in partitioning of volume among daughter cells, and not surprisingly, this process is well regulated among microbes. Moreover, depending on the dominant noise mechanism, different size-control strategies (or a combination of them) provide efficient buffering of size variations. We further explore mixer models of size control, where a timer phase precedes/follows an adder, as has been proposed in Caulobacter crescentus. Although mixing a timer and an adder can sometimes attenuate size variations, it invariably leads to higher-order moments growing unboundedly over time. This results in a power-law distribution for the cell size, with an exponent that depends inversely on the noise in the timer phase. Consistent with theory, we find evidence of power-law statistics in the tail of C. crescentus cell-size distribution, although there is a discrepancy between the observed power-law exponent and that predicted from the noise parameters. The discrepancy, however, is removed after data reveal that the size added by individual newborns in the adder phase itself exhibits power-law statistics. Taken together, this study provides key insights into the role of noise mechanisms in size homeostasis, and suggests an inextricable link between timer-based models of size control and heavy-tailed cell-size distributions.

Biphasic growth dynamics control cell division in Caulobacter crescentus

Cell size is specific to each species and impacts cell function. Various phenomenological models for cell size regulation have been proposed, but recent work in bacteria has suggested an 'adder' model, in which a cell increments its size by a constant amount between each division. However, the coupling between cell size, shape and constriction remains poorly understood. Here, we investigate size control and the cell cycle dependence of bacterial growth using multigenerational cell growth and shape data for single Caulobacter crescentus cells. Our analysis reveals a biphasic mode of growth: a relative timer phase before constriction where cell growth is correlated to its initial size, followed by a pure adder phase during constriction. Cell wall labelling measurements reinforce this biphasic model, in which a crossover from uniform lateral growth to localized septal growth is observed. We present a mathematical model that quantitatively explains this biphasic 'mixer' model for cell size control.

Mycobacteria Modify Their Cell Size Control under Sub-Optimal Carbon Sources

The decision to divide is the most important one that any cell must make. Recent single cell studies suggest that most bacteria follow an “adder” model of cell size control, incorporating a fixed amount of cell wall material before dividing. Mycobacteria, including the causative agent of tuberculosis Mycobacterium tuberculosis, are known to divide asymmetrically resulting in heterogeneity in growth rate, doubling time, and other growth characteristics in daughter cells. The interplay between asymmetric cell division and adder size control has not been extensively investigated. Moreover, the impact of changes in the environment on growth rate and cell size control have not been addressed for mycobacteria. Here, we utilize time-lapse microscopy coupled with microfluidics to track live Mycobacterium smegmatis cells as they grow and divide over multiple generations, under a variety of growth conditions. We demonstrate that, under optimal conditions, M. smegmatis cells robustly follow the adder principle, with constant added length per generation independent of birth size, growth rate, and inherited pole age. However, the nature of the carbon source induces deviations from the adder model in a manner that is dependent on pole age. Understanding how mycobacteria maintain cell size homoeostasis may provide crucial targets for the development of drugs for the treatment of tuberculosis, which remains a leading cause of global mortality.

Phenomenology of stochastic exponential growth

Stochastic exponential growth is observed in a variety of contexts, including molecular autocatalysis, nuclear fission, population growth, inflation of the universe, viral social media posts, and financial markets. Yet literature on modeling the phenomenology of these stochastic dynamics has predominantly focused on one model, geometric Brownian motion (GBM), which can be described as the solution of a Langevin equation with linear drift and linear multiplicative noise. Using recent experimental results on stochastic exponential growth of individual bacterial cell sizes, we motivate the need for a more general class of phenomenological models of stochastic exponential growth, which are consistent with the observation that the mean-rescaled distributions are approximately stationary at long times. We show that this behavior is not consistent with GBM, instead it is consistent with power-law multiplicative noise with positive fractional powers. Therefore, we consider this general class of phenomenological models for stochastic exponential growth, provide analytical solutions, and identify the important dimensionless combination of model parameters, which determines the shape of the mean-rescaled distribution. We also provide a prescription for robustly inferring model parameters from experimentally observed stochastic growth trajectories.

Coordination of size-control, reproduction and generational memory in freshwater planarians

Uncovering the mechanisms that control size, growth, and division rates of organisms reproducing through binary division means understanding basic principles of their life cycle. Recent work has focused on how division rates are regulated in bacteria and yeast, but this question has not yet been addressed in more complex, multicellular organisms. We have, over the course of several years, assembled a unique large-scale data set on the growth and asexual reproduction of two freshwater planarian species, Dugesia japonica and Girardia tigrina, which reproduce by transverse fission and succeeding regeneration of head and tail pieces into new planarians. We show that generation-dependent memory effects in planarian reproduction need to be taken into account to accurately capture the experimental data. To achieve this, we developed a new additive model that mixes multiple size control strategies based on planarian size, growth, and time between divisions. Our model quantifies the proportions of each strategy in the mixed dynamics, revealing the ability of the two planarian species to utilize different strategies in a coordinated manner for size control. Additionally, we found that head and tail offspring of both species employ different mechanisms to monitor and trigger their reproduction cycles. Thus, we find a diversity of strategies not only between species but between heads and tails within species. Our additive model provides two advantages over existing 2D models that fit a multivariable splitting rate function to the data for size control: firstly, it can be fit to relatively small data sets and can thus be applied to systems where available data is limited. Secondly, it enables new biological insights because it explicitly shows the contributions of different size control strategies for each offspring type.

Image analysis driven single-cell analytics for systems microbiology

BACKGROUND

Time-lapse microscopy is an essential tool for capturing and correlating bacterial morphology and gene expression dynamics at single-cell resolution. However state-of-the-art computational methods are limited in terms of the complexity of cell movies that they can analyze and lack of automation. The proposed Bacterial image analysis driven Single Cell Analytics (BaSCA) computational pipeline addresses these limitations thus enabling high throughput systems microbiology.

RESULTS

BaSCA can segment and track multiple bacterial colonies and single-cells, as they grow and divide over time (cell segmentation and lineage tree construction) to give rise to dense communities with thousands of interacting cells in the field of view. It combines advanced image processing and machine learning methods to deliver very accurate bacterial cell segmentation and tracking (F-measure over 95%) even when processing images of imperfect quality with several overcrowded colonies in the field of view. In addition, BaSCA extracts on the fly a plethora of single-cell properties, which get organized into a database summarizing the analysis of the cell movie. We present alternative ways to analyze and visually explore the spatiotemporal evolution of single-cell properties in order to understand trends and epigenetic effects across cell generations. The robustness of BaSCA is demonstrated across different imaging modalities and microscopy types.

CONCLUSIONS

BaSCA can be used to analyze accurately and efficiently cell movies both at a high resolution (single-cell level) and at a large scale (communities with many dense colonies) as needed to shed light on e.g. how bacterial community effects and epigenetic information transfer play a role on important phenomena for human health, such as biofilm formation, persisters' emergence etc. Moreover, it enables studying the role of single-cell stochasticity without losing sight of community effects that may drive it.

Relevant parameters in models of cell division control

A recent burst of dynamic single-cell data makes it possible to characterize the stochastic dynamics of cell division control in bacteria. Different models were used to propose specific mechanisms, but the links between them are poorly explored. The lack of comparative studies makes it difficult to appreciate how well any particular mechanism is supported by the data. Here, we describe a simple and generic framework in which two common formalisms can be used interchangeably: (i) a continuous-time division process described by a hazard function and (ii) a discrete-time equation describing cell size across generations (where the unit of time is a cell cycle). In our framework, this second process is a discrete-time Langevin equation with simple physical analogues. By perturbative expansion around the mean initial size (or interdivision time), we show how this framework describes a wide range of division control mechanisms, including combinations of time and size control, as well as the constant added size mechanism recently found to capture several aspects of the cell division behavior of different bacteria. As we show by analytical estimates and numerical simulations, the available data are described precisely by the first-order approximation of this expansion, i.e., by a "linear response" regime for the correction of size fluctuations. Hence, a single dimensionless parameter defines the strength and action of the division control against cell-to-cell variability (quantified by a single "noise" parameter). However, the same strength of linear response may emerge from several mechanisms, which are distinguished only by higher-order terms in the perturbative expansion. Our analytical estimate of the sample size needed to distinguish between second-order effects shows that this value is close to but larger than the values of the current datasets. These results provide a unified framework for future studies and clarify the relevant parameters at play in the control of cell division.

Membranes, energetics, and evolution across the prokaryote-eukaryote divide

The evolution of the eukaryotic cell marked a profound moment in Earth’s history, with most of the visible biota coming to rely on intracellular membrane-bound organelles. It has been suggested that this evolutionary transition was critically dependent on the movement of ATP synthesis from the cell surface to mitochondrial membranes and the resultant boost to the energetic capacity of eukaryotic cells. However, contrary to this hypothesis, numerous lines of evidence suggest that eukaryotes are no more bioenergetically efficient than prokaryotes. Thus, although the origin of the mitochondrion was a key event in evolutionary history, there is no reason to think membrane bioenergetics played a direct, causal role in the transition from prokaryotes to eukaryotes and the subsequent explosive diversification of cellular and organismal complexity.

Over time, life on Earth has evolved into three large groups: archaea, bacteria, and eukaryotes. The most familiar forms of life – such as fungi, plants and animals – all belong to the eukaryotes. Bacteria and archaea are simpler, single-celled organisms and are collectively referred to as prokaryotes.

The hallmark feature that distinguishes eukaryotes from prokaryotes is that eukaryotic cells contain compartments called organelles that are surrounded by membranes. Each organelle supports different activities in the cell. Mitochondria, for example, are organelles that provide eukaryotes with most of their energy by producing energy-rich molecules called ATP. Prokaryotes lack mitochondria and instead produce their ATP on their cell surface membrane.

Some researchers have suggested that mitochondria might actually be one of the reasons that eukaryotic cells are typically larger than prokaryotes and more varied in their shape and structure. The thinking is that producing ATP on dedicated membranes inside the cell, rather than on the cell surface, boosted the amount of energy available to eukaryotic cells and allowed them to diversify more. However, other researchers are not convinced by this view. Moreover, some recent evidence suggested that eukaryotes are no more efficient in producing energy than prokaryotes.

Lynch and Marinov have now used computational and comparative analysis to compare the energy efficiency of different organisms including prokaryotes and eukaryotes grown under defined conditions. To do the comparison, the results were scaled based on cell volume and the total surface area deployed in energy production.

From their findings, Lynch and Marinov concluded that mitochondria did not enhance how much energy eukaryotes could produce per unit of cell volume in any substantial way. Although the origin of mitochondria was certainly a key event in evolutionary history, it is unlikely to have been responsible for the diversity and complexity of today’s life forms.

Rapid, precise quantification of bacterial cellular dimensions across a genomic-scale knockout library

BACKGROUND

The determination and regulation of cell morphology are critical components of cell-cycle control, fitness, and development in both single-cell and multicellular organisms. Understanding how environmental factors, chemical perturbations, and genetic differences affect cell morphology requires precise, unbiased, and validated measurements of cell-shape features.

RESULTS

Here we introduce two software packages, Morphometrics and BlurLab, that together enable automated, computationally efficient, unbiased identification of cells and morphological features. We applied these tools to bacterial cells because the small size of these cells and the subtlety of certain morphological changes have thus far obscured correlations between bacterial morphology and genotype. We used an online resource of images of the Keio knockout library of nonessential genes in the Gram-negative bacterium Escherichia coli to demonstrate that cell width, width variability, and length significantly correlate with each other and with drug treatments, nutrient changes, and environmental conditions. Further, we combined morphological classification of genetic variants with genetic meta-analysis to reveal novel connections among gene function, fitness, and cell morphology, thus suggesting potential functions for unknown genes and differences in modes of action of antibiotics.

CONCLUSIONS

Morphometrics and BlurLab set the stage for future quantitative studies of bacterial cell shape and intracellular localization. The previously unappreciated connections between morphological parameters measured with these software packages and the cellular environment point toward novel mechanistic connections among physiological perturbations, cell fitness, and growth.

Suppression of Beneficial Mutations in Dynamic Microbial Populations

Quantitative predictions for the spread of mutations in bacterial populations are essential to interpret evolution experiments and to improve the stability of synthetic gene circuits. We derive analytical expressions for the suppression factor for beneficial mutations in populations that undergo periodic dilutions, covering arbitrary population sizes, dilution factors, and growth advantages in a single stochastic model. We find that the suppression factor grows with the dilution factor and depends nontrivially on the growth advantage, resulting in the preferential elimination of mutations with certain growth advantages. We confirm our results by extensive numerical simulations.

Spatiotemporal Models of the Asymmetric Division Cycle of Caulobacter crescentus

The spatial localization of proteins within the cytoplasm of bacteria is an underappreciated but critical aspect of cell cycle regulation for many prokaryotes. In Caulobacter crescentus-a model organism for the study of asymmetric cell reproduction in prokaryotes-heterogeneous localization of proteins has been identified as the underlying cause of asymmetry in cell morphology, DNA replication, and cell division. However, significant questions remain. Firstly, the mechanisms by which proteins localize in the organelle-free prokaryotic cytoplasm remain obscure. Furthermore, how variations in the spatial and temporal dynamics of cell fate determinants regulate signaling pathways and orchestrate the complex programs of asymmetric cell division and differentiation are subjects of ongoing research. In this chapter, we review current efforts in investigating these two questions. We describe how mathematical models of spatiotemporal protein dynamics are being used to generate and test competing hypotheses and provide complementary insight about the control mechanisms that regulate asymmetry in protein localization and cell division.

Effects of temperature variability on community structure in a natural microbial food web

Climate change research has demonstrated that changing temperatures will have an effect on community-level dynamics by altering species survival rates, shifting species distributions, and ultimately, creating mismatches in community interactions. However, most of this work has focused on increasing temperature, and still little is known about how the variation in temperature extremes will affect community dynamics. We used the model aquatic community held within the leaves of the carnivorous plant, Sarracenia purpurea, to test how food web dynamics will be affected by high temperature variation. We tested the community response of the first (bacterial density), second (protist diversity and composition), and third trophic level (predator mortality), and measured community respiration. We collected early and late successional stage inquiline communities from S. purpurea from two North American and two European sites with similar average July temperature. We then created a common garden experiment in which replicates of these communities underwent either high or normal daily temperature variation, with the average temperature equal among treatments. We found an impact of temperature variation on the first two, but not on the third trophic level. For bacteria in the high-variation treatment, density experienced an initial boost in growth but then decreased quickly through time. For protists in the high-variation treatment, alpha-diversity decreased faster than in the normal-variation treatment, beta-diversity increased only in the European sites, and protist community composition tended to diverge more in the late successional stage. The mortality of the predatory mosquito larvae was unaffected by temperature variation. Community respiration was lower in the high-variation treatment, indicating a lower ecosystem functioning. Our results highlight clear impacts of temperature variation. A more mechanistic understanding of the effects that temperature, and especially temperature variation, will have on community dynamics is still greatly needed.

Relative Rates of Surface and Volume Synthesis Set Bacterial Cell Size

Many studies have focused on the mechanisms underlying length and width determination in rod-shaped bacteria. Here, we focus instead on cell surface area to volume ratio (SA/V) and demonstrate that SA/V homeostasis underlies size determination. We propose a model whereby the instantaneous rates of surface and volume synthesis both scale with volume. This model predicts that these relative rates dictate SA/V and that cells approach a new steady-state SA/V exponentially, with a decay constant equal to the volume growth rate. To test this, we exposed diverse bacterial species to sublethal concentrations of a cell wall biosynthesis inhibitor and observed dose-dependent decreases in SA/V. Furthermore, this decrease was exponential and had the expected decay constant. The model also quantitatively describes SA/V alterations induced by other chemical, nutritional, and genetic perturbations. We additionally present evidence for a surface material accumulation threshold underlying division, sensitizing cell length to changes in SA/V requirements.

To grow is not enough: impact of noise on cell environmental response and fitness

Quantitative single cell measurements have shown that cell cycle duration (the time between cell divisions) for diverse cell types is a noisy variable. The underlying distribution is mean scalable with a universal shape for many cell types in a variety of environments. Here we explore through both experiment and theory the response of these distributions to large environmental perturbations. In particular, we discuss how the stochasticity of the ensemble may be related to the response. Our findings show that slow growing, noisy populations are more adaptive than those which are fast growing. We suggest that even non-cooperative cells in exponential growth phase may not optimize fitness through growth rate alone, but also optimize adaptability to changing conditions. In this work, we wish to emphasize that in a manner similar to genetic evolution, noise in biochemical processes may be important to allow for cells to adapt to rapid to environmental changes.

Redefining the roles of the FtsZ-ring in bacterial cytokinesis

In most bacteria, cell division relies on the functions of an essential protein, FtsZ. FtsZ polymerizes at the future division site to form a ring-like structure, termed the Z-ring, that serves as a scaffold to recruit all other division proteins, and possibly generates force to constrict the cell. The scaffolding function of the Z-ring is well established, but the force generating function has recently been called into question. Additionally, new findings have demonstrated that the Z-ring is more directly linked to cell wall metabolism than simply recruiting enzymes to the division site. Here we review these advances and suggest that rather than generating a rate-limiting constrictive force, the Z-ring's function may be redefined as an orchestrator of septum synthesis.

Role of Proteome Physical Chemistry in Cell Behavior

We review how major cell behaviors, such as bacterial growth laws, are derived from the physical chemistry of the cell’s proteins. On one hand, cell actions depend on the individual biological functionalities of their many genes and proteins. On the other hand, the common physics among proteins can be as important as the unique biology that distinguishes them. For example, bacterial growth rates depend strongly on temperature. This dependence can be explained by the folding stabilities across a cell’s proteome. Such modeling explains how thermophilic and mesophilic organisms differ, and how oxidative damage of highly charged proteins can lead to unfolding and aggregation in aging cells. Cells have characteristic time scales. For example, E. coli can duplicate as fast as 2–3 times per hour. These time scales can be explained by protein dynamics (the rates of synthesis and degradation, folding, and diffusional transport). It rationalizes how bacterial growth is slowed down by added salt. In the same way that the behaviors of inanimate materials can be expressed in terms of the statistical distributions of atoms and molecules, some cell behaviors can be expressed in terms of distributions of protein properties, giving insights into the microscopic basis of growth laws in simple cells.

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.

Asymmetric Damage Segregation Constitutes an Emergent Population-Level Stress Response

Asymmetric damage segregation (ADS) is a mechanism for increasing population fitness through non-random, asymmetric partitioning of damaged macromolecules at cell division. ADS has been reported across multiple organisms, though the measured effects on fitness of individuals are often small. Here, we introduce a cell-lineage-based framework that quantifies the population-wide effects of ADS and then verify our results experimentally in E. coli under heat and antibiotic stress. Using an experimentally validated mathematical model, we find that the beneficial effect of ADS increases with stress. In effect, low-damage subpopulations divide faster and amplify within the population acting like a positive feedback loop whose strength scales with stress. Analysis of protein aggregates shows that the degree of asymmetric inheritance is damage dependent in single cells. Together our results indicate that, despite small effects in single cell, ADS exerts a strong beneficial effect on the population level and arises from the redistribution of damage within a population, through both single-cell and population-level feedback.

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.

Growth against entropy in bacterial metabolism: the phenotypic trade-off behind empirical growth rate distributions in E. coli

The solution space of genome-scale models of cellular metabolism provides a map between physically viable flux configurations and cellular metabolic phenotypes described, at the most basic level, by the corresponding growth rates. By sampling the solution space of E. coli's metabolic network, we show that empirical growth rate distributions recently obtained in experiments at single-cell resolution can be explained in terms of a trade-off between the higher fitness of fast-growing phenotypes and the higher entropy of slow-growing ones. Based on this, we propose a minimal model for the evolution of a large bacterial population that captures this trade-off. The scaling relationships observed in experiments encode, in such frameworks, for the same distance from the maximum achievable growth rate, the same degree of growth rate maximization, and/or the same rate of phenotypic change. Being grounded on genome-scale metabolic network reconstructions, these results allow for multiple implications and extensions in spite of the underlying conceptual simplicity.

Cell-Size Homeostasis and the Incremental Rule in a Bacterial Pathogen

How populations of growing cells achieve cell-size homeostasis remains a major question in cell biology. Recent studies in rod-shaped bacteria support the "incremental rule" where each cell adds a constant length before dividing. Although this rule explains narrow cell-size distributions, its mechanism is still unknown. We show that the opportunistic pathogen Pseudomonas aeruginosa obeys the incremental rule to achieve cell-length homeostasis during exponential growth but shortens its cells when entering the stationary phase. We identify a mutant, called frik, which has increased antibiotic sensitivity, cells that are on average longer, and a fraction of filamentous cells longer than 10 μm. When growth slows due to entry in stationary phase, the distribution of frik cell sizes decreases and approaches wild-type length distribution. The rare filamentous cells have abnormally large nucleoids, suggesting that a deficiency in DNA segregation prevents cell division without slowing the exponential elongation rate.

Optimal Stochastic Restart Renders Fluctuations in First Passage Times Universal

Stochastic restart may drastically reduce the expected run time of a computer algorithm, expedite the completion of a complex search process, or increase the turnover rate of an enzymatic reaction. These diverse first-passage-time (FPT) processes seem to have very little in common but it is actually quite the other way around. Here we show that the relative standard deviation associated with the FPT of an optimally restarted process, i.e., one that is restarted at a constant (nonzero) rate which brings the mean FPT to a minimum, is always unity. We interpret, further generalize, and discuss this finding and the implications arising from it.

Cell-shape homeostasis in Escherichia coli is driven by growth, division, and nucleoid complexity

Analysis of recently published high-throughput measurements of wild-type Escherichia coli cells growing at a wide range of rates demonstrates that cell width W, which is constant at any particular growth rate, is related (with a CV = 2.4%) to the level of nucleoid complexity, expressed as the amount of DNA in genome equivalents that is associated with chromosome terminus (G/terC). The relatively constant (CV = 7.3%) aspect ratio of newborn cells (Lb/W) in populations growing at different rates indicates existence of cell-shape homeostasis. Enlarged W of thymine-limited thyA mutants growing at identical rates support the hypothesis that nucleoid complexity actively affects W. Nucleoid dynamics is proposed to transmit a primary signal to the peptidoglycan-synthesizing system through the transertion mechanism, i.e., coupled transcription/translation of genes encoding membrane proteins and inserting these proteins into the membrane.

The Biokinetic Spectrum for Temperature

We identify and describe the distribution of temperature-dependent specific growth rates for life on Earth, which we term the biokinetic spectrum for temperature. The spectrum has the potential to provide for more robust modeling in thermal ecology since any conclusions derived from it will be based on observed data rather than using theoretical assumptions. It may also provide constraints for systems biology model predictions and provide insights in physiology. The spectrum has a Δ-shape with a sharp peak at around 42°C. At higher temperatures up to 60°C there was a gap of attenuated growth rates. We found another peak at 67°C and a steady decline in maximum rates thereafter. By using Bayesian quantile regression to summarise and explore the data we were able to conclude that the gap represented an actual biological transition between mesophiles and thermophiles that we term the Mesophile-Thermophile Gap (MTG). We have not identified any organism that grows above the maximum rate of the spectrum. We used a thermodynamic model to recover the Δ-shape, suggesting that the growth rate limits arise from a trade-off between activity and stability of proteins. The spectrum provides underpinning principles that will find utility in models concerned with the thermal responses of biological processes.

Shape dynamics of growing cell walls

We introduce a general theoretical framework to study the shape dynamics of actively growing and remodeling surfaces. Using this framework we develop a physical model for growing bacterial cell walls and study the interplay of cell shape with the dynamics of growth and constriction. The model allows us to derive constraints on cell wall mechanical energy based on the observed dynamics of cell shape. We predict that exponential growth in cell size requires a constant amount of cell wall energy to be dissipated per unit volume. We use the model to understand and contrast growth in bacteria with different shapes such as spherical, ellipsoidal, cylindrical and toroidal morphologies. Coupling growth to cell wall constriction, we predict a discontinuous shape transformation, from partial constriction to cell division, as a function of the chemical potential driving cell wall synthesis. Our model for cell wall energy and shape dynamics relates growth kinetics with cell geometry, and provides a unified framework to describe the interplay between shape, growth and division in bacterial cells.

In Vivo study of naturally deformed Escherichia coli bacteria

A combination of light-microscopy and image processing has been applied to study naturally deformed Escherichia coli under in vivo condition and at the order of sub-pixel high-resolution accuracy. To classify deflagellated non-dividing E. coli cells to the rod-shape and bent-shape, a geometrical approach has been applied. From the analysis of the geometrical data which were obtained of image processing, we estimated the required effective energy for shaping a rod-shape to a bent-shape with the same size. We evaluated the energy of deformation in the naturally deformed bacteria with minimum cell manipulation, under in vivo condition, and with minimum influence of any external force, torque and pressure. Finally, we have also elaborated on the possible scenario to explain how naturally deformed bacteria are formed from initial to final-stage.

Noise-driven growth rate gain in clonal cellular populations

Cellular populations in both nature and the laboratory are composed of phenotypically heterogeneous individuals that compete with each other resulting in complex population dynamics. Predicting population growth characteristics based on knowledge of heterogeneous single-cell dynamics remains challenging. By observing groups of cells for hundreds of generations at single-cell resolution, we reveal that growth noise causes clonal populations of Escherichia coli to double faster than the mean doubling time of their constituent single cells across a broad set of balanced-growth conditions. We show that the population-level growth rate gain as well as age structures of populations and of cell lineages in competition are predictable. Furthermore, we theoretically reveal that the growth rate gain can be linked with the relative entropy of lineage generation time distributions. Unexpectedly, we find an empirical linear relation between the means and the variances of generation times across conditions, which provides a general constraint on maximal growth rates. Together, these results demonstrate a fundamental benefit of noise for population growth, and identify a growth law that sets a "speed limit" for proliferation.

Adder and a coarse-grained approach to cell size homeostasis in bacteria

Cell size control and homeostasis is a long-standing subject in biology. Recent experimental work provides extensive evidence for a simple, quantitative size homeostasis principle coined adder (as opposed to sizer or timer). The adder principle provides unexpected insights into how bacteria maintain their size without employing a feedback mechanism. We review the genesis of adder and recent cell size homeostasis study on evolutionarily divergent bacterial organisms and beyond. We propose new coarse-grained approaches to understand the underlying mechanisms of cell size control at the whole cell level.

Individuality and universality in the growth-division laws of single E. coli cells

The mean size of exponentially dividing Escherichia coli cells in different nutrient conditions is known to depend on the mean growth rate only. However, the joint fluctuations relating cell size, doubling time, and individual growth rate are only starting to be characterized. Recent studies in bacteria reported a universal trend where the spread in both size and doubling times is a linear function of the population means of these variables. Here we combine experiments and theory and use scaling concepts to elucidate the constraints posed by the second observation on the division control mechanism and on the joint fluctuations of sizes and doubling times. We found that scaling relations based on the means collapse both size and doubling-time distributions across different conditions and explain how the shape of their joint fluctuations deviates from the means. Our data on these joint fluctuations highlight the importance of cell individuality: Single cells do not follow the dependence observed for the means between size and either growth rate or inverse doubling time. Our calculations show that these results emerge from a broad class of division control mechanisms requiring a certain scaling form of the "division hazard rate function," which defines the probability rate of dividing as a function of measurable parameters. This "model free" approach gives a rationale for the universal body-size distributions observed in microbial ecosystems across many microbial species, presumably dividing with multiple mechanisms. Additionally, our experiments show a crossover between fast and slow growth in the relation between individual-cell growth rate and division time, which can be understood in terms of different regimes of genome replication control.

Single-Cell Analysis of Growth in Budding Yeast and Bacteria Reveals a Common Size Regulation Strategy

To maintain a constant cell size, dividing cells have to coordinate cell-cycle events with cell growth. This coordination has long been supposed to rely on the existence of size thresholds determining cell-cycle progression [1]. In budding yeast, size is controlled at the G1/S transition [2]. In agreement with this hypothesis, the size at birth influences the time spent in G1: smaller cells have a longer G1 period [3]. Nevertheless, even though cells born smaller have a longer G1, the compensation is imperfect and they still bud at smaller cell sizes. In bacteria, several recent studies have shown that the incremental model of size control, in which size is controlled by addition of a constant volume (in contrast to a size threshold), is able to quantitatively explain the experimental data on four different bacterial species [4-7]. Here, we report on experimental results for the budding yeast Saccharomyces cerevisiae, finding, surprisingly, that cell size control in this organism is very well described by the incremental model, suggesting a common strategy for cell size control with bacteria. Additionally, we argue that for S. cerevisiae the "volume increment" is not added from birth to division, but rather between two budding events.

Self-Consistent Examination of Donachie's Constant Initiation Size at the Single-Cell Level

How growth, the cell cycle, and cell size are coordinated is a fundamental question in biology. Recently, we and others have shown that bacterial cells grow by a constant added size per generation, irrespective of the birth size, to maintain size homeostasis. This "adder" principle raises a question as to when during the cell cycle size control is imposed. Inspired by this question, we examined our single-cell data for initiation size by employing a self-consistency approach originally used by Donachie. Specifically, we assumed that individual cells divide after constant C + D minutes have elapsed since initiation, independent of the growth rate. By applying this assumption to the cell length vs. time trajectories from individual cells, we were able to extract theoretical probability distribution functions for initiation size for all growth conditions. We found that the probability of replication initiation shows peaks whenever the cell size is a multiple of a constant unit size, consistent with the Donachie's original analysis at the population level. Our self-consistent examination of the single-cell data made experimentally testable predictions, e.g., two consecutive replication cycles can be initiated during a single cell-division cycle.

Universal protein distributions in a model of cell growth and division

Protein distributions measured under a broad set of conditions in bacteria and yeast were shown to exhibit a common skewed shape, with variances depending quadratically on means. For bacteria these properties were reproduced by temporal measurements of protein content, showing accumulation and division across generations. Here we present a stochastic growth-and-division model with feedback which captures these observed properties. The limiting copy number distribution is calculated exactly, and a single parameter is found to determine the distribution shape and the variance-to-mean relation. Estimating this parameter from bacterial temporal data reproduces the measured distribution shape with high accuracy and leads to predictions for future experiments.

Single-cell protein dynamics reproduce universal fluctuations in cell populations

Protein variability in single cells has been studied extensively in populations, but little is known about temporal protein fluctuations in a single cell over extended times. We present here traces of protein copy number measured in individual bacteria over multiple generations and investigate their statistical properties, comparing them to previously measured population snapshots. We find that temporal fluctuations in individual cells exhibit the same properties as those previously observed in populations. Scaled fluctuations around the mean of each trace exhibit the universal distribution shape measured in populations under a wide range of conditions and in two distinct microorganisms; the mean and variance of the traces over time obey the same quadratic relation. Analyzing the individual protein traces reveals that within a cell cycle protein content increases exponentially, with a rate that varies from cycle to cycle. This leads to a compact description of the trace as a 3-variable stochastic process -exponential rate, cell cycle duration and value at the cycle start- sampled once a cycle. This description is sufficient to reproduce both universal statistical properties of the protein fluctuations. Our results show that the protein distribution shape is insensitive to sub-cycle intracellular microscopic details and reflects global cellular properties that fluctuate between generations.

Chromosome replication, cell growth, division and shape: a personal perspective

The origins of Molecular Biology and Bacterial Physiology are reviewed, from our personal standpoints, emphasizing the coupling between bacterial growth, chromosome replication and cell division, dimensions and shape. Current knowledge is discussed with historical perspective, summarizing past and present achievements and enlightening ideas for future studies. An interactive simulation program of the bacterial cell division cycle (BCD), described as "The Central Dogma in Bacteriology," is briefly represented. The coupled process of transcription/translation of genes encoding membrane proteins and insertion into the membrane (so-called transertion) is invoked as the functional relationship between the only two unique macromolecules in the cell, DNA and peptidoglycan embodying the nucleoid and the sacculus respectively. We envision that the total amount of DNA associated with the replication terminus, so called "nucleoid complexity," is directly related to cell size and shape through the transertion process. Accordingly, the primary signal for cell division transmitted by DNA dynamics (replication, transcription and segregation) to the peptidoglycan biosynthetic machinery is of a physico-chemical nature, e.g., stress in the plasma membrane, relieving nucleoid occlusion in the cell's center hence enabling the divisome to assemble and function between segregated daughter nucleoids.

Size sensors in bacteria, cell cycle control, and size control

Bacteria proliferate by repetitive cycles of cellular growth and division. The progression into the cell cycle is admitted to be under the control of cell size. However, the molecular basis of this regulation is still unclear. Here I will discuss which mechanisms could allow coupling growth and division by sensing size and transmitting this information to the division machinery. Size sensors could act at different stages of the cell cycle. During septum formation, mechanisms controlling the formation of the Z ring, such as MinCD inhibition or Nucleoid Occlusion (NO) could participate in the size-dependence of the division process. In addition or alternatively, the coupling of growth and division may occur indirectly through the control of DNA replication initiation. The relative importance of these different size-sensing mechanisms could depend on the environmental and genetic context. The recent demonstration of an incremental strategy of size control in bacteria, suggests that DnaA-dependent control of replication initiation could be the major size control mechanism limiting cell size variation.

Intergenerational continuity of cell shape dynamics in Caulobacter crescentus

We investigate the intergenerational shape dynamics of single Caulobacter crescentus cells using a novel combination of imaging techniques and theoretical modeling. We determine the dynamics of cell pole-to-pole lengths, cross-sectional widths, and medial curvatures from high accuracy measurements of cell contours. Moreover, these shape parameters are determined for over 250 cells across approximately 10000 total generations, which affords high statistical precision. Our data and model show that constriction is initiated early in the cell cycle and that its dynamics are controlled by the time scale of exponential longitudinal growth. Based on our extensive and detailed growth and contour data, we develop a minimal mechanical model that quantitatively accounts for the cell shape dynamics and suggests that the asymmetric location of the division plane reflects the distinct mechanical properties of the stalked and swarmer poles. Furthermore, we find that the asymmetry in the division plane location is inherited from the previous generation. We interpret these results in terms of the current molecular understanding of shape, growth, and division of C. crescentus.

Growth rate and cell size: a re-examination of the growth law

Research into the mechanisms regulating bacterial cell size has its origins in a single paper published over 50 years ago. In it Schaechter and colleagues made the observation that the chemical composition and size of a bacterial cell is a function of growth rate, independent of the medium used to achieve that growth rate, a finding that is colloquially referred to as the growth law. Recent findings hint at unforeseen complexity in the growth law, and suggest that nutrients rather than growth rate are the primary arbiter of size. The emerging picture suggests that size is a complex, multifactorial phenomenon mediated through the varied impacts of central carbon metabolism on cell cycle progression and biosynthetic capacity.

Cell-size control and homeostasis in bacteria

How cells control their size and maintain size homeostasis is a fundamental open question. Cell-size homeostasis has been discussed in the context of two major paradigms: "sizer," in which the cell actively monitors its size and triggers the cell cycle once it reaches a critical size, and "timer," in which the cell attempts to grow for a specific amount of time before division. These paradigms, in conjunction with the "growth law" [1] and the quantitative bacterial cell-cycle model [2], inspired numerous theoretical models [3-9] and experimental investigations, from growth [10, 11] to cell cycle and size control [12-15]. However, experimental evidence involved difficult-to-verify assumptions or population-averaged data, which allowed different interpretations [1-5, 16-20] or limited conclusions [4-9]. In particular, population-averaged data and correlations are inconclusive as the averaging process masks causal effects at the cellular level. In this work, we extended a microfluidic "mother machine" [21] and monitored hundreds of thousands of Gram-negative Escherichia coli and Gram-positive Bacillus subtilis cells under a wide range of steady-state growth conditions. Our combined experimental results and quantitative analysis demonstrate that cells add a constant volume each generation, irrespective of their newborn sizes, conclusively supporting the so-called constant Δ model. This model was introduced for E. coli [6, 7] and recently revisited [9], but experimental evidence was limited to correlations. This "adder" principle quantitatively explains experimental data at both the population and single-cell levels, including the origin and the hierarchy of variability in the size-control mechanisms and how cells maintain size homeostasis.

Cell-size maintenance: universal strategy revealed

How cells maintain a stable size has fascinated scientists since the beginning of modern biology, but has remained largely mysterious. Recently, however, the ability to analyze single bacteria in real time has provided new, important quantitative insights into this long-standing question in cell biology.