Generality of the growth kinetics of the average individual cell in different bacterial populations

Protection of bacteriophage-sensitive Escherichia coli by lysogens

SignificanceSome viruses that infect bacteria, temperate bacteriophages, can confer immunity to infection by the same virus. Here we report λ-immune bacteria could protect λ-sensitive bacteria from killing by phage λ in mixed culture. The protection depended on the extent to which the immune bacteria were able to adsorb the phage. Reconciling modeling with experiment led to identifying a decline in protection as bacteria stopped growing. Adsorption of λ was compromised by inhibition of bacterial energy metabolism, explaining the loss of protection as bacterial growth ceased.

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

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

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

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.

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.

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.

Threshold effect of growth rate on population variability of Escherichia coli cell lengths

A long-standing question in biology is the effect of growth on cell size. Here, we estimate the effect of Escherichia coli growth rate (r) on population cell size distributions by estimating the coefficient of variation of cell lengths (CV_L) from image analysis of fixed cells in DIC microscopy. We find that the CV_L is constant at growth rates less than one division per hour, whereas above this threshold, CV_L increases with an increase in the growth rate. We hypothesize that stochastic inhibition of cell division owing to replication stalling by a RecA-dependent mechanism, combined with the growth rate threshold of multi-fork replication (according to Cooper and Helmstetter), could form the basis of such a threshold effect. We proceed to test our hypothesis by increasing the frequency of stochastic stalling of replication forks with hydroxyurea (HU) treatment and find that cell length variability increases only when the growth rate exceeds this threshold. The population effect is also reproduced in single-cell studies using agar-pad cultures and 'mother machine'-based experiments to achieve synchrony. To test the role of RecA, critical for the repair of stalled replication forks, we examine the CV_L of E. coli ΔrecA cells. We find cell length variability in the mutant to be greater than wild-type, a phenotype that is rescued by plasmid-based RecA expression. Additionally, we find that RecA-GFP protein recruitment to nucleoids is more frequent at growth rates exceeding the growth rate threshold and is further enhanced on HU treatment. Thus, we find growth rates greater than a threshold result in increased E. coli cell lengths in the population, and this effect is, at least in part, mediated by RecA recruitment to the nucleoid and stochastic inhibition of division.

Threshold effect of growth rate on population variability of Escherichia coli cell lengths

A long-standing question in biology is the effect of growth on cell size. Here, we estimate the effect of Escherichia coli growth rate (r) on population cell size distributions by estimating the coefficient of variation of cell lengths (CV_L) from image analysis of fixed cells in DIC microscopy. We find that the CV_L is constant at growth rates less than one division per hour, whereas above this threshold, CV_L increases with an increase in the growth rate. We hypothesize that stochastic inhibition of cell division owing to replication stalling by a RecA-dependent mechanism, combined with the growth rate threshold of multi-fork replication (according to Cooper and Helmstetter), could form the basis of such a threshold effect. We proceed to test our hypothesis by increasing the frequency of stochastic stalling of replication forks with hydroxyurea (HU) treatment and find that cell length variability increases only when the growth rate exceeds this threshold. The population effect is also reproduced in single-cell studies using agar-pad cultures and 'mother machine'-based experiments to achieve synchrony. To test the role of RecA, critical for the repair of stalled replication forks, we examine the CV_L of E. coli ΔrecA cells. We find cell length variability in the mutant to be greater than wild-type, a phenotype that is rescued by plasmid-based RecA expression. Additionally, we find that RecA-GFP protein recruitment to nucleoids is more frequent at growth rates exceeding the growth rate threshold and is further enhanced on HU treatment. Thus, we find growth rates greater than a threshold result in increased E. coli cell lengths in the population, and this effect is, at least in part, mediated by RecA recruitment to the nucleoid and stochastic inhibition of division.

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.

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.

Scaling laws governing stochastic growth and division of single bacterial cells

Uncovering the quantitative laws that govern the growth and division of single cells remains a major challenge. Using a unique combination of technologies that yields unprecedented statistical precision, we find that the sizes of individual Caulobacter crescentus cells increase exponentially in time. We also establish that they divide upon reaching a critical multiple (≈ 1.8) of their initial sizes, rather than an absolute size. We show that when the temperature is varied, the growth and division timescales scale proportionally with each other over the physiological temperature range. Strikingly, the cell-size and division-time distributions can both be rescaled by their mean values such that the condition-specific distributions collapse to universal curves. We account for these observations with a minimal stochastic model that is based on an autocatalytic cycle. It predicts the scalings, as well as specific functional forms for the universal curves. Our experimental and theoretical analysis reveals a simple physical principle governing these complex biological processes: a single temperature-dependent scale of cellular time governs the stochastic dynamics of growth and division in balanced growth conditions.

Heterogeneous structure of stem cells dynamics: statistical models and quantitative predictions

Understanding stem cell (SC) population dynamics is essential for developing models that can be used in basic science and medicine, to aid in predicting cells fate. These models can be used as tools e.g. in studying patho-physiological events at the cellular and tissue level, predicting (mal)functions along the developmental course, and personalized regenerative medicine. Using time-lapsed imaging and statistical tools, we show that the dynamics of SC populations involve a heterogeneous structure consisting of multiple sub-population behaviors. Using non-Gaussian statistical approaches, we identify the co-existence of fast and slow dividing subpopulations, and quiescent cells, in stem cells from three species. The mathematical analysis also shows that, instead of developing independently, SCs exhibit a time-dependent fractal behavior as they interact with each other through molecular and tactile signals. These findings suggest that more sophisticated models of SC dynamics should view SC populations as a collective and avoid the simplifying homogeneity assumption by accounting for the presence of more than one dividing sub-population, and their multi-fractal characteristics.

Timing the start of division in E. coli: a single-cell study

We monitor the shape dynamics of individual E. coli cells using time-lapse microscopy together with accurate image analysis. This allows measuring the dynamics of single-cell parameters throughout the cell cycle. In previous work, we have used this approach to characterize the main features of single-cell morphogenesis between successive divisions. Here, we focus on the behavior of the parameters that are related to cell division and study their variation over a population of 30 cells. In particular, we show that the single-cell data for the constriction width dynamics collapse onto a unique curve following appropriate rescaling of the corresponding variables. This suggests the presence of an underlying time scale that determines the rate at which the cell cycle advances in each individual cell. For the case of cell length dynamics a similar rescaling of variables emphasizes the presence of a breakpoint in the growth rate at the time when division starts, tau(c). We also find that the tau(c) of individual cells is correlated with their generation time, tau(g), and inversely correlated with the corresponding length at birth, L(0). Moreover, the extent of the T-period, tau(g) - tau(c), is apparently independent of tau(g). The relations between tau(c), tau(g) and L(0) indicate possible compensation mechanisms that maintain cell length variability at about 10%. Similar behavior was observed for both fast-growing cells in a rich medium (LB) and for slower growth in a minimal medium (M9-glucose). To reveal the molecular mechanisms that lead to the observed organization of the cell cycle, we should further extend our approach to monitor the formation of the divisome.

Measuring diameters of rod-shaped bacteria in vivo with polarized light scattering

The angular function for elements of the Mueller matrix for polarized light scattering from suspensions of microorganisms is known to be reproducible for different growths of a given bacterial strain in the log (or exponential) phase of growth. The reason for this, the stability of the size and shape distribution for cells, is briefly discussed. Experiments were performed using suspensions of two different strains of Escherichia coli cells in log phase and measuring the angular dependence of the Mueller matrix ratio S34/S11. Calculations were then performed using the coupled dipole approximation to model electromagnetic scattering from particles where the shape of an individual cell was approximated by a cylinder capped with hemispheres of the same radius as the cylinder. Using previously measured values for the length distribution and index of refraction of the cells, the calculated scattering curve was found to fit the measured curve very well. The values obtained for the cell diameters were quite close to diameters previously measured by optical microscopy. Thus this method provides a rapid and convenient method for monitoring bacterial diameters in vivo even when there is an appreciable distribution of bacterial lengths in the population.

The growth kinetics of B. subtilis

There has been considerable discussion by Kubitschek and Cooper concerning the growth rate of cells of E. coli throughout the cell cycle. Consequently, it is relevant to test Kubitschek's linear model against the exponential model espoused by Cooper (and many others) with another organism and another technique. Burdett et al. measured, by electron microscopy and computer analysis of the microphotographs, the distribution of lengths of a population of cells of Bacillus subtilis grown in 0.4% succinate in a minimal medium. The data were fitted to the extended Collins-Richmond method of Kirkwood & Burdett which subdivided the cell cycle into several phases. I have taken their results and compared them with the linear and exponential growth models for the entire cell cycle after applying correction to the data for the shape of completed and forming poles; i.e., to put the data on a cell-volume basis instead of a cell-length basis. Most of the correction involves no arbitrary assumptions. The conclusion is that global volume growth rate is nearly proportional to cell volume; i.e. growth of Bacillus subtilis is nearly exponential for almost every cell in the growing culture.

Influence of low-temperature storage and glucose starvation on growth recovery in Escherichia coli relA and relA+ strains

To study the influence of microgravity on bacterial growth behavior during a space mission, the special experimental conditions and the hardware environment necessitate storage of cells at low temperature, and permit a relatively short experimental period. Before this experimental period, cells have to recover their condition of steady-state growth, because it is only in this condition that the growth behavior of the flight and ground populations can be adequately compared. To meet these requirements and to obtain cells which recover rapidly their steady-state growth, we analyzed the size and shape of Escherichia coli cells during storage at 4 degrees C, with and without previous glucose starvation of the cells. It appeared that cells stored at low temperature in the presence of glucose continued to increase in average mass and assumed ovoid shapes. In addition, upon restoration of maximal growth rate at 37 degrees C, they continued to increase in size and showed a transient overshoot of their final steady-state value, which was reached after about 5 h. Cells previously starved for glucose, however, maintained their average size and rod-shape during low-temperature storage. Recovery of the starved cells was most rapid in the relA+ strain which, contrary to the isogenic relA strain, showed no overshoot and reached its final steady-state size within 2 h.

Topology of penicillin-binding protein 1b of Escherichia coli and topography of four antigenic determinants studied by immunocolabeling electron microscopy

A method has been developed to study the orientation of proteins in the cytoplasmic membrane of Escherichia coli. Vesicles from sonicated cells were incubated in droplets on electron microscope support grids in sequence with a monoclonal antibody (MAb) against a protein with an unknown orientation (PBP 1b) followed by a MAb against a periplasmic component (peptidoglycan). The different MAbs were made visible with 5- and 10-nm gold-conjugated secondary antibodies, respectively. PBP 1b appeared to colabel with peptidoglycan. The labeling of PBP 1b in membrane vesicles with MAbs against four different epitopes was further used to estimate the number of PBP 1b molecules per cell. Approximately 1,400 PBP 1b molecules per cell grown in broth were labeled. The spatial distribution of the epitopes of the MAbs was studied by immunocolabeling of pairs of MAbs and by competitive antibody-binding inhibition. It could be tentatively concluded that the four epitopes form a cluster of antigenic determinants which occupy less than half of the surface of PBP 1b.

Rate and topography of cell wall synthesis during the division cycle of Salmonella typhimurium

The rates of synthesis of peptidoglycan and protein during the division cycle of Salmonella typhimurium have been measured by using the membrane elution technique and differentially labeled diaminopimelic acid and leucine. The cells were labeled during unperturbed exponential growth and then bound to a nitrocellulose membrane by filtration. Newborn cells were eluted from the membrane with fresh medium. The radioactivity in the newborn cells in successive fractions was determined. As the cells are eluted from the membrane as a function of their cell cycle age at the time of labeling, the rate of incorporation of the different radioactive compounds as a function of cell cycle age can be determined. During the first part of the division cycle, the ratio of the rates of protein and peptidoglycan synthesis was constant. During the latter part of the division cycle, there was an increase in the rate of peptidoglycan synthesis relative to the rate of protein synthesis. These results support a simple, bipartite model of cell surface increase in rod-shaped cells. Before the start of constriction, the cell surface increased only by cylindrical extension. After cell constriction started, the cell surface increased by both cylinder and pole growth. The increase in surface area was partitioned between the cylinder and the pole so that the volume of the cell increased exponentially. No variation in cell density occurred because the increase in surface allowed a continuous exponential increase in cell volume that accommodated the exponential increase in cell mass. Protein was synthesized exponentially during the division cycle. The rate of cell surface increase was described by a complex equation which is neither linear nor exponential.

Elongation and surface extension of individual cells of Escherichia coli B/r: comparison of theoretical and experimental size distributions

The way individual cells grow and divide uniquely determines the (time-invariant) cell size distribution of populations in steady-state exponential growth. In the preceding article, theoretical distributions were derived for two exponential and six linear models containing a small number of adjustable parameters but no assumptions other than that all cells obey the same growth law. The linear models differ from each other with respect to the timing of the presumptive doubling in their growth rate, the exponential models--according to whether there is or is not a part of the cell that does not contribute to the growth rate. Here we compared the size distributions predicted by each of these models with those of cell length and surface area measured by electron microscopy; the quality of the fit, as determined by the mean-square successive-differences test and the chi 2 goodness-of-fit test, was taken as a measure of the adequacy of the model. The actual data came from two slow-growing E. coli B/r cultures, an A strain (pi = 125 min) and a K strain (pi = 106 min), and a correction was introduced in each to account for the distortion caused by the finite size of the picture frame. The parameter estimates produced by the various models are quite reliable (cv less than 0.1%); we discuss them briefly and compare their values in the two strains. All the length extension models were rejected outright whereas most of the surface growth versions were not. When the same models were tested on A-strain data from a faster growing culture (tau = 21 min), those models that provided an adequate fit to the cell surface area data proved equally satisfactory in the case of cell length. These findings are evaluated and shown to be consistent with cell surface area rather than cell length being the dimension under active control. Three surface area models, all linear, are rejected--those in which doubling of the growth rate occurs with a constant probability from cell birth, at a particular cell age, and precisely at cell division. The evidence in the literature that appears to contradict this last result, rejection of the simple linear surface growth model, is shown to be faulty. The 16 original models are here reduced to five, two involving exponential surface growth and three linear, and possible reasons are presented for our inability to discriminate further at this stage.

Effect of growth rate and cell shape on the peptidoglycan composition in Escherichia coli

The muropeptide composition of peptidoglycan from Escherichia coli W7 cultivated at different growth rates in chemostat cultures was compared by using high-pressure liquid chromatography. At a low growth rate (D = 0.1 h-1), about 40% more covalently bound lipoprotein and at least twofold more diaminopimelyl-diaminopimelic acid cross-bridges were found than at a high growth rate (D = 0.8 h-1). The total degree of cross-linkage was only slightly increased, and the fraction of trimeric muropeptides and the average length of the glycan chains were not changed significantly. Analysis of the peptidoglycan from a morphological variant strain of W7 revealed that the altered peptidoglycan composition in slowly growing W7 cells was not correlated with the observation that these cells, due to their decreased cell length, were relatively enriched in polar material. In fact, our results suggested that peptidoglycan forming cell poles is chemically identical to that forming lateral wall.

Growth kinetics of individual Bacillus subtilis cells and correlation with nucleoid extension

The growth rate of individual cells of Bacillus subtilis (doubling time, 120 min) has been calculated by using a modification of the Collins-Richmond principle which allows the growth rate of mononucleate, binucleate, and septate cells to be calculated separately. The standard Collins-Richmond equation represents a weighted average of the growth rate calculated from these three major classes. Both approaches strongly suggest that the rate of length extension is exponential. By preparing critical-point-dried cells, in which major features of the cell such as nucleoids and cross-walls can be seen, it has also been possible to examine whether nucleoid extension is coupled to length extension. Growth rates for nucleoid movement are parallel to those of total length extension, except possibly in the case of septate cells. Furthermore, by calculating the growth rate of various portions of the cell surface, it appears likely that the limits of the site of cylindrical envelope assembly lie between the distal tips of the nucleoid; the old poles show zero growth rate. Coupling of nucleoid extension with increase of cell length is envisaged as occurring through an exponentially increasing number of DNA-surface attachment sites occupying most of the available surface.

Estimating the mode of growth of individual microbial cells from cell volume distributions

Two new methods are derived for inferring the mode of growth of individual microbial cells from measurements made of the volume distributions of populations. One is based on statistics of the observed distribution and has the particular advantage that it is very easy to use. The second, which requires gradient centrifugation, yields the mode of growth directly, rather than by comparison with theoretically derived distributions. Both methods have been found to be more sensitive than those previously suggested.

Cell elongation and division probability during the Escherichia coli growth cycle

A new method is presented for determining the growth rate and the probability of cell division (separation) during the cell cycle, using size distributions of cell populations grown under steady-state conditions. The method utilizes the cell life-length distribution, i.e., the probability that a cell will have any specific size during its life history. This method was used to analyze cell length distributions of six cultures of Escherichia coli, for which doubling times varied from 19 to 125 min. The results for each culture are in good agreement with a single model of growth and division kinetics: exponential elongation of cells during growth phase of the cycle, and normal distributions of length at birth and at division. The average value of the coefficient of variation was 13.5% for all strains and growth rates. These results, based upon 5,955 observations, support and extend earlier proposals that growth and division patterns of E. coli are similar at all growth rates and, in addition, identify the general growth pattern of these cells to be exponential.

Effects of temperature on the size and shape of Escherichia coli cells

Two substrains of Escherichia coli B/r were grown to steady-state in batch cultures at temperatures between 22 and 42 degrees C in different growth media. The size and shape of the cells were measured from light and electron micrographs and with the Coulter channelizer. The results indicate that cells are shorter and somewhat thicker at the lower temperatures, especially in rich growth media; cell volume is then slightly smaller. A lower temperature was further found to increase the relative duration of the constriction period. The shapes of the cell size distributions are indistinguishable, indicating that the pattern of growth of the cells is the same at all temperatures. The adaptation of the cells to a temperature shift lasted several generations, indicating that the morphological effects of temperature are mediated by the cell's physiology.