How does a bacterium grow during its cell cycle?

Rod-shaped bacteria such as Escherichia coli and Bacillus subtilis appear to extend continuously in length between divisions. However, the kinetics of growth of the individual cell in the steady state is still unknown. A brief, critical account of the main approaches used to determine the pattern of surface extension is given. In general, these approaches are of three types. Firstly, attempts have been made to relate average cell size to growth rate of the culture and to determine possible stages in the cell cycle at which the rate of length extension might change. Secondly, comparisons have been made between the measured length distribution of cells and theoretical distributions, based on three primary hypotheses (linear, bilinear and exponential growth). Thirdly, the principle of Collins and Richmond, involving the calculation of growth rate from the length distributions of extant, separating and new-born cells, is described. It is emphasized that there is a strong element of variation in size at different stages of the cell cycle. This variation imposes severe limitations on models which utilize only average cellular dimensions. We conclude that the Collins-Richmond principle affords the most powerful approach to the analysis of bacterial growth kinetics. However, we propose that the method be modified to permit calculation of separate rates of growth of cells between discernible events in the cell cycle, as well as simply between birth and division.     

Predicted steady-state cell size distributions for various growth models

The question of how an individual bacterial cell grows during its life cycle remains controversial. In 1962 Collins and Richmond derived a very general expression relating the size distributions of newborn, dividing and extant cells in steady-state growth and their growth rate; it represents the most powerful framework currently available for the analysis of bacterial growth kinetics. The Collins-Richmond equation is in effect a statement of the conservation of cell numbers for populations in steady-state exponential growth. It has usually been used to calculate the growth rate from a measured cell size distribution under various assumptions regarding the dividing and newborn cell distributions, but can also be applied in reverse--to compute the theoretical cell size distribution from a specified growth law. This has the advantage that it is not limited to models in which growth rate is a deterministic function of cell size, such as in simple exponential or linear growth, but permits evaluation of far more sophisticated hypotheses. Here we employed this reverse approach to obtain theoretical cell size distributions for two exponential and six linear growth models. The former differ as to whether there exists in each cell a minimal size that does not contribute to growth, the latter as to when the presumptive doubling of the growth rate takes place: in the linear age models, it is taken to occur at a particular cell age, at a fixed time prior to division, or at division itself; in the linear size models, the growth rate is considered to double with a constant probability from cell birth, with a constant probability but only after the cell has reached a minimal size, or after the minimal size has been attained but with a probability that increases linearly with cell size. Each model contains a small number of adjustable parameters but no assumptions other than that all cells obey the same growth law. In the present article, the various growth laws are described and rigorous mathematical expressions developed to predict the size distribution of extant cells in steady-state exponential growth; in the following paper, these predictions are tested against high-quality experimental data.     

Cell cycle dynamics inferred from the static properties of cells in balanced growth

The duration of a morphological phase of the cell cycle is reflected in the steady state distribution of the sizes of cells in that phase. Relationships presented here provide a method for estimating the timing and variability of any cell cycle phase. It is shown that the mean size of cells initiating and finishing any phase can be estimated from (1) the frequency of cells exhibiting the distinguishing morphological or autoradiographic features of the phase; (2) the mean size of cells in the phase; and (3) their coefficient of variation. The calculations are based on a submodel of the Koch-Schaechter Growth Controlled Model which assumes that (i) the distribution of division sizes is Gaussian; (ii) there is no correlation in division sizes between successive generations; and (iii) every cell division gives rise to two daughter cells of equal size. The calculations should be useful for a wider range of models, however, because the extrapolation factors are not sensitive to the chosen model. Criteria are proposed to allow the user to check the method's applicability for any experimental case. The method also provides a more efficient test of the dependence of growth on cell size than does the Collins-Richmond method. This is because the method uses the mean and coefficient of variation of the size of the total population, in conjunction with those of the cells in a final phase of the cell cycle, to test potential growth laws. For Escherichia coli populations studied by electron microscopy, an exponential growth model provided much better agreement than did a linear growth model. The computer simulations were used to generate rules for three types of cell phases: those that end at cell division, those that start at cell division, and those totally contained within a single cell cycle. For the last type, additional criteria are proposed to establish if the phase is well enough contained for the formulae and graphs to be used. The most useful rule emerging from these computer studies is that the fraction of the cell cycle time occupied by a phase is the product of the frequency of the phase and the ratio of the mean size of cells in that phase to the mean size of all cells in the population. A further advantage of the techniques presented here is that they use the 'extant' distributions that were actually measured, and not hypothesized distributions nor the special distributions needed for Collins-Richmond method that can only be calculated from the observed distributions of dividing or newborn cells on the basis of an assumed growth law.     

The distributions of cell size and generation time in a model of the cell cycle incorporating size control and random transitions

A deterministic/probabilistic model of the cell division cycle is analysed mathematically and compared to experimental data and to other models of the cell cycle. The model posits a random-exiting phase of the cell cycle and a minimum-size requirement for entry into the random-exiting phase. By design, the model predicts exponential "beta-curves", which are characteristic of sister cell generation times. We show that the model predicts "alpha-curves" with exponential tails and hyperbolic-sine-like shoulders, and that these curves fit observed generation-time data excellently. We also calculate correlation coefficients for sister cells and for mother-daughter pairs. These correlation coefficients are more negative than is generally observed, which is characteristic of all size-control models and is generally attributed to some unknown positive correlation in growth rates of related cells. Next we compare theoretical size distributions with observed distributions, and we calculate the dependence of average cell mass on specific growth rate and show that this dependence agrees with a well-known relation in bacteria. In the discussion we argue that unequal division is probably not the source of stochastic fluctuations in deterministic size-control models, transition-probability models with no feedback from cell size cannot account for the rapidity with which the new, stable size distribution is established after perturbation, and Kubitschek's rate-normal model is not consistent with exponential beta-curves.     

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.     

Variation in precursor pool size during the division cycle of Escherichia coli: further evidence for linear cell growth.

The magnitudes of several pools of radioactively labeled precursors for RNA and protein synthesis were determined as a function of cell age during the division cycle of Escherichia coli 15 THU. Uracil, histidine, and methionine pools increased from low initial values for cells at birth to maxima during midcycle and then subsided again. These pools were small or nonexistent at the beginning and the end of the cycle, and their average values during the cycle were less than 4% of the total cellular radioactivity. The results are consistent with a linear pattern of growth for cells during the division cycle and provide strong evidence against exponential or bilinear growth of E. coli cells.     

The rate and topography of cell wall synthesis during the division cycle of Escherichia coli using N-acetylglucosamine as a peptidoglycan label

The rates of synthesis of peptidoglycan and protein during the division cycle of Escherichia coli were measured by the membrane elution technique using cells differentially labelled with N-acetylglucosamine and leucine. During the first part of the division cycle the ratio of the rates of protein and peptidoglycan synthesis was constant. The rate of peptidoglycan synthesis, relative to the rate of protein synthesis, increased during the latter part of the division cycle. These results support a simple, bipartite model of cell surface increase in rod-shaped cells. Prior to the start of constriction the cell surface increases only by lateral wall extension. After cell constriction starts, the cell surface increases by both lateral wall and pole growth. The increase in surface area is partitioned between the lateral wall and the pole so that the volume of the cell increases exponentially. No variation in cell density occurs, because the increase in surface allows a continuous exponential increase in cell volume that accommodates the exponential increase in cell mass. The results are consistent with the constant density of the growing cell and the surface stress model for the regulation of cell surface synthesis. In addition, the elution pattern suggests that the membrane elution method does work by having the cells effectively bound to the membrane by their poles.     

Minimal requirements for robust cell size control in eukaryotic cells

Most cell types living in a stable environment tend to keep a constant characteristic size over successive generations. Size homeostasis requires that cells exert a tight control over the size at which they divide. Cell size control is not only robust against various noises, but also highly flexible since cell sizes can vary tremendously, notably as a function of nutrient levels. We formulated a minimal mathematical model of the eukaryotic cell cycle in which the cell size control operates through a cell growth-dependent bifurcation in the cell cycle dynamics. Such a bifurcation mechanism can readily explain the occurrence of a minimum critical size at division under limiting growth conditions. However, it also predicts that cells should become progressively larger and larger under prolific growth conditions. We argue that the cell size control can be reinforced at fast growth rates by adding a new cell cycle inhibitory activity whose strength would increase with the cell growth rate. We further show that various sources of noise may also generate a large variability in cell size at division and interdivision time that exhibit characteristic exponential tail distributions, without compromising the robustness of the cell size control.     

Increase in cell mass during the division cycle of Escherichia coli B/rA.

Increase in the mean cell mass of undivided cells was determined during the division cycle of Escherichia coli B/rA. Cell buoyant densities during the division cycle were determined after cells from an exponentially growing culture were separated by size. The buoyant densities of these cells were essentially independent of cell age, with a mean value of 1.094 g ml-1. Mean cell volume and buoyant density were also determined during synchronous growth in two different media, which provided doubling times of 40 and 25 min. Cell volume and mass increased linearly at both growth rates, as buoyant density did not vary significantly. The results are consistent with only one of the three major models of cell growth, linear growth, which specifies that the rate of increase in cell mass is constant throughout the division cycle.     

Buoyant density constancy during the cell cycle of Escherichia coli

Cell buoyant densities were determined in exponentially growing cultures of Escherichia coli B/r NC32 and E. coli K-12 PAT84 by equilibrium centrifugation in Percoll gradients. Distributions within density bands were measured as viable cells or total numbers of cells. At all growth rates, buoyant densities had narrow normal distributions with essentially the same value for the coefficient of variation, 0.15%. When the density distributions were determined in Ficoll gradients, they were more than twice as broad, but this increased variability was associated with the binding of Ficoll to the bacteria. Mean cell volumes and cell lengths were independent of cell densities in Percoll bands, within experimental errors, both in slowly and in rapidly growing cultures. Buoyant densities of cells separated by size, and therefore by age, in sucrose gradients also were observed to be independent of age. The results make unlikely any stepwise change in mean buoyant density of 0.1% or more during the cycle. These results also make it unlikely that signaling functions for cell division or for other cell cycle events are provided by density variations.     

Replication of prophage P1 is cell-cycle specific.

P1 prophage replication during the Escherichia coli division cycle has been analyzed by using the membrane-elution technique to produce cells labelled at different times during the division cycle and scintillation counting for quantitative analysis of radioactive prophage DNA. P1 prophage replicates during a restricted portion of the bacterial division cycle, like the minichromosome, but at a time during the division cycle different than the time at which the minichromosome replicates in the same cell. A high-copy mini-R6K plasmid present in the same cell replicates throughout the division cycle. Over a wide range of growth rates, the P1 prophage replicates approximately one-half generation after the minichromosome replicates. Thus, the mechanisms underlying P1 replication are similar to those for the F plasmid and the chromosome. Replication occurs when some property related to cell size or cell mass reaches a constant value per origin.     

Growth Pattern of Single Fission Yeast Cells Is Bilinear and Depends on Temperature and DNA Synthesis

Cell growth and division have to be tightly coordinated to keep the cell size constant over generations. Changes in cell size can be easily studied in the fission yeast Schizosaccharomyces pombe because these cells have a cylindrical shape and grow only at the cell ends. However, the growth pattern of single cells is currently unclear. Linear, exponential, and bilinear growth models have been proposed. Here we measured the length of single fission yeast cells with high spatial precision and temporal resolution over the whole cell cycle by using time-lapse confocal microscopy of cells with green fluorescent protein-labeled plasma membrane. We show that the growth profile between cell separation and the subsequent mitosis is bilinear, consisting of two linear segments separated by a rate-change point (RCP). The change in growth rate occurred at the same relative time during the cell cycle and at the same relative extension for different temperatures. The growth rate before the RCP was independent of temperature, whereas the growth rate after the RCP increased with an increase in temperature, leading to clear bilinear growth profiles at higher temperatures. The RCP was not directly related to the initiation of growth at the new end (new end take-off). When DNA synthesis was inhibited by hydroxyurea, the RCP was not detected. This result suggests that completion of DNA synthesis is required for the increase in growth rate. We conclude that the growth of fission yeast cells is not a simple exponential growth, but a complex process with precise rates regulated by the events during the cell cycle.     

Analysis of protein distribution in budding yeast

Flow cytometry is a fast and sensitive method that allows monitoring of different cellular parameters on large samples of a population. Protein distributons give relevant information on growth dynamics, since they are related to the age distribution and depend on the law of growth of the population and the law of protein accumulation during the cell cycle. We analyzed protein distributions to evaluate alternative growth models for the budding yeast Saccharomyces cerevisiae and to monitor the changes in population dynamics that result from environmental modifications; such an analysis could potentially give parameters useful in the control of biotechnological processes. Theoretical protein distributions (taking into account the unequal division of yeast cells and the exponential law of protein accumulation during a cell cycle) quantitatively fit experimental distributions, once appropriate variability sources are introduced. Best fits are obtained when the protein threshold required for bud emergence increases at each new generation of parent cells.     

Cell cycle analysis of F''lac replication in Escherichia coli B/r.

The timing and control of replication of an F'lac plasmid was investigated in two substrains of Escherichia coli B/r lac/F'lac growing at a variety of rates. The cellular content of covalently closed circular F'lac deoxyribonucleic acid and the cellular mass at the time of F'lac replication both increased as a function of growth rate. The timing of plasmid replication during the division cycle was determined by measuring the inducibility of beta-galactosidase in cells of different ages in exponentially growing cultures. At all growth rates, the rate of induced beta-galactosidase synthesis increased in a step-wise fashion during the division cycle, indicating that the F'lac plasmid replicated at a discrete time in the cycle. At growth rates greater than one doubling per h, the cell age at F'lac replication was indistinguishable from the cell age at chromosomal lac+ replication in an isogenic F- parent. The ratio of plasmids to chromosomal origins decreased from about 0.7 to 0.4 between growth rates of 1.0 to 2.5 doublings per h. These observations are all consistent with replication of F'lac at about the same time in the division cycle as replication of the homologous chromosomal region at these growth rates. This similarity in timing of replication of homologous deoxyribonucleic acid regions was not evident in slower-growing cells.     

Coordinate initiation of chromosome and minichromosome replication in Escherichia coli.

Escherichia coli minichromosomes harboring as little as 327 base pairs of DNA from the chromosomal origin of replication (oriC) were found to replicate in a discrete burst during the division cycle of cells growing with generation times between 25 and 60 min at 37 degrees C. The mean cell age at minichromosome replication coincided with the mean age at initiation of chromosome replication at all growth rates, and furthermore, the age distributions of the two events were indistinguishable. It is concluded that initiation of replication from oriC is controlled in the same manner on minichromosomes and chromosomes over the entire range of growth rates and that the timing mechanism acts within the minimal oriC nucleotide sequence required for replication.     

Midcycle doubling of uptake rates of adenine and serine in Saccharomyces cerevisiae.

Rates of uptake of serine and of adenine were measured as a function of cell size, and therefore age, in asynchronous, exponential phase cultures of diploid Saccharomyces cerevisiae strain Y55. In both cases, uptake rates were constant during the initial third of the cell cycle and doubled during the S period in the middle part of the cycle to a constant value during the final third. Cell size and age at mid-step doubling were indistinguishable for serine and adenine uptake, and occurred during the period of DNA synthesis. The results extend an earlier hypothesis of constancy of cell growth rates (mass accumulation rates) and rates of uptake of all or almost all compounds into cells in exponential phase growth to one of piecewise constancy, with an abrupt doubling of growth and uptake rates during DNA synthesis.     

Chromosome replication during the division cycle in slowly growing, steady-state cultures of three Escherichia coli B/r strains.

The period of DNA synthesis C during the cell cycle was determined over a broad range of generation times in slowly growing, steady-state batch cultures in the exponential phase and in chemostat cultures of three strains of Escherichia coli, strains B/r A, B/r K, and B/r TT, utilizing measurements of average amounts of DNA per cell and cell survival after radioactive decay of 125I incorporated into the DNA of synthesizing cells. At each growth rate, values for cell survival and for C periods were the same within experimental errors for the three strains. The length of the DNA synthesis period increased linearly with generation (doubling) time T of the culture and approached a limiting value of C = 0.36T at very long generation times. In very slowly growing cultures, DNA replication was limited almost entirely to the final third of the cell cycle. D periods, between termination of DNA replication and cell division, were found to be relatively short at all growth rates for each strain. Average amounts of DNA per cell measured in slowly growing cultures of strains B/r A and B/r TT were indistinguishable from results for strain B/r K at the same growth rates. Amounts of DNA per cell calculated from the cell survival values alone are completely consistent with the measured DNA per cell.     

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.     

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

The kinetics of growth of all the cells in a population is reflected in the shape of the size distribution of the population. To ascertain whether the kinetics of growth of the average individual cell is similar for different strains or growth conditions, we compared the shape of normalized size distributions obtained from steady-state populations. Significant differences in the size distributions were found, but these could be ascribed either to the precision achieved at division or to a constriction period which is long relative to the total cell cycle time. The remaining difference is quite small. Thus, without establishing the pattern itself, it is concluded that the basic course of growth is very similar for the various Escherichia coli strains examined and probably also for other rod-shaped bacteria. The effects of differences in culture technique (batch or chemostat culture), growth rate, and differences among strains were not found to influence the shape of the size distributions and hence the growth kinetics in a direct manner; small differences were found, but only when the precision at division or the fraction of constricted cells (long constriction period) were different as well.