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.     

Division Delay Induced in Escherichia coli by Near-Ultraviolet Radiation

Beams of near-ultraviolet radiation at several principal emission lines of a mercury arc were isolated with a grating monochromator and directed upon cell suspensions. During subsequent incubation at room temperature in Nutrient Broth, the population was studied by removing samples and obtaining cell numbers and cell size distributions with an electronic cell counter. Division delay without lethality was observed. The shapes of the dose-response curves for induction, the doses of near-ultraviolet radiation required, and the action spectrum for division delay were found to be similar to those for growth delay (in broth) and for photoprotection. These findings indicate that all three effects, division delay, growth delay, and photoprotection, are induced by a common type of critical event. Changes in cell size distribution in the culture during incubation in Nutrient Broth after near-ultraviolet irradiation are very similar for control and irradiated populations, although these changes occur at a much later time in the irradiated population. This indicates that, in Nutrient Broth, the population recovers completely from the inhibition of growth and division, thus justifying use of the term "delay," and suggesting that the damage is nongenetic.     

Morphological analysis of the division cycle of two Escherichia coli substrains during slow growth.

Morphological parameters of the cell division cycle have been examined in Escherichia coli B/r A and K. Whereas the shape factor (length of newborn cell/width) of the two strains was the same at rapid growth (doubling time, tau, less than 60 min), with decreasing growth rate the dimensions of the two strains did change so that B/r A cells became more rounded and B/r K cells became more elongated. The process of visible cell constriction (T period) lasted longer in B/r A than in B/r K during slow growth, reaching at tau = 200 min values of 40 and 17 min, respectively. The time between termination of chromosome replication and cell division (D period) was found to be longer in B/r A than in B/r K. As a result, in either strain completion of chromosome replication seemed always to occur before initiation of cell constriction. Nucleoplasmic separation did not coincide with termination as during rapid growth but occurred in both strains within the T period, about 10 min before cell division.     

Production of cells without deoxyribonucleic acid during thymidine starvation of lexA- cultures of Escherichia coli K-12.

When thymidine-requiring lexA- strains were starved for thymidine, the kinetics of survival were similar to those of a nearly isogenic lexA+ strain. The size distribution of cells in the lexA- and lexA+ cultures were, however, quite different. Whereas most of the cells in the starved lexA+ cultures grew into long filamentous forms (longer than 4.0 mum), many of the lexA- cells were found to have a normal rod shape (4.0 mum or shorter). It was shown that lexA- cells undergo more divisions during thymidine starvation than lexA+ cells. Furthermore, using an autoradiographic method to analyze deoxyribonucleic acid (DNA) distribution in the starved cells, we demonstrated that cells without DNA are produced in both normal and starved lexA- cultures at a much higher frequency than in lexA+ cultures. Some of these cells may be produced by breakdown of DNA, but we favor the hypothesis that they result from an abnormal cell division process. Since lexA mutations are dominant, we conclude that a diffusible product decreases the synthesis or activity of an inhibitor of cell division in lexA- strains when DNA synthesis is blocked by thymidine starvation.     

Division behavior and shape changes in isogenic ftsZ, ftsQ, ftsA, pbpB, and ftsE cell division mutants of Escherichia coli during temperature shift experiments.

Isogenic ftsZ, ftsQ, ftsA, pbpB, and ftsE cell division mutants of Escherichia coli were compared with their parent strain in temperature shift experiments. To improve detection of phenotypic differences in division behavior and cell shape, the strains were grown in glucose-minimal medium with a decreased osmolality (about 100 mosM). Already at the premissive temperature, all mutants, particularly the pbpB and ftsQ mutants, showed an increased average cell length and cell mass. The pbpB and ftsQ mutants also exhibited a prolonged duration of the constriction period. All strains, except ftsZ, continued to initiate new constrictions at 42 degrees C, suggesting the involvement of FtsZ in an early step of the constriction process. The new constrictions were blunt in ftsQ and more pronounced in ftsA and pbpB filaments, which also had elongated median constrictions. Whereas the latter strains showed a slow recovery of cell division after a shift back to the permissive temperature, ftsZ and ftsQ filaments recovered quickly. Recovery of filaments occurred in all strains by the separation of newborn cells with an average length of two times LO, the length of newborn cells at the permissive temperature. The increased size of the newborn cells could indicate that the cell division machinery recovers too slowly to create normal-sized cells. Our results indicate a phenotypic resemblance between ftsA and pbpB mutants and suggest that the cell division gene products function in the order FtsZ-FtsQ-FtsA, PBP3. The ftsE mutant continued to constrict and divide at 42 degrees C, forming short filaments, which recovered quickly after a shift back to the permissive temperature. After prolonged growth at 42 degree C, chains of cells, which eventually swelled up, were formed. Although the ftsE mutant produced filaments in broth medium at the restrictive temperature, it cannot be considered a cell division mutant under the presently applied conditions.     

On the precision and accuracy achieved by Escherichia coli cells at fission about their middle

Length and width of each of the prospective siblings of constricted Escherichia coli cells from different strains and culture conditions were measured from electron micrographs. The data were statistically analyzed to investigate how equally the length and volume of one cell was divided into two. The analysis showed that, for all cultures, bipartition is unbiased or very nearly so, i.e. sibling cells were on the average equally long and large. The precision of bipartition attained by the cells was usually high; it was related to the average cell shape (length/width): slender E. coli cells divided into two less precisely than squat cells. Absolute size, growth rate and strain specificity affected the precision of bipartition only indirectly, i.e. in as much as they influenced cell shape.     

Cell size and growth characteristics of cultured fibroblasts isolated from normal and keloid tissue.

No differences in appearance or in cell size distribution were observed between cultured fibroblasts derived from normal skin, mature scars, or keloids. Artifactual differences in cell size distributions between strains can result when populations are compared at different cell densities. Keloid derived fibroblasts remain euploid in culture, and they have the same growth rate and same degree of density-dependent growth inhibition as cultured normal human fibroblasts.     

Individual-based modeling of phytoplankton: evaluating approaches for applying the cell quota model

Present phytoplankton models typically use a population-level (lumped) modeling (PLM) approach that assumes average properties of a population within a control volume. For modern biogeochemical models that formulate growth as a nonlinear function of the internal nutrient (e.g. Droop kinetics), this averaging assumption can introduce a significant error. Individual-based (agent-based) modeling (IBM) does not make the assumption of average properties and therefore constitutes a promising alternative for biogeochemical modeling. This paper explores the hypothesis that the cell quota (Droop) model, which predicts the population-average specific growth or cell division rate, based on the population-average nutrient cell quota, can be applied to individual algal cells and produce the same population-level results. Three models that translate the growth rate calculated using the cell quota model into discrete cell division events are evaluated, including a stochastic model based on the probability of cell division, a deterministic model based on the maturation velocity and fraction of the cell cycle completed (maturity fraction), and a deterministic model based on biomass (carbon) growth and cell size. The division models are integrated into an IBM framework (iAlgae), which combines a lumped system representation of a nutrient with an individual representation of algae. The IBM models are evaluated against a conventional PLM (because that is the traditional approach) and data from a number of steady and unsteady continuous (chemostat) and batch culture laboratory experiments. The stochastic IBM model fails the steady chemostat culture test, because it produces excessive numerical randomness. The deterministic cell cycle IBM model fails the batch culture test, because it has an abrupt drop in cell quota at division, which allows the cell quota to fall below the subsistence quota. The deterministic cell size IBM model reproduces the data and PLM results for all experiments and the model parameters (e.g. maximum specific growth rate, subsistence quota) are the same as those for the PLM. In addition, the model-predicted cell age, size (carbon) and volume distributions are consistent with those derived analytically and compare well to observations. The paper discusses and illustrates scenarios where intra-population variability in natural systems leads to differences between the IBM and PLM models.     

Relationship between size of parent at cell division and relative size of its progeny in Escherichia coli

This article examines the empirical basis for the assumption of independence between the relative size (length or surface area) of a newborn cell w and the absolute size of its mother at cell division. Random samples from two strains of Escherichia coli B/r cells in steady-state exponential growth, covering a range of doubling times, were fixed in osmium tetroxide and prepared for electron microscopy by agar filtration. Length and diameter of over 3000 constricted cells were measured from the electron micrographs and cell surface area computed by assuming an idealized geometry of right circular cylinders with hemispherical polar caps. In general, these strains were found to divide into two daughter cells with a precision that is independent of the size of the mother. In addition, both a normal and a symmetrical beta-distribution were shown to fit the observed size distributions of w rather well; theoretical grounds for preferring the latter are discussed.     

Overinitiation of replication of the Escherichia coli chromosome from an integrated runaway-replication derivative of plasmid R1.

A 16-base-pair fragment, deletion of which completely inactivated oriC, was replaced by a temperature-dependent runaway-replication derivative (the copy number of which increases with temperature) of the IncFII plasmid R1. The constructed strains were temperature sensitive, and flow cytometry revealed a severalfold increase in the DNA/mass ratio following shifts to nonpermissive temperatures. The cell size distribution was broader in the constructed strains relative to that in the wild type because of asynchrony between the chromosome replication and cell division cycles. This difference was more pronounced for counterclockwise initiation of chromosomal replication, in which small DNA-less cells and long filaments were abundant. Following a temperature shift the cell size distributions became even more broad, showing that changes in the frequency of chromosomal replication affect cell division and emphasizing the interplay between these two processes.     

The size distribution of conspecific populations: the peoples of New Guinea

The size distribution of the language populations in New Guinea, which represent over 15% of the world's languages, is analysed using models analogous to the resource division models of species abundance distribution in ecological communities. A model distribution of resource segments reflecting population size is created by repeated selection of an existing resource segment and its division into two. We found that any dependency of the selection probability on the size of the segment generated negatively skewed abundance distributions after log transformation. Asymmetric segment division further exacerbated the negative skewness. Size-independent selection produced lognormal abundance distributions, irrespective of the segment division method. Size-dependent selection and asymmetric division were deemed reasonable assumptions since large language populations are more likely to generate isolates, which develop into new populations, than small ones, and these isolates are likely to be small relative to the progenitor population. A negatively skewed distribution of the log-transformed population sizes was therefore expected. However, the observed distributions were lognormal, scale invariant for areas containing between 100 and over 1000 language populations. The dynamics of language differentiation, as reflected by the models, may therefore be unimportant relative to the effect of variable growth rates among populations. All lognormal distributions from resource division models had a higher variance than the observed one, where half of the 1053 populations had between 350 and 3000 individuals. The possible mechanisms maintaining such a low variance around a modal population size of 1000 are discussed.     

Division Cycle of Myxococcus xanthus III. Kinetics of Cell Growth and Protein Synthesis

The kinetics of cell growth and protein synthesis during the division cycle of Myxococcus xanthus was determined. The distribution of cell size for both septated and nonseptated bacteria was obtained by direct measurement of the lengths of 8,000 cells. The Collins-Richmond equation was modified to consider bacterial growth in two phases: growth and division. From the derived equation, the growth rate of individual cells was computed as a function of size. Nondividing cells (growth phase) comprised 91% of the population and took up 87% of the time of the division cycle. The absolute and specific growth rates of nondividing cells were observed to increase continually throughout the growth phase; the growth rate of dividing cells could not be determined accurately by this technique because of changes in the geometry of cells between the time of septation and physical separation. The rate of protein synthesis during the division cycle was measured by pulselabeling an exponential-phase culture with radio-active valine or arginine and then preparing the cells for quantitative autoradiography. By measuring the size of individual cells as well as the number of grains, the rate of protein synthesis as a function of cell size was obtained. Nondividing cells showed an increase in both the absolute and specific rates of protein synthesis throughout the growth phase; the specific rate of protein synthesis for dividing cells was low when compared to growthphase cells. Cell growth and protein synthesis are compared to the previously reported kinetics of deoxyribonucleic acid and ribonucleic acid synthesis during the division cycle.     

Timing of FtsZ Assembly in Escherichia coli

The timing of the appearance of the FtsZ ring at the future site of division in Escherichia coli was determined by in situ immunofluorescence microscopy for two strains grown under steady-state conditions. The strains, B/rA and K-12 MC4100, differ largely in the duration of the D period, the time between termination of DNA replication and cell division. In both strains and under various growth conditions, the assembly of the FtsZ ring was initiated approximately simultaneously with the start of the D period. This is well before nucleoid separation or initiation of constriction as determined by fluorescence and phase-contrast microscopy. The durations of the Z-ring period, the D period, and the period with a visible constriction seem to be correlated under all investigated growth conditions in these strains. These results suggest that (near) termination of DNA replication could provide a signal that initiates the process of cell division.     

Continuous cultivation of fission yeast: Analysis of single-cell protein synthesis kinetics

A fundamental problem in microbial reactor analysis is identification of the relationship between environment and individual cell metabolic activity. Population balance equations provide a link between experimental measurements of composition frequency functions in microbial populations on the one hand and macromolecular synthesis kinetics and cell division control parameters for single cells on the other. Flow microfluorometry measurements of frequency functions for single-cell protein content in Schizosaccharomyces pombe in balanced exponential growth have been analyzed by two different methods. One approach utilizes the integrated form of the population balance equation known as the Collins-Richmond equation, and the other method involves optimization of parameters in assumed kinetic and cell division functional forms in order to best fit measured frequency functions with corresponding model solutions. Both data interpretation techniques indicate that rates of protein synthesis increase most in small protein content cells as the population specific growth rate increases, leading to parabolic single-cell protein synthesis kinetics at large specific growth rates. Utilization of frequency function data for an asynchronous population is shown in this case to be a far more sensitive method for determination of single-cell kinetics than is monitoring the metabolic dynamics of a single cell or, equivalently, synchronous culture analyses.     

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° 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.     

Division synchrony and the dynamics of microbial populations: A size-specific model

A discrete, environmentally coupled, size-specific model of microbial population dynamics in continuous culture is presented. It is mathematically simpler than other models based on similar assumptions and lends itself to numerical and analytic solutions. It displays several phenomena which have been reported in the experimental literature but which are not well understood; specifically, a loose relationship between biomass and numbers (i.e., a time lag between mass growth and cell division) and a critical damping of biomass while numbers continue to oscillate. In addition, the model provides several new predictions: The stable biomass distribution is independent of the environmental factors considered in the model and uniformly distributes the biomass among the size classes. The rate of approach to stability and the frequency of waves through the size distributions are a function of the flow rate and the variance in rate of growth and size at division. The model should provide a useful basis for studying the effects of size specificity on the dynamics of microbial populations cultured in chemostats.     

Unequal cell division, growth regulation and colony size of mammalian cells: a mathematical model and analysis of experimental data.

This work describes mathematically the dynamics of expansion of cell populations from the initial division of single cells to colonies of several hundred cells. This stage of population growth is strongly influenced by stochastic (random) elements including, among others, cell death and quiescence. This results in a wide distribution of colony sizes. Experimental observations of the NIH3T3 cell line as well as for the NIH3T3 cell line transformed with the ras oncogene were obtained for this study. They include the number of cells in 4-day-old colonies initiated from single cells and measurements of sizes of sister cells after division, recorded in the 4-day-old colonies. The sister cell sizes were recorded in a way which enabled investigation of their interdependence. We developed a mathematical model which includes cell growth and unequal cell division, with three possible outcomes of each cell division: continued cell growth and division, quiescence, and cell death. The model is successful in reproducing experimental observations. It provides good fits to colony size distributions for both NIH3T3 mouse fibroblast cells and the same cells transformed with the rasEJ human cancer gene. The difference in colony size distributions could be fitted by assuming similar cell lifetimes (12-13 hr) and similar probabilities of cell death (q = 0.15), but using different probabilities of quiescence, r = 0 for the ras oncogene transformed cells and r = 0.1 for the non-transformed cells. The model also reproduces the evolution of distributions of sizes of cells in colonies, from a single founder cell of any specified size to the stable limit distribution after eight to ten cell divisions. Application of the model explains in what way both random events and deterministic control mechanisms strongly influence cell proliferation at early stages in the expansion of colonies.     

Cell cycle characteristics of crenarchaeota: unity among diversity

The hyperthermophilic archaea Acidianus hospitalis, Aeropyrum pernix, Pyrobaculum aerophilum, Pyrobaculum calidifontis, and Sulfolobus tokodaii representing three different orders in the phylum Crenarchaeota were analyzed by flow cytometry and combined phase-contrast and epifluorescence microscopy. The overall organization of the cell cycle was found to be similar in all species, with a short prereplicative period and a dominant postreplicative period that accounted for 64 to 77% of the generation time. Thus, in all Crenarchaeota analyzed to date, cell division and initiation of chromosome replication occur in close succession, and a long time interval separates termination of replication from cell division. In Pyrobaculum, chromosome segregation overlapped with or closely followed DNA replication, and further genome separation appeared to occur concomitant with cellular growth. Cell division in P. aerophilum took place without visible constriction.     

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.     

POLYGONAL ASPECTS OF CELL FACES. III. CELL SIZE,CELL DIVISION,AND CELL FACE DISTRIBUTIONS

Wheeler , George E. (Brooklyn Coll., Brooklyn, New York.) Polygonal aspects of cell faces. III. Cell size, cell division, and cell face distributions. Amer. Jour. Bot. 50(8): 747–754. Illus. 1963.—The effects of cell size differences on cell face (polygon type) distributions, and the relationship of cell division to these effects were investigated, using published and original data. Only “relative” size was considered, i.e., the size of a body compared with the sizes of contiguous bodies. A preliminary study of mixtures including 2 sizes of nonliving bodies (bubbles or shot) showed that, with increase in number of smaller bodies relative to larger, 4- and 5-gons decrease and 6-gons increase on both large and small bodies (except for 5-gons on small bubbles). Since the cited papers on cells generally include no volume measurements, cell size was assumed to be proportional to number of faces; therefore, the cells from each sample were pooled in 3 groups of different ranges as to number of faces per cell. This device made evident the 4- and 5-gon decreases and the 6-gon increases with increasing size, but only in certain samples;- marked exceptions occurred among others. New data from 2 original samples showed, in general, the correlations described above, but again then; were exceptions. Discrepancies were found to arise from variable division patterns which affect face types both directly, and indirectly through division effects on cell size. It was concluded that relative size is generally less important in determining cell face distributions than are the more direct cell division events, and that size difference effects can be detected only if the other events operate minimally.