N.B. Grover

Dimensional rearrangement of rod-shaped bacteria following nutritional shift-up. II. Experiments with Escherichia coliBr☆

The dimensions of Escherichia coliBr (strain H266) in transition between two states of balanced growth, were determined from electron micrographs of fixed cells by sampling the culture at various times following nutritional shift-up from a doubling time of 72 min to one of 24 min. Mean cell length rises immediately and overshoots its final steady-state value, cell diameter increases monotonically; both approach their asymptotic levels only after several hours. The results are compared with the dimensions predicted by each of two models of cell growth and morphogenesis in rod-shaped bacteria. The first attributes cell elongation to circular zones that double in number at a particular time during the cell cycle and which act at rates proportional to the growth rate; the second is similar, except that it considers surface growth rather than length extension as the active process, length being determined passively. Two possibilities are examined, that the zonal growth rate adjusts immediately to the new growth conditions, and that it does so gradually. The experimental data appear consistent with the gradual response version of the surface growth model.

Elongation of rod-shaped bacteria.

Three models relating cell length to generation time are considered for rod-shaped bacteria growing under steady-state conditions; all three presuppose linear elongation. The first model assumes that the rate of elongation is proportional to the instantaneous number of chromosome replication forks per cell; the others, that it is inversely related to the generation time and doubles a fixed time prior to cell division. One of these (model 2) treats this relationship as continuous, with the doubling occurring during the last division cycle (at chromosome termination), while the other is a discrete model in which the doubling in rate takes place at chromosome initiation. Expressions are derived for mean cell length and length at birth in each case. Comparison with experimental data on E. coli B/r using non-linear least-squares techniques results in an excellent fit for model 2 and unsatisfactory ones for the others, the best estimate for the time at which the rate doubles being 15·3 min prior to cell division and for the minimum length at birth (i.e., as the growth rate of the culture tends to zero), 1·47 μm. The functional relationship between cell radius and generation time implied by model 2 is also presented. This model again produces a good fit to the experimental data and provides, for the first time, a direct estimate of the volume/origin ratio at initiation of chromosome replication 0·35 ± 0·05 μm3 (s.e.). The results obtained here are compared with various qualitative observations reported in the literature and with such numerical data as are available.

Dimensional regulation of cell-cycle events in Escherichia coli during steady-state growth.

Two opposing models have been put forward in the literature to describe the changes in the shape of individual Escherichia coli cells in steady-state growth that take place during the cell cycle: the Length model, which maintains that the regulating dimension is cell length, and the Volume model, which asserts it to be cell volume. In addition, the former model envisages cell diameter as decreasing with length up to constriction whereas the latter sees it as being constrained by the rigid cell wall. These two models differ in the correlations they predict between the various cellular dimensions (diameter, length, volume) not only across the entire population of bacteria but also, and especially, within subpopulations that define specific cell-cycle events (division, for example, or onset of constriction); the coefficients of variation at these specific events are also expected to be very different. Observations from cells prepared for electron microscopy (air-dried) and for phase-contrast microscopy (hydrated) appeared qualitatively largely in accordance with the predictions of the Length model. To obtain a more quantitative comparison, simulations were carried out of populations defined by each of the models; again, the results favoured the Length model. Finally, in age-selected cells using membrane elution, the diameter-length and diameter-volume correlations were in complete agreement with the Length model, as were the coefficients of variation. It is concluded that, at least with respect to cell-cycle events such as onset of constriction and cell division, length rather than volume is the controlling dimension.

Dimensional rearrangement of rod-shaped bacteria following nutritional shift-up. I. Theory☆

Abstract Theoretical expressions are derived for two models that describe average length and radius of a population of rod-shaped bacteria as a function of time following their transfer to a medium that supports a higher growth rate. The first attributes cell elongation to circular zones produced at a particular time during the cell cycle and which act thereafter at rates proportional to the growth rate; the second is formally identical but considers surface growth rather than length extension. Two possibilities are considered, that the zonal growth rate adjusts immediately to the transition, and that it does so gradually. The results are also displayed graphically, covering a broad range of each of the various parameters involved; values are chosen to permit a direct comparison between the models. Average cell length is seen to undergo a large overshoot and to approach its steady-state value from above, while cell radius remains almost constant or even decreases somewhat before increasing monotonically towards its asymptotic level; both require a considerable period of time to reach steady state. The transient behavior predicted by the two models is found to be quite different even when the steady-state dimensions are identical; the differences between immediate and gradual response of the zonal growth rate are even greater. It is shown that using a dimensionless measure of cell geometry, the aspect ratio, can facilitate selection of the appropriate model.

Surface Growth in Rod-shaped Bacteria

Abstract Various models advanced to explaintherelationship between cell dimensions and generation time are compared for rod-shaped bacteria growing under steady-state conditions. Equations are developed for three such models based on the linear extension of surface area. The first assumes that the rate of envelope synthesis is proportional to the instantaneous number of chromosome replication forks per cell; the second, that it is inversely related to the generation time and doubles a fixed time d prior to cell division; the third, that it is constant and doubles at initiation of chromosome replication. Non-linear least-squares analysis is used to fit the theoretical expressions for mean surface area to values calculated from experimental measurements of length and width by assuming the geometry of a right circular cylinder with hemispherical polar caps. The functions describing area at birth are all discontinuous and cannot be solved by accepted techniques; they can, however, be used to test the internal consistency of each model. Model 1 is consistent only when lateral extension and septum formation are not considered as independent processes. Model 2 provides a very satisfactory fit, the best estimate for d being 49 ± 4 min. In both cases, the values of the parameters obtained are statistically indistinguishable from those predicted on the basis of a much simpler geometry: a circular cylinder with plane parallel ends. Model 3 is unsuitable and can be rejected. Sources of experimental error and some possible consequences of the simplifications used in constructing the models, are considered. A detailed comparison is made between the control of length extension proposed previously and control of envelope synthesis. The implications of the results are discussed, and a more promising way of discriminating among the remaining models is suggested.

Dimensional rearrangement of Escherichia coli B/r cells during a nutritional shift-down

In a search for the mechanism underlying dimensional changes in bacteria, the glucose analogue methyl alpha-D-glucoside was used to effect a rapid reduction in the mass growth rate of Escherichia coli by competitively inhibiting glucose uptake, a so-called nutritional shift-down. The new steady-state cell mass and volume were reached after 1 h, during which the rate of cell division was maintained; rearrangement of the linear dimensions (cell length, diameter), however, required an additional 2 h and caused an undershoot in cell length, consistent with the view that E. coli is slow to modify its diameter. The results are compared with the overshoot in cell length that occurs following nutritional shift-up.

Elongation and surface extension of individual cells of Escherichia coli B/r: Comparision 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.

Initiation of DNA replication in bacteria: Analysis of an autorepressor control model*

The precise mechanism by which the initiation of chromosome replication in bacteria is controlled has not yet been established, and several theoretical models have been proposed in an attempt to provide a conceptual framework for the accumulated experimental evidence. The present article contains a detailed quantitative analysis, using computer simulation, of the control model first put forward schematically by Sompayrac & Maaløe in 1973, in which a single operon codes for both the initiator protein and an autorepressor. By comparing the predictions of the model with what is known about the physiology and molecular biology of Escherichia coli under different growth conditions, we are able to delineate the characteristics that such a control system would need to possess in order to be capable of regulating chromosome replication: the control operon has to lie fairly near the origin of replication and contain a moderate to strong promoter and an operator that competes for its repressor with other equally specific binding sites along the chromosome in an interaction that is somewhat weaker than usual; in addition, the messenger molecules encoded for by the repressor gene must have a relatively ineffective ribosome binding site and not too long a halflife.

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.

Characterization of cell-cycle-specific events in synchronous cultures of Escherichia coli: a theoretical evaluation.

Synchronous growth studies are often used to assess the presence, timing and duration of periodic phenomena in the bacterial cell cycle. In an effort to evaluate the quality and quantity of information on cycle-specific events that can reasonably be expected from such inquiries, a model was constructed of a synchronous culture of Escherichia coli cells as would be derived from a growing population immobilized on a surface, and applied to the case of one stable and one unstable cellular component. The results indicated that, while the presence of cycle-specific events may be easily detectable, their timing and duration are very difficult to establish in synchronous growth experiments. Furthermore, differences in timing can be misconstrued as differences in duration, and vice versa, when interpretations are based on the qualitative analysis of the data.