Cell Growth and Size Homeostasis in Silico

Cell growth in size is a complex process coordinated by intrinsic and environmental signals. In a research work performed by a different group, size distributions of an exponentially growing population of mammalian cells were used to infer cell-growth rate in size. The results suggested that cell growth was neither linear nor exponential, but subject to size-dependent regulation. To explain the observed growth pattern, we built a mathematical model in which growth rate was regulated by the relative amount of mRNA and ribosomes in a cell. Under the growth model and a stochastic division rule, we simulated the evolution of a population of cells. Both the sampled growth rate and size distribution from this in silico population agreed well with experimental data. To explore the model space, alternative growth models and division rules were studied. This work may serve as a starting point to understand the mechanisms behind cell growth and size regulation using predictive models.     

Modeling the impact of single-cell stochasticity and size control on the population growth rate in asymmetrically dividing cells

Microbial populations show striking diversity in cell growth morphology and lifecycle; however, our understanding of how these factors influence the growth rate of cell populations remains limited. We use theory and simulations to predict the impact of asymmetric cell division, cell size regulation and single-cell stochasticity on the population growth rate. Our model predicts that coarse-grained noise in the single-cell growth rate λ decreases the population growth rate, as previously seen for symmetrically dividing cells. However, for a given noise in λ we find that dividing asymmetrically can enhance the population growth rate for cells with strong size control (between a “sizer” and an “adder”). To reconcile this finding with the abundance of symmetrically dividing organisms in nature, we propose that additional constraints on cell growth and division must be present which are not included in our model, and we explore the effects of selected extensions thereof. Further, we find that within our model, epigenetically inherited generation times may arise due to size control in asymmetrically dividing cells, providing a possible explanation for recent experimental observations in budding yeast. Taken together, our findings provide insight into the complex effects generated by non-canonical growth morphologies.     

Morphologically-structured models of growing budding yeast populations

It has been well recognized that many key aspects of cell cycle regulation are encoded into the size distributions of growing budding yeast populations due to the tight coupling between cell growth and cell division present in this organism. Several attempts have been made to model the cell size distribution of growing yeast populations in order to obtain insight on the underlying control mechanisms, but most were based on the age structure of asymmetrically dividing populations. Here we propose a new framework that couples a morphologically-structured representation of the population with population balance theory to formulate a dynamic model for the size distribution of growing yeast populations. An advantage of the presented framework is that it allows derivation of simpler models that are directly identifiable from experiments. We show how such models can be derived from the general framework and demonstrate their utility in analyzing yeast population data. Finally, by employing a recently proposed numerical scheme, we proceed to integrate numerically the full distributed model to provide predictions of dynamics of the cell size structure of growing yeast populations.     

Cell size homeostasis: Metabolic control of growth and cell division

Joint regulation of growth rate and cell division rate determines cell size. Here we discuss how animal cells achieve cell size homeostasis potentially involving multiple signaling pathways converging at metabolic regulation of growth rate and cell cycle progression. While several models have been developed to explain cell size control, comparison of the two predominant models shows that size homeostasis is dependent on the ability to adjust cellular growth rate based on cell size. Consequently, maintenance of size homeostasis requires that larger cells can grow slower than small cells in relative terms. We review recent experimental evidence showing that such size adjustment occurs primarily at or immediately before the G1/S transition of the cell cycle. We further propose that bidirectional feedback between growth rate and size results in cell size sensing and discuss potential mechanisms how this may be accomplished.     

Estimation of parameters in population models for Schizosaccharomyces pombe from chemostat data

The integro-differential growth model of Eakman, Fredriekson, and Tsuehiya has been employed to fit cell size distribution data for Schizosaccharomyces pombe grown in a chemostat under severe product inhibition by ethanol. The distributions were obtained with a Coulter aperture and an electronic system patterned after that of Harvey and Marr. Four parameters—mean cell division size, cell division size standard deviation, daughter cell size standard deviation, and a growth rate coefficient—were calculated for models where the cell growth rate was inversely proportional to size, constant, and proportional to size. A fourth model, one where sigmoidal growth behavior was simulated by two linear growth segments, was also investigated. Linear and sigmoidal models fit the distribution data best. While the mean cell division size remained relatively constant at all growth rates, standard deviation of division size distribution increased with increasing holding times. Standard deviation of the daughter size distribution remained small at all dilution rates. Unlike previous findings with other organisms, the average cell size of Schizosaccharomyces pobme increased at low growth rates.     

INFLUENCE OF ENVIRONMENTAL FACTORS AND POPULATION COMPOSITION ON THE TIMING OF CELL DIVISION IN THALASSIOSIRA FLUVIATILIS (BACILLARIOPHYCEAE) GROWN ON LIGHT/DARK CYCLES1

Cell division patterns in Thalassiosira fluviatilis grown in a cyclostat were analyzed as a function of temperature, photoperiod, nutrient limitation and average cell size of the population. Typical cell division patterns in populations doubling more than once per day had multiple peaks in division rate each day, with the lowest rates always being greater than zero. Division bursts occurred in both light and dark periods with relative intensities depending on growth conditions. Multiple peaks in division rate were also found, when population growth rates were reduced to less than one doubling per day by lowering temperature, nutrients, or photoperiod and the degree of division phasing was not enhanced. Temperature and nutrient limitation shifted the timing of the major division burst relative to the light/dark cycle. Average cell volume of the inoculum was found to be a significant determinant of the average population growth rate and the timing and magnitude of the peaks in division rate. The results are interpreted in the context of a cell cycle model in which generation times are “quantized” into values separated by a constant time interval.     

Sloppy size control of the cell division cycle

In an asynchronous, exponentially proliferating cell culture there is a great deal of variability among individual cells in size at birth, size at division and generation time (= age at division). To account for this variability we assume that individual cells grow according to some given growth law and that, after reaching a minimum size, they divide with a certain probability (per unit time) which increases with increasing cell size. This model is called sloppy size control because cell division is assumed to be a random process with size-dependent probability. We derive general equations for the distribution of cell size at division, the distribution of generation time, and the correlations between generation times of closely related cells. Our theoretical results are compared in detail with experimental results (obtained by Miyata and coworkers) for cell division in fission yeast, Schizosaccharomyces pombe. The agreement between theory and experiment is superior to that found for any other simple models of the coordination of cell growth and division.     

Stability of the steady-state size distribution in a model of cell growth and division

The approach to steady-state size distribution is studied for a growing population of cells. The model incorporates cell growth at a linear rate and division into two equal daughters after a random time composed of an exponentially distributed phase and a constant deterministic phase.This work was supported by the National Science Foundation under Grant No. MCS 8300559This work was supported by the National Science Foundation under Grant No. MCS 8301104     

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.     

A comprehensive continuous-time model for the appearance of CGH signal due to chromosomal missegregations during mitosis

Aneuploidy, the gain or loss of large regions of the genome, is a common feature in cancer cells. Irregularities in chromosomal copy number caused by missegregations of chromosomes during mitosis can be visualized by cytogenetic techniques including fluorescence in situ hybridization (FISH), spectral karyotyping (SKY) and comparative genomic hybridization (CGH). In the current work, we consider the propagation of irregular copy numbers throughout a cell population as the individual cells progress through ordinary mitotic cell cycles. We use an algebraic model to track the different copy numbers as states in a stochastic process, based on the model of chromosome instability of Gusev, Kagansky, and Dooley, and consider the average copy number of a particular chromosome within a cell population as a function of the cell division rate. We review a number of mathematical models for determining the length of the cell cycle, including the Smith-Martin transition probability model and the 'sloppy size' model of Wheals, Tyson and Diekmann. The program MITOSIM simulates the growth of a population of cells using the aforementioned models of the cell cycle. MITOSIM allows the cell population to grow, with occasional resampling, until the average copy number of a given chromosome in the population reaches a preset threshold signifying a positive copy number alteration in this region. MITOSIM calculates the relationship between the missegregation rate and the growth rate of the cell population. This allows the user to test hypotheses regarding the effect chromosomal aberrations have upon the cell cycle, cell growth rates, and time to population dominance.     

Modelling and analysis of time-lags in some basic patterns of cell proliferation

 In this paper, we present a systematic approach for obtaining qualitatively and quantitatively correct mathematical models of some biological phenomena with time-lags. Features of our approach are the development of a hierarchy of related models and the estimation of parameter values, along with their non-linear biases and standard deviations, for sets of experimental data. We demonstrate our method of solving parameter estimation problems for neutral delay differential equations by analyzing some models of cell growth that incorporate a time-lag in the cell division phase. We show that these models are more consistent with certain reported data than the classic exponential growth model. Although the exponential growth model provides estimates of some of the growth characteristics, such as the population-doubling time, the time-lag growth models can additionally provide estimates of: (i) the fraction of cells that are dividing, (ii) the rate of commitment of cells to cell division, (iii) the initial distribution of cells in the cell cycle, and (iv) the degree of synchronization of cells in the (initial) cell population. Received: 15 September 1997/Revised version: 1 April 1998     

Downward regulation of macronuclear DNA content in Paramecium tetraurelia : Effects of excess DNA on the subsequent DNA and protein content and the cell growth rate

Paramecium cells were selected which received the entire parental macronucleus at fission and thus started the cell cycle with twice the normal post-fission DNA content. During each of the subsequent two cell cycles the cells synthesized approximately as much DNA as did control cells. The amount of excess macronuclear DNA was consequently halved during each cell cycle. The minimum pre-fission DNA content was just larger than the mean post-replication DNA amount, confirming that a similar amount of DNA, approximately equal to the mean post-fission DNA content of the non-selected population, was synthesized in macronuclei, regardless of the post-fission DNA content. These observations confirm a model for DNA content regulation previously devised for Paramecium and are inconsistent with DNA content regulation schemes proposed for other ciliates. The increased DNA content has no effect either on the subsequent total protein content of pre-fission cells, or on the rate of cell growth. This suggests that the rate of cell growth is limited by the size of the cell when the macronuclear gene-dosage is equal to or greater than that in normal cells. The results also suggest that the amount of DNA synthesized within an interfission period is also limited by the size of the cell and is proportional to the cell mass. Paramecium does not require a fixed nucleocy oplasmic ratio as a pre-condition either for cell division, or, by inference, for initiation of DNA synthesis.     

Control of cell size at division in fission yeast by a growth-modulated size control over nuclear division

In the fission yeast Schizosaccharomyces pombe, nutritional reduction of growth rate by supplying poor nitrogen, carbon or phosphate sources causes a decrease in cell size. The effect on cell division following three different nutritional shifts-up has been investigated. In all cases, about 20% of the cells divide at the original cell length, and then cell division stops for a period. Cell division then resumes at the new faster rate, cell length at division being characteristic of the new medium. Further investigation reveals that the first effect of the shift is to inhibit nuclear division rapidly and completely. These results are strongly suggestive of the operation of a cell size requirement for entry into nuclear division. The cell size necessary for nuclear division is set, or modulated, by the prevailing growth conditions. This model is confirmed by a nutritional shift-down, where nuclear division and cell division are stimulated after the shift. Cell length at division falls rapidly until the new shorter length is attained, when a new steady state is assumed at a slower growth rate. The control system is compared with that in bacteria, and its implications for various models proposed for the control of timing of mitosis are discussed.     

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.     

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.     

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.     

Regulation of cell size in the yeast Saccharomyces cerevisiae.

For cells of the yeast Saccharomyces cerevisiae, the size at initiation of budding is proportional to growth rate for rates from 0.33 to 0.23 h-1. At growth rates lower than 0.23 h-1, cells displayed a minimum cell size at bud initiation independent of growth rate. Regardless of growth rate, cells displayed an increase in volume each time budding was initiated. When abnormally small cells, produced by starvation for nitrogen, were placed in fresh medium containing nitrogen but with different carbon sources, they did not initiate budding until they had grown to the critical size characteristic of that medium. Moreover, when cells were shifted from a medium supporting a low growth rate and small size at bud initiation to a medium supporting a higher growth rate and larger size at bud initiation, there was a transient accumulation of cells within G1. These results suggest that yeast cells are able to initiate cell division at different cell sizes and that regulation of cell size occurs within G1.     

Role of timer and sizer in regulation of Chlamydomonas cell cycle

To estimate the role that time and size had in controlling the Chlamydomonas cell cycle, we used a new on-chip single-cell microcultivation system, which involved the direct observation of single cells captured in microchambers made on a thin glass slide. The dependence of the pattern of energy supply for cells on its cell cycle was examined through a series of different intensities of continuous illumination in a minimal medium, and we found that cell division occurred when cells reached the critical size, which was 2.2 times larger than that of the newly created cells. When illumination stopped before cells reached the critical size, even though growth had stopped, they continued dividing during the delay time, which was shorter when cells were larger. With re-illumination after darkness, cells began to grow again and the timing of cell division was again controlled by the critical size. This indicates that the co-existence of two cell cycle regulation mechanisms and the sizer mechanism had a stronger influence than the timer.     

Epidermal Patterning Genes Impose Non-cell Autonomous Cell Size Determination and have Additional Roles in Root Meristem Size Control

The regulation of cellular growth is of vital importance for embryonic and postembryonic patterning. Growth regulation in the epidermis has importance for organ growth rates in roots and shoots, proposing epidermal cells as an interesting model for cellular growth regulation. Here we assessed whether the root epidermis is a suitable model system to address cell size determination. In Arabidopsis thaliana L., root epidermal cells are regularly spaced in neighbouring tricho- (root hair) and atrichoblast (non-hair) cells, showing already distinct cell size regulation in the root meristem. We determined cell sizes in the root meristem and at the onset of cellular elongation, revealing that not only division rates but also cellular shape is distinct in tricho- and atrichoblasts. Intriguingly, epidermal-patterning mutants, failing to define differential vacuolization in neighbouring epidermal cell files, also display non-differential growth. Using these epidermal-patterning mutants, we show that polarized growth behaviour of epidermal tricho- and atrichoblast is interdependent, suggesting non-cell autonomous signals to integrate tissue expansion. Besides the interweaved cell-type-dependent growth mechanism, we reveal an additional role for epidermal patterning genes in root meristem size and organ growth regulation. We conclude that epidermal cells represent a suitable model system to study cell size determination and interdependent tissue growth.