Cell Growth and Division: IV. Determination of Volume Growth Rate and Division Probability

Volume growth rate and division probability functions for mammalian cells have been determined as functions of cell volume with good reproducibility and statistical precision using Coulter volume spectrometry and the equations of the Bell model. Results are compared with independent measurements on synchronous cultures. The slow rate of volume dispersion requires that the growth rate F(tau, V) be closely proportional to volume for cells of a given age. However, when F(tau, V) is averaged over the age distribution of a population in balanced exponential growth to give the growth rate function f(V), the latter may rise more steeply than V.     

Cell Growth and Division: I. A Mathematical Model with Applications to Cell Volume Distributions in Mammalian Suspension Cultures

A mathematical model is formulated for the development of a population of cells in which the individual members may grow and divide or die. A given cell is characterized by its age and volume, and these parameters are assumed to determine the rate of volume growth and the probability per unit time of division or death. The initial value problem is formulated, and it is shown that if cell growth rate is proportional to cell volume, then the volume distribution will not converge to a time-invariant shape without an added dispersive mechanism. Mathematical simplications which are possible for the special case of populations in the exponential phase or in the steady state are considered in some detail. Experimental volume distributions of mammalian cells in exponentially growing suspension cultures are analyzed, and growth rates and division probabilities are deduced. It is concluded that the cell volume growth rate is approximately proportional to cell volume and that the division probability increases with volume above a critical threshold. The effects on volume distribution of division into daughter cells of unequal volumes are examined in computer models.     

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.     

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.     

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.     

A note on the dispersionless growth law for single cells

A population of initially synchronized cells is considered wherein each cell grows according to a dispersionless growth law and the probability of cell division is determined by cell age. The first and second moments of the distribution of birth volumes are considered as functions of time and it is shown that it is impossible for both moments to approach finite, nonzero limits ast→∞. This implies that the volume distribution of the population will not approach a limiting distribution on any finite, nonzero volume interval and that the population will not attain balanced exponential growth. An illustrative example is worked out in detail. The distribution of birth volumes is also analyzed as a function of generation number and it is found that the logarithm of the birth volume in thejth generation is normally distributed asj→∞, with an unbounded variance. Generalizations and implications of these results are briefly discussed. Work supported by the U.S. Atomic Energy Commission.     

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.     

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.     

Modes of Growth in Mammalian Cells

The increase of cell volume as a function of time was studied throughout the generation cycle in synchronous cultures of Chinese hamster cells using a Coulter aperture and a multichannel analyzer calibrated against known cell volumes. The experimental results were compared to a mathematical model of cell volume increase which considered the effect of the distribution of individual cell generation times on the progress of the population. Several modes of volume increase, including linear and exponential, were considered. The mean volume vs. time curve was rounded at the ends of the cycle even when linear growth was assumed. The experimental results show that cell volume increased in a smooth fashion as a function of time, with no discontinuities in rate detectable at periods when cells may have been undergoing metabolic shifts as, for example, through the phases associated with DNA synthesis, G₁, S, G₂. A statistical test on the comparison of the modal cell volume vs. time data to the predictions of linear and exponential growth models accepted both hypotheses within the resolution of these experiments. However, exponential growth was favored over linear growth in one cell line. Volume dispersion was almost constant with time in both sublines which is also consistent with exponential growth. Limitations of the electronic technique of volume measurement and indications for future experiments are discussed.     

Growth During the Bacterial Cell Cycle: Analysis of Cell Size Distribution

Cell volume distributions were determined electronically for steady-state cultures of Escherichia coli, Bacillus megaterium, Bacillus subtilis, and Salmonella typhimurium by use of a Coulter transducer-multichannel analyzer system of good resolution. All of the cell volume distributions had the same general shape, even though cultures were grown at widely different rates. Some results were independent of any particular growth model. Both the variability in the volumes of dividing cells and the fraction of constricted and unseparated doublet cells increased with growth rate. The greater separation to single cells at slow growth rates is in agreement with the general finding that filamentous and hyphal forms are greatly reduced in slowly growing chemostat cultures. The distributions were fitted equally well by simple models which assumed that cell growth was either linear or exponential throughout the entire cell cycle. It is concluded that methods of determining growth rate by analysis of distributions of bacterial volumes do not yet have sufficient resolution to distinguish between a variety of alternative models for growth of bacteria.     

Growth in Volume of Euglena gracilis During the Division Cycle

The distribution of volumes of Euglena gracilis cells was measured conductimetrically. The volume spectrum of cultures in balanced growth was analyzed by the method of Collins and Richmond. The kinetics of volume increase of Euglena is neither linear nor exponential; the growth rate of small and large cells is low, but intermediate size cells show the largest growth rate.     

Steady size distributions for cells in one-dimensional plant tissues

We consider a population of cells growing and dividing steadily without mortality, so that the total cell population is increasing, but the proportion of cells in any size class remains constant. The cell division process is non-deterministic in the sense that both the size at which a cell divides, and the proportions into which it divides, are described by probability density functions. We derive expressions for the steady size/birth-size distribution (and the corresponding size/age distribution) in terms of the cell birth-size distribution, in the particular case of one-dimensional growth in plant organs, where the relative growth rate is the same for all cells but may vary with time. This birth-size distribution is shown to be the principal eigenfunction of a Fredholm integral operator. Some special cases of the cell birth-size distribution are then solved using analytical techniques, and in more realistic examples, the eigen-function is found using a simple, generally applicable numerical iteration.     

Asymmetrical division of Saccharomyces cerevisiae.

The unequal division model proposed for budding yeast (L. H. Hartwell and M. W. Unger, J. Cell Biol. 75:422-435, 1977) was tested by bud scar analyses of steady-state exponential batch cultures of Saccharomyces cerevisiae growing at 30 degrees C at 19 different rates, which were obtained by altering the carbon source. The analyses involved counting the number of bud scars, determining the presence or absence of buds on at least 1,000 cells, and independently measuring the doubling times (gamma) by cell number increase. A number of assumptions in the model were tested and found to be in good agreement with the model. Maximum likelihood estimates of daughter cycle time (D), parent cycle time (P), and the budded phase (B) were obtained, and we concluded that asymmetrical division occurred at all growth rates tested (gamma, 75 to 250 min). D, P, and B are all linearly related to gamma, and D, P, and gamma converge to equality (symmetrical division) at gamma = 65 min. Expressions for the genealogical age distribution for asymmetrically dividing yeast cells were derived. The fraction of daughter cells in steady-state populations is e-alpha P, and the fraction of parent cells of age n (where n is the number of buds that a cell has produced) is (e-alpha P)n-1(1-e-alpha P)2, where alpha = IN2/gamma; thus, the distribution changes with growth rate. The frequency of cells with different numbers of bud scars (i.e., different genealogical ages) was determined for all growth rates, and the observed distribution changed with the growth rate in the manner predicted. In this haploid strain new buds formed adjacent to the previous buds in a regular pattern, but at slower growth rates the pattern was more irregular. The median volume of the cells and the volume at start in the cell cycle both increased at faster growth rates. The implications of these findings for the control of the cell cycle are discussed.     

The effect of cellular differentiation on the development of permanent drug resistance

A stem cell compartment model is utilized to simulate the growth of human tumors. This model is used to explore the effect of cell differentiation and loss on the development of spontaneous drug resistance. Cellular differentiation is found to increase the rate of development of single drug resistance, although this is balanced by the likehood that such resistant cells will subsequently become extinct. Overall the probability that singly resistant cells will develop and persist is found to be independent of the rate of cellular differentiation. Conversely, when two drugs are available, the probability that cells resistant to both drugs will persist is proportional to the rate of cellular differentiation. Approximate formulae relating the net overall mutation rate to the intrinsic mutation rates and net growth rates of the stem cell compartment are developed.     

Dependency of size of Saccharomyces cerevisiae cells on growth rate.

The mean size and percentage of budded cells of a wild-type haploid strain of Saccharomyces cerevisiae grown in batch culture over a wide range of doubling times (tau) have been measured using microscopic measurements and a particle size analyzer. Mean size increased over a 2.5-fold range with increasing growth rate (from tau = 450 min to tau = 75 min). Mean size is principally a function of growth rate and not of a particular carbon source. The duration of the budded phase increased at slow growth rates according to the empirical equation, budded phase = 0.5 tau + 27 (all in minutes). Using a recent model of the cell cycle in which division is thought to be asymmetric, equations have been derived for mean cell age and mean cell volume. The data are consistent with the notion that initiation of the cell cycle occurs at "start" after attainment of a critical cell size, and this size is dependent on growth rate, being, at slow growth rates, 63% of the volume of fast growth rates. Previous reports are reanalyzed in the light of the unequal division model and associated population equations.     

Identification of age-structured models: cell cycle phase transitions

A methodology is developed that determines age-specific transition rates between cell cycle phases during balanced growth by utilizing age-structured population balance equations. Age-distributed models are the simplest way to account for varied behavior of individual cells. However, this simplicity is offset by difficulties in making observations of age distributions, so age-distributed models are difficult to fit to experimental data. Herein, the proposed methodology is implemented to identify an age-structured model for human leukemia cells (Jurkat) based only on measurements of the total number density after the addition of bromodeoxyuridine partitions the total cell population into two subpopulations. Each of the subpopulations will temporarily undergo a period of unbalanced growth, which provides sufficient information to extract age-dependent transition rates, while the total cell population remains in balanced growth. The stipulation of initial balanced growth permits the derivation of age densities based on only age-dependent transition rates. In fitting the experimental data, a flexible transition rate representation, utilizing a series of cubic spline nodes, finds a bimodal G(0)/G(1) transition age probability distribution best fits the experimental data. This resolution may be unnecessary as convex combinations of more restricted transition rates derived from normalized Gaussian, lognormal, or skewed lognormal transition-age probability distributions corroborate the spline predictions, but require fewer parameters. The fit of data with a single log normal distribution is somewhat inferior suggesting the bimodal result as more likely. Regardless of the choice of basis functions, this methodology can identify age distributions, age-specific transition rates, and transition-age distributions during balanced growth conditions.     

Cell Growth and Division: II. Experimental Studies of Cell Volume Distributions in Mammalian Suspension Cultures

Experimental proof is given that the volume distribution spectrum of mammalian cells in suspension culture can be determined accurately with a Coulter spectrometer. Stable spectra corresponding to the predictions of a mathematical model are observed under favorable conditions of growth. Cell volume spectrometry appears to be a useful method for diagnosing the state of the culture with respect to past uniformity of growth rate and present population age distribution. In addition, it offers a method for quantitative study of the laws governing cell growth and division.     

Control of Cell Growth in Bacteria: Experiments with Thymine Starvation

When cells of Escherichia coli THU were starved for thymine, they continued to grow without division for at least two successive volume doublings at their initial rate. Within experimental error this average rate of volume increase, 0.21 mum(3) per hr, was identical with that observed in control cultures during two generations of growth in the presence of thymine. This growth rate was also independent of the age of the cells at the time of starvation. These results are consistent with the hypothesis, proposed earlier, that growth rates are controlled by uptake sites for binding, transport, or accumulation of compounds into the cell, that the number of these sites is constant throughout most of the cell cycle, and that this number doubles near or at cell division.     

Linear Cell Growth in Escherichia coli

Growth was studied in synchronous cultures of Escherichia coli, using three strains and several rates of cell division. Synchrony was obtained by the Mitchison-Vincent technique. Controls gave no discernible perturbation in growth or rate of cell division. In all cases, mean cell volumes increased linearly (rather than exponentially) during the cycle except possibly for a small period near the end of the cycle. Linear volume growth occurred in synchronous cultures established from cells of different sizes, and also for the first volume doubling of cells prevented from division by a shift up to a more rapid growth rate. As a model for linear kinetics, it is suggested that linear growth represents constant uptake of all major nutrient factors during the cycle, and that constant uptake in turn is established by the presence of a constant number of functional binding or accumulation sites for each growth factor during linear growth of the cell.     

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.