A growth curve of cell numbers in the neural retina of embryonic chicks

The growth of cell numbers in a normal embryonic population of neural retinal cells is described. The numbers were estimated from a time shortly after the neural retina first becomes recognizable to a time when numbers of retinal cells have become steady. Cell numbers were estimated in preparations of an entire neural retina dispersed into a suspension of single nuclei which were then counted in a Coulter counter. The growth curve of the ln numbers of cells has three phases of growth: an exponential phase during which there is steady-state exponential growth, a differentiative phase during which cell proliferation ceases and an end phase when no further change in cell numbers can be detected. The variances of the ln numbers of cells were highest during the exponential phase. The variances decreased during the differentiative phase and were at their lowest during the end phase. For variances to decrease requires mechanisms which control the final numbers of cells in the neural retina very precisely. The implications of mechanisms which operate by controlling cell lineages are explored.     

A Growth Curve of Cell Numbers In the Neural Retina of Embryonic Chicks

ABSTRACT The growth of cell numbers in a normal embryonic population of ncural retinal cells is described. the numbers were estimated from a time shortly after the neural retina first becomes recognizable to a time when numbers of retinal cells have become steady. Cell numbers were estimated in preparations of an entire neural retina dispersed into a suspension of single nuclei which were then counted in a Coulter counter. the growth curve of the In numbers of cells has three phases of growth: an exponential phase during which there is steady-state exponential growth, a differentiative phase during which cell proliferation ceases and an end phase when no further change in cell numbers can be detected. the variances of the In numbers of cells were highest during the exponential phase. the variances decreased during the differentiative phase and were at their lowest during the end phase. For variances to decrease requires mechanisms which control the final numbers of cells in the neural retina very precisely. the implications of mechanisms which operate by controlling cell lineages are explored.     

Growth of cell populations with arbitrarily distributed cycle durations. I. basic model

A discrete model is proposed describing the growth of cell populations with arbitrary frequency distributions of cycle durations. The model assumes that each cell divides into two cells at the end of its cycle, and that each new cell is assigned an individual cycle duration according to a probability distribution that can be arbitrarily defined. The increase in the cell number is calculated, either from the numbers of cells at earlier time points or from the initial conditions of the population, by a recurrence formula; it is also approximated by the optimal exponential function, whose parameters are determined by the initial conditions. The appropriate average cycle duration is shown not to be the arithmetic or geometric mean, but rather the solution to a more complex equation. Age distributions are calculated and compared with those found in the literature. The results of the model calculations are compared with computer simulations and with observed data on populations of the ciliate Tetrahymena geleii.     

Modeling growth and telomere dynamics in Saccharomyces cerevisiae

A general branching process is proposed to model a population of cells of the yeast Saccharomyces cerevisiae following loss of telomerase. Previously published experimental data indicate that a population of telomerase-deficient cells regain exponential growth after a period of slowing due to critical telomere shortening. The explanation for this phenomenon is that some cells engage telomerase-independent pathways to maintain telomeres that allow them to become “survivors.” Our model takes into account random variation in individual cell cycle times, telomere length, finite replicative lifespan of mother cells, and survivorship. We identify and estimate crucial parameters such as the probability of an individual cell becoming a survivor, and compare our model predictions to experimental data.     

Fission yeast cells grow approximately exponentially

How the rate of cell growth is influenced by cell size is a fundamental question of cell biology. The simple model that cell growth is proportional to cell size, based on the proposition that larger cells have proportionally greater synthetic capacity than smaller cells, leads to the prediction that the rate of cell growth increases exponentially with cell size. However, other modes of cell growth, including bilinear growth, have been reported. The distinction between exponential and bilinear growth has been explored in particular detail in the fission yeast Schizosaccharomyces pombe. We have revisited the mode of fission yeast cell growth using high-resolution time-lapse microscopy and find, as previously reported, that these two growth models are difficult to distinguish both because of the similarity in shapes between exponential and bilinear curves over the two-fold change in length of a normal cell cycle and because of the substantial biological and experimental noise inherent to these experiments. Therefore, we contrived to have cells grow more than twofold, by holding them in G2 for up to 8 h. Over this extended growth period, in which cells grow up to 5.5-fold, the two growth models diverge to the point that we can confidently exclude bilinear growth as a general model for fission yeast growth. Although the growth we observe is clearly more complicated than predicted by simple exponential growth, we find that exponential growth is a robust approximation of fission yeast growth, both during an unperturbed cell cycle and during extended periods of growth.     

A stochastic branching model with formation of subunits applied to the growth of intestinal crypts

The intestinal epithelium is one of the most rapidly regenerating tissues in mammals. Cell production takes place in the intestinal crypts which contain about 250 cells. Only a minority of 1-60 proliferating cells are able to maintain a crypt over a long period of time. However, so far attempts to identify these stem cells were unsuccessful. Therefore, little is known about their cellular growth and selfmaintenance properties. On the other hand, the crypts appear to exhibit a life cycle which starts by fission of existing crypts and ends by fission or extinction. Data on these processes have recently become available. Here, we demonstrate how these data on the life cycle of the macroscopic crypt structure can be used to derive a quantitative model of the microscopic process of stem cell growth. The model assumptions are: (1) stem cells undergo a time independent supracritical Markovian branching process (Galton-Watson process); (2) a crypt divides if the number of stem cells exceeds a given threshold and the stem cells are distributed to both daughter crypts according to binomial statistics; (3) the size of the crypt is proportional to the stem cell number. This model combining two different stochastic branching processes describes a new class of processes whose stationary stability and asymptotic behavior are examined. This model should be applicable to various growth processes with formation of subunits (e.g. population growth with formation of colonies in biology, ecology and sociology). Comparison with crypt data shows that intestinal stem cells have a probability of over 0.8 of dividing asymmetrically and that the threshold number should be 8 or larger.     

A model with 'growth retardation' for the kinetic heterogeneity of tumour cell populations

In the present paper we propose a continuous cell population model based on Shackney's idea of growth retardation. Cells are characterized by two state variables: the cell maturity x, 0 < or = x < or = 1, and a state variable T that identifies the rate of maturation along cell cycle. During their life span, cells can change T at random by jump transitions to T values corresponding to slower maturation rates, while at each jump the maturity x is conserved. Both the time evolution of the population and the exponential stationary solution are numerically computed. The distribution of the cell cycle transit time in asynchronous exponential growth is investigated by Monte Carlo simulation. An approximated formula for the distribution of cell cycle time is also provided.     

An analysis of the growth of the retinal cell population in embryonic chicks yielding proliferative ratios, numbers of proliferative and non-proliferative cells and cell-cycle times for successive generations of cell cycles

Growth curves of the retinal cell population of embryonic chicks were fitted by a branching-process model of cell population growth, thereby estimating the proliferative ratios and mean cell-cycle times of the generations of cell cycles that underlie retinal growth. The proliferative ratio determines the proportion of cells that divides in the next generation, so the numbers of proliferative and non-proliferative cells in each generation of cell cycles were obtained. The mean cell-cycle times determine the times over which the generations are extant. Assuming growth starts from one cell in generation 0, the proliferative cells reach 3.6 × 10⁶ and the non-proliferative cells reach 1.1 × 10⁶ by generation 23. The next four generations increase the proliferative cell numbers to 13.9 × 10⁶ and produce 20.1 × 10⁶ non-proliferative cells. In the next five generations in the end phase of growth, non-proliferative cells are produced in large numbers at an average of 13.9 × 10⁶ cells per generation as the retinal lineages are completed. The retinal cell population reaches a maximum estimated here at 98.2 × 10⁶ cells. The mean cell-cycle time estimates range between 6.8 and 10.1 h in generations before the end phase of growth and between 10.6 and 17.2 h in generations in the end phase. The retinal cell population growth is limited by the depletion of the proliferative cell population that the production of non-proliferative cells entails. The proliferative ratios and the cell-cycle-time distribution parameters are the likely determinants of retinal growth rates. The results are discussed in relation to other results of spatial and temporal patterns of the cessation of cell cycling in the embryonic chick retina.     

The growth kinetics ofB. subtilis

There has been considerable discussion by Kubitschek and Cooper concerning the growth rate of cells ofE. coli throughout the cell cycle. Consequently, it is relevant to test Kubitschek's linear model against the exponential model espoused by Cooper (and many others) with another organism and another technique.Burdett et al. measured, by electron microscopy and computer analysis of the microphotographs, the distribution of lengths of a population of cells ofBacillus subtilis grown in 0.4% succinate in a minimal medium. The data were fitted to the extended Collins-Richmond method of Kirkwood & Burdett which subdivided the cell cycle into several phases. I have taken their results and compared them with the linear and exponential growth models for the entire cell cycle after applying correction to the data for the shape of completed and forming poles; i.e., to put the data on a cell-volume basis instead of a cell-length basis. Most of the correction involves no arbitrary assumptions. The conclusion is that global volume growth rate is nearly proportional to cell volume; i.e. growth ofBacillus subtilis is nearly exponential for almost every cell in the growing culture.     

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

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

Cell cycles and growth laws: the CCC model

In the cell-cycle-with-control model (CCC model), cells have to satisfy a condition before they are allowed to pass a control point during G1. Different cycle durations within a cell population are explained by individual time spans needed to satisfy the passing condition. If the distribution of cycle durations is time invariant, the population will grow exponentially. However, if the average cycle duration becomes longer, while the population grows, non-exponential population growth results. Simple functions for the lengthening of the average cycle duration, like linear or exponential ones, yield the well-known growth laws found in the biological literature. The same functions can be represented by an "S-system" differential equation that was derived earlier as an approximation for biochemical systems with many fast reactions (metabolism) and one slow process (e.g. ageing).     

A checkpoint-oriented cell cycle simulation model

Modeling and in silico simulations are of major conceptual and applicative interest in studying the cell cycle and proliferation in eukaryotic cells. In this paper, we present a cell cycle checkpoint-oriented simulator that uses agent-based simulation modeling to reproduce the dynamics of a cancer cell population in exponential growth. Our in silico simulations were successfully validated by experimental in vitro supporting data obtained with HCT116 colon cancer cells. We demonstrated that this model can simulate cell confluence and the associated elongation of the G1 phase. Using nocodazole to synchronize cancer cells at mitosis, we confirmed the model predictivity and provided evidence of an additional and unexpected effect of nocodazole on the overall cell cycle progression. We anticipate that this cell cycle simulator will be a potential source of new insights and research perspectives.     

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     

A model of proliferating cell populations with inherited cycle length

A mathematical model of cell population growth introduced by J. L. Lebowitz and S. I. Rubinow is analyzed. Individual cells are distinguished by age and cell cycle length. The cell cycle length is viewed as an inherited property determined at birth. The density of the population satisfies a first order linear partial differential equation with initial and boundary conditions. The boundary condition models the process of cell division of mother cells and the inheritance of cycle length by daughter cells. The mathematical analysis of the model employs the theory of operator semigroups and the spectral theory of linear operators. It is proved that the solutions exhibit the property of asynchronous exponential growth.     

A stochastic model of a cell population with quiescence

A cell population in which cells are allowed to enter a quiescent (nonproliferating) phase is analyzed using a stochastic approach. A general branching process is used to model the population which, under very mild conditions, exhibits balanced exponential growth. A formula is given for the asymptotic fraction of quiescent cells, and a numerical example illustrates how convergence toward the asymptotic fraction exhibits a typical oscillatory pattern. The model is compared with deterministic models based on semigroup analysis of systems of differential equations.     

A stochastic model of a cell population with quiescence

A cell population in which cells are allowed to enter a quiescent (nonproliferating) phase is analyzed using a stochastic approach. A general branching process is used to model the population which, under very mild conditions, exhibits balanced exponential growth. A formula is given for the asymptotic fraction of quiescent cells, and a numerical example illustrates how convergence toward the asymptotic fraction exhibits a typical oscillatory pattern. The model is compared with deterministic models based on semigroup analysis of systems of differential equations.     

Proliferation of anchorage-dependent contact-inhibited cells: I. Development of theoretical models based on cellular automata

We report the development of new class of discrete models that can accurately describe the contact-inhibited proliferation of anchorage-dependent cells. The models are based on cellular automata, and they quantitatively account for contact inhibition phenomena occurring during all stages of the proliferation process: (a) the initial stage of "exponential" growth of cells without contact inhibition; (b) the second stage where cell colonies form and grow with few colony mergings; and (c) the final stage where proliferation rates are dominated by colony merging events. Model prediction are presented and analyzed to study the complicated dynamics of large cell populations and determine how the initial spatial cell distribution, the seeding density, and the geometry of the growth surface affect the observed proliferation rates. Finally, we present a model variant that can simulate contact-inhibited proliferation of asynchronous cell populations with arbitrary cell cycle-time distribution. The latter model can also compute the percentage of cells that are in a specific phase of their division cycle at a given time.     

Embryonic cell commitment by a simultaneous alteration in nucleo/cytoplasmic ratio in sibling cells between subsequent cell cycles

Studies on murine embryonal carcinoma (EC) cell lines have revealed a mechanism for commitment of early embryonic cells. By means of a particular intraclonal cell surface glycoprotein and intermitotic time heterogeneity found in the EC lines used, we have devised a cell cycle model and written a computer program for cell cycle stimulation. In the present investigation experimental tissue culture data obtained from the EC lines were inserted into the computer program and the simulations represent a good fit to experimental data. It is shown that the dynamics of the driving forces in the 'cell growth cycle' and the 'DNA-division cycle', when assumed to be loosely coupled and analyzed in subsequent cell cycles, reveal a mechanism that can commit the mother cell to impel her daughter cells into the next stage in embryonic development by altering the relationship between those two uncoupled subcycles. Thereby the nucleo/cytoplasmic ratio (DNA/mass) is altered in a similar way in the two daughter cells. The simulations increase the reliability of the model and open up possibilities to test other embryonic cell systems.