Effects of asymmetric division on a stochastic model of the cell division cycle

The stochastic model of cell division formulated by Alt and Tyson is generalized to the case of imprecise binary fission. Closed-form expressions are derived for the generation-time distribution, the birth-size and division-size distributions, the beta curve, and the correlation coefficient of generation times of sister cells. The theoretical results are compared to observations of cell division statistics in a culture of fission yeast.     

Age-dependent cell cycle models.

This paper presents age-dependent cell cycle models i.e., models where cell generation time is a random variable given by some distribution function, and the probability of cell division per unit time is a function only of cell age (not, for example, of cell mass). It is shown that there does not exist a stable mass distribution if the cells grow exponentially. In the case of linear growth, conditions for stability of the mass distribution are derived. To show these, the methods different from those considered up till now in the literature, are used. It is also shown that one can consider the cell mass growth as a linear dynamical system with a stochastic perturbation. The sister cell model as an improvement of the Transition Probability Model is derived. Statistical data are obtained for that model, and comparisons are made with some experimental data. As a verification tool, alpha and beta curves, are used.     

Analysis of the cell cycle duration of cells of permanent line L-929

The direct measurement of the cell cycle duration in L-929 cells was performed using time-lapse photography. The cell cycle duration was 15.77 +/- 0.08 h with a standard deviation of 1.54 +/- 0.06 h. The experimental value fit to a normal distribution with a correlation coefficient 0.999. High homogeneity of this parameter and a wide range of variability of the karyotype (58-66 chromosomes) indicate that there is no correlation between these characteristics of L-929 cells. It is also shown that the difference between cell cycle durations of daughter cells tent to zero and fits by an exponent.     

Deterministic model of steady-state cell proliferation

A new approach to the kinetics of cell proliferation, based on the postulated restriction of the number of cell divisions in an organism gives the possibility to determine the individual lifetimes of cells. In the model, a necessary condition for a steady-state population is that two sister cells have distinct lifetimes. A steady state was obtained as a consequence of constant rate of cell production in each generation, when sister cell divisions alternated. The mean value of the generation time of cells is in the direct proportion to the number of cells in each generation and connected (with a coefficient of 2) with the generation number. In consequence, we attach great importance to the identification of cells belonging to distinct generations. Corresponding mathematical method to determine the cell population parameters is given and conclusions about stem cells' organization have been drawn.     

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.     

Growth of cell populations with arbitrarily distributed cycle durations. II. extended model for correlated cycle durations of mother and daughter cells

A model is proposed that describes the growth of cell populations, in which the cycle durations of mother and daughter and of sister cells can be correlated. The model accounts for arbitrary frequency distributions of cycle durations and for arbitrary correlations. Depending on the mother-daughter correlations, the frequency distribution of cycle durations either remains the same or changes from one cell generation to the next one. Both phenomena are described in the literature for different cell populations. Sister-sister correlations are shown to influence only numerical values in the model but not the model's structure. Model calculations with different types of correlations are compared with growth data on the ciliate Tetrahymena geleii.     

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.     

Investigation into the experimental kinetic support of the two-state model of the cell cycle

Five previously published cell generation-time distribution functions have been examined in an effort to elucidate the parameters of the two-state model of the cell cycle. These parameters are the fractional number of cells that bypass the G0 state, the probability of exit from G0, and the distribution of traversal times through the active state. To explain observed beta-curve behavior of cell populations, it is necessary to define the parameters in terms of pairwise behavior of newborn sister cells. From the beta-curve, we demonstrate that at least 50% of the cells must pass through the G0 state. The alpha-curve is consistent with any positive fraction of newborn cells passing through the G0 state, and provides no further information. We explore a possible method for resolving the remaining indeterminacy regarding the number of cells bypassing the G0 state, namely, examination of the generation-time distribution functions of fast sister cells only. Such an approach, although theoretically attractive, presents formidable experimental difficulties, however. If it should turn out that indeed only 50% of the cells are apparently passing through a random-exiting phase of the cell cycle, then an alterative plausible biological mechanism for the observed variability in generation times is supplied by Prescott's hypothesis: variability is a consequence of the inequality in the metabolic content of sister cells at birth.     

Investigation into the experimental kinetic support of the two-state model of the cell cycle

Five previously published cell generation-time distribution functions have been examined in an effort to elucidate the parameters of the two-state model of the cell cycle. These parameters are the fractional number of cells that bypass the G₀ state, the probability of exit from G₀ and the distribution of traversal times through the active state. To explain observed β-curve behavior of cell populations, it is necessary to define the parameters in terms of pairwise behavior of newborn sister cells. From the β-curve, we demonstrate that at least 50% of the cells must pass through the G₀ state. The α-curve is consistent with any positive fraction of newborn cells passing through the G₀ state, and provides no further information. We explore a possible method for resolving the remaining indeterminacy regarding the number of cells bypassing the G₀ state, namely, examination of the generation-time distribution functions of fast sister cells only. Such an approach, although theoretically attractive, presents formidable experimental difficulties, however. If it should turn out that indeed only 50% of the cells are apparently passing through a randomexiting phase of the cell cycle, then an alternative plausible biological mechanism for the observed variability in generation times is supplied by Prescott's hypothesis: variability is a consequence of the inequality in the metabolic content of sister cells at birth.     

The duration of cell cycle phases studied using 5-bromodeoxycytidine

The quantitative method is suggested to estimate cell cycle phase durations and dispersions of progress through the phases for population of cells. The method is based on the analysis of frequency of cells with different staining of sister chromatids by means of 5-bromodeoxycytidine. The process of cell population progress is described by the Gauss probability integral. The durations of the cell cycle phases are determined for cell culture of Chinese hamster.     

Cell cycle studies in chorionic villi

Summary We have studied the cell cycle of cells obtained from chorionic villi in direct and culture preparations by incorporation of the thymidine analogue BrdU to produce latelabelling or sister chromatid differentiation patterns. We have, therefore, been able to estimate the duration of the cell cycle and, more specifically, the length of some of its phases. While results for chorionic villus sample cells in culture resembled those obtained for fibroblasts, data for the spontaneously dividing trophoblastic cells in direct preparations were different. Villi exposed to BrdU immediately after sampling showed a slight delay in the incorporation of the analogue and a lower percentage of labelled cells compared to villi treated after an overnight incubation, probably due to a temporary effect of the sampling technique. Results from semi-direct protocols suggest that cells have a G2 of no more than 4h, and a mid-S phase of 10–16h. The G1 period is very variable. After 48 h incubation with BrdU, only 4% of cells reach their second generation, whereas this percentage increases up to 70% after 72h, indicating that under these experimental conditions most cells have a cell cycle of approximately 36 h. The average number of sister chromatid exchanges was similar in both direct preparations and cultures: 5.2±2.1 SCE per cell.     

The cell cycle,cell death,and cell morphology during retinoic acid-induced differentiation of embryonal carcinoma cells

Time-lapse films were made of PC13 embryonal carcinoma cells, synchronized by mitotic shake off, in the absence and presence of retinoic acid. Using a method based on the transition probability model, cell cycle parameters were determined during the first five generations following synchronization. In undifferentiated cells, cell cycle parameters remained identical for the first four generations, the generation time being 11–12 hr. In differentiating cells, with retinoic acid added at the beginning of the first cycle, the first two generations were the same as controls. The duration of the third generation, however, was increased to 15.7 hr while the fourth and fifth generation were approximately 20 hr, the same as in exponentially growing, fully differentiated cells. The increase in generation time of dividing cells was principally due to an increase in the length of S phase. Cell death induced by retinoic acid also occurred principally in the third and subsequent generations. Cell population growth was then significantly less than that expected from the generation time derived from cycle analysis of dividing cells. Cells lysed frequently as sister pairs suggesting susceptibility to retinoic acid toxicity determined in a generation prior to death. Morphological differentiation, as estimated by the area of substrate occupied by cells, was shown to begin in the second cell cycle after retinoic acid addition. These results demonstrate that as in the early mammalian embryo, differentiation of embryonal carcinoma cells to an endoderm-like cell is also accompanied by a decrease in growth rate but that this is preceded by acquisition of the morphology characteristic of the differentiated progeny.     

Time-lapse cinematography studies of cell cycle and mitosis duration

The progenies of 44 cells (EMT6 cell line) have been studied in vitro by time-lapse cinematography for up to eight generations. It has been found that the mean mitotic and intermitotic times vary significantly with the age of the culture and that they are positively correlated. There are correlations between mother and daughter parameters and between sister cells. All these correlations are higher when the age of the culture is greater.     

Global asymptotic stability of the size distribution in probabilistic models of the cell cycle

Probabilistic models of the cell cycle maintain that cell generation time is a random variable given by some distribution function, and that the probability of cell division per unit time is a function only of cell age (and not, for instance, of cell size). Given the probability density, f(t), for time spent in the random compartment of the cell cycle, we derive a recursion relation for _n(x), the probability density for cell size at birth in a sample of cells in generation n. For the case of exponential growth of cells, the recursion relation has no steady-state solution. For the case of linear cell growth, we show that there exists a unique, globally asymptotically stable, steady-state birth size distribution, _*(x). For the special case of the transition probability model, we display _*(x) explicitly.This work was supported by the National Science Foundation under grants MCS8301104 (to J.J.T.) and MCS8300559 (to K.B.H.), and by the National Institutes of Health under grant GM27629 (to J.J.T.).     

Sister chromatid exchange frequency,cellular replication and relative cloning efficiency in human teratocarcinoma-derived cells

To determine the relationships between the induction of specific biological responses and exposure to DNA-damaging agents, human teratocarcinoma-derived cells were exposed to either ethyl methanesulfonate or to methyl methanesulfonate, and sister chromatid exchange, cellular proliferation and relative cloning ability measured. SCE increased while cellular proliferation and relative cloning ability each decreased in a concentration-dependent manner. Methyl methanesulfonate was consistently more efficient in inducing biological responses than was ethyl methanesulfonate. When the individual responses were compared, the decrease in cellular proliferation paralleled the reduction in cloning efficiency. A strong correlation was also observed between the reduction in relative cloning ability and sister chromatid exchange frequency. Because these relationships are similar to those previously described in other mammalian cell lines, the observations in our study suggest that the P3 cell line is an appropriate choice for modeling effects of toxicant exposure in human cells.Abbreviations AGT average generation time - BUdR 5-bromodeoxyuridine - CHO Chinese hamster ovary - EMS ethyl methanesulfonate - ENU N-ethyl-N-nitrosourea - MMS methyl methanesulfonate - MNU N-methyl-Nnitrosourea - SCE sister chromatid exchange     

On the distribution of cell cycle generation times

The problem of whether the cell cycle is a deterministic or probabilistic process is widely discussed in the current literature (P. Nurse, Nature, 286, pp. 9–10, 1980). In this report the question of fluctuations of cell cycle period is treated in the limits of the membrane model of cell division regulation. The parametric analysis of the equations set both for normal and tumour cells is carried out. We describe the bifurcation parameters in the neighbourhood of which the system can amplify the small fluctuations. The presence of white noise in parameters describing the lipids and antioxidants influxes into membrane is examined by methods of Marcovian processes and also by direct stochastic computer simulation. The equation for the distribution function of generation times is obtained and the increase of dispersion and mean cycle time during the changes of those parameters which would be connected with cell culture density is calculated.The influence of parameter fluctuations upon the cycle period for both normal and tumour cells is compared in the framework of model assumptions. The ratio of dispersion of generation time distribution to mean period value for an ensemble of tumour cells is shown to be several times greater than that for normal ones.In the discussion the problem of the presence of a premitotical (G₀₂) resting state and of the possibility of its experimental detection is considered.     

Temperature compensation in the mammalian cell cycle

Random and synchronous V79 cells were shifted from 37.5 °C to temperatures between 29 ° and 41 °C. Intermitotic time determinations of random cultures showed an increase in generation time and a broadening in the distribution of generation times in cells whose cycle spanned the temperature shift, but only a slight increase in generation time after one generation at temperatures between 34 °–40 °C. At 33.5 °C and below there was a stepwise increase in generation time. When cells grown at non-standard temperatures were allowed to habituate for 48 h at the altered temperature prior to analysis, the increase in median intermitotic time was slightly less in comparison to analyses done after only one generation following the temperature step. The Q₁₀ for cell division of cells growing at temperatures from 34 ° to 40 °C was between 1.15 and 1.26, suggesting that the mammalian cell cycle is temperature compensated over a limited (6–7 °C) temperature span. Mammalian cells in culture appear to have the same capacity for temperature compensation in their cell cycle as do unicellular eukaryotes. The fact that cycle time at lower temperatures increases in a discrete manner is taken as evidence for a quantal clock.     

Characterization of mitoses with sister chromatid differentiation (SCD) and consequences for the analysis of proliferation kinetics and sister chromatid exchanges in asynchronously growing cells

Summary Labeling cells with bromodeoxyuridine (BrdU) permits the differentiation of mitoses of the first, second, and third generation after the addition of BrdU. The term second mitoses is used for those cells which have incorporated BrdU for two-S-phases and which exhibit sister chromatid differentiation (SCD). However, SCD can also be obtained if the cell was in S-phase at the time of BrdU-addition and had already replicated part of its DNA. Such cells with incomplete BrdU-substitution in the first S-phase can only be differentiated from completely substituted ones by the quality of the SCD and are usually also grouped as second mitoses in the evaluation of experiments. Due to the heterogeneity of the evaluated second mitoses, the determination of proliferation delay and the incidence of sister chromatid exchange-induction can depend on the time of chromosome preparation.     

Simultaneous estimation of sister chromatid exchange and evaluation of cell cycle delay

The frequency of sister chromatid exchange and cell cycle duration were evaluated simultaneously. This approach is based on the analysis of distribution of cells with differential staining of sister chromatids after treatment of cells with 5-bromodeoxyuridine. The treatment of cells with thiotepa caused no changes in cell cycle duration, while the combination of thiotepa and hydroxyurea (HU) or cytosine-beta-D-arabinofuranoside (ARA-C) was observed to prolong cell cycle duration. Furthermore, it has been shown that caffeine, HU, ARA-C do not increase frequency of sister chromatid exchange in control cells and in cell treated with thiotepa.     

Mathematical description and analysis of cell cycle kinetics and the application to Ehrlich ascites tumor

The linear and nonlinear aspects of the dynamics of the cell cycle kinetics of cell populations are studied. The dynamics are represented by difference equations. The characteristics of cell population systems are analyzed by applying the model to Ehrlich ascites tumor. The model applied for the simulations of the growth of Ehrlich ascites tumor cells incorporates processes of cell division, cell death, transition of cells to resting states and clearance of dead cells. Comparison of the results obtained with the model and the experimental data suggests that the duration of the mean generation time of the proliferating EAT cells increases with aging of the tumor. An attempt is made to relate the prolongation of cell mean generation time with processes of cell death and dead cell clearance. Studying the transition of cells to the resting states, it becomes apparent that in fact transition of proliferating cells to the resting states occurs somewhere close to the end of the cell cycle and with a rate that varies with the age of the tumor. Time course behavior of the cell age, cell size, and cell DNA distribution with aging of the tumor are obtained. Variations in average size and average DNA contents are determined.