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.).     

A Generalised Age- and Phase-Structured Model of Human Tumour Cell Populations Both Unperturbed and Exposed to a Range of Cancer Therapies

We develop a general mathematical model for a population of cells differentiated by their position within the cell division cycle. A system of partial differential equations governs the kinetics of cell densities in certain phases of the cell division cycle dependent on time t (hours) and an age-like variable τ (hours) describing the time since arrival in a particular phase of the cell division cycle. Transition rate functions control the transfer of cells between phases. We first obtain a theoretical solution on the infinite domain −∞ < t < ∞. We then assume that age distributions at time t=0 are known and write our solution in terms of these age distributions on t=0. In practice, of course, these age distributions are unknown. All is not lost, however, because a cell line before treatment usually lies in a state of asynchronous balanced growth where the proportion of cells in each phase of the cell cycle remain constant. We assume that an unperturbed cell line has four distinct phases and that the rate of transition between phases is constant within a short period of observation (‘short’ relative to the whole history of the tumour growth) and we show that under certain conditions, this is equivalent to exponential growth or decline. We can then gain expressions for the age distributions. So, in short, our approach is to assume that we have an unperturbed cell line on t ≤ 0, and then, at t=0 the cell line is exposed to cancer therapy. This corresponds to a change in the transition rate functions and perhaps incorporation of additional phases of the cell cycle. We discuss a number of these cancer therapies and applications of the model.     

Mixed-stock aging analysis reveals variable sea turtle maturity rates in a recovering population

Quantifying demographic parameters and variable vital rates, such as somatic growth rates, time to maturity, and reproductive longevity, is important for effective management of threatened and endangered populations such as sea turtles (Cheloniidae). To address these knowledge gaps, we applied skeletochronology to analyze and compare somatic growth rates and variation in life-history traits such as age and size at sexual maturity for 65 green turtles (Chelonia mydas) in the eastern Pacific Ocean (EP), along the west coast of the United States; turtles belonged to ≥2 nesting subpopulations that differed in body size (mean nesting size). Green turtles in the EP spend approximately 5 years in the oceanic stage before recruiting to nearshore habitats, males may be smaller and younger than females at maturation (x̅ = 17.7 ± 5.5 yr vs. 28.0 ± 8.2 yr), and younger age at sexual maturity was associated with smaller size at sexual maturity, suggesting that mean nesting body size may be reflective of maturation timing for subpopulations. Smaller body sizes for females nesting at Michoacán, Mexico (continental) rookeries, yielded a younger predicted age at sexual maturity (x̅ = ~17 yr) compared to females from Revillagigedo Islands, Mexico rookeries, which displayed larger body sizes and older age at sexual maturity (x̅ = ~30 yr). We consider possible mechanisms driving the observed divergence in life-history traits, including the possibility that earlier maturation (reduced generation length) for turtles in the Michoacán nesting subpopulation may be a response to intense harvesting in the past 50 years, and consideration of such anthropogenic impacts is warranted by population managers. Finally, our results indicate green turtles moved into nearshore neritic habitats at a young age (4–6 yr), emphasize the importance of protecting neritic habitats along the southwestern United States and northwestern Mexican coasts, and encourage the incorporation of variable maturation time in population recovery assessments.     

Influence of the existence of a resting state on the probability of cell division in culture

A cell cycle model developed by Smith and Martin is generalized to allow for the possibility that the duration of the B phase is not fixed. The B phase is the equivalent of the traditional S, G₂, and M phases of the cell cycle. The duration of the B phase is represented by a Gaussian probability distribution; the duration of the resting or A state which replaces the traditional G₁ phase is represented by a decaying exponential distribution. A doubling time distribution, termed the CEG distribution, is obtained by convolution of the A state and B phase distributions. Like the reciprocal normal, rate normal, and log normal distributions, it is a rounded unimodal peak that is skewed to the right. None of the three former distributions is associated with a cell cycle model that includes a resting state. However the CEG distribution, which is so associated, bears little resemblance to the delayed exponential distribution which results when the duration of the B phase is fixed and the duration of the A state is random. Consequently, it would be difficult to use the doubling time distribution to determine whether or not a resting state exists in a particular cell population.     

Alternative modes of population growth inhibition in a human lymphoid cell line growing in suspension

Using a human lymphoid cell line grown under continuous culture conditions, two distinct plateau states were induced, either by lack of sufficient medium-supplied nutrient, or by other unknown mechanisms dependent on cell density. Flow microfluorometric measurements show that growth arrest due to nutritional insufficiency results in an accumulation of cells with G1 DNA content. In contrast, growth arrest due to high cell density is not associated with an altered distribution of cells with respect to DNA content as the population progresses from exponential to plateau state growth. Cell size decreases with progression of the plateau state induced by either type of growth arrest. Cells in a plateau state induced by high cell density utilize glucose and incorporate exogenous amino acid into protein at approximately the same rate as exponential cells. Proliferating, high cell density, plateau state cells have cell cycle phase durations similar to exponential cells. The stable, plateau state cell density is maintained by cell loss. No stable, unbound growth inhibitory factor was found in the medium of density-inhibited plateau state cultures.     

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.     

Cell density related gene expression: SV40 large T antigen levels in immortalized astrocyte lines

Background

Gene expression is affected by population density. Cell density is a potent negative regulator of cell cycle time during exponential growth. Here, we asked whether SV40 large T antigen (Tag) levels, driven by two different promoters, changed in a predictable and regular manner during exponential growth in clonal astrocyte cell lines, immortalized and dependent on Tag.

Results

Expression and cell cycle phase fractions were measured and correlated using flow cytometry. T antigen levels did not change or increased during exponential growth as a function of the G₁ fraction and increasing cell density when Tag was transcribed from the Moloney Murine Leukemia virus (MoMuLV) long terminal repeat (LTR). When an Rb-binding mutant T antigen transcribed from the LTR was tested, levels decreased. When transcribed from the herpes thymidine kinase promoter, Tag levels decreased. The directions of change and the rates of change in Tag expression were unrelated to the average T antigen levels (i.e., the expression potential).

Conclusions

These data show that Tag expression potential in these lines varies depending on the vector and clonal variation, but that the observed level depends on cell density and cell cycle transit time. The hypothetical terms, expression at zero cell density and expression at minimum G₁ phase fraction, were introduced to simplify measures of expression potential.

     

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.     

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.     

Cadmium cytotoxicity and variation in nuclear content of DNA in Euglena gracilis

Cultures of Euglena gracilis (strain Z from French CNRS collection) can be made cadmium resistant if grown in a medium with 5x10^-4M cadmium chloride. This resistance is reflected by the appearance of a second exponential growth phase. The development of this resistance was studied at the cellular level by determining the relative content of DNA at different stages of the cell cycle in an asynchronously grown culture. The culture was followed until the second, cadmium resistant, growth phase had reached its stationary state. During the first exponential growth phase, cells were mostly in the late period of DNA synthesis (stage S of the cell cycle), or in the gap preceding mitosis (stage G₂ of the cell cycle). In addition, some cells contained high multiples of the normal amount of DNA. In the beginning of the second exponential growth phase, a few cells were again in G₁ (the post mitotic stage of the cell cycle preceding DNA synthesis). These G₁ cells were predominant at the end of the second growth period. During the second stationary phase the DNA content of the cadmium treated cells was similar to the stationary phase of the control culture. Cells had stopped growing in G₁ with an unreplicated genome. The implications of these data are discussed.     

The effect of growth rate,size, and season on oocyte development and maturity of Atlantic cod (Gadus morhua L.)

The effect of growth rate, body weight, age, and season on ovarian development and maturation was investigated for Atlantic cod (Gadus morhua L.) reared in the laboratory over 10 months, for each of two consecutive years, 1978–1980. Cod were also collected from the Gulf of St. Lawrence, the Scotian Shelf, Georges Bank, the Flemish Cap, and the N.E. Newfoundland Shelf.The state of maturity was recognized by oocyte size. Stage I oocytes did not vary in size with growth rate, season, age, or maturity, while the size of stage II oocytes was positively correlated with maturity and negatively correlated with maximum stage of development achieved. Ovarian wall thickness was positively correlated with age and maturity.The frequency distribution of stage I and II oocytes distinguished the state of maturation, with cod that would mature by the next spawning season having a minimum of 20% ($\text{x}\text{?}\text{= 42\%}$) of their oocytes at stage II or greater level of development.Maturing 3-yr-old cod had greater life specific growth rates than immature 3-yr-olds, but growth rates during the third year itself were not significantly different. A hypothesis of a three-part density-dependent mechanism controlling fecundity is postulated. Future reductions in partial recruitment and total fecundity are predicted for the Gulf of St. Lawrence cod stock based on calculated growth rates for Gulf cod in 1979.     

The effects of in vitro age and culture state on histone variant synthesis in human diploid fibroblasts

Histone variant synthesis patterns from human diploid fibroblast-like cells of different in vitro ages were determined during exponential growth, at confluence, and during low serum arrest. The results are reported as the ratios of H2A variant synthesis (H2A.1 and H2A.2/H2A.x and H2A.z) and H3 variant synthesis (H3.1 and H3.2/H3.3) that have been used to characterize individual cell cycle states. Hydroxyurea was employed in some experiments to reduce S phase cells. The results indicate that high population doubling level (PDL) cells move through the G1 phase of the division cycle during exponential growth and exist in the G0 cell cycle state at confluence and during low serum arrest. Low PDL cells, however, exist in the G1 cell cycle state at confluence and revert to a G0 state only after maintenance as quiescent populations. This would suggest that when stimulated high PDL cells cannot enter into S phase, they revert to a GO cell cycle state.     

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.     

Theory of distributed quiescent state in the cell cycle

A generalization of the familiar two-compartment or G₀ model of the cell cycle is described. Instead of reserving the quiescent state strictly to newly born cells, it is distributed throughout the cell cycle. A cell may cease its proliferative activities anywhere in the cycle with a probability depending on its maturity. The probability of returning to cycle is also a function of maturity. Analytical expressions for cycle time distributions, growth rates, wave frequency and relative damping rates are derived for certain cases. A stable, diffusion-free numerical algorithm is used to work out some examples.     

CHANGES IN CELL PROLIFERATION KINETICS OCCURRING DURING THE LIFE HISTORY OF MONOLAYER CULTURES OF A MOUSE TUMOUR CELL LINE

Following inoculation of monolayer cultures of EMT6 mouse tumour cells at 10⁵ cells, a short lag is followed by 3 days of exponential growth with a population doubling time of 12 hr. A plateau cell number is reached between days 4 and 5 and is maintained for at least 8 days. During exponential growth, the pulse ³H-TdR labelling index is 55–60%, all cells are in cycle, and the median cycle time is 11–12 hr. For the first 3 days of plateau phase, the labelling index is about 25 % and there is considerable cell loss. The cell cycle is 32–40 hr, and S-phase is very long. Later in plateau phase, the labelling index falls to <2 % and there is little cell loss. The changes in kinetics occurring in EMT6 cultures are discussed with reference to reported changes occurring in other cell lines.     

温度对萼花臂尾轮虫卵的发育、种群增长和生产量的影响

1981—1983年,在不同的培养温度下,观察了萼花臂尾轮虫(Brachionus calyciflorus)卵的发育时间、种群的增长并用3种不同方法测算生产量。在5—30℃的培养温度下,轮虫卵的发育时间(D)随温度(T)升高而缩短,其曲线迴归方程为: LnD=2.0539+0.1097LnT-0.3046(LnT)2 在10,15,20,25℃的培养温度下,从休眠卵孵化出来的孤雌生殖雌体,其繁殖的种群增长曲线都呈“S”形,或称逻辑斯蒂曲线(Logistic curve)。不同的温度,种群达到高峰所需的时间有所不同,温度高者短,低者长;容纳量(carrying capacity)却随温度增加而有所增加。用线性和指数方法计算轮虫种群的生产量所得的结果相似;而与世代时间方法计算所获得的结果相比,差距很大。这种差距随着温度增加而增加。根据本文的研究结果和文献中报道的数据,获得了在0.6—35.2℃温度范围内,卵的发育时间(D)与温度(T)之间的迴归方程: LnD=2.1869-0.1919LnT-0.2218(LnT)2       

Influence of the existence of a resting state on the decay of synchronization in cell culture

General relationships between the distribution of cell doubling times and the growth pattern of an initially synchronized cell population are applied to the model proposed by Smith and Martin (1973) in which the mitotic cycle or "B" phase is preceded by a random-exit resting "A" state. Results show that culture synchronization decays so rapidly as to be virtually unobservable unless the time spent by a cell in the B phase is at least equal to that spent in the A state. If synchronization persists over several mitotic cycles, the growth pattern is determined to a much greater extent by variation in the duration of the B phase than by the probability of exit from the A state. Accordingly the growth pattern of a cell population, like the doubling time distribution which governs the pattern, is of limited usefulness in detecting the existence of a resting state.     

Clonal heterogeneity in specific growth rate of Succhuromyces cerevisiue cells

The cell volume increase in individual clones of cells of the yeast Saccharomyces cerevisiae has been measured using time lapse cinematography in populations showing steady state balanced exponential growth. There were significant differences in clonal specific growth rates within the population in each of 10 experiments using different strains on different media supporting different growth rates. The results suggest that specific growth rates of cells which are either genetically identical or very closely related can be different and this difference can be propagated over at least three generations. Since the proliferation rate in yeast is determined by growth rate, these observed differences provide an additional source of cell cycle variability for yeast cells that has not been considered before. The implications for the theoretical analysis of cell cycle kinetics are examined.     

Seasonal dynamics of the acanthocephalan Pomphorhynchm laevis (Müller, 1776) in its intermediate and preferred definitive hosts

No seasonal cycle was found in either the prevalence or the intensity of natural Pomphorhynchus laevis infections in Leuciscus cephalus . There was a slight seasonal change in female maturity distribution but only irregular fluctuations in the size structure of the adult parasite population throughout the year. Cystacanths were available in all seasons. Rates of parasite growth, maturation and mortality, but not establishment, increased with water temperature (or factors indirectly associated with elevated water temperature) in laboratory-infected Salmo gairdneri . Increased rates of parasite growth and maturation mask any marked shifts in the size and maturity structure of the adult parasite population which might otherwise be due to the higher turnover of adult parasites in the summer months.     

Genetic and social control of male maturation in Phallichthys quadripunctatus (Pisces: Poeciliidae)

Age and size at maturity can have significant fitness consequences. Selection often favors early-maturing individuals because of their higher survival to maturity and greater relative contribution to population growth rate, but it may also favor delayed maturation if fitness increases with age or size at maturity. Males of several poeciliid fishes exhibit variation in age and size at maturity primarily controlled by a sex-linked gene called the P-locus. Wild-caught Phallichthys quadripunctatus males show a bimodal size distribution, which is often associated with a P-locus polymorphism in other poeciliids. We conducted two experiments to evaluate the inheritance of male age and size at maturity and the influence of social environment (presence of mature or juvenile males during development) on these traits. We specifically tested the hypothesis that male age and size at maturity in P. quadripunctatus are governed by a single Y-linked locus, and modified by the social environment. Although our results imply both a genetic and an environmental component to the dimorphism in maturation, both large and small males were able to sire both large and small sons, allowing us to reject the hypothesis that age and size at maturity in this species are controlled by a single, Y-linked locus. Our data do not conform adequately to any of the genetic mechanisms described to date for maturation polymorphism in poeciliids. We suggest alternative mechanisms that may operate in P. quadripunctatus.