A model of cell cycle behavior dominated by kinetics of a pathway stimulated by growth factors

A modified version of a previously developed mathematical model [Obeyesekere et al., Cell Prolif. (1997)] of the G1-phase of the cell cycle is presented. This model describes the regulation of the G1-phase that includes the interactions of the nuclear proteins, RB, cyclin E, cyclin D, cdk2, cdk4 and E2F. The effects of the growth factors on cyclin D synthesis under saturated or unsaturated growth factor conditions are investigated based on this model. The solutions to this model (a system of nonlinear ordinary differential equations) are discussed with respect to existing experiments. Predictions based on mathematical analysis of this model are presented. In particular, results are presented on the existence of two stablesolutions, i. e., bistability within the G1-phase. It is shown that this bistability exists under unsaturated growth factor concentration levels. This phenomenon is very noticeable if the efficiency of the signal transduction, initiated by the growth factors leading to cyclin D synthesis, is low. The biological significance of this result as well as possible experimental designs to test these predictions are presented.     

Methylmercury effects on cell cycle kinetics

Methylmercury (MeHg) effects on cell cycle kinetics were investigated to help identify its mechanisms of action. Flow cytometric analysis of normal human fibroblasts grown in vitro in the presence of BrdU allowed quantitation of the proportion of cells in G1, S, G2 and the next G1 phase. This technique provides a rapid and easily performed method of characterizing phase lengths and transition rates for the complete cell cycle. After first exposure to MeHg the cell cycle time was lengthened due to a prolonged G1. At 3 microM MeHg the G1 phase length was 25% longer than the control. The G1/S transition rate was also decreased in a dose-related manner. Confluent cells exposed to MeHg and replated with MeHg respond in the same way as cells which have not been exposed to MeHg before replating. Cells exposed for long times to MeHg lost a detectable G1 effect, and instead showed an increase in the G2 percentage, which was directly related to MeHg concentration and length of exposure. After 8 days at 5 microM MeHg, 45% of the population was in G2. The G2 accumulation was reversible up to 3 days, but at 6 days the cells remained in G2 when the MeHg was removed. Cell counts and viability indicated that there was not a selective loss of cells from the MeHg. MeHg has multiple effects on the cell cycle which include a lengthened G1 and decreased transition probability after short term exposure of cycling cells, and a G2 accumulation after a longer term exposure. There were no detectable S phase effects. It appears that mitosis (the G2 accumulation) and probably synthesis of some macromolecules in G1 (the lengthened G1 and lowered transition probability) are particularly susceptible to MeHg.     

A simple mathematical model based on the cancer stem cell hypothesis suggests kinetic commonalities in solid tumor growth

Background

The Cancer Stem Cell (CSC) hypothesis has gained credibility within the cancer research community. According to this hypothesis, a small subpopulation of cells within cancerous tissues exhibits stem-cell-like characteristics and is responsible for the maintenance and proliferation of cancer.

Methodologies/Principal Findings

We present a simple compartmental pseudo-chemical mathematical model for tumor growth, based on the CSC hypothesis, and derived using a “chemical reaction” approach. We defined three cell subpopulations: CSCs, transit progenitor cells, and differentiated cells. Each event related to cell division, differentiation, or death is then modeled as a chemical reaction. The resulting set of ordinary differential equations was numerically integrated to describe the time evolution of each cell subpopulation and the overall tumor growth. The parameter space was explored to identify combinations of parameter values that produce biologically feasible and consistent scenarios.

Conclusions/Significance

Certain kinetic relationships apparently must be satisfied to sustain solid tumor growth and to maintain an approximate constant fraction of CSCs in the tumor lower than 0.01 (as experimentally observed): (a) the rate of symmetrical and asymmetrical CSC renewal must be in the same order of magnitude; (b) the intrinsic rate of renewal and differentiation of progenitor cells must be half an order of magnitude higher than the corresponding intrinsic rates for cancer stem cells; (c) the rates of apoptosis of the CSC, transit amplifying progenitor (P) cells, and terminally differentiated (D) cells must be progressively higher by approximately one order of magnitude. Simulation results were consistent with reports that have suggested that encouraging CSC differentiation could be an effective therapeutic strategy for fighting cancer in addition to selective killing or inhibition of symmetric division of CSCs.     

Modulating effects of drugs on cell cycle kinetics

The effects of individual drugs on cell cycle progression can often be determined by analyzing the DNA distribution of cultured cells at appropriate times after drug administration. In addition, to cell counts, RNA and/or protein content, the alterations in cell cycle distribution of drug-treated cells can yield information on the potential cell cycle phase specificity of the drug. However single parameter DNA analysis, as currently used, can give inaccurate information when drug effects are associated with cytokinesis perturbations, in spite of various statistical and mathematical analyses. Scanning flow cytometry could be an interesting alternative for the detection of binucleate cells.     

Tuning the cell cycle: a model based on averaging

Abstract. The ability of intercellular communication and the basement membrane to revert the phenotypic behaviour of malignant cells suggests that such cells can be tuned to behave more benignly. In addition, the large variation in cell doubling times observed in tumour cells poses the question of whether or not cell doubling times, and hence, patient survival, can be lengthened by therapeutic intervention. In both cases, the understanding may be enhanced by obtaining a parsimonious and tractable model of the cell cycle which behaves appropriately and suggests a philosophical framework for addressing these complex issues. We introduce a simple two-dimensional model based on averaging cyclin and maturation promotion factor over a fast oscillating subsystem that exhibits the basic features of cellular division, and discuss the ramifications of the model.     

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.     

A stochastic model of cell cycle desynchronization

A general branching process model is suggested to describe cell cycle desynchronization. Cell cycle phase times are modeled as random variables and a formula for the expected fraction of cells in S phase as a function of time is established. The model is compared to data from the literature and is also compared to previously suggested deterministic and stochastic models.     

A deterministic model of the cell cycle

The variability of the duration of the cell cycle is explained by the phenomenon of sensitive dependence upon initial conditions; as may occur in deterministic non-linear systems. Chaotic dynamics of a system is the result of this sensitive dependence. First a deterministic system is formulated that is equivalent to the Smith-Martin transition probability model of the cell cycle. Next the model is extended to a dynamic process that ranges over the cell generations. A deterministic non-linear relationship between the cycle time of the mother and daughter cell is established. It clarifies the variability of mother-daughter correlation for the different cell types. The model is fitted to two different cell cultures; it shows that the graph of the non-linear relation has the same shape for different cell types.     

A stochastic model in tumor growth

A stochastic model of solid tumor growth based on deterministic Gompertz law is presented. Tumor cells evolution is described by a one-dimensional diffusion process limited by two absorbing boundaries representing healing threshold and patient death (carrying capacity), respectively. Via a numerical approach the first exit time problem is analysed for the process inside the region restricted by the boundaries. The proposed model is also implemented to simulate the effects of a time-dependent therapy. Finally, some numerical results are obtained for the specific case of a parathyroid tumor.     

Linking cell division to cell growth in a spatiotemporal model of the cell cycle

Cell division must be tightly coupled to cell growth in order to maintain cell size, yet the mechanisms linking these two processes are unclear. It is known that almost all proteins involved in cell division shuttle between cytoplasm and nucleus during the cell cycle; however, the implications of this process for cell cycle dynamics and its coupling to cell growth remains to be elucidated. We developed mathematical models of the cell cycle which incorporate protein translocation between cytoplasm and nucleus. We show that protein translocation between cytoplasm and nucleus not only modulates temporal cell cycle dynamics, but also provides a natural mechanism coupling cell division to cell growth. This coupling is mediated by the effect of cytoplasmic-to-nuclear size ratio on the activation threshold of critical cell cycle proteins, leading to the size-sensing checkpoint (sizer) and the size-independent clock (timer) observed in many cell cycle experiments.     

A unified model of sigmoid tumour growth based on cell proliferation and quiescence

Abstract.   Objectives : A class of sigmoid functions designated generalized von Bertalanffy, Gompertzian and generalized Logistic has been used to fit tumour growth data. Various models have been proposed to explain the biological significance and foundations of these functions. However, no model has been found to fully explain all three or the relationships between them. Materials and Methods : We propose a simple cancer cell population dynamics model that provides a biological interpretation for these sigmoids' ability to represent tumour growth. Results and Conclusions : We show that the three sigmoids can be derived from the model and are in fact a single solution subject to the continuous variation of parameters describing the decay of the proliferation fraction and/or cell quiescence. We use the model to generate proliferation fraction profiles for each sigmoid and comment on the significance of the differences relative to cell cycle-specific and non-cell cycle-specific therapies.     

Hydrocarbon fermentations using Candida lipolytica. II. A model for cell growth kinetics

A mechanistic model is presented for the growth kinetics of a yeast grown by submerged aerobic fermentation using a liquid hydrocarbon as sole carbon source. The model is based on the assumption that cell growth is governed by the extent of probable cell attachment at the hydrocarbon oil-droplet surfaces in a four-phase dispersion. An analytical expression has been developed for the model. It is shown that for the case of relatively small oil droplets, the model predicts the present and previous experimental data for growth of yeasts (Candida species) in n-alkane systems. The model is further examined for maximal growth in terms of substrate dilution rate and agitation power consumption for a continuous fermentation process.     

A simplified mathematical model of tumor growth

A one-dimensional model of tumor tissue growth is presented in which the source of mitotic inhibitor is nonuniformly distributed within the tissue (in contrast to many earlier models). As a result, stable and unstable regimes of growth become significantly modified from the uniform-source case, indicating that the model, schematic though it is, is very sensitive to the type of source term assumed, and this has implications for experimental and theoretical comparisons in more realistic geometries.     

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.     

A simple model describing the kinetics of the xanthophyll cycle.

A new kinetic model of the xanthophyll cycle is proposed. The model is based on the assumption that the light-dependent interconversion of the so-called available and unavailable violaxanthin constitutes the rate-limiting process of the cycle at intermediate, non-saturating light intensities. This assumption, together with the known properties of violaxanthin de-epoxidase, explains all specific features of the experimental facts.