A stochastic model for cellular senescence. Part I. Theoretical considerations

Closed-form stochastic solutions for non-equilibrium dynamics and inheritance of cellular components over many cell divisions

Stochastic dynamics govern many important processes in cellular biology, and an underlying theoretical approach describing these dynamics is desirable to address a wealth of questions in biology and medicine. Mathematical tools exist for treating several important examples of these stochastic processes, most notably gene expression and random partitioning at single-cell divisions or after a steady state has been reached. Comparatively little work exists exploring different and specific ways that repeated cell divisions can lead to stochastic inheritance of unequilibrated cellular populations. Here we introduce a mathematical formalism to describe cellular agents that are subject to random creation, replication and/or degradation, and are inherited according to a range of random dynamics at cell divisions. We obtain closed-form generating functions describing systems at any time after any number of cell divisions for binomial partitioning and divisions provoking a deterministic or random, subtractive or additive change in copy number, and show that these solutions agree exactly with stochastic simulation. We apply this general formalism to several example problems involving the dynamics of mitochondrial DNA during development and organismal lifetimes.

The two-process model of cellular aging

To understand the mechanism of aging at the cellular level, cellular senescence has been extensively studied as an experimental model of aging in vitro. Although several hypotheses have been proposed for the mechanism of cellular senescence, none of them could give a comprehensive framework to the mechanism. In this study, we showed our results of extensive computer simulation designed to identify possible molecular models of cellular senescence. By examining representative cases of various molecular models, we elucidated the requirements for the plausible mechanism of cellular senescence. Based on these simulation results, we proposed a new molecular model of cellular senescence--the two-process model. In this model, we assumed that two independent, but time-aligned regulatory processes functioned in individual cells. We defined these two processes as S- and C-processes. The S-process mainly determines the rate of decline in the proliferative potential of the cell population. The simulation results suggested that the growth-inhibitory cell-to-cell interaction was required to drive the S-process. The C-process determines the latent proliferative potential of individual cells. The effector genes for the C-process are suggested to be regulated by a certain threshold-type mechanism. Both growth kinetics and senescence-associated gene expression were generated with high accuracy by the combined effect of these two processes. We also succeeded in simulating the effects of simian virus 40 large T antigen and its inducible variant on cellular senescence. From these theoretical considerations, we discuss the validity of the two-process model and the possible involvement of the heterochromatin structure as a determinant of the replicative lifespan of cells.

Time-continuous branching walk models of unstable gene amplification

We consider a stochastic mechanism of the loss of resistance of cancer cells to cytotoxic agents, in terms of unstable gene amplification. Two models being different versions of a time-continuous branching random walk are presented. Both models assume strong dependence in replication and segregation of the extrachromosomal elements. The mathematical part of the paper includes the expression for the expected number of cells with a given number of gene copies in terms of modified Bessel functions. This adds to the collection of rare explicit solutions to branching process models. Original asymptotic expansions are also demonstrated. Fitting the model to experimental data yields estimates of the probabilities of gene amplification and deamplification. The thesis of the paper is that purely stochastic mechanisms may explain the dynamics of reversible drug resistance of cancer cells. Various stochastic approaches and their limitations are discussed.

Cellular aging (the Hayflick limit) and species longevity: a unification model based on clonal succession

A model is presented which proposes a specific cause-and-effect relationship between a limited cell division potential and the maximum lifespan of humans and other mammals. It is based on the clonal succession hypothesis of Kay which states that continually replicating cell beds (e.g. bone marrow, intestinal crypts, epidermis) could be composed of cells with short, well-defined division potentials. In this model, the cells of these beds are proposed to exist in an ordered hierarchy which establishes a specific sequence for cell divisions throughout the organism's lifespan. The depletion of division potential at all hierarchical levels leads to a loss of bed function and sets an intrinsic limit to species longevity. A specific hierarchy for cell proliferation is defined which allows the calculation of time to bed depletion and, ultimately, to organism mortality. The model allows the existence of a small number (n) of critical cell beds within the organism and defines organism death as the inability of any one of these beds to produce cells. The model is consistent with all major observations related to cellular and organismic aging. In particular, it links the PDLs (population doubling limit) observed for various species to their mean lifespan; it explains the slow decline in PDL as a function of age of the donor; it establishes a thermodynamically stable maximum lifespan for a disease-free population; and it can explain why tissue transplants outlive donors or hosts.

A return to time, cells, systems, and aging: III. Gompertzian models of biological aging and some possible roles for critical elements

In this paper, I continue my investigation into the modeling of senescence in biological hierarchies. Making use of my previous discussion on non-reestablishable biological components, I derive a mathematical model which has Gompertzian-like dynamics. I show how this model may be approximated, in certain instances, by a Gompertzian equation. I then demonstrate how our approach yields a biological interpretation for the parameters in the Gompertzian equation. I then demonstrate how changes in the parameter values may be interpreted in light of the biology. Subsequently, I review the literature on the allometry of aging, and I demonstrate how my reliability model may be used to obtain--in a qualitative manner--some of the lifespan curves found in the literature. I close my discussion by constructing a more complex reliability model which incorporates the deterministic failure of biological components with stochastic aspects of senescence.

Differentiation of cultured pre-adipose cells: a probability model

Some cells of the established preadipose cell line, 3T3-L1, synthesize triglyceride after becoming confluent and quiescent. An analysis of the distribution of clusters of lipid-containing cells was consistent with a commitment event during exponential growth followed by clonal growth of committed cells. Experiments were designed to determine if the final clonal pattern of fat among nonfat cells could be described by a probability model. Undifferentiated cells (fibroblastic cells with no detectable accumulation of triglycerides) were plated at various cell numbers so that the total number of cell divisions to confluence could be controlled. Cells were passaged by trypsinization and replating, or trypsinization followed by passage through a narrow-bore needle before replating. Passing cells through a 22G needle seems to eliminate already committed cells from the population. We determined the percentage of fat cells and the range of clone sizes in cultures in which clone sizes depended upon the number of allowed cell divisions. Patterns of clone sizes in experimental cultures were compared to expected patterns obtained by computer simulations of several programmed and stochastic commitment models. Both the observed range of clone sizes and pattern of clones can be approximated by a simple stochastic model, suggesting that commitment to fat production in 3T3-L1 cells is a random process occurring with a fixed probability in single cells in exponential growth, followed by division of both committed and uncommitted cells. The probability of commitment was essentially constant at each cell division. The number of cells committed during each passage is just large enough to replace "terminally differentiated" lipid-containing cells that have been lost, thereby maintaining a constant percentage of fat cells in any given culture of 3T3-L1.