Investigating the link between epithelial-mesenchymal transition and the cancer stem cell phenotype: A mathematical approach

Under the cancer stem cell (CSC) hypothesis, sustained metastatic growth requires the dissemination of a CSC from the primary tumour followed by its re-establishment in a secondary site. The epithelial-mesenchymal transition (EMT), a differentiation process crucial to normal development, has been implicated in conferring metastatic ability on carcinomas. Balancing these two concepts has led researchers to investigate a possible link between EMT and the CSC phenotype—indeed, recent evidence indicates that, following induction of EMT in human breast cancer and related cell lines, stem cell activity increased, as judged by the presence of cells displaying the CD44^high/CD24^low phenotype and an increase in the ability of cells to form mammospheres. We mathematically investigate the nature of this increase in stem cell activity. A stochastic model is used when small number of cells are under consideration, namely in simulating the mammosphere assay, while a related continuous model is used to probe the dynamics of larger cell populations. Two scenarios of EMT-mediated CSC enrichment are considered. In the first, differentiated cells re-acquire a CSC phenotype—this model implicates fully mature cells as key subjects of de-differentiation and entails a delay period of several days before de-differentiation occurs. In the second, pre-existing CSCs experience accelerated division and increased proportion of self-renewing divisions; a lack of perfect CSC biomarkers and cell sorting techniques requires that this model be considered, further emphasizing the need for better characterization of the mammary (cancer) stem cell hierarchy. Additionally, we suggest the utility of comparing mammosphere data to computational mammosphere simulations in elucidating the growth characteristics of mammary (cancer) stem cells.     

Stochastic branching model for hemopoietic progenitor cell differentiation

We present algebraic expressions describing the predictions of a stochastic branching model for differentiation of hemopoietic progenitor cells. The model assumes that there is a fixed probability, p (0 less than or equal to p less than or equal to 1), that commitment to a differentiative event occurs per progenitor cell division for each daughter cell. The model describes properties of in vitro hemopoietic cell differentiation including the population structure at the time the first progenitor cell becomes committed, the number of committed progenitor cells engendered by a single progenitor cell, and the probability of eventual commitment of all daughter cells derived from a single progenitor or stem cell. Application of the model to experimental data obtained from erythroid cultures suggests that the observed data can be explained by the stochastic branching model alone without making the deterministic assumption that there is a differentiative hierarchy in the lineage of the progenitors of erythropoiesis (BFU-E). The qualitative and quantitative aspects of the proposed stochastic model are discussed in conjunction with other analogous stochastic branching models.     

A computational model for telomere-dependent cell-replicative aging

Telomere shortening provides a molecular basis for the Hayflick limit. Recent data suggest that telomere shortening also influence mitotic rate. We propose a stochastic growth model of this phenomena, assuming that cell division in each time interval is a random process which probability decreases linearly with telomere shortening. Computer simulations of the proposed stochastic telomere-regulated model provides good approximation of the qualitative growth of cultured human mesenchymal stem cells.     

Modeling Cell Dynamics in Colon and Intestinal Crypts: The Significance of Central Stem Cells in Tumorigenesis

Colon and intestinal crypts have been widely chosen to study cell dynamics because of their fairly simple structures. In the colon and intestinal crypts, stem cells (SCs) are located at very bottom of the crypt, fully differentiated cells (FDs) are located in the top of the crypt, and transit-amplifying cells (TAs) are in the middle of the crypt between FDs and SCs. Recently, it has been discovered that there are two types of stem cells in the intestinal crypts: central stem cells (CeSCs) and border stem cells. To investigate dynamics of mutants in colon and intestinal crypts, we develop a four-compartmental stochastic model, which includes two SC compartments, and TAs and FDs compartments. We calculate the probability of the progeny of marked or mutant cells located at each of these compartments taking over the entire crypt or being washed out from the crypt. We found that the progeny of CeSCs will take over the entire crypt with a probability close to one. Interestingly, the progeny of advantageous mutant TAs and FDs will be washed out faster than disadvantageous mutants. Saliently, the model predicts that the time that the progeny of wild-type central stem cells will take over the mouse intestinal crypt is around 60 days, which is in perfect agreement with an experimental observation.     

A biological and computational model of megakaryocyte development as a stochastic branching process

The purpose of this paper is to describe a model of megakaryocytopoiesis as a branching process with stochastic processes regulating critical control points of differentiation along the stem cell megakaryocyte platelet axis. Progress of cells through these critical control points are regulated by transitional probabilities, which in turn are regulated by influences such as growth factors. The critical control points include transition of resting megakaryocytic stem cells (CFU-meg) into proliferating stem cells, the cessation of cytokinesis, and the cessation of DNA synthesis. A computerized computational method has been developed for directly fitting the stochastic branching model to colony growth data. The computational model has allowed transitional probabilities to be derived from colony size data. The model provides a unifying explanation for much of the heterogeneity of stages of maturation within populations of megakaryocytes and is fully compatible with historical data supporting the stochastic nature of hematopoietic stem cell regulation and with modern molecular concepts about control of the cell cycle.     

Myogenic stem cell commitment probability remains constant as a function of organismal and mitotic age

Chicken myogenic stem cells can undergo symmetric and asymmetric cell divisions. Symmetric divisions produce two stem cells or two cells committed to terminal muscle differentiation. Asymmetric divisions produce one stem cell and one committed cell. Committed cells undergo four divisions, and their progeny differentiate into postmitotic, biochemically distinct muscle cells, which can be identified immunocytochemically. The control of stem cell commitment was investigated in vitro by means of cell cloning and subcloning experiments, and computer modeling. We found that stem cell commitment is a process which can be modeled as a stochastic event, with a central tendency or probability of 0.2 +/- 0.1. This value is independent of organismal or mitotic age of the stem cells, cell density, or growth in a mitogen-poor environment. Myogenic stem cells stop dividing after approximately 30 divisions in vitro. Since the probability of commitment to terminal differentiation remains below 0.5, clonal senescence and terminal differentiation are separate processes in this system.     

The rate of multi-step evolution in Moran and Wright-Fisher populations

Several groups have recently modeled evolutionary transitions from an ancestral allele to a beneficial allele separated by one or more intervening mutants. The beneficial allele can become fixed if a succession of intermediate mutants are fixed or alternatively if successive mutants arise while the previous intermediate mutant is still segregating. This latter process has been termed stochastic tunneling. Previous work has focused on the Moran model of population genetics. I use elementary methods of analyzing stochastic processes to derive the probability of tunneling in the limit of large population size for both Moran and Wright-Fisher populations. I also show how to efficiently obtain numerical results for finite populations. These results show that the probability of stochastic tunneling is twice as large under the Wright-Fisher model as it is under the Moran model.     

Predicted steady-state cell size distributions for various growth models

The question of how an individual bacterial cell grows during its life cycle remains controversial. In 1962 Collins and Richmond derived a very general expression relating the size distributions of newborn, dividing and extant cells in steady-state growth and their growth rate; it represents the most powerful framework currently available for the analysis of bacterial growth kinetics. The Collins-Richmond equation is in effect a statement of the conservation of cell numbers for populations in steady-state exponential growth. It has usually been used to calculate the growth rate from a measured cell size distribution under various assumptions regarding the dividing and newborn cell distributions, but can also be applied in reverse--to compute the theoretical cell size distribution from a specified growth law. This has the advantage that it is not limited to models in which growth rate is a deterministic function of cell size, such as in simple exponential or linear growth, but permits evaluation of far more sophisticated hypotheses. Here we employed this reverse approach to obtain theoretical cell size distributions for two exponential and six linear growth models. The former differ as to whether there exists in each cell a minimal size that does not contribute to growth, the latter as to when the presumptive doubling of the growth rate takes place: in the linear age models, it is taken to occur at a particular cell age, at a fixed time prior to division, or at division itself; in the linear size models, the growth rate is considered to double with a constant probability from cell birth, with a constant probability but only after the cell has reached a minimal size, or after the minimal size has been attained but with a probability that increases linearly with cell size. Each model contains a small number of adjustable parameters but no assumptions other than that all cells obey the same growth law. In the present article, the various growth laws are described and rigorous mathematical expressions developed to predict the size distribution of extant cells in steady-state exponential growth; in the following paper, these predictions are tested against high-quality experimental data.     

Symmetric vs. Asymmetric Stem Cell Divisions: An Adaptation against Cancer?

Traditionally, it has been held that a central characteristic of stem cells is their ability to divide asymmetrically. Recent advances in inducible genetic labeling provided ample evidence that symmetric stem cell divisions play an important role in adult mammalian homeostasis. It is well understood that the two types of cell divisions differ in terms of the stem cells'' flexibility to expand when needed. On the contrary, the implications of symmetric and asymmetric divisions for mutation accumulation are still poorly understood. In this paper we study a stochastic model of a renewing tissue, and address the optimization problem of tissue architecture in the context of mutant production. Specifically, we study the process of tumor suppressor gene inactivation which usually takes place as a consequence of two “hits”, and which is one of the most common patterns in carcinogenesis. We compare and contrast symmetric and asymmetric (and mixed) stem cell divisions, and focus on the rate at which double-hit mutants are generated. It turns out that symmetrically-dividing cells generate such mutants at a rate which is significantly lower than that of asymmetrically-dividing cells. This result holds whether single-hit (intermediate) mutants are disadvantageous, neutral, or advantageous. It is also independent on whether the carcinogenic double-hit mutants are produced only among the stem cells or also among more specialized cells. We argue that symmetric stem cell divisions in mammals could be an adaptation which helps delay the onset of cancers. We further investigate the question of the optimal fraction of stem cells in the tissue, and quantify the contribution of non-stem cells in mutant production. Our work provides a hypothesis to explain the observation that in mammalian cells, symmetric patterns of stem cell division seem to be very common.     

An enteroendocrine cell-based model for a quiescent intestinal stem cell niche

We have shown that the kinetics of conversion of intestinal crypt cell populations to a partially or wholly mutant phenotype are consistent with a model in which each crypt contains an infrequently dividing 'deep' stem cell that is the progenitor of several more frequently dividing 'proximate' stem cells. An assumption of our model is that each deep stem cell exists in a growth inhibitory niche. We have used information from the literature to develop a model for a quiescent intestinal stem cell niche. This niche is postulated to be primarily defined by an enteroendocrine cell type that maintains stem cell quiescence by secretion of growth inhibitory peptides such as somatostatin and guanylin/uroguanylin. Consistent with this model, there is evidence that the proteins postulated as defining a growth-inhibitory stem cell niche can act as intestinal tumour suppressors. Confirmation that a growth-inhibitory niche does exist would have important implications for our understanding of intestinal homeostasis and tumorigenesis.     

Temporal autocorrelation and stochastic population growth

How much does environmental autocorrelation matter to the growth of structured populations in real life contexts? Interannual variances in vital rates certainly do, but it has been suggested that between‐year correlations may not. We present an analytical approximation to stochastic growth rate for multistate Markovian environments and show that it is accurate by testing it in two empirically based examples. We find that temporal autocorrelation has sizeable effect on growth rates of structured populations, larger in many cases than the effect of interannual variability. Our approximation defines a sensitivity to autocorrelated variability, showing how demographic damping and environmental pattern interact to determine a population's stochastic growth rate.     

Application of Bayesian inference to characterize risks associated with low doses of low-LET radiation

Improved risk characterization for stochastic biological effects of low doses of low-LET radiation is important for protecting nuclear workers and the public from harm from radiation exposure. Here we present a Bayesian approach to characterize risks of stochastic effects from low doses of low-LET radiation. The stochastic effect considered is neoplastic transformation of cells because it relates closely to cancer induction. We have used a published model of neoplastic transformation called NEOTRANS1. It is based on two different classes of cellular sensitivity for asynchronous, exponentially growing populations (in vitro). One sensitivity class is the hypersensitive cell; the other is the resistant cell. NEOTRANS1 includes the effects of genomic damage accumulation, DNA repair during cell cycle arrest, and DNA misrepair (non-lethal repair errors). The model-associated differential equations are solved for conditions of in vitro irradiation at a fixed rate. Previously published solutions apply only to high dose rates and were incorrectly assumed to apply to only high-LET radiation. Solutions provided here apply to any fixed dose rate and to both high- and low-LET radiations. Markov chain Monte Carlo methods are used to carry out the Bayesian inference of the low-dose risk for neoplastic transformation of aneuploid C3H 10T1/2 cells for X-ray doses from 0 to 1000 mGy. We have assumed that for this low-dose range only the hypersensitive fraction of the cells are affected. Our results indicate that the initial slope of the risk vs dose relationship for neoplastic transformation is as follows: (1) directly proportional to the fraction, f1, of hypersensitive cells; (2) directly proportional to the radiosensitivity of the genomic target; and (3) inversely proportional to the rate at which hypersensitive cells with radiation-induced damage are committed to undergo correct repair of genomic damage. Further, our results indicate that very fast molecular events are associated with the commitment of cells to the correct repair pathway. Results also indicate a relatively large probability for misrepair that leads to genomic instability. Our results are consistent with the view that for very low doses, dose rate is not an important variable for characterizing low-LET radiation risks so long as age-related changes in sensitivity do not occur during irradiation.     

A stochastic model for cancer risk

We propose a simple stochastic model based on the two successive mutations hypothesis to compute cancer risks. Assume that only stem cells are susceptible to the first mutation and that there are a total of D stem cell divisions over the lifetime of the tissue with a first mutation probability mu(1) per division. Our model predicts that cancer risk will be low if m = mu(1)D is low even in the case of very advantageous mutations. Moreover, if mu(1)D is low the mutation probability of the second mutation is practically irrelevant to the cancer risk. These results are in contrast with existing models but in agreement with a conjecture of Cairns. In the case where m is large our model predicts that the cancer risk depends crucially on whether the first mutation is advantageous or not. A disadvantageous or neutral mutation makes the risk of cancer drop dramatically.     

Stochastic exit from mitosis in budding yeast

Unlike many mutants that are completely viable or inviable, the CLB2-dbΔ clb5Δ mutant of Saccharomyces cerevisiae is inviable in glucose but partially viable on slower growth media such as raffinose. On raffinose, the mutant cells can bud and divide but in each cycle there is a chance that a cell will fail to divide (telophase arrest), causing it to exit the cell cycle. This effect gives rise to a stochastic phenotype that cannot be explained by a deterministic model. We measure the inter-bud times of wild type and mutant cells growing on raffinose and compute statistics and distributions to characterize the mutant’s behavior. We convert a detailed deterministic model of the budding yeast cell cycle to a stochastic model and determine the extent to which it captures the stochastic phenotype of the mutant strain. Predictions of the mathematical model are in reasonable agreement with our experimental data and suggest directions for improving the model. Ultimately, the ability to accurately model stochastic phenotypes may prove critical to understanding disease and therapeutic interventions in higher eukaryotes.     

De-differentiation Response of Cultured Astrocytes to Injury Induced by Scratch or Conditioned Culture Medium of Scratch-Insulted Astrocytes

Our previous reports indicated that astrocytes (ASTs) in injured adult rat spinal cord underwent a process of de-differentiation, and may acquire the potential of neural stem cells (NSCs). However, the AST de-differentiation and transitional rejuvenation process following injury is still largely unclear. The aim of the present study was to determine whether injured in vitro ASTs can re-enter the multipotential-like stem cell pool and regain NSC characteristics, and to further understand the mechanism of AST de-differentiation. We used an in vitro scratch-wound model to evoke astrocytic response to mechanical injury. GFAP and nestin double-labeled indirect immunofluorescence were carried out to characterize these scratched cells at various periods. Western-blot analysis was used to determine the changes of GFAP and nestin expression following injury. Furthermore, the rate of proliferation was determined by immunocytochemical detection of BrdU incorporating cells. These scratch-wound ASTs were cultured with stem cells medium to explore their ability to generate neurospheres and examine the self-renewal and multi-potency of such neurospheres. Moreover, scratched AST culture supernatant as conditioned cultured medium (ACM) was used to investigate if some diffusible factors derived from injured ASTs could induce de-differentiation of AST. The results showed: (1) the nestin positivity first appeared in GFAP-positive cells at the edge of the scratch, subsequently, disseminated into un-insulted zone. The expression of nestin in AST was increased with longer culture, while that of GFAP was decreased. Furthermore, these nestin-immunoreactive ASTs could generate neurospheres, which showed self-renewal and could be differentiated into neurons, ASTs and oligodendrocytes. (2) Scratched ASTs culture supernatant can induce astrocytic proliferation and de-differentiation. These results reveal that the in vitro injured ASTs can de-differentiate into nestin-positive stem/precursor cells, the process of de-differentiation may arise from direct injury or some diffusible factors released from injured ASTs.     

On the utility of a compartmental population kinetics model of intestinal epithelial stem cell proliferation and differentiation

Background

The small intestinal epithelium is a dynamic system with specialized cell types. The various cell populations of this tissue are continually renewed and replenished from stem cells that reside in the small intestinal crypt. The cell types and their locations in the crypt and villus are well known, but the details of the kinetics of stem cell division, and precursor cell proliferation and differentiation into mature enterocytes and secretory cells are still being studied. These proliferation and differentiation events have been extensively modeled with a variety of computational approaches in the past.

Methods

A compartmental population kinetics model, incorporating experimentally measured proliferation rates for various intestinal epithelial cell types, is implemented for a previously reported scheme for the intestinal cell dynamics. A sensitivity analysis is performed to determine the effect that varying the model parameters has upon the model outputs, the steady-state cell populations.

Results

The model is unable to reproduce the experimentally known timescale of renewal of the intestinal epithelium if literature values for the proliferation rates of stem cells and transit amplifying cells are employed. Unphysically large rates of proliferation result when these parameters are allowed to vary to reproduce this timescale and the steady-state populations of terminally differentiated intestinal epithelial cells. Sensitivity analysis reveals that the strongest contributor to the steady-state populations is the transit amplifying cell proliferation rate when literature values are used, but that the differentiation rate of transit amplifying cells to secretory progenitor cells dominates when all parameters are allowed to vary.

Conclusions

A compartmental population kinetics model of proliferation and differentiation of cells of the intestinal epithelium can provide a simplifying means of understanding a complicated multistep process. However, when literature values for proliferation rates of the crypt based columnar and transit amplifying cell populations are employed in the model, it cannot reproduce the experimentally known timescale of intestinal epithelial renewal. Nevertheless, it remains a valuable conceptual tool, and its sensitivity analysis provides important clues for which events in the process are the most important in controlling the steady-state populations of specialized intestinal epithelial cells.

     

Stochastic model of autocrine and paracrine signals in cell culture assays

Autocrine signaling systems are commonly studied under cell culture conditions. In a typical cell culture assay, a layer of liquid medium covers a random two-dimensional dispersion of cells, which secrete ligands. In a growing number of experiments, it is important to characterize the spatial range of autocrine and paracrine cell communication. Currently, the spatial distribution of diffusing signals can be analyzed only indirectly, from their effects on the intracellular signaling or physiological responses of autocrine cells. To directly characterize the spatial range of secreted ligands, we propose a stochastic model for autocrine cell cultures and analyze it using a combination of analytical and computational tools. The two main results derived within the framework of this model are 1), an expression for the fraction of autocrine trajectories, i.e., the probability for a ligand to be trapped by the same cell from which it has been secreted; and 2), an expression for the spatial distribution of trapping points of paracrine trajectories. We test these analytical results by stochastic simulations with efficient Brownian dynamics code and apply our model to analyze the spatial operation of autocrine epidermal growth factor receptor systems.