Analytic formulas for discrete stochastic models of cell populations with both differentiation and de-differentiation

Cell differentiation often appears to be a stochastic process particularly in the hemopoietic system. One of the earliest stochastic models for the growth of stem cell populations was proposed by Till et al. in 1964. In this model there are just two cell types: stem cells and specialized cells. At each time step there is a fixed probability that a stem cell differentiates into a specialized cell and a fixed probability that it undergoes mitosis to produce two stem cells. Even though this model is conceptually simple the myriad of possible outcomes has made it difficult to analyse. We present original closed-form expressions for the probability functions and a fast algorithm for computing them. Renewed interest in stem cells has raised questions about the effect de-differentiation has on stem cell populations. We have extended the stochastic model to include de-differentiation and show that even a small amount of de-differentiation can have a large effect on stem cell population growth.