The Cyton Model for Lymphocyte Proliferation and Differentiation

We discuss a stochastic model for lymphocyte population dynamics based on the interaction of sub-cellular mechanisms responsible for cell death, division and differentiation. Competition between these mechanisms determines the fate of an individual, and their stochastic nature allows a range of outcomes, generating the large-scale diversity that is characteristic of an immune response. Mathematically the function of each mechanism can be expressed as a probability density function for the time for the particular event to occur, and it is the parameters that define these distributions that are the free parameters of the model. We formulate the problem as a branching process, and derive an expression for the probability generating function for the number of live cells of each type in all divisions, as a function of time. This allows the calculation of arbitrary moments for the live cell numbers, giving quantification not just of the mean system behaviour, but of the random fluctuations that are a consequence of the stochasticity of the underlying mechanisms.