Perturbation theory analysis of competition in a heterogeneous population

A physiologically structured population model of T cell populations at the end of an immune response is proposed and analysed. The proposed model describes a T cell population with a structured death rate mediated by cytokine receptor surface expression. This provides a structuring that determines the ability of a T cell to compete for survival cytokines. Homogeneous differentiation into memory combined with heterogeneous survival abilities leads to a system of nonlinear first-order integro differential equations. The mathematical model is studied analytically by a perturbation approach which agrees well with numerical integrations. We find that the structured death rate leads to selection of T cells expressing high cytokine receptor levels. Selection in this model is independent of the initial T cell distribution which allows us to define a universal selection curve, a property lost under receptor downregulation (i.e. loss of cytokine receptors from the surface). Furthermore, we examine the effects of a population dependent memory differentiation rate and of cytokine receptor downregulation on selection. A population dependent memory differentiation rate increases selection significantly while putting an upper bound on the number of T cells differentiating into memory. Under receptor downregulation a dramatic change of the memory T cell distribution is induced. We find that there exists a critical ratio of the receptor downregulation rate to the average T cell death rate for which selection becomes maximal. Our analysis reveals that this maximum property is the result of an interplay between single cell dynamics and population dynamics. We propose that the selection studied in this paper plays an important role in the selection of optimal T cells for the memory pool.