Some deterministic and stochastic mathematical models of naive T-cell homeostasis

Humans live for decades, whereas mice live for months. Over these long timescales, naïve T cells die or divide infrequently enough that it makes sense to approximate death and division as instantaneous events. The population of T cells in the body is naturally divided into clonotypes; a clonotype is the set of cells that have identical T‐cell receptors. While total numbers of cells, such as naïve CD4+ T cells, are large enough that ordinary differential equations are an appropriate starting point for mathematical models, the numbers of cells per clonotype are not. Here, we review a number of basic mathematical models of the maintenance of clonal diversity. As well as deterministic models, we discuss stochastic models that explicitly track the integer number of naïve T cells in many competing clonotypes over the lifetime of a mouse or human, including the effect of waning thymic production. Experimental evaluation of clonal diversity by bulk high‐throughput sequencing has many difficulties, but the use of single‐cell sequencing is restricted to numbers of cells many orders of magnitude smaller than the total number of T cells in the body. Mathematical questions associated with extrapolating from small samples are therefore key to advances in understanding the diversity of the repertoire of T cells. We conclude with some mathematical models on how to advance in this area.