Some deterministic and stochastic mathematical models of naïve 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.