Predicting the optimal number of MHC proteins against positive and negative T-cell selection via clonotype-based modelling of T-cell repertoire formation
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We adapt a previous mathematical model of T cell development with the goal of describing the dynamics of cells that have same fate during thymic selection. The original model considers the survival of a random T cell, using averages of experimental data taken from homozygous mice describing T cell counts at various stages of thymic selection. In the new `clonotype' model we split the original model into cells surviving selection, those not surviving negative selection, and those not surviving positive selection. Solving the fractions of cells in each category from the steady state of the original model, we obtain estimates for the fraction of clonotypes surviving at each stage. These new estimates are input into another updated model that links the number of MHC molecule types to T cell survival. From this, we predict a lower and upper bound for the optimal number of MHC types to maximise survival. The true number of MHC types observed for heterozygous mice in vivo (≈ 12) falls comfortably in this estimated range (6.5 < M < 15). This suggests that the total number of types of MHC molecule present in an individual is inuenced by a selection pressure to maximise survival of T cells during positive and negative selection. More generally, the methods used also represent a novel framework by which ODE models of populations may be split into a system of equations for distinct groups with separate outcomes.