Thymic selection of T-cell receptors as an extreme value problem

T lymphocytes (T cells) orchestrate adaptive immune responses upon activation. T-cell activation requires sufficiently strong binding of T-cell receptors on their surface to short peptides (p) derived from foreign proteins, which are bound to major histocompatibility gene products (displayed on antigen-presenting cells). A diverse and self-tolerant T-cell repertoire is selected in the thymus. We map thymic selection processes to an extreme value problem and provide an analytic expression for the amino acid compositions of selected T-cell receptors (which enable its recognition functions).