Thymic Selection of T Cells as Diffusion with Intermittent Traps

T cells orchestrate adaptive immune responses by recognizing short peptides derived from pathogens, and by distinguishing them from self-peptides. To ensure the latter, immature T cells (thymocytes) diffuse within the thymus gland, where they encounter an ensemble of self-peptides presented on (immobile) antigen presenting cells. Potentially autoimmune T cells are eliminated if the thymocyte binds sufficiently strongly with any such antigen presenting cell. We model thymic selection of T cells as a random walker diffusing in a field of immobile traps that intermittently turn "on" and "off". The escape probability of potentially autoimmune T cells is equivalent to the survival probability of such a random walker. In this paper we describe the survival probability of a random walker on a d-dimensional cubic lattice with randomly placed immobile intermittent traps, and relate it to the result of a well-studied problem where traps are always "on". Additionally, when switching between the trap states is slow, we find a peculiar caging effect for the survival probability.