  Regulatory T cells (Tregs) play a crucial role in mediating immune response.
Yet an algorithmic understanding of the role of Tregs in adaptive immunity
remains lacking. Here, we present a biophysically realistic model of Treg
mediated self-tolerance in which Tregs bind to self-antigens and locally
inhibit the proliferation of nearby activated T cells. By exploiting a duality
between ecological dynamics and constrained optimization, we show that Tregs
tile the potential antigen space while simultaneously minimizing the overlap
between Treg activation profiles. We find that for sufficiently high Treg
diversity, Treg mediated self-tolerance is robust to fluctuations in
self-antigen concentrations but lowering the Treg diversity results in a sharp
transition -- related to the Gardner transition in perceptrons -- to a regime
where changes in self-antigen concentrations can result in an auto-immune
response. We propose a novel experimental test of this transition in
immune-deficient mice and discuss potential implications for autoimmune
diseases.
