I will discuss joint work with H. van den Berg and N. Burroughs about how the exquisite antigen specificity of T cells can be reconciled with the low affinity of the binding between TCR and MHC-peptide complex and the complexity of antigen presentation profiles.  A new mathematical model of T cell signalling and activation is used to obtain a working T
cell repertoire which will with high probability both recognise foreign antigen and tolerate self and do this in a timely and effective fashion. The success of our approach relies on novel applications of mathematical results about large deviations which suggest new insights about the relationship between antigen presentation profile and the activation level
of the T cell, the effect of thymic selection on the repertoire, and the efficacy of recognition and the probability of tolerance.

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