Dynamical analysis of a degenerate primary and secondary humoral immune response

Lymphocyte receptor response to antigen is degenerate. Each receptor can have a high affinity to more than one antigen. The optimal level of degeneracy was previously modeled using different methods; all showing that the degeneracy level should be inversely proportional to the probability that an antigen belongs to the self repertoire. Here we develop a new formalism, reproducing the results of previous models, which enables us to study the relation between receptor degeneracy and the pathogen-immune cell interaction dynamics, in primary and secondary response. We begin by developing a general formalism and reproducing the results obtained by Nemazee: (1) that an optimal immune system will have a capacity which is inversely proportional to the fraction of self-antigens and (2) that the number of self-reactive cells that the body destroys is tuned by this capacity optimization to be 63%. We then use our extended framework to relate the minimal number of B cell precursor required to mount an immune response to the naive B cell production rate. Finally, we analyze the dynamics of the interaction between the immune system and a pathogen and show that memory cells may be used as the first line of defense, while newly created cells are used later to refine the immune response.