Deriving Shape Space Parameters from Immunological Data for a Model of Cross-Reactive Memory

Cross-reactive memory responses occur when the immune system is primed by one strain of a pathogen and challenged with a related, but different strain. Much of the nature of a cross-reactive response is determined by the quantity and distribution of the memory cells, raised to the primary antigen, that cross-react with the secondary antigen. B cells with above threshold affinity for an antigen lie in a region of shape space that we call a {\it ball of stimulation}. In a cross-reactive response, the intersection of balls of stimulation of the primary and secondary antigens contains the cross-reactive B cells and thus determines the degree of cross-reactivity between the antigens. We derive formulas for the volume of intersection of balls of stimulation in different shape spaces, and show that the parameters of shape space, such as its dimensionality, have a large impact on the number of B cells in the intersection. We present a method for deriving shape space parameters that are consistent with immunological data, and illustrate the method by deriving shape space parameters for a model of cross-reactive memory. For Hamming shape spaces, the method indicates that twenty to twenty-five dimensions, a three or four letter alphabet, and balls of stimulation of radius five or six, are choices that match the experimental data. For Euclidean shape spaces, five to eight dimensions and balls of stimulation with radius about twenty percent of the radius of the whole space, match the experimental data.