Deriving Shape Space Parameters from Immunological Data

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. 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 ball of stimulation. In a cross-reactive response, the intersection of the 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 tha..