Group theoretical and combinatorial analysis of histocompatibility and switching algebra

Abstract

A combination of mathematical tools are brought together to study the problem of the reduction of a certain class of antibody-antigen reaction data to understand the fundamental interactions between antibodies and antigens. Algebraic methods analogous to those used in computer switching theory are developed for the purpose of defining the functional nature of the antibody-antigen data contained in the reaction matrices. Extensive use is made of group theoretic and combinatorial techniques in obtaining in closed form analytic expressions which permit a determination of the number of equivalence classes and the number of reaction matrices belonging to each class without the need for extensive enumeration. The effect of cross reactions on the antibody-antigen reaction data is considered, and the mathematical analysis is applied to the case where the reaction strengths are constrained to integral values. However, this constraint is a convenience rather than a necessity and it is shown how the constraint may be relaxed. This analysis points out certain fundamental characteristics inherent in antibody-antigen interactions which make them amenable to quantitative analysis.