The geometry of shape space: application to influenza

Shape space was proposed over 20 years ago as a conceptual formalism in which to represent antibody/antigen binding. It has since played a key role in computational immunology. Antigens and antibodies are considered to be points in an abstract "shape space", where coordinates of points in this space represent generalized physico-chemical properties associated with various (unspecified) physical properties related to binding, such as geometric shape, hydrophobicity, charge, etc. Distances in shape space between points representing antibodies and (the shape complement) of antigens are assumed to be related to their affinity, with small distances corresponding to high affinity. In this paper, we provide algorithms, related to metric and ordinal multidimensional scaling algorithms first developed in the mathematical psychology literature, which construct explicit, quantitative coordinates for points in shape space given experimental data such as hemagglutination inhibition assays, or other general affinity assays. Previously, such coordinates had been conceptual constructs and totally implicit. The dimension of shape space deduced from hemagglutination inhibition assays for influenza is low, approximately five dimensional. The deduction of the explicit geometry of shape space given experimental affinity data provides new ways to quantify the similarity of antibodies to antibodies, antigens to antigens, and the affinity of antigens to antibodies. This has potential utility in, e.g. strain selection decisions for annual influenza vaccines, among other applications. The analysis techniques presented here are not restricted to the analysis of antibody-antigen interactions and are generally applicable to affinity data resulting from binding assays.