Robert Farber, Alan Lapedes

Paper #: 00-01-006

Shape space was proposed by Perelson and Oster 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 thought of as 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. Up to now, coordinates of points in shape space have been purely implicit. 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 assays, or other general affinity assays. An application of these algorithms to panels of hemagglutination inhibition assays for influenza show that the dimension of immunological shape space is low, roughly dimension five, in accord with Oster's and Perelson's earlier qualitative estimates. Explicit numerical values are provided by the algorithms for coordinates of molecules in shape space, whereas previously such coordinates have been a conceptual construct and totally implicit. 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 analysis of antibody-antigen interactions and are generally applicable to affinity data resulting from binding assays.