A geometric and algebraic view of MHC-peptide complexes and their binding properties

Abstract

Background: Major histocompatibility complex (MHC) molecules present peptides to T lymphocytes. It is of critical biological and medical importance to elucidate how different MHC alleles bind to a specific set of peptides. Method: In this study we approach the problem from the algebraic and geometric point of view to analyse MHC-peptide-binding data accumulated over the years. The space of sequence properties (having a particular amino acid at a particular position) of MHC-peptide complexes conveys a geometric structure to these sequence properties in the form of a distance measure, which reveals the peptide binding requirements imposed by the polymorphic sequence characteristics of the MHC molecules. Results: Comparison of the results of this study with our current knowledge of MHC-peptide binding constraints leads to robust agreement. This study provides the tools to quantitate these binding constraints giving a more detailed account of them and opening the way to make peptide binding predictions for MHC alleles for which there is no peptide elution data. In addition, the geometric representation of MHC-peptide complex sequence data gives a distance measure between amino acids in reference to their ability to meet MHC binding requirements. Conclusions: The algebraic and geometric view of amino acid sequences provides a theoretical framework to study the function of proteins when there is enough variation in this sequence to account for the variation in their function, as it is the case with MHC molecules in regard to their ability to present peptides.