Symmetries in proteins: A knot theory approach

Whereas the symmetries of small molecules are described by the methods of group theory, there is no corresponding way to describe the complex symmetries in proteins. We develop a quantitative method to define and classify symmetries in compact polymers, based on the mathematical theory of graphs and knots. We represent different chain folds by their "polymer graphs," equivalent to contact maps. We transform those graphs into mathematical knots to give a parsing of different possible chain folds into conformational taxonomies. We use Alexander-Conway knot polynomials to characterize the knots. We find that different protein structures with the same tertiary fold, e.g., a βαβ motif with different lengths of α helix and β sheet, can be described in terms of the different powers of the propagation matrices of the knot polynomial. This identifies a fundamental type of topological length invariance in proteins, "elongatable" symmetries. For example, "helix," "sheet," "helix-turn-helix," and other secondary, supersecondary, and tertiary structures define structures of any chain length. Possibly the nine superfolds identified by Thornton et al. have eloneatable symmetries.