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Protein similarity from knot theory and geometric convolution

Shape similarity is one of the most elusive and intriguing questions of nature and mathematics. Proteins provide a rich domain in which to test theories of shape similarity. Proteins can match at different scales and in different arrangements. Sometimes the detection of common local structure is sufficient to infer global alignment of two proteins; at other times it provides false information. Proteins with very low sequence identity may share large substructures, or perhaps just a central core. There are even examples of proteins with nearly identical primary sequence in which α-helices have become Β-sheets.Shape similarity can be formulated (i) in terms of global metrics, such as RMSD or Hausdorff distance, (ii) in terms of subgraph isomorphisms, such as the detection of shared substructures with similar relative locations, or (iii) purely topologically, in terms of the cohomology induced by structure preserving transformations. Existing protein structure detection programs are built on the first two types of similarity. The third forms the foundations of knot theory.The thesis of this paper is: Protein similarity detection leads naturally to an algorithm operating at the metric, relational, and homotopic scales. The paper introduces a definition of similarity based on atomic motions that preserve local backbone topology without incurring significant distance errors. Such motions are motivated by the physical requirements for rearranging subsequences of a protein. Similarity detection then seeks rigid body motions able to overlay pairs of substructures, each related by a substructure-preserving motion, without necessarily requiring global structure preservation. This definition is general enough to span a wide range of questions: One can ask for full rearrangement of one protein into another while preserving global topology, as in drug design; or one can ask for rearrangements of sets of smaller substructures, each of which preserves local but not global topology, as in protein evolution.In the appendix, we exhibit an algorithm for answering the general question. That algorithm has the complexity of robot motion planning. In the text, we consider a more common case in which one seeks protein similarity by rearrangements of relatively short peptide segments. We exhibit an algorithm based on writhing numbers that runs in time O(n2) to O(n4). We define and use a new datastructure, called geometric self-convolution, within this algorithm.Contributions: We believe that this is the first paper to consider carefully the need for combining metric and homotopic qualities in seeking protein similarity. We provide a parameterized definition of similarity that leads naturally to a metric in protein space. We exhibit algorithms for computing the metric and detecting similarity. We report results obtained with three pairs of proteins, each pair exhibiting different typical characteristics.