A Topological Characterization of Protein Structure

We develop an objective characterization of protein structure based entirely on the geometry of its parts. The three-dimensional alpha complex filtration of the protein represented as a union of balls (one per residue) captures all the relevant information about the geometry and topology of the molecule. The neighborhood of a strand of contiguous alpha carbon atoms along the back-bone chain is defined as a “tube” which is a sub-complex of the original complex that has been sub-divided. We then define a retraction for the tube to another complex that is guaranteed to be a 2-manifold with boundary. We capture the topology of the retracted tube by computing the most persistent connected components and holes in the entire filtration. A “motif” for a 3D structure is characterized by the number of persistent 0- and 1-cycles, and the relative persistences of these cycles in the filtration of the “tube” complex. These motifs represent non-random, recurrent, tertiary interactions between parts of the protein back-bone chain that characterize the overall structure of the protein. A basis set of 1300 motifs are identified by analyzing the alpha complex filtrations of several proteins. Any test protein is represented by the number of times each motif from the basis set occurs in it. Preliminary results from the discrimination of protein families using this representation are provided.