Conformational transitions and principal geodesic analysis on the positive semidefinite matrix manifold

Abstract

Given an initial and final protein conformation, generating the intermediate conformations provides important insight into the protein's dynamics. We represent a protein conformation by its Gram matrix, which is a point on the rank 3 positive semidefinite matrix manifold, and show matrices along the geodesic linking an initial and final Gram matrix can be used to generate a feasible pathway for the protein's structural change. This geodesic is based on a particular quotient geometry. If a protein is known to contain domains or groups of atoms that act as rigid clusters, facial reduction can be used to decrease the size of the Gram matrices before calculating the geodesic. The geodesic between two conformations is only one path a protein's Gram matrix can follow; principal geodesic analysis (PGA) is one possible strategy to find other geodesics. © 2014 Springer International Publishing Switzerland.