Fractal Structure of Shortest Interaction Paths in Native Proteins and Determination of Residues on a Given Shortest Path

Abstract

Fractal structure of shortest paths depends strongly on interresidue interaction cutoff distance. The dimensionality of shortest paths is calculated as a function of interaction cutoff distance. Shortest paths are self similar with a fractal dimension of 1.12 when calculated with step lengths larger than 6.8 {\AA}. Paths are multifractal below 6.8 {\AA}. The number of steps to traverse a shortest path is a discontinuous function of cutoff size at short cutoff values, showing abrupt decreases to smaller values as cutoff distance increases. As information progresses along the direction of a shortest path a large set of residues are affected because they are interacting neighbors to the residues of the shortest path. Thus, several residues are involved diffusively in information transport which may be identified with the present model. An algorithm is introduced to determine the residues of a given shortest path. The shortest path residues are the highly visited residues during information transport. These paths are shown to lie on the high entropy landscape of the protein where entropy is taken to increase with abundance of visits to nodes during signal transport.