Torsional random-walk statistics on lattices using convolution on crystallographic motion groups

This paper presents a new algorithm for generating the conformational statistics of lattice polymer models. The inputs to the algorithm are the distributions of poses (positions and orientations) of reference frames attached to sequentially proximal bonds in the chain as it undergoes all possible torsional motions in the lattice. If z denotes the number of discrete torsional motions allowable around each of the n bonds, our method generates the probability distribution in end-to-end pose corresponding to all of the zn independent lattice conformations in O(nD+1) arithmetic operations for lattices in D-dimensional space. This is achieved by dividing the chain into short segments and performing multiple generalized convolutions of the pose distribution functions for each segment. The convolution is performed with respect to the crystallographic space group for the lattice on which the chain is defined. The formulation is modified to include the effects of obstacles (excluded volumes) and to calculate the frequency of the occurrence of each conformation when the effects of pairwise conformational energy are included. In the latter case (which is for three dimensional lattices only) the computational cost is O(z4n4). This polynomial complexity is a vast improvement over the O(zn) exponential complexity associated with the brute-force enumeration of all conformations. The distribution of end-to-end distances and average radius of gyration are calculated easily once the pose distribution for the full chain is found. The method is demonstrated with square, hexagonal, cubic and tetrahedral lattices.