A combinatorial algorithm for effective generation of long maximally compact lattice chains

Abstract

Effects of long range interactions on the conformational statistics of short polypeptide chains generated by a Monte Carlo methodWe investigate the problem of generation of maximally compact lattice chains which are useful in understanding folding of model proteins. The term, maximally compact chain, refers to a lattice self-avoiding walk that visits every lattice site. Generation of a representative sample of compact conformations is extremely difficult by conventional simulation methods such as static growth methods or dynamic Monte Carlo techniques. Growing a random walk is ineffective for generating long walks in a compact shape because a large number of walks are rejected due to overlap ͑attrition͒. In the interest of an unbiased sample, one needs to enumerate all possible compact conformations that are realizable or produce a representative sample, the former of which is intractable for long chains. In this paper a method is proposed for generation of compact chains on a lattice based on a mathematical programming approach. The method, which we refer to as the Hamiltonian path generation method, generates a random sample of lattice filling self-avoiding walks. A detailed description of a randomized generation algorithm is presented, which is effective for producing a static sample of compact lattice chains. There is a statistical evidence of fair generation of conformations from the conformational space using this scheme. This method generates a compact conformation on a 60ϫ60ϫ60 cubic lattice in forty minutes on a Sparc-2 workstation.

A dynamic Monte Carlo algorithm for exploration of dense conformational spaces in heteropolymers

“…We refer to this composite move as the Hamiltonian path rearrangement ͑HPR͒ move, since this was motivated by the HPG method for the generation of compact chains. 19 As described in our earlier work on the HPG method, there exist situations where a four-cycle move may not be feasible. When this happens, we use the branching exactly as described in that paper.…”

Design of toy proteins capable of rearranging conformations in a mechanical fashion

“…In this case, a special procedure called two-matching is applied. 26 During this and the following steps of the algorithm, some randomly chosen edges can be removed from the lattice and changed by others. However, we impose the condition that the edges forming switching elements cannot be removed at any step of conformation design.…”

Conformational properties of compact polymers

“…every allowed conformation is sampled with equal probability. While other proposed algorithms for sampling Hamiltonian paths have been shown to produce biased results [14,16], Mansfield proved in his paper [15] ergodicity for small lattices by exact enumeration and devised a method providing strong evidence that the algorithm is ergodic for even larger lattices.…”

“…In the last few years several Monte Carlo algorithms have been proposed to study compact polymers based on the idea of Hamiltonian paths [14,15,16]. A Hamiltonian path on some graph G with set of vertices V and edges E is defined as a path which visits each vertex V ∈ V exactly once.…”

“…One important aspect of such an algorithm is that sampling of conformations is done in an unbiased way. Two algorithms have been shown to fail this test [14,16]. Recently, Mansfield proposed an algorithm which is shown to produce unbiased samples to a high degree of certainty [15].…”