Fractal and Statistical Properties of Large Compact Polymers: A Computational Study

TL;DR

This study introduces an improved algorithm for generating lattice protein conformations, analyzes their fractal properties, and investigates the relationship between topology and compactness in Hamiltonian loops.

Contribution

It presents a new combinatorial algorithm reducing bias in sampling Hamiltonian walks and explores the fractal and topological properties of compact polymer conformations.

Findings

The new algorithm significantly reduces statistical bias in conformation sampling.

Chain pieces follow Gaussian statistics below the globule size, with a larger crossover scale.

Probability of knots increases rapidly with loop length, affecting chain compactness.

Abstract

We propose a novel combinatorial algorithm for efficient generation of Hamiltonian walks and cycles on a cubic lattice, modeling the conformations of lattice toy proteins. Through extensive tests on small lattices (allowing complete enumeration of Hamiltonian paths), we establish that the new algorithm, although not perfect, is a significant improvement over the earlier approach by Ramakrishnan et. al., as it generates the sample of conformations with dramatically reduced statistical bias. Using this method, we examine the fractal properties of typical compact conformations. In accordance with Flory theorem celebrated in polymer physics, chain pieces are found to follow Gaussian statistics on the scale smaller than the globule size. Cross-over to this Gaussian regime is found to happen at the scales which are numerically somewhat larger than previously believed. We further used Alexander and Vassiliev degrees 2 and 3 topological invariants to identify the trivial knots among the Hamiltonian loops. We found that the probability of being knotted increases with loop length much faster than it was previously thought, and that chain pieces are consistently more compact than Gaussian if the global loop topology is that of a trivial knot.