Fractional Brownian motion meets topology: statistical and topological properties of globular macromolecules with volume interactions

TL;DR

This study explores the statistical and topological characteristics of fractional Brownian polymer chains with volume interactions, revealing how fractal dimension influences conformational properties and knot complexity in globular macromolecules.

Contribution

It introduces a simplified model for collapsed globular conformations using fractal paths, combining numerical simulations with Flory theory to analyze their properties.

Findings

Higher fractal dimension leads to more territorial and less knotted conformations.

Distribution of knot complexity varies with fractal dimension, indicating a link between topology and statistics.

The model effectively mimics the conformational behavior of crumpled globules without topological constraints.

Abstract

In the paper we investigate statistical and topological properties of fractional Brownian polymer chains, equipped with the short-range volume interactions. The attention is paid to statistical properties of collapsed conformations with the fractal dimension $D_f\ge 2$ in the three-dimensional space, which are analyzed both numerically and \textit{via} the mean-field Flory approach. Our study is motivated by an attempt to mimic the conformational statistics of collapsed unknotted polymer rings, which are known to form compact hierarchical crumpled globules (CG) with $D_f=3$ at large scales. Replacing the topologically-stabilized CG state by a self-avoiding fractal path adjusted to the fractal dimension $D_f=3$ we tremendously simplify the problem of generating compact self-avoiding conformations since we wash out the topological constraints from the consideration. We make use of the Monte-Carlo simulations to prepare the equilibrium ensemble of swollen chains with various fractal dimensions. A combination of the Flory arguments with statistical analysis of the conformations from simulations allows one to infer the dependence of the critical exponent of the swollen chains on the fractal dimension of the seed chain. We show that with the increase of $D_f$, typical conformations become more territorial and less knotted. Distributions of the knot complexity, $P(\chi)$ for various fractal dimensions of the swollen chains suggest a close relationship between statistical and topological properties of fractal paths with volume interactions.