Configurational entropy of randomly double-folding ring polymers

TL;DR

This paper derives the exact count of tightly double-folded configurations of ring polymers with tree-like structures, validated by Monte Carlo simulations, providing insights into their topological entropy.

Contribution

It introduces a novel coding scheme and exact combinatorial formulas to quantify the configurational entropy of double-folded ring polymers.

Findings

Exact enumeration of double-folded ring configurations.

Validation of theoretical results with Monte Carlo simulations.

Agreement between combinatorial formulas and simulation data.

Abstract

Topologically constrained genome-like polymers often double-fold into tree-like configurations. Here we calculate the exact number of tightly double-folded configurations available to a ring polymer in ideal conditions. For this purpose, we introduce a scheme which allows us to define a ``code'' specifying how a ring wraps a randomly branching tree and calculate the number of admissible wrapping codes via a variant of Bertrand's ballot theorem. As a validation, we demonstrate that data from Monte Carlo simulations of an elastic lattice model of non-interacting tightly double-folded rings with controlled branching activity are in excellent agreement with exact expressions for branch-node and tree size statistics that can be derived from our expression for the ring entropy.