Statistical Mechanical Model for a closed loop plectoneme with weak helix specific forces

TL;DR

This paper introduces a statistical mechanical model for closed loop plectonemes that incorporates weak helix-specific forces, revealing asymmetries in supercoiling structure and providing a systematic way to enforce topological constraints.

Contribution

It develops a variational approximation framework that includes helix-dependent forces and topological constraints, offering new insights into plectoneme structure and energetics.

Findings

Left-handed supercoils are looser than right-handed ones.

The model predicts a slight asymmetry in supercoiling structure.

Calculated free energies suggest lower energy for right-handed supercoils.

Abstract

We develop a statistical mechanical framework, based on a variational approximation, to describe closed loop plectonemes. This framework incorporates weak helix structure dependent forces into the determination of the free energy and average structure of a plectoneme. Notably, due to their chiral nature, helix structure dependent forces break the symmetry between left and right handed supercoiling. The theoretical approach, presented here, also provides a systematic way of enforcing the topological constraint of closed loop supercoiling in the variational approximation. At large plectoneme lengths, by considering correlation functions in an expansion in terms of the spatial mean twist density about its thermally averaged value, it can be argued that topological constraint may be approximated with by replacing twist and writhe by their thermal averages. A Lagrange Multiplier, containing the sum of average twist and writhe, can be added to the free energy to conveniently inforce this result. The average writhe can be calculated through the thermal average of Gauss integral in the variational approximation. Furthermore, this approach allows for a possible way to calculate finite size corrections due to the topological constraint. Using interaction energy terms from the mean-field Kornyshev-Leikin theory, for parameter values that correspond to weak helix dependent forces, we calculate the free energy, fluctuation magnitudes and mean geometric parameters for the plectoneme. We see a slight asymmetry, where interestingly enough left handed supercoils have looser structure than right handed ones, although a lower free energy, unlike what previous ground state calculations would suggest.