A statistical mechanical model of closed loop plectoneme supercoiling and its variational approximation

TL;DR

This paper develops a statistical mechanical model for closed loop plectonemes under supercoiling, incorporating helix-dependent interactions and variational approximations to analyze their structure and energetics.

Contribution

It introduces a novel variational approximation framework for modeling supercoiled closed loop plectonemes with helix-dependent interactions.

Findings

Model effectively separates end loops and braided section contributions.

Variational approximations for strong and weak helix interactions are developed.

Method for computing average writhe and potential for Monte Carlo simulations are discussed.

Abstract

Presented here, is a technical manuscript that may form the basis of later published work. In it, we develop a statistical mechanical model to describe a closed loop plectoneme, applicable for when the closed loop is sufficiently supercoiled. The model divides the system up into end loops and a braided section; the end loops are assumed to contribute little to the super-coil writhe. Within the braided section, the model incorporates interactions that depend on the structure of the molecule; in particular, we consider those that depend on helical structure. A method for approximating the steric interactions is utilized that we had previously used in other publications. We go on to construct variational approximations for our closed loop plectoneme model in two cases. The first case is where helix dependent interactions are strong, and in the second case they are considered weak. In developing these approximations, we approximate the Fuller-White condition by replacing, in all expressions that depend on twist, writhe with average writhe, valid when the braided section is sufficiently long. How this approximation is made and the conditions when this approximation is valid are also discussed. The approximation allows for a Legendre transformation of the free energy, which with the introduction of moment (or torque), effectively allowing for twist and average writhe to be treated independently in the transformed (Gibbs like) free energy. Next, we then show how one may compute the average writhe of the braided section. Lastly, we discuss how some of the approximations considered may be relaxed, and discuss how the resulting model free energy might be computed by MC simulation.