Free energy of a folded polymer under cylindrical confinement

TL;DR

This study uses Monte Carlo simulations to analyze the free energy of folded polymers in cylindrical confinement, revealing scaling behaviors and discrepancies with simple models, especially for complex structures like S-loops.

Contribution

It provides detailed scaling laws for the free energy of confined polymers, including flexible, semiflexible, and complex folded structures, highlighting finite-size effects and deviations from simple theories.

Findings

Scaling of free energy gradient for flexible polymers close to theoretical predictions.

Measured free energy cost of hairpin formation aligns with recent theories.

Scaling behavior for S-loops differs more significantly from predictions.

Abstract

Monte Carlo computer simulations are used to study the conformational free energy of a folded polymer confined to a long cylindrical tube. The polymer is modeled as a hard-sphere chain. Its conformational free energy $F$ is measured as a function of $\lambda$, the end-to-end distance of the polymer. In the case of a flexible linear polymer, $F(\lambda)$ is a linear function in the folded regime with a gradient that scales as $f\equiv |dF/d\lambda| \sim N^0 D^{-1.20\pm 0.01}$ for a tube of diameter $D$ and a polymer of length $N$. This is close to the prediction $f \sim N^0 D^{-1}$ obtained from simple scaling arguments. The discrepancy is due in part to finite-size effects associated with the de-Gennes blob model. A similar discrepancy was observed for the folding of a single arm of a three-arm star polymer. We also examine backfolding of a semiflexible polymer of persistence length $P$ in the classic Odijk regime. In the overlap regime, the derivative scales $f \sim N^0 D^{-1.72\pm 0.02} P^{-0.35\pm 0.01}$, which is close to the prediction $f \sim N^0 D^{-5/3} P^{-1/3}$ obtained from a scaling argument that treats interactions between deflection segments at the second virial level. In addition, the measured free energy cost of forming a hairpin turn is quantitatively consistent with a recent theoretical calculation. Finally, we examine the scaling of $F(\lambda)$ for a confined semiflexible chain in the presence of an S-loop composed of two hairpins. While the predicted scaling of the free energy gradient is the same as that for a single hairpin, we observe a scaling of $f \sim D^{-1.91\pm 0.03} P^{-0.36\pm 0.01}$. Thus, the quantitative discrepancy between this measurement and the predicted scaling is somewhat greater for S-loops than for single hairpins.