Free Energy of a Knotted Polymer Confined to Narrow Cylindrical and Conical Channels

TL;DR

This study uses Monte Carlo simulations to analyze how knots in semiflexible polymers behave and how their free energy varies in narrow cylindrical and conical channels, revealing effects of topology, channel shape, and size.

Contribution

It provides new insights into the free energy and conformational behavior of knotted polymers in confined geometries, extending theoretical models to include conical channels.

Findings

Knot complexity increases the typical size of the knot.

Conformational behavior is independent of topology at low extension.

Free energy and knot span scale according to theoretical predictions.

Abstract

Monte Carlo simulations are used to study the conformational behavior of a semiflexible polymer confined to cylindrical and conical channels. The channels are sufficiently narrow that the conditions for the Odijk regime are marginally satisfied. For cylindrical confinement, we examine polymers with a single knot of topology $3_1$, $4_1$, or $5_1$, as well as unknotted polymers that are capable of forming S-loops. We measure the variation of the free energy $F$ with the end-to-end polymer extension length $X$ and examine the effect of varying the polymer topology, persistence length $P$ and cylinder diameter $D$ on the free energy functions. Similarly, we characterize the behavior of the knot span along the channel. We find that increasing the knot complexity increases the typical size of the knot. In the regime of low $X$, where the knot/S-loop size is large, the conformational behavior is independent of polymer topology. In addition, the scaling properties of the free energy and knot span are in agreement with predictions from a theoretical model constructed using known properties of interacting polymers in the Odijk regime. We also examine the variation of $F$ with position of a knot in conical channels for various values of the cone angle $\alpha$. The free energy decreases as the knot moves in a direction where the cone widens, and it also decreases with increasing $\alpha$ and with increasing knot complexity. The behavior is in agreement with predictions from a theoretical model in which the dominant contribution to the change in $F$ is the change in the size of the hairpins as the knot moves to the wider region of the channel.