Average Structures of a Single Knotted Ring Polymer

TL;DR

This study uses Brownian dynamics simulations to analyze average structures of single knotted ring polymers, revealing how knot localization varies with the number of segments and identifying a transition from double to single loop structures.

Contribution

It introduces a novel method to define and compare average structures of knotted ring polymers, accounting for segment re-labeling and rotation, and uncovers the localization-delocalization transition of knots.

Findings

Knot delocalizes for small N and localizes as N increases.

Average structure transitions from double to single loop with increasing N.

Method accounts for segment relabeling in structural averaging.

Abstract

Two types of average structures of a single knotted ring polymer are studied by Brownian dynamics simulations. For a ring polymer with N segments, its structure is represented by a 3N -dimensional conformation vector consisting of the Cartesian coordinates of the segment positions relative to the center of mass of the ring polymer. The average structure is given by the average conformation vector, which is self-consistently defined as the average of the conformation vectors obtained from a simulation each of which is rotated to minimize its distance from the average conformation vector. From each conformation vector sampled in a simulation, 2N conformation vectors are generated by changing the numbering of the segments. Among the 2N conformation vectors, the one closest to the average conformation vector is used for one type of the average structure. The other type of the averages structure uses all the conformation vectors generated from those sampled in a simulation. In thecase of the former average structure, the knotted part of the average structure is delocalized for small N and becomes localized as N is increased. In the case of the latter average structure, the average structure changes from a double loop structure for small N to a single loop structure for large N, which indicates the localization-delocalization transition of the knotted part.