Under-knotted and Over-knotted Polymers: Unrestricted Loops

TL;DR

This study uses computer simulations to analyze how the topology of closed polymers affects their size distribution, revealing that knots influence the polymers' entropic rigidity and that distributions tend to a universal shape for large lengths.

Contribution

It provides new insights into the probability distributions of gyration radii for knotted polymers and compares their behavior to phantom loops, highlighting topological effects.

Findings

Gyration radius distributions are narrower for knotted loops than phantom loops.

Mean square gyration radius scales similarly for trivial knots and self-avoiding walks.

Distributions approach a universal shape for large N beyond the characteristic length N_0.

Abstract

We present computer simulations to examine probability distributions of gyration radius for the no-thickness closed polymers of N straight segments of equal length. We are particularly interested in the conditional distributions when the topology of the loop is quenched to be a certain knot, K. The dependence of probability distribution on length, N, as well as topological state K are the primary parameters of interest. Our results confirm that the mean square average gyration radius for trivial knots scales with N in the same way as for self-avoiding walks, where the cross-over length to this "under-knotted" regime is the same as the characteristic length of random knotting, N_0. Probability distributions of gyration radii are somewhat more narrow for topologically restricted under-knotted loops compared to phantom loops, meaning knots are entropically more rigid than phantom polymers. We also found evidence that probability distributions approach a universal shape at N>N_0 for all simple knots.