Knots in Macromolecules in Constraint Space

TL;DR

This paper investigates the knotting properties of macromolecules, revealing power-law relationships for knot-monomers and knot size, and analyzing crossing numbers in different polymer models, confirming theoretical laws and providing new constants.

Contribution

It introduces new power-law relationships for knot characteristics in macromolecules and compares crossing number laws across models with and without geometric constraints.

Findings

Power law for knot-monomers with exponent ~0.39

Average knot size scales as N^{0.20}

Data supports N log N + bN law for crossing numbers

Abstract

We find a power law for the number of knot-monomers with an exponent $0.39 \pm0.13$ in agreement with previous simulations. For the average size of a knot we also obtain a power law $N_m=2.56\cdot N^{0.20\pm0.04}$. We further present data on the average number of knots given a certain chain length and confirm a power law behaviour for the number of knot-monomers. Furthermore we study the average crossing number for random and self-avoiding walks as well as for a model polymer with and without geometric constraints. The data confirms the $aN\log N + bN$ law in the case of without excluded volume and determines the constants $a$ and $b$ for various cases. For chains with excluded volume the data for chains up to N=1500 is consistent with $aN\log N + bN$ rather than the proposed $N^{4/3}$ law. Nevertheless our fits show that the $N^{4/3}$ law is a suitable approximation.