Geometry and topology of knotted ring polymers in an array of obstacles

TL;DR

This paper explores how knotted ring polymers behave in obstacle arrays, revealing a crossover from localized to delocalized knots and branched structures, with implications for gel electrophoresis.

Contribution

It provides a detailed analysis of the geometrical and topological transitions of knotted polymers in confined environments, highlighting the regimes of localization, branching, and delocalization.

Findings

Knot localization depends on the polymer's size relative to obstacle spacing.

Polymer geometry transitions from linear to branched as size increases.

Knot delocalization occurs when the knot size exceeds obstacle spacing.

Abstract

We study knotted polymers in equilibrium with an array of obstacles which models confinement in a gel or immersion in a melt. We find a crossover in both the geometrical and the topological behavior of the polymer. When the polymers' radius of gyration, $R_G$, and that of the region containing the knot, $R_{G,k}$, are small compared to the distance b between the obstacles, the knot is weakly localised and $R_G$ scales as in a good solvent with an amplitude that depends on knot type. In an intermediate regime where $R_G > b > R_{G,k}$, the geometry of the polymer becomes branched. When $R_{G,k}$ exceeds b, the knot delocalises and becomes also branched. In this regime, $R_G$ is independent of knot type. We discuss the implications of this behavior for gel electrophoresis experiments on knotted DNA in weak fields.