Topological Disentanglement of Linear Polymers under Tension.

Michele Caraglio,Enzo Orlandini,Boris Marcone,Attilio L Stella,Fulvio Baldovin

doi:10.3390/polym12112580

Abstract

We develop a theoretical description of the topological disentanglement occurring when torus knots reach the ends of a semiflexible polymer under tension. These include decays into simpler knots and total unknotting. The minimal number of crossings and the minimal knot contour length are the topological invariants playing a key role in the model. The crossings behave as particles diffusing along the chain and the application of appropriate boundary conditions at the ends of the chain accounts for the knot disentanglement. Starting from the number of particles and their positions, suitable rules allow reconstructing the type and location of the knot moving on the chain Our theory is extensively benchmarked with corresponding molecular dynamics simulations and the results show a remarkable agreement between the simulations and the theoretical predictions of the model.

Highlights

From tying shoelaces to maneuvers in sailing or mountaineering, knots are useful in our everyday lives
The paper is organized as follows: in Section 2 we present the molecular dynamics (MD)
In cases in which the knot decays into a simpler one, like for an initially tied torus knot 51, one can report the probability that at time t the knot is already present in the simpler form (31 in this example) resulting from topological decay

Summary

Introduction

From tying shoelaces to maneuvers in sailing or mountaineering, knots are useful in our everyday lives. Knots move and fluctuate in size permanently (equilibrium state) but if tied in open chains they do not represent a genuine topological state of the system and can disappear, form again, or change the underlined topological complexity (knot type) [9,10,11] These “physical knots", act as long-lived constraints and can, for instance, affect the metric and mechanical properties of the hosting chain, interfere with the elongation processes induced by either confinement [11,12,13] or tensile forces [14,15], hinder the ejection dynamics of viral. In this work the statistical properties of unknotting are shown to be well described by a simple model of nk random walks interacting via hard-core exclusion in one spatial dimension (single-file diffusion), with nk being the number of essential crossings associated to the knot type k This analysis, is restricted to the simpler case of quasi-2D chains in which the position of the crossings do not depend on the projection.

Numerical Simulations

Strong Tension Limit

Free Diffusion Limit

Results

Conclusions