Knotted Globular Ring Polymers: How Topology Affects Statistics and Thermodynamics

Abstract

The statistical mechanics of a long knotted collapsed polymer is determined by a free-energy with a knot-dependent subleading term, which is linked to the length of the shortest polymer that can hold such knot. The only other parameter depending on the knot kind is an amplitude such that relative probabilities of knots do not vary with the temperature T , in the limit of long chains. We arrive at this conclusion by simulating interacting self-avoiding rings at low T on the cubic lattice, both with unrestricted topology and with setups where the globule is divided by a slip link in two loops (preserving their topology) which compete for the chain length, either in contact or separated by a wall as for translocation through a membrane pore. These findings suggest that in macromolecular environments there may be entropic forces with a purely topological origin, whence portions of polymers holding complex knots should tend to expand at the expense of significantly shrinking other topologically simpler portions. 1

Molecular Crowding Increases Knots Abundance in Linear Polymers

“…The remaining knots population consists about entirely of 4 1 or figure-eight knots. The fact that the increased abundance of knots is not accompanied by a substantial increase of their complexity is intriguing because it is different from what has been observed in simulations of polymers that are compactified by three-dimensional spatial confinement or under poor-solvency conditions. , In such cases, in fact, the knotting enhancement is paralleled by a complication of the knot spectrum. An equal footing quantitative comparison of these cases with the presently discussed crowding effects cannot be made because of the differences either in the nature of the model chains, their lengths, and degree of compactness.…”

Topological Disentanglement Dynamics of Torus Knots on Open Linear Polymers

“…43,44 Indeed, while knots are weakly localized in good solvent, 21,24 in globular polymers they are believed to be delocalized 45,46 and their statistical properties can be interpreted in terms of the essential crossings behaving as point-like particles that compete for a share of the polymer backbone along which they are diffusing. 43,44 Finally, the role of n k has been shown to be crucial also in the nonequilibrium dynamics of granular knotted chains flattened into 2D by gravity. 25 Because of the genuine 3D character of our polymer dynamics, the n k particles have to encode subtle features.…”

“…First, n k is a knot invariant for closed loops and it univocally identifies the topology of simple torus knots . Moreover, n k has been shown to play a key role in the equilibrium thermodynamics of compact ring polymers, when constrained either by slip links or to move across pores. , Indeed, while knots are weakly localized in good solvent, , in globular polymers they are believed to be delocalized , and their statistical properties can be interpreted in terms of the essential crossings behaving as point-like particles that compete for a share of the polymer backbone along which they are diffusing. , Finally, the role of n k has been shown to be crucial also in the nonequilibrium dynamics of granular knotted chains flattened into 2D by gravity …”

Two Adhesive Sites Can Enhance the Knotting Probability of DNA

“…These molecules, in fact, can undergo the same fate of an everyday piece of rope: they can be knotted. Characterized by a wealth of three-dimensional conformations and traditionally described in structural terms, biopolymers have demonstrated to entail a similarly rich variety of topological features, which largely affect their behavior [ 15 , 16 ]. Knotted protein folds, for example, have long been associated only with folding errors [ 17 , 18 ], whether in vivo or in silico .…”

“…The well established biology paradigm sequence → structure → function [ 25 ] is then enriched with topology, and it is of the greatest importance to understand the interplay between these four instances. To shed light on this conundrum, a body of work has been carried out especially by means of numerical simulations, with focus on different aspects of the problem, for example knotted protein folding [ 4 , 5 , 26 – 28 ], DNA [ 10 – 13 , 23 ], knot-specific sequences in model polymers [ 29 – 31 ], and equilibrium properties of knotted chains [ 15 , 16 , 21 , 22 , 32 – 35 ].…”