Total curvature and total torsion of knotted random polygons in confinement

Abstract

Knots in nature are typically confined spatially. The confinement affects the possible configurations, which in turn affects the spectrum of possible knot types as well as the geometry of the configurations within each knot type. The goal of this paper is to determine how confinement, length, and knotting affect the total curvature and total torsion of random polygons. Previously published papers have investigated these effects in the unconstrained case. In particular, we analyze how the total curvature and total torsion are affected by (1) varying the length of polygons within a fixed confinement radius and (2) varying the confinement radius of polygons with a fixed length. We also compare the total curvature and total torsion of groups of knots with similar complexity (measured as crossing number). While some of our results fall in line with what has been observed in the studies of the unconfined random polygons, a few surprising results emerge from our study, showing some properties that are unique due to the effect of knotting in confinement.

A statistical approach to knot confinement via persistent homology

“…Interestingly, unlike in the case of the ACN, we do not find any significant correlation between either the curvature or torsion of the knots and the integral I. This should be compared with other more ad hoc approaches, such as [34][35][36].…”

Knotting spectrum of polygonal knots in extreme confinement

“…In this section, we argue that the cylindrical knot model is a good approximation of knotting under spherical confinement with a radius significantly smaller than one. We have studied knotting of equilateral random polygons under spherical confinement [11][12][13][14][15][16][17][18] concentrating on how the confinement radius R and/or the polygon length L (i.e. the number of edges) influence knot complexity.…”

“…In [11][12][13][14][15][16][17][18], the authors (most of whom are authors on this paper) explored a direct method for generating knots in spherical confinement at levels of compaction that would be impractical using Monte Carlo methods. The idea was to explore how the confinement condition alone affects the knotting spectrum and the structure of the configurations as a function of length and confinement radius.…”

Knotted proteins: Tie etiquette in structural biology

“…More recently, statistical analysis of large samples of random polygons bounded within spheres of variable radius was employed (65)(66)(67) to explore in this context the effect of confinement on knotting probability and knot type, as well as on the relation between knot type and average geometrical properties, such as total curvature and total torsion. Progress towards more realistic physical models was obtained with off-lattice numerical simulations of a simplified model of polyethylene (68), represented by chains of interacting monomers connected by springs, parametrized so as to reproduce experimental data.…”