The linking number and the writhe of uniform random walks and polygons in confined spaces

Abstract

Random walks and polygons are used to model polymers. In this paper we consider the extension of writhe, self-linking number and linking number to open chains. We then study the average writhe, self-linking and linking number of random walks and polygons over the space of configurations as a function of their length. We show that the mean squared linking number, the mean squared writhe and the mean squared self-linking number of oriented uniform random walks or polygons of length n, in a convex confined space, are of the form O(n 2 ). Moreover, for a fixed simple closed curve in a convex confined space, we prove that the mean absolute value of the linking number between this curve and a uniform random walk or polygon of n edges is of the form O( √ n). Our numerical studies confirm those results. They also indicate that the mean absolute linking number between any two oriented uniform random walks or polygons, of n edges each, is of the form O(n). Equilateral random walks and polygons are used to model polymers in θ-conditions. We use numerical simulations to investigate how the self-linking and linking number of equilateral random walks scale with their length.

Models of random knots

“…Consider the linking number L mn of a random two-component link with n and m segments. It is known (Arsuaga et al 2007a;Flapan and Kozai 2016) that its variance is HðnmÞ, and it is conjectured that L mn = ffiffiffiffiffiffi nm p converges in distribution to a Gaussian (Panagiotou et al 2010;Karadayi 2010). Based on our analysis of the Petaluma model (Even-Zohar et al 2016) we tend to doubt this conjecture.…”

The linking number in systems with Periodic Boundary Conditions

“…The Gauss linking number is a traditional measure of the algebraic entanglement of two disjoint oriented closed curves that extends directly to disjoint oriented open chains [43,42,27]. Definition 2.…”

“…For applications, where the chains are simulated by open or closed polygonal chains which satisfy some conditions (for example restrictions on bond angles and length), the Gauss linking number can be used to compute an average absolute linking number over the space of configurations [44,48,42]. This quantity then is of special interest, since it is independent of any particular configuration and can be related to other physical properties of the system [27].…”

“…For pairs of "frozen" open chains, or for a mixed frozen pair, the Gauss linking integral can be applied to calculate an average linking number. For open or mixed pairs, the calculated quantity is a real number that is characteristic of the conformation and changes continuously under continuous deformations of the constituent chains [42]. Thus, the application of the Gauss linking integral to open chains is very clearly not a topological invariant, but a quantity that depends on the specific geometry of the chains.…”

“…Exactly the same considerations apply for the linking number and the self-linking number. Indeed, computer experiments indicate that the linking number and the writhe are effective indirect measures of whatever one might call "entanglement", especially in systems of "random" filaments [41,27,43,44,45,46,47,48,42,49,50,51,52,53,54] and it has been shown that they can provide information relevant to the tube model [52,53].…”