The second Vassiliev measure of uniform random walks and polygons in confined space

Abstract

Biopolymers, like chromatin, are often confined in small volumes. Confinement has a great effect on polymer conformations, including polymer entanglement. Polymer chains and other filamentous structures can be represented by polygonal curves in three-space. In this manuscript, we examine the topological complexity of polygonal chains in three-space and in confinement as a function of their length. We model polygonal chains by equilateral random walks in three-space and by uniform random walks (URWs) in confinement. For the topological characterization, we use the second Vassiliev measure. This is an integer topological invariant for polygons and a continuous functions over the real numbers, as a function of the chain coordinates for open polygonal chains. For URWs in confined space, we prove that the average value of the Vassiliev measure in the space of configurations increases as O(n ²) with the length of the walks or polygons. We verify this result numerically and our numerical results also show that the mean value of the second Vassiliev measure of equilateral random walks in three-space increases as O(n). These results reveal the rate at which knotting of open curves and not simply entanglement are affected by confinement.