Statistical topology of closed curves: Some applications in polymer physics

Abstract

Topological entanglement in polymers and biopolymers is a topic that involves different fields of science such as chemistry, biology, physics, and mathematics. One of the main issues in this topic is to understand how the entanglement complexity can depend on factors such as the degree of polymerization, the quality of the solvent, and the temperature or the degree of confinement of the macromolecule. In this respect a statistical approach to the problem is natural and in the last few years there has been a lot of work on the study of the entanglement complexity of polymers within the statistical mechanics framework. A review on this topic is given here stressing the main results obtained and describing the tools most used with this approach

Knotted Globular Ring Polymers: How Topology Affects Statistics and Thermodynamics

“…8(b). Here again we have seen that the fluctuations of P(l 1 /N) are broad, yet the average loop lengths are determined on the basis of the minimal crossing numbers of the competing knots, as 3 10 N and 7 10 N. A further confirmation of the fact that n c alone determines the loop statistics, is given by the competition 3 1 + 3 1 vs 6 1 . In this case [ Fig.…”

“…In the first panel of Fig. 4 we report, for five different temperatures below T Θ , the behavior with 1/N of the topological free energy correction applying to the knot 3 1 .…”

“…Given the reported increasingly broad distributions of loop lengths, it is not straightforward to predict the scaling of their averages l 1 and l 2 (a) (b) Figure 6: Sketch of a polymer ring with a slip-link separating two loops, the first holding a 3 1 knot and the second a 4 1 . Each knot is confined in its loop because the holes in the slip-link are narrow enough to prevent the passage of a blob containing more than one monomer.…”

Models of random knots

“…There is plenty of literature on the modeling of knots in thread-like molecules. Find some expositions and surveys in Cozzarelli (1986), Vologodskii (1992), Sumners (1992), Sumners (1995), Grosberg et al (1997), Bates and Maxwell (2005), Orlandini and Whittington (2007), McLeish (2008), Fenlon (2008, Buck (2009), Sumners et al (2009), Micheletti et al (2011 and Lim and Jackson (2015).…”

Probability Theory of Random Polygons from the Quaternionic Viewpoint

“…Orlandini et al [18] provide an overview of the standard methods, many of which depend on establishing Markov chains which are ergodic on equilateral closed polygon space and then iterating the chain until the resulting distribution on polygon space is close to uniform. Grosberg and Moore [17] discuss some potential difficulties with these iterative methods, and give a method for explicitly sampling random equilateral closed polygons for small numbers of edges by computing the conditional probability distribution of the n + 1-st edge based on the choice of the first n edges (see [7], [6], and [8] for conditional probability methods applied to the even more difficult problem of sampling equilateral closed polygons confined to a sphere, and [20] for an alternate approach to generating ensembles of equilateral closed polygons).…”