Universality of random knotting

Abstract

Knotting probability ͓ P K (N)͔ is defined by the probability of an N-noded random polygon being topologically equivalent to a given knot K. For several nontrivial knots we numerically evaluate the knotting probabilities for Gaussian and rod-bead models. We find that they are well approximated by the following formula: P K (N)ϭC(K)͓Ñ /N(K)͔ m(K) exp͓ϪÑ/N(K)͔ where Ñ ϭNϪN ini (K), and that the fitting parameters C(K), N(K), and N ini (K) are model dependent, while m(K) is not. We suggest that given a knot K, the exponent m(K) should be universal: it is independent of models of random polygon and is determined only by the knot K. ͓S1063-651X͑97͒09004-1͔ PACS number͑s͒: 05.40.ϩj, 61.41.ϩe, 36.20.Ey Recently knotted ring polymers such as knotted DNA molecules are synthesized in various experiments in chemistry and biology ͓1-3͔. In particular, the formation of knotted species on random ring closure of DNA was observed and their fractions were measured ͓4,5͔. In statistical mechanics, the topological constraint that a ring polymer does not change its topology under any thermal fluctuation leads to a great reduction in the available volume of the configuration space ͓6͔. The topological constraint, or the selfentanglement effect, is derived from the fact that any bond between neighboring monomers in the ring polymer is not disconnected when the bonding energy is very large.In the 1960s Delbrück formulated a fundamental question about the self-entanglement of a ring polymer: What fraction of permissible configurations of a chain of given length will contain a knot ͓7,8͔? The fraction of knotted ring polymers has been studied from the following three approaches: numerical experiments using certain knot invariants ͓9-18͔, mathematical discussion of the self-avoiding polygon ͓19-21͔, and biological experiments using DNAs ͓4,5͔.Let us assume that a model of an N-noded random polygon describes a ring polymer with N bonds. Given a knot K, we define knotting probability P K (N) to the model by the fraction of those configurations of the random polygon that have the same knot type K. The main questions in this paper are how the knotting probability P K (N) behaves as a function of step number N for each knot K, and how it depends on models of a random polygon.For the trivial knot (Kϭ0) we call the knotting probability P 0 (N) the unknotting probability. It has been evaluated for several different models of random polygons with different lengths N less than about 2000 ͓9-14͔. The exponential decay of P 0 (N) with respect to N has been discussed for the molecular dynamical model of ring polymers by Michels and Wiegel, and for the rod-bead model by Koniaris and Muthukumar ͓11,14͔.For nontrivial knots, however, the knotting probabilities have been evaluated only for short polygons with NϽ200, where the graphs of P K (N) versus N can be approximated by linear functions of N ͓9͔. When we calculate knot invariants for polygons with large N, there are two technical difficulties: memory-size and computation-time problems ͓22͔.For pol...

The abundance of unknots in various models of polymer loops

“…It was also noticed [17] that the same exponential law, with the same decay parameter N 0 , also describes the large N asymptotical tail of the abundance of any other particular knot -although for complex knots exponential decay starts only at sufficiently large N (as soon as the given knot can be identified as an underknot [24]). An alternative view of formula (1), useful in the context of thermodynamics, implies that the removal of all knots from the loop is associated with thermodynamically additive (linear in N ) entropy loss of 1/N 0 per segment; in other words, at the temperature T , untying all knots would require mechanical work of at least k B T /N 0 per segment.…”

“…In other words, using professional parlance of the field, what is the probability that random loop is a trivial knot (unknot) Most of what we know about these "probabilistic topology" questions is learned from computer simulations. In particular, it has been observed by many authors over the last 3 decades [15,16,17,18,19] that the trivial knot probability depends on the length of the loop, decaying exponentially with the number of segments in the loop, N :…”

“…First, the constant's value was invariably found to be quite large, around 300 for all examined models of "thin" loops with no excluded volume, or no self-avoidance [15,16,17,23,24]. Second, it is known that knots are dramatically suppressed for "thick" self-avoiding polymers, which means that N 0 rapidly increases with the radius of self-avoidance [17,19]. The latter issue is also closely connected to the probabilities of knots in lattice models, where the non-zero effective self-avoidance parameter is automatically set by the lattice geometry.…”

Winding angle distribution for planar random walk, polymer ring entangled with an obstacle, and all that: Spitzer–Edwards–Prager–Frisch model revisited

“…In fact, as we see from Eqs. (27,28) all these expansions are ones over the inverse powers of ln(a/b), and we keep the leading terms only. Incorporating, as explained above, the factor 2/ ln(ta 2 /b 2 ) ≃ 1/ ln(a/b) to establish the normalization ∞ −∞ W (θ)dθ = 1, we finally get…”

“…• The typical annealed topology question is that about ring closure experiment and knot probabilities [25,26,27,28,29]: having a linear polymer with "sticky" ends, what is the probability to obtain a certain type of a knot upon first meeting of the two ends [19,20]? A similar question for the winding model is this: what is the probability that a random walk on the plane links number n (or winding angle 2πn) with an obstacle?…”