On the dominance of trivial knots among SAPs on a cubic lattice

Abstract

The knotting probability is defined by the probability with which an N -step self-avoiding polygon (SAP) with a fixed type of knot appears in the configuration space. We evaluate these probabilities for some knot types on a simple cubic lattice. For the trivial knot, we find that the knotting probability decays much slower for the SAP on the cubic lattice than for continuum 1 models of the SAP as a function of N . In particular the characteristic length of the trivial knot that corresponds to a 'half-life' of the knotting probability is estimated to be 2.5 × 10 5 on the cubic lattice.

Topological Information Embodied in Local Juxtaposition Geometry Provides a Statistical Mechanical Basis for Unknotting by Type-2 DNA Topoisomerases

“…For n = 500 unconstrained loops, p K = 0.00174, which agrees with the previously determined value of 0.00151 ± 0.00028. 53 The knot probability for loops with a hooked juxtaposition (I) is higher than that with a half-hooked juxtaposition (IV) (for example, p K | I = 0.0139 and p K | IV = 0.00683 for n = 500), even though the half-hooked juxtaposition generates more knots and samples more knot types for small loops ( Table 1). The most likely explanation for the increased number of knots and increased knot complexity is that the half-hooked juxtaposition places less restriction on conformational freedom.…”

“…These efforts have made important advances. [50][51][52][53][54][55][56] Here, we consider single-loop (one ring polymer) conformations configured on simple cubic lattices. Each conformation consists of n beads, and a set of n bonds joining the beads together to form a closed circuit, which can be knotted or unknotted.…”

“…In this respect, our methodology is distinct, yet complementary to other coarse-grained modeling investigations of polymer entanglement. [50][51][52][53][54][55][56][62][63][64][65][66][67][68][69][70][71] Here, for a range of small loop sizes, Table 1 gives the exact numbers of conformations that resulted from each of these five preformed juxtapositions. Each count in the table corresponds to the number of one-loop conformations with at least one instance of the given juxtaposition.…”

Ranking Knots of Random, Globular Polymer Rings

“…On one hand, it allows us to argue the rate at which the amplitude of the spectrum of different knot topologies grows with increasing ring length. At the same time, the emerging scenario confirms that knots are delocalized and clarifies how topology controls the statistics of the globular state.A cornerstone in topological polymer statistics has been the realization that, for self-avoiding polygons (SAPs), unknotted configurations are entropically disfavored, so that their probability approaches zero exponentially with growing chain length [11,14,15]. However, after this important step, progress in the statistical analysis of knot complexity was hindered by the circumstance that unknotted configurations in swollen, good solvent regimes remain overwhelmingly dominant even for relatively very long chains.…”

Crossing-sign discrimination and knot-reduction for a lattice model of strand passage

“…In order to calculate the numerator ratio in eqn (1), one would need to consider all possible knot types (other than the unknot) as an initial knot type for a -SAP. However, for polygon lengths 5000 and the non-interacting -SAP model studied, the predominant knot type observed at equilibrium is the trefoil (see Table 4 in [30], Table 1 in [31], [32], and Tables 4 and 5 and Conjecture in [33]). Hence the numerator ratio can be approximated by replacing K with 3 1 as indicated above.…”