The abundance of unknots in various models of polymer loops

Abstract

Abstract. A veritable zoo of different knots is seen in the ensemble of looped polymer chains, whether created computationally or observed in vitro. At short loop lengths, the spectrum of knots is dominated by the trivial knot (unknot). The fractional abundance of this topological state in the ensemble of all conformations of the loop of N segments follows a decaying exponential form, ∼ exp (−N/N 0 ), where N 0 marks the crossover from a mostly unknotted (ie topologically simple) to a mostly knotted (ie topologically complex) ensemble. In the present work we use computational simulation to look closer into the variation of N 0 for a variety of polymer models. Among models examined, N 0 is smallest (about 240) for the model with all segments of the same length, it is somewhat larger (305) for Gaussian distributed segments, and can be very large (up to many thousands) when the segment length distribution has a fat power law tail.

Topology of Light’s Darkness

“…where T % 30 for threading any by vortex line and T % 185 for threading by another loop. This apparent exponential decay of topological isolation is similar to that found in the literature for other systems of random tangles, such as polymers [18,21]. It is numerically observed, across a range of models of random polymer loops, that the probability that a loop of length L is unknotted is expðÀL=APÞ where P is the persistence length of the polymer and A is a dimensionless number that depends on the details of the specific model.…”

Symmetry-breaking in cumulative measures of shapes of polymer models

“…[1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] While one can generate millions of independent configurations and visualize them individually, difficulty arises when one wants to study the aggregation of many configurations to obtain cumulative measures of polymer shape. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] While one can generate millions of independent configurations and visualize them individually, difficulty arises when one wants to study the aggregation of many configurations to obtain cumulative measures of polymer shape.…”

A Monte Carlo Study of Knots in Long Double-Stranded DNA Chains

“…Remarkably, more than ≈68% contain complex knots with more than three crossings or composite knots. The transition from a mostly unknotted to a mostly knotted ensemble of DNA chains is indicated by the base pair count B 0 at which the probability of obtaining an unknotted conformation is 1/ e ≈0.37 [ 25 ]. B 0 ≈250,000bp(N≈19,000 beads) also characterizes the regime where knots with higher crossing number (≥4) become more abundant than trefoil knots.…”