Knotting probabilities after a local strand passage in unknotted self-avoiding polygons

Abstract

We investigate the knotting probability after a local strand passage is performed in an unknotted self-avoiding polygon on the simple cubic lattice. We assume that two polygon segments have already been brought close together for the purpose of performing a strand passage, and model this using Theta-SAPs, polygons that contain the pattern Theta at a fixed location. It is proved that the number of n-edge Theta-SAPs grows exponentially (with n) at the same rate as the total number of n-edge unknotted self-avoiding polygons, and that the same holds for subsets of n-edge Theta-SAPs that yield a specific after-strand-passage knot-type. Thus the probability of a given after-strand-passage knot-type does not grow (or decay) exponentially with n, and we conjecture that instead it approaches a knot-type dependent amplitude ratio lying strictly between 0 and 1. This is supported by critical exponent estimates obtained from a new maximum likelihood method for Theta-SAPs that are generated by a composite (aka multiple) Markov Chain Monte Carlo BFACF algorithm. We also give strong numerical evidence that the after-strand-passage knotting probability depends on the local structure around the strand passage site. Considering both the local structure and the crossing-sign at the strand passage site, we observe that the more "compact" the local structure, the less likely the after-strand-passage polygon is to be knotted. This trend is consistent with results from other strand-passage models, however, we are the first to note the influence of the crossing-sign information. Two measures of "compactness" are used: the size of a smallest polygon that contains the structure and the structure's "opening" angle. The opening angle definition is consistent with one that is measurable from single molecule DNA experiments.