Revisiting the Non-monotonic Dependence of Polymer Knotting Probability on the Bending Stiffness

Abstract

Knots can spontaneously form in polymers. How knotting affects polymer behavior depends on polymer knotting probability, pknot. An intriguing result about pknot in recent studies is that pknot exhibits a non-monotonic dependence on the bending stiffness and is maximized at Lp ≈ 8a, where Lp is the persistence length and a is the hardcore diameter of the monomer. In this work, we propose a new explanation for the non-monotonic behavior of pknot based on the fact that polymer knots are typically localized. We find that the non-monotonic behavior results from the competition of a special entropic effect arising from the variation in the sizes of localized knots and an effect arising from the variation in the free-energy cost of forming a localized knot on a fragment of a polymer. The first effect refers to the situation that shrinking the knot size for a polymer with a fixed length essentially increases the number of “slots” for knot formation and enhances pknot. Based on this explanation, we derive an approximate analytic equation that captures the non-monotonic behavior of pknot. Overall, this work provides new insights into pknot beyond previous studies, in particular, unifying the effect of the knot size on pknot and the effect of the polymer length on pknot. The results can be applied to understand DNA knotting, considering that the effective Lp/a for DNA can be widely varied by the ionic strength.