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

Knots can spontaneously form in polymers. How knotting affects polymer behavior depends on polymer knotting probability, p_knot. An intriguing result about p_knot in recent studies is that p_knot exhibits a non-monotonic dependence on the bending stiffness and is maximized at L_p ≈ 8a, where L_p 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 p_knot 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 p_knot. Based on this explanation, we derive an approximate analytic equation that captures the non-monotonic behavior of p_knot. Overall, this work provides new insights into p_knot beyond previous studies, in particular, unifying the effect of the knot size on p_knot and the effect of the polymer length on p_knot. The results can be applied to understand DNA knotting, considering that the effective L_p/a for DNA can be widely varied by the ionic strength.