Contents lists available at ScienceDirect Reactive and Functional Polymers journal homepage: www.elsevier.com/locate/react Bimodality in the knotting probability of semifexible rings suggested by mapping with self-avoiding polygons Erica Uehara a , Lucia Coronel b , Cristian Micheletti b, ⁎ , Tetsuo Deguchi a a Department of Physics, Faculty of Core Research, Ochanomizu University, 2-1-1 Ohtsuka, Bunkyo-ku, Tokyo 112-8610, Japan b SISSA, Via Bonomea 265, I-34014 Trieste, Italy ARTICLEINFO Keywords: Knots Self-avoiding polygons Semifexible rings Knotting probability Model mapping ABSTRACT We use a simple physical mapping to adapt the known asymptotic expressions for the knotting probabilities of self-avoiding polygons to the case of semifexible rings of beads. We thus obtain analytical expressions that approximate the abundance of the simplest knots as a function of the length and bending rigidity of the rings. We validate the predictions against previously published data from stochastic simulations of rings of beads showing that they reproduce the intriguing non-monotonic dependence of knotting probability on bending rigidity. The mapping thus provides a useful theoretical tool not only for a physically-transparent interpretation of previous results, but especially to predict the knotting probabilities for previously unexplored combinations of chain lengths and bending rigidities. In particular, our mapping suggests that for rings longer than 20,000 beads, the rigidity-dependent knotting probability profle switches from unimodal to bimodal. 1. Introduction Polymers in canonical equilibrium are inevitably prone to becoming entangled and form knots [1–5]. The specifc type and abundance of these knots they can form, the so-called knot spectrum, depend on ex- ternally imposed conditions, such as molecular crowding [6] or spatial confnement [7–10], as well as on intrinsic physical properties of the polymers themselves [11–13]. These primarily include the chain con- tour length, the persistence length, the length and thickness of the monomeric units and the details of their interactions [14–18]. Intrinsic features, such as the fexural rigidity, can vary signifcantly across naturally-occurring polymers [19]. For example polysaccharides, proteins, and single stranded nucleic-acids in high salt concentrations have a persistence length comparable to the size of their constitutive monomers [20–22], about 1 nm, while double-stranded DNA flaments have a persistence length of about 50 nm [23], which largely exceeds their thickness (2.5 nm) and the spacing of the base-pairs (0.34 nm). Diferent models have been introduced to capture the salient phy- sical properties of these diverse systems with the fewest possible de- grees of freedom. For instance, chains of beads with a bending rigidity are the model of choice for intrinsically-fexible polymers, while freely- jointed chains of cylinders have been largely used to model DNA. As a matter of fact, rings of cylinders, also termed self-avoiding polygons, have been frst introduced precisely for modelling the spontaneous knotting probability of DNA rings of various lengths and comparing it with experiments [11,24]. These studies have then moti- vated various systematic surveys of how the topological spectrum de- pends on the number of cylinders and on their diameter [25–27], for which there are now accurate parametric expressions that have proved useful for understanding various knotting phenomena in polymer chains. Analogous studies have been carried out for rings or chains of beads as a function of contour length and, more limitedly, of bending rigidity too [18,28,29]. The latter dependence presents particularly intriguing aspects. In fact, the recent systematic study of ref. [18] showed that at fxed contour length the abundance of knots varies non-monotonically with the bending rigidity of the rings. Here we use a simple physical mapping to relate the knotting probabilities of these two reference models, self-avoiding polygons and semifexible rings of beads. As we discuss, the mapping provides a novel perspective on how and why the knotting probability of semi-fexible rings varies non-monotonically with bending rigidity. More im- portantly, it predicts that this non-monotonicity is more complex than previously thought, and changes from unimodal to multimodal for sufciently long rings. https://doi.org/10.1016/j.reactfunctpolym.2018.11.008 Received 28 April 2018; Received in revised form 29 September 2018; Accepted 12 November 2018 ⁎ Corresponding author. E-mail address: michelet@sissa.it (C. Micheletti). Reactive and Functional Polymers 134 (2019) 141–149 Available online 17 November 2018 1381-5148/ © 2018 Elsevier B.V. All rights reserved. T