Bending rigidity, supercoiling and knotting of ring polymers: models and simulations

The first part of the thesis was focussed on the interplay between knotting propensity and bending rigidity of equilibrated rings polymers. We found a surprising result: the equilibrium incidence of knots has a strongly non- monotonic dependence on bending, with a maximum at intermediate flexural rigidities. We next provided a quantitative framework, based on the balance of bending energy and configurational entropy, that allowed for rationalizing this counter-intuitive effect. We next extended the investigation to rings of much larger number of beads, via an heuristic model mapping between our semiflexible rings of beads and self-avoiding rings of cylinder. By the mapping, we not only confirmed the unimodal knotting profile for chains of 1,000 beads, but further found that chains of > 20,000 beads are expected to feature a bi-modal profile. We believe it would be most interesting to direct future efforts to confirm this transition from uni- to bi-modality using advanced sampling techniques for very long polymer rings. The second part of the thesis focused on the interplay of DNA knots and su- percoiling which are typically simultaneously present in vivo. We first studied this interplay by using oxDNA, an accurate mesoscopic DNA model and using it to study ings of thousands of base pairs tied in complex knots and with or without negative supercoiling (as appropriate for bacterial plasmids). By monitoring the dynamics of the DNA rings we found that the simultaneous presence of knots and supercoiling, and only their simultaneous presence, leads to a dramatic slowing down of the system reconfiguration dynamics. In particular, the essential tangles in the knotted region acquire a very long-lived character that, we speculate, could aid their recognition and simplification by topoisomerase. Finally, motivated by the recent experimental breakthrough that detected knots in eukaryotic DNA, we investigated the relationship between the compactness, writhe and knotting probability. The model was tuned to capture some of the salient properties of yeast minichromosomes, which were shown experimentally to become transiently highly knotted during transcription.