Equilibrium and kinetic properties of knotted ring polymers: a computational approach

We provide hereafter a summary of the Thesis organization. Chapter 1 contains a short introduction to the mathematical theory of knots. Starting from the mathematical definition of knotting, we introduce the fundamental concepts and knot properties used throughout this Thesis. In chapter 2 we tackle the problem of measuring the degree of localization of a knot. This is in general a very challenging task, involving the assignment of a topological state to open arcs of the ring. To assign a topological state to an open arc, one must first close it into a ring whose topological state can be assessed using the tools introduced in chapter 1. Consequently, the resulting topological state may depend on the specific closure scheme that is followed. To reduce this ambiguity we introduce a novel closure scheme, the minimally-interfering closure. We prove the robustness of the minimally-interfering closure by comparing its results against several standard closure schemes. We further show that the identified knotted portion depends also on the search algorithm adopted to find it. The knot search algorithms adopted in literature can be divided in two general categories: bottom-up searches and top-down searches. We show that bottom-up and knot-down searches give in general different results for the length of a knot, the difference increasing with increasing length of the polymer rings. We suggest that this systematic difference can explain the discrepancies between previous numerical results on the scaling behaviour of the knot length with increasing length of polymer rings in good solvent. In chapter 3 we investigate the mutual entanglement between multiple prime knots tied on the same ring. Knots like these, which can be decomposed into simpler ones, are called composite knots and dominate the knot spectrum of sufficiently long polymers [131]. Since prime knots are expected to localize to point-like decorations for asymptotically large chain lengths, it is expected that composite knots should factorize into separate prime components [101, 82, 43, 11]. Therefore the asymptotic properties of composite knots should merely depend on the number of prime components (factor knots) by which they are formed [101, 82, 43, 11] and the properties of the single prime components should be largely independent from the presence of other knots on the ring. We show that this factorization into separate prime components is only partial for composite knots which are dominant in an equilibrium population of Freely Jointed Rings. As a consequence the properties of those prime knots which are found as separate along the chain depend on the number of knots tied on it. We further show that these results can be explained using a transparent one-dimensional model in which prime knots are substituted with paraknots. Chapters 4, 5 and 6 are dedicated to investigate the interplay between topological entanglement and geometrical entanglement produced either by surrounding rings in a dense solution or spherical confinement. In chapter 4 we investigate the equilibrium and kinetic properties of solutions of model ring polymers, modulating the interplay of inter- and intra-chain entanglement by varying both solution density (from infinite dilution up to 40% volume occupancy) and ring topology (by considering unknotted and trefoil-knotted chains). The equilibrium metric properties of rings with either topology are found to be only weakly affected by the increase of solution density. Even at the highest density, the average ring size, shape anisotropy and length of the knotted region differ at most by 40% from those of isolated rings. Conversely, kinetics are strongly affected by the degree of inter-chain entanglement: for both unknots and trefoils the characteristic times of ring size relaxation, reorientation and diffusion change by one order of magnitude across the considered range of concentrations. Yet, significant topology-dependent differences in kinetics are observed only for very dilute solutions (much below the ring overlap threshold). For knotted rings, the slowest kinetic process is found to correspond to the diffusion of the knotted region along the ring backbone. In chapter 5 we study the interplay of geometrical and topological entanglement in semiflexible knotted polymer rings under spherical confinement. We first characterize how the top-down knot length lk depends on the ring contour length, Lc and the radius of the confining sphere, Rc. In the no- and strong-confinement cases we observe weak knot localization and complete knot delocalization, respectively. We show that the complex interplay of lk, Lc and Rc that seamlessly bridges these two limits can be encompassed by a simple scaling argument based on deflection theory. We then move to study the behaviour of the bottom-up knot length lsk under the same conditions and observe that it follows a qualitatively different behaviour from lk, decreasing upon increasing confinement. The behaviour of lsk is rationalized using the same argument based on deflection theory. The qualitative difference between the two knot lengths highlights a multiscale character of the entanglement emerging upon increasing confinement. Finally, in chapter 6 we adopt a complementary approach, using topological analysis (the properties of the knot spectrum) to infer the physical properties of packaged bacteriophage genome. With their m long dsDNA genome packaged inside capsids whose diameter are in the 50 80 nm range, bacteriophages bring the highest level of compactification and arguably the simplest example of genome organization in living organisms [31, 40]. Cryo-em studies showed that DNA in bacteriophages epsilon-15 and phi-29 is neatly ordered in concentric shells close to the capsid wall, while an increasing level of disorder was measured when moving away from the capsid internal surface. On the other hand the detected spectrum of knots formed by DNA that is circularised inside the P4 viral capsid showed that DNA tends to be knotted with high probability, with a knot spectrum characterized by complex knots and biased towards torus knots and against achiral ones. Existing coarse-grain DNA models, while being capable of reproducing the salient physical aspects of free, unconstrained DNA, are not able to reproduce the experimentally observed features of packaged viral DNA. We show, using stochastic simulation techniques, that both the shell ordering and the knot spectrum can be reproduced quantitatively if one accounts for the preference of contacting DNA strands to juxtapose at a small twist angle, as in cholesteric liquid crystals.