Probing the entanglement and locating knots in ring polymers: a comparative study of different arc closure schemes

The interplay between the topological and geometrical properties of a polymer ring can be clarified by establishing the entanglement trapped in any portion (arc) of the ring. The task requires to close the open arcs into a ring, and the resulting topological state may depend on the specific closure scheme that is followed. To understand the impact of this ambiguity in contexts of practical interest, such as knot localization in a ring with non trivial topology, we apply various closure schemes to model ring polymers. The rings have the same length and topological state (a trefoil knot) but have different degree of compactness. The comparison suggests that a novel method, termed the minimally-interfering closure, can be profitably used to characterize the arc entanglement in a robust and computationally-efficient way. This closure method is finally applied to the knot localization problem which is tackled using two different localization schemes based on top-down or bottom-up searches.Comment: 9 pages, 7 figures. Submitted to Progress of Theoretical Physic

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

Multiscale entanglement in ring polymers under spherical confinement

The interplay of geometrical and topological entanglement in semiflexible knotted polymer rings confined inside a spherical cavity is investigated using advanced numerical methods. By using stringent and robust algorithms for locating knots, we characterize how the 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. The same argument is used to rationalize the multiscale character of the entanglement that emerges with increasing confinement.Comment: 9 pages 9 figure

KymoKnot: A web server and software package to identify and locate knots in trajectories of linear or circular polymers

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Viral RNA as a branched polymer

Myriad viruses use positive-strand RNA molecules as their genomes. Far from being only a repository of genetic material, viral RNA performs numerous other functions mediated by its physical structure and chemical properties. In this chapter, we focus on its structure and discuss how long RNA molecules can be treated as branched polymers through planar graphs. We describe the major results that can be obtained by this approach, in particular the observation that viral RNA genomes have a characteristic compactness that sets them aside from similar random RNAs. We also discuss how different parameters used in the current RNA folding software influence the resulting structures and how they can be related to experimentally observable quantities. Finally, we show how the connection to branched polymers can be extended to take advantage of known results from polymer physics and can be further moulded to include additional interactions, such as excluded volume or electrostatics.Comment: 24 pages, 9 figure

Dynamic and Facilitated Binding of Topoisomerase Accelerates Topological Relaxation

: How type 2 Topoisomerase (TopoII) proteins relax and simplify the topology of DNA molecules is one of the most intriguing open questions in genome and DNA biophysics. Most of the existing models neglect the dynamics of TopoII which is expected of proteins searching their targets via facilitated diffusion. Here, we show that dynamic binding of TopoII speeds up the topological relaxation of knotted substrates by enhancing the search of the knotted arc. Intriguingly, this in turn implies that the timescale of topological relaxation is virtually independent of the substrate length. We then discover that considering binding biases due to facilitated diffusion on looped substrates steers the sampling of the topological space closer to the boundaries between different topoisomers yielding an optimally fast topological relaxation. We discuss our findings in the context of topological simplification in vitro and in vivo

Dynamic and Facilitated Binding of Topoisomerase Accelerates Topological Relaxation

How type 2 Topoisomerase (TopoII) proteins relax and simplify the topology of DNA molecules is one of the most intriguing open questions in biophysics. Most of the existing models neglect the dynamics of TopoII which is characteristics for proteins searching their targets via facilitated diffusion. Here, we show that dynamic binding of TopoII speeds up the topological relaxation of knotted substrates by enhancing the search of the knotted arc. Intriguingly, this in turn implies that the timescale of topological relaxation is virtually independent of the substrate length. We then discover that considering binding biases due to facilitated diffusion on looped substrates steers the sampling of the topological space closer to the boundaries between different topoisomers yielding an optimally fast topological relaxation. We discuss our findings in the context of topological simplification in vitro and in vivo

Comparing equilibration schemes of high-molecular-weight polymer melts with topological indicators.

Recent theoretical studies have demonstrated that the behaviour of molecular knots is a sensitive indicator of polymer structure. Here, we use knots to verify the ability of two state-of-the-art algorithms-configuration assembly and hierarchical backmapping-to equilibrate high-molecular-weight (MW) polymer melts. Specifically, we consider melts with MWs equivalent to several tens of entanglement lengths and various chain flexibilities, generated with both strategies. We compare their unknotting probability, unknotting length, knot spectra, and knot length distributions. The excellent agreement between the two independent methods with respect to knotting properties provides an additional strong validation of their ability to equilibrate dense high-MW polymeric liquids. By demonstrating this consistency of knotting behaviour, our study opens the way for studying topological properties of polymer melts beyond time and length scales accessible to brute-force molecular dynamics simulations