Multiscale entanglement in ring polymers under spherical confinement

📝 Abstract

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.

💡 Analysis

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.

📄 Content

Multiscale entanglement in ring polymers under spherical confinement Luca Tubiana1, Enzo Orlandini2, Cristian Micheletti1 1 SISSA - Via Bonomea 265 - I-34136, Trieste - Italy 2 Dipartimento di Fisica “G. Galilei” and Sezione INFN - Via Marzolo 8 - I-35100 Padova, Italy 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. If we tie a knot in a piece of rope and pull the latter at its extremities, the knotted part will become readily distinguishable from the rest of the rope because it localizes. Indeed, for a very long rope only a neglible fraction of its contour length will be required to accommodate the knot [1]. In this intuitive example of knot localization the topological (global) entanglement embodied by the knot does not interfere with the geometrical (local) entanglement of the rest of the rope. This is hardly the case for polymers that circularize in equilibrium. In such rings, knots are abundant [2] and the interplay between topological and geometrical entanglement impacts significantly the molecules’ dynamical, mechanical and metric properties [3–5]. Understanding how and to what extent global and local entangle- ment are related is a major open issue in polymer physics [6] with ramifications in key biological contexts[7], especially those related to genome organization in eukaryotes, bacteria and viruses [8–12]. A first breakthrough in the problem could be made by establishing what fraction of the polymer ring is occupied by the knot(s) and whether this measure is sufficiently robust or depends instead on the geometrical entanglement degree. As a prototypical context to examine this problem we consider semi-flexible, self-avoiding rings of cylinders with the simplest knotted topology, a 31 (trefoil) knot, and subject to isotropic spatial confinement. In such a system, by varying the size of the confining region, the degree of geometrical complexity can be changed and related to the equilibrium size of the knotted ring portion. For definiteness, the rings properties are set to match dsDNA. Specifically, the cylinder diameter and long axis are set equal to d = 2.5nm and b = 10nm, respectively. The latter quantity is ten times smaller than the DNA Kuhn length (equal to twice the persistence length, lp = 50nm) thus ensuring a fine discretization of the model DNA. The system energy includes steric hindrance of non consecutive cylinders plus a bending potential: Eb = −KbT lp b N X i=1 ⃗ti · ⃗ti+1 (1) where ⃗ti is the orientation of the axis of the ith cylinder, ⃗tN+1 ≡⃗t1 and the temperature T is set to 300K. We considered rings of N = 50, …250 cylinders, corresponding to countour lengths, Lc = Nb ranging from 500nm to 2.5 µm. This range allows for probing changes in knot localization going from semiflexible to fully-flexible rings [13] as well as examing the effect of the interplay between Lc, lp and the radius of the confining sphere, Rc. For simplicity of notation in the following all lengthscales are expressed in units of b. Because compact ring configurations are entropically disfavoured with respect to unconstrained ones, simple stochas- tic sampling schemes cannot be effectively used to generate spatially confined rings [6]. An analogous entropic attrition works against having a sizeable population of knots of a given type, trefoils in our case, at all levels of confinement [6]. To overcome these two difficulties we used a biased Multiple-Markov-chain sampling scheme [12] with 24 Markovian replicas. For each replica, ring configurations are evolved using crankshaft and hedgehog Monte Carlo moves [14]. The moves preserve the length of the rings, but not the topology, consistently with ergodicity requirements [6, 15]. In fact, even when the configurations before and after the move are self-avoiding, the move itself may involve self-crossings of the chain. A newly generated ring, Γ, is accepted according to the standard Metropolis criterion with canonical weight, exp [−U(Γ)/KBT], where the potential energy, U is suitably chosen to enhance the populations of compacts trefoil rings. Specifically, if Γ is non self-avoiding or has a too complicated topology (8 or more minimal crossings according to the Alexander polynomial [6]) then U is set equal to infinity, otherwise U(Γ) = Eb(Γ) + p Rc(Γ). In the latter express

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