Linking of Ring Polymers in Slit-Like Confinement

Spatial Organization of Slit-Confined Melts of Ring Polymers with Nonconserved Topology: A Lattice Monte Carlo Study

We present Monte Carlo computer simulations for melts of semiflexible randomly knotted and randomly concatenated ring polymers on the fcc lattice and in slit confinement. Through systematic variation of the slit width at fixed melt density, we explore the influence of confinement on single-chain conformations and interchain interactions. We demonstrate that confinement makes chains globally larger and more elongated while enhancing both contacts and knottedness propensities. As for multichain properties, we show that ring-ring contacts decrease with the confinement, yet neighboring rings overlap more as confinement grows. These aspects are accompanied by a marked decrease in the links formed between pairs of neighboring rings. In connection with the quantitative relation between links and entanglements in polymer melts recently established by us [Ubertini M. A.; Rosa A.Macromolecules2023, 56, 3354-3362], we propose that confinement can be used to set polymer networks that act softer under mechanical stress and suggest a viable experimental setup to validate our results.

Percolation and dissolution of Borromean networks

Inspired by experiments on topologically linked DNA networks, we consider the connectivity of Borromean networks, in which no two rings share a pairwise-link, but groups of three rings form inseparable triplets. Specifically, we focus on square lattices at which each node is embedded a loop which forms a Borromean link with pairs of its nearest neighbors. By mapping the Borromean link network onto a lattice representation, we investigate the percolation threshold of these networks (the fraction of occupied nodes required for a giant component), as well as the dissolution properties: the spectrum of topological links that would be released if the network were dissolved to varying degrees. We find that the percolation threshold of the Borromean square lattice occurs when approximately 60.75% of nodes are occupied, slightly higher than the 59.27% typical of a square lattice. Compared to the dissolution of Hopf-linked networks, a dissolved Borromean network will yield more isolated loops, and fewer isolated triplets per single loop. Our simulation results may be used to predict experiments from Borromean structures produced by synthetic chemistry.

Flatness and intrinsic curvature of linked-ring membranes

Recent experiments have elucidated the physical properties of kinetoplasts, which are chain-mail-like structures found in the mitochondria of trypanosome parasites formed from catenated DNA rings. Inspired by these studies, we use Monte Carlo simulations to examine the behavior of two-dimensional networks ("membranes") of linked rings. For simplicity, we consider only identical rings that are circular and rigid and that form networks with a regular linking structure. We find that the scaling of the eigenvalues of the shape tensor with membrane size are consistent with the behavior of the flat phase observed in self-avoiding covalent membranes. Increasing ring thickness tends to swell the membrane. Remarkably, unlike covalent membranes, the linked-ring membranes tend to form concave structures with an intrinsic curvature of entropic origin associated with local excluded-volume interactions. The degree of concavity increases with increasing ring thickness and is also affected by the type of linking network. The relevance of the properties of linked-ring model membranes to those observed in kinetoplasts is discussed.

Topological and physical links in soft matter systems

Linking, or multicomponent topological entanglement, is ubiquitous in soft matter systems, from mixtures of polymers and DNA filaments packedin vivoto interlocked line defects in liquid crystals and intertwined synthetic molecules. Yet, it is only relatively recently that theoretical and experimental advancements have made it possible to probe such entanglements and elucidate their impact on the physical properties of the systems. Here, we review the state-of-the-art of this rapidly expanding subject and organize it as follows. First, we present the main concepts and notions, from topological linking to physical linking and then consider the salient manifestations of molecular linking, from synthetic to biological ones. We next cover the main physical models addressing mutual entanglements in mixtures of polymers, both linear and circular. Finally, we consider liquid crystals, fluids and other non-filamentous systems where topological or physical entanglements are observed in defect or flux lines. We conclude with a perspective on open challenges.

Sliding dynamics of multi-rings on a semiflexible polymer in poly[n]catenanes

The sliding dynamics of one- or multi-ring structures along a semiflexible cyclic polymer in radial poly[n]catenanes is investigated using molecular dynamics simulations. The fixed and fluctuating (non-fixed) semiflexible central cyclic polymers are considered, respectively. With increasing bending energy of the central cyclic polymer, for the fixed case, the diffusion coefficient increases monotonically due to the reduction of the tortuous sliding path, while for the fluctuating case, the diffusion coefficient decreases. This indicates that the contribution of the polymer fluctuation is suppressed by a further increase in the stiffness of the central cyclic chain. Compared with the one ring case, the mean-square displacement of the multiple rings exhibits a unique sub-diffusive behavior at intermediate time scales due to the repulsion between two neighboring rings. In addition, for the multi-ring system, the whole set of rings exhibit relatively slower diffusion, but faster local dynamics of threading rings and rotational diffusion of the central cyclic polymer arise. These results may help us to understand the diffusion motion of rings in radial poly[n]catenanes from a fundamental point of view and control the sliding dynamics in molecular designs.

Effects of topological constraints on linked ring polymers in solvents of varying quality

We investigate the effects of topological constraints in catenanes composed of interlinked ring polymers on their size in a good solvent as well as on the location of their θ-point when the solvent quality is worsened. We mainly focus on poly[n]catenanes consisting of n ring polymers each of length m interlocked in a linear fashion. Using molecular dynamics simulations, we study the scaling of the poly[n]catenane's radius of gyration in a good solvent, assuming in general that Rg∼mμnν and we find that μ = 0.65 ± 0.02 and ν = 0.60 ± 0.01 for the range of n and m considered. These findings are further rationalized with the help of a mean-field Flory-like theory yielding the values of μ = 16/25 and ν = 3/5, consistent with the numerical results. We show that individual rings within catenanes feature a surplus swelling due to the presence of NL topological links. Furthermore, we consider poly[n]catenanes in solvents of varying quality and we demonstrate that the presence of topological links leads to an increase of its θ-temperature in comparison to isolated linear and ring chains with the following ordering: T > T > T. Finally, we show that the presence of links similarly raises the θ-temperature of a single linked ring in comparison to an unlinked one, bringing its θ-temperature close to the one of a poly[n]catenane.

Characterising knotting properties of polymers in nanochannels

Using a lattice model of polymers in a tube, we define one way to characterise different configurations of a given knot as either "local" or "non-local", based on a standard approach for measuring the "size" of a knot within a knotted polymer chain. The method involves associating knot-types to subarcs of the chain, and then identifying a knotted subarc with minimal arclength; this arclength is then the knot-size. If the resulting knot-size is small relative to the whole length of the chain, then the knot is considered to be localised or "local"; otherwise, it is "non-local". Using this definition, we establish that all but exponentially few sufficiently long self-avoiding polygons (closed chains) in a tubular sublattice of the simple cubic lattice are "non-locally" knotted. This is shown to also hold for the case when the same polygons are subject to an external tensile force, as well as in the extreme case when they are as compact as possible (no empty lattice sites). We also provide numerical evidence for small tube sizes that at equilibrium non-local knotting is more likely than local knotting, regardless of the strength of the stretching or compressing force. The relevance of these results to other models and recent experiments involving DNA knots is also discussed.

Mechanical Pulling of Linked Ring Polymers: Elastic Response and Link Localisation

By using Langevin dynamics simulations, we study how semiflexible rings that are topologically linked respond to mechanical stretching. We use both constant-force and constant-velocity pulling protocols and map out how the mechanical tension affects observables related to metric quantities such as the longitudinal extension or span, and topology-related ones such as the length of the linked portion. We find that the average extension of linked rings, once divided by that of a single equivalent ring, is nonmonotonic in the applied force. We show that this remarkable feature becomes more prominent as the link complexity is increased, and originates from the different stretching compliance of the linked portion and the rest of the rings’ contour. By comparing the results of different pulling protocols, we also establish the best one for telling apart different types of links from their tensile response.