Deformation Response of Catenated DNA Networks in a Planar Elongational Field

Unraveling the Influence of Topology and Spatial Confinement on Equilibrium and Relaxation Properties of Interlocked Ring Polymers

We use Langevin dynamics simulations to study linked ring polymers in channel confinement. We address the in- and out-of-equilibrium behavior of the systems for varying degrees of confinement and increasing topological and geometrical complexity of the interlocking. The main findings are three. First, metric observables of different link topologies collapse onto the same master curve when plotted against the crossing number, revealing a universal response to confinement. Second, the relaxation process from initially stretched states is faster for more complex links. We ascribe these properties to the interplay of several effects, including the dependence of topological friction on the link complexity. Finally, we show that transient forms of geometrical entanglement purposely added to the initial stressed state can leave distinctive signatures in force-spectroscopy curves. The insight provided by the findings could be leveraged in single-molecule nanochannel experiments to identify geometric entanglement within topologically linked rings.

Effects of Linking Topology on the Shear Response of Connected Ring Polymers: Catenanes and Bonded Rings Flow Differently

We perform computer simulations of mechanically linked (poly[2]catenanes, PC) and chemically bonded (bonded rings, BR) pairs of self-avoiding ring polymers in steady shear. We find that BRs develop a novel motif, termed gradient tumbling, rotating around the gradient axis. For the PCs the rings are stretched and display another new pattern, termed slip tumbling. The dynamics of BRs is continuous and oscillatory, whereas that of PCs is intermittent between slip-tumbling attempts. Our findings demonstrate the interplay between topology and hydrodynamics in dilute solutions of connected polymers.

Circular Polycatenanes: Supramolecular Structures with Topologically Tunable Properties

Polycatenanes, macrochains of topologically interlocked rings with unique physical properties have recently gained considerable interest in supramolecular chemistry, biology, and soft matter. Most of the work has been, so far, focused on linear chains and on their variety of conformational properties compared to standard polymers. Here we go beyond the linear case and show that, by circularizing such macrochains, one can exploit the topology of the local interlockings to store twist in the system, significantly altering its metric and local properties. Moreover, by properly defining the twist (Tw) and writhe (Wr) of these macrorings we show the validity of a relation equivalent to the Călugăreanu-White-Fuller theorem Tw+Wr=const, originally proved for ribbonlike structures such as double stranded DNA. Our results suggest that circular polycatenanes with storable and tunable twist can form a new category of highly designable multiscale structures with potential applications in supramolecular chemistry and material science.

Random Knotting in Fractal Ring Polymers

Many ring polymer systems of physical and biological interest exhibit both pronounced topological effects and nontrivial self-similarity, but the relationship between these two phenomena has not yet been clearly established. Here, we use theory and simulation to formulate such a connection by studying a fundamental topological property—the random knotting probability—for ring polymers with varying fractal dimension, d_f. Using straightforward scaling arguments, we generalize a classic mathematical result, showing that the probability of a trivial knot decays exponentially with chain size, N, for all fractal dimensions: P₀(N) ∝ exp(−N/N₀). However, no such simple considerations can account for the dependence of the knotting length, N₀, on d_f, necessitating a more involved analytical calculation. This analysis reveals a complicated double-exponential dependence, which is well supported by numerical data. By contrast, functional forms typical of simple scaling theories fail to adequately describe the observations. These findings are equally valid for two-dimensional ring polymer systems, where “knotting” is defined as the intersection of any two segments.

Polycatenanes: synthesis, characterization, and physical understanding

Chemical composition and architecture are two key factors that control the physical and material properties of polymers. Some of the more unusual and intriguing polymer architectures are the polycatenanes, which are a class of polymers that contain mechanically interlocked rings. Since the development of high yielding synthetic routes to catenanes, there has been an interest in accessing their polymeric counterparts, primarily on account of the unique conformations and degrees of freedom offered by non-bonded interlocked rings. This has lead to the synthesis of a wide variety of polycatenane architectures and to studies aimed at developing structure-property relationships of these interesting materials. In this review, we provide an overview of the field of polycatenanes, exploring synthesis, architecture, properties, simulation, and modelling, with a specific focus on some of the more recent developments.

Topological Friction and Relaxation Dynamics of Spatially Confined Catenated Polymers

We study catenated ring polymers confined inside channels and slits with Langevin dynamics simulations and address how the contour position and size of the interlocked or physically linked region evolve with time. We show that the catenation constraints generate a drag, or topological friction, that couples the contour motion of the interlocked regions. Notably, the coupling strength decreases as the interlocking is made tighter, but also shorter, by confinement. Though the coupling strength differs for channel and slit confinement, the data outline a single universal curve when plotted against the size of the linked region. Finally, we study how the relaxation kinetics changes after one of the rings is cut open and conclude that considering interlocked circular polymers is key for isolating the manifestations of topological friction. The results ought to be relevant for linked biomolecules in experimental or biological confining conditions.

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.

Tunable Knot Segregation in Copolyelectrolyte Rings Carrying a Neutral Segment

We use Langevin dynamics simulations to study the knotting properties of copolyelectrolyte rings carrying neutral segments. We show that by solely tuning the relative length of the neutral and charged blocks, one can achieve different combinations of knot contour position and size. Strikingly, the latter is shown to vary nonmonotonically with the length of the neutral segment; at the same time, the knot switches from being pinned at the block's edge to becoming trapped inside it. Model calculations relate both effects to the competition between two adversarial mechanisms: the energy gain of localizing one or more of the knot's essential crossings on the neutral segment and the entropic cost of such localization. Tuning the length of the neutral segment sets the balance between the two mechanisms and hence the number of localized essential crossings, which in turn modulates the knot's size. This general principle ought to be useful in more complex systems, such as multiblock copolyelectrolytes, to achieve a more granular control of topological constraints.

Phase Transition of Catenated DNA Networks in Poly(ethylene glycol) Solutions

Conformational phase transitions of macromolecules are an important class of problems in fundamental polymer physics. While the conformational phase transitions of linear DNA have been extensively studied, this feature of topologically complex DNA remains unexplored. We report herein the polymer-and-salt-induced (Ψ) phase transition of 2D catenated DNA networks, called kinetoplasts, using single-molecule fluorescence microscopy. We observe that kinetoplasts can undergo a reversible transition from the flat phase to the collapsed phase in the presence of NaCl as a function of the crowding agent poly(ethylene glycol). The nature of this phase transition is tunable through varying ionic strengths. For linear DNA, the coexistence of coil and globule phases was attributed to a first order phase transition associated with a double well potential in the transition regime. Kinetoplasts, however, navigate from the flat to the collapsed phase by passing through an intermediate regime, characterized by the coexistence of a multipopulation with varying shapes and sizes. Conformations of individual molecules in the multipopulation are long-lived, which suggests a rugged energy landscape.

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.

Equilibrium Conformation of Catenated DNA Networks in Slitlike Confinement

A kinetoplast is a planar network of catenated DNA rings with topology that resembles that of chain mail armor. In this work, we use single-molecule experiments to probe the conformation of kinetoplasts confined to slits. We find that the in-plane size of kinetoplasts increases with degree of confinement, akin to the slitlike confinement of linear DNA. The change in kinetoplast size with channel height is consistent with the scaling prediction from a Flory-type approach for a 2D polymer. With an increase in extent of confinement, the kinetoplasts appear to unfold and take on more uniform circular shapes, in contrast to the broad range of conformations observed for kinetoplasts in bulk.

Hydrodynamic interactions in topologically linked ring polymers

Despite decades of interdisciplinary research on topologically linked ring polymers, their dynamics remain largely unstudied. These systems represent a major scientific challenge as they are often subject to both topological and hydrodynamic interactions (HI), which render dynamical solutions either mathematically intractable or computationally prohibitive. Here we circumvent these limitations by preaveraging the HI of linked rings. We show that the symmetry of ring polymers leads to a hydrodynamic decoupling of ring dynamics. This decoupling is valid even for nonideal polymers and nonequilibrium conditions. Physically, our findings suggest that the effects of topology and HI are nearly independent and do not act cooperatively to influence polymer dynamics. We use this result to develop highly efficient Brownian dynamics algorithms that offer enormous performance improvements over conventional methods and apply these algorithms to simulate catenated ring polymers at equilibrium, confirming the independence of topological effects and HI. The methods developed here can be used to study and simulate large systems of linked rings with HI, including kinetoplast DNA, Olympic gels, and poly[n]catenanes.