How molecular knots can pass through each other

Decoding chirality in circuit topology of a self entangled chain through braiding

Circuit topology employs fundamental units of entanglement, known as soft contacts, for constructing knots from the bottom up, utilizing circuit topology relations, namely parallel, series, cross, and concerted relations. In this article, we further develop this approach to facilitate the analysis of chirality, which is a significant quantity in polymer chemistry. To achieve this, we translate the circuit topology approach to knot engineering into a braid-theoretic framework. This enables us to calculate the Jones polynomial for all possible binary combinations of contacts in cross or concerted relations and to show that, for series and parallel relations, the polynomial factorises. Our results demonstrate that the Jones polynomial provides a powerful tool for analysing the chirality of molecular knots constructed using circuit topology. The framework presented here can be used to design and engineer a wide range of entangled chain with desired chiral properties, with potential applications in fields such as materials science and nanotechnology.

Influence of ionic conditions on knotting in a coarse-grained model for DNA

We investigate knotting probabilities of long double-stranded DNA strands in a coarse-grained Kratky-Porod model for DNA with Monte Carlo simulations. Various ionic conditions are implemented by adjusting the effective diameter of monomers. We find that the occurrence of knots in DNA can be reinforced considerably by high salt conditions and confinement between plates. Likewise, knots can almost be dissolved completely in a low salt scenario. Comparisons with recent experiments confirm that the coarse-grained model is able to capture and quantitatively predict topological features of DNA and can be used for guiding future experiments on DNA knots.

Expansion of Single Chains Released from a Spherical Cavity

A two-stage model is developed to explain the phenomena of chain expansion, released from a confining cavity. In the first stage, the chain is assumed to expand as a sphere, while in the second stage it expands like a coil. The kinetic equations for the variation of chain size are derived in the two stages by balancing the rate of the free energy change with the rate of the energy dissipation. Langevin dynamics simulations are then performed to examine the theory. We find that the expansion process is dominated by the second stage and the evolution of chain size follows, mainly, the predicted curve for coil expansion, which depends on the chain length and is not sensitive to the confining volume fraction. It permits to define the expansion time for the process. Further study reveals that the chain does undergo a spherical expansion in the first stage with the characteristic time much shorter than the one for the second stage. As a consequence, the first-stage variation of chain size can be regarded as an add-on to the principal curve of expansion designated by the second stage. The scaling behaviors and the associated scaling exponents are analyzed in details. The simulation results well support the theory.

Knot formation of dsDNA pushed inside a nanochannel

Recent experiments demonstrated that knots in single molecule dsDNA can be formed by compression in a nanochannel. In this manuscript, we further elucidate the underlying molecular mechanisms by carrying out a compression experiment in silico, where an equilibrated coarse-grained double-stranded DNA confined in a square channel is pushed by a piston. The probability of forming knots is a non-monotonic function of the persistence length and can be enhanced significantly by increasing the piston speed. Under compression knots are abundant and delocalized due to a backfolding mechanism from which chain-spanning loops emerge, while knots are less frequent and only weakly localized in equilibrium. Our in silico study thus provides insights into the formation, origin and control of DNA knots in nanopores.

Interactions between two knots in nanochannel-confined DNA molecules

Experimental data on the interaction between two knots in deoxyribonucleic acid (DNA) confined in nanochannels produced two particular behaviors of knot pairs along the DNA molecules: (i) widely separated knots experience an attractive interaction but only remain in close proximity for several seconds and (ii) knots tend to remain separated until one of the knots unravels at the chain end. The associated free energy profile of the knot-knot separation distance for an ensemble of DNA knots exhibits a global minimum when knots are separated, indicating that the separated knot state is more stable than the intertwined knot state, with dynamics in the separated knot state that are consistent with independent diffusion. The experimental observations of knot-knot interactions under nanochannel confinement are inconsistent with previous simulation-based and experimental results for stretched polymers under tension wherein the knots attract and then stay close to each other. This inconsistency is postulated to result from a weaker fluctuation-induced attractive force between knots under confinement when compared to the knots under tension, the latter of which experience larger fluctuations in transverse directions.

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.

Investigation of Ring and Star Polymers in Confined Geometries: Theory and Simulations

The calculations of the dimensionless layer monomer density profiles for a dilute solution of phantom ideal ring polymer chains and star polymers with f=4 arms in a Θ-solvent confined in a slit geometry of two parallel walls with repulsive surfaces and for the mixed case of one repulsive and the other inert surface were performed. Furthermore, taking into account the Derjaguin approximation, the dimensionless layer monomer density profiles for phantom ideal ring polymer chains and star polymers immersed in a solution of big colloidal particles with different adsorbing or repelling properties with respect to polymers were calculated. The density-force relation for the above-mentioned cases was analyzed, and the universal amplitude ratio B was obtained. Taking into account the small sphere expansion allowed obtaining the monomer density profiles for a dilute solution of phantom ideal ring polymers immersed in a solution of small spherical particles, or nano-particles of finite size, which are much smaller than the polymer size and the other characteristic mesoscopic length of the system. We performed molecular dynamics simulations of a dilute solution of linear, ring, and star-shaped polymers with N=300, 300 (360), and 1201 (4 × 300 + 1-star polymer with four arms) beads accordingly. The obtained analytical and numerical results for phantom ring and star polymers are compared with the results for linear polymer chains in confined geometries.

Can Soft Models Describe Polymer Knots?

Similar to macroscopic ropes and cables, long polymers create knots. We address the fundamental question whether and under which conditions it is possible to describe these intriguing objects with crude models that capture only mesoscale polymer properties. We focus on melts of long polymers which we describe by a model typical for mesoscopic simulations. A worm-like chain model defines the polymer architecture. To describe nonbonded interactions, we deliberately choose a generic "soft" repulsive potential that leads to strongly overlapping monomers and coarse local liquid structure. The soft model is parametrized to accurately reproduce mesoscopic structure and conformations of reference polymer melts described by a microscopic model. The microscopically resolved samples retain all generic features affecting polymer topology and provide, therefore, reliable reference data on knots. We compare characteristic knotting properties in mesoscopic and microscopically resolved melts for different cases of chain stiffness. We conclude that mesoscopic models can reliably describe knots in those melts, where the length scale characterizing polymer stiffness is substantially larger than the size of monomer-monomer excluded volume. In this case, simplified local liquid structure influences knotting properties only marginally. In contrast, mesoscopic models perform poorly in melts with flexible chains. We qualitatively explain our findings through a free energy model of simple knots available in the literature.

Topological Disentanglement of Linear Polymers under Tension

We develop a theoretical description of the topological disentanglement occurring when torus knots reach the ends of a semiflexible polymer under tension. These include decays into simpler knots and total unknotting. The minimal number of crossings and the minimal knot contour length are the topological invariants playing a key role in the model. The crossings behave as particles diffusing along the chain and the application of appropriate boundary conditions at the ends of the chain accounts for the knot disentanglement. Starting from the number of particles and their positions, suitable rules allow reconstructing the type and location of the knot moving on the chain Our theory is extensively benchmarked with corresponding molecular dynamics simulations and the results show a remarkable agreement between the simulations and the theoretical predictions of the model.

Complex DNA knots detected with a nanopore sensor

Equilibrium knots are common in biological polymers-their prevalence, size distribution, structure, and dynamics have been extensively studied, with implications to fundamental biological processes and DNA sequencing technologies. Nanopore microscopy is a high-throughput single-molecule technique capable of detecting the shape of biopolymers, including DNA knots. Here we demonstrate nanopore sensors that map the equilibrium structure of DNA knots, without spurious knot tightening and sliding. We show the occurrence of both tight and loose knots, reconciling previous contradictory results from different experimental techniques. We evidence the occurrence of two quantitatively different modes of knot translocation through the nanopores, involving very different tension forces. With large statistics, we explore the complex knots and, for the first time, reveal the existence of rare composite knots. We use parametrized complexity, in concert with simulations, to test the theoretical assumptions of the models, further asserting the relevance of nanopores in future investigation of knots.

Topological Disentanglement Dynamics of Torus Knots on Open Linear Polymers

We simulate and study the topological disentanglement occurring when torus knots reach the ends of a semiflexible open polymer (decay into simpler knots or unknotting). Through a rescaling procedure and the application of appropriate boundary conditions, we show that the full unknotting process can be understood in terms of point-like particles representing essential crossings, diffusing on the support [0, 1]. We address the bending and configurational free energy drives on the diffusion process, together with the scaling properties of the effective diffusion and friction coefficients. Agreement with simulations suggests universal features for these two model parameters.

Topological entanglement of interlocked knotted-unknotted polymer rings

Topological entanglements in biopolymers could drive them to certain internal statics and dynamics with important implications for biological functions. In this study, by means of molecular dynamics simulations, we demonstrate that the minimal crossing pattern of a braid plays a major role in its structural and dynamical properties; the braid consists of a knotted ring and an interlocked entwined unknotted polymer ring. In particular, we show that depending on the bending rigidity of the chains, the conformational energy of the braid can be either lower or higher than the unlocked polymer rings. Additionally, we find that a non-identical crossing pattern in the braid could distinctly enforce concerted internal conformational fluctuations between the interlocked rings.

Securing a Supramolecular Architecture by Tying a Stopper Knot

We report on a rotaxane-like architecture secured by the in situ tying of an overhand knot in the tris(2,6-pyridyldicarboxamide) region of the axle through complexation with a lanthanide ion (Lu3+ ). The increase in steric bulk caused by the knotting locks a crown ether onto the thread. Removal of the lutetium ion unties the knot, and when the axle binding site for the ring is deactivated, the macrocycle spontaneously dethreads. When the binding interaction is switched on again, the crown ether rethreads over the 10 nm length of the untangled strand. The overhand knot can be retied, relocking the threaded structure, by once again adding lutetium ions.

Mapping onto Ideal Chains Overestimates Self-Entanglements in Polymer Melts

We analyze the occurrence of knots, their spectrum, and sizes in polymer melts. Surprisingly, the number of knots in melt conformations is much lower than expected from a mapping to a random walk with the same Kuhn segment length. The effective random walk severely overrates the occurrence of knots and their complexity, particularly when compared to melts of flexible chains, indicating that nontrivial effects due to remnants of self-avoidance play a role for the chain lengths considered in this numerical study. For melt chains with higher persistence length, the effect is less pronounced. In addition, we find that chains in a melt have a knot structure very similar to dilute single chains close to the collapse transition. We finally show that typical equilibration procedures are well-suited to relax the topology in melts.

A nanofluidic knot factory based on compression of single DNA in nanochannels

Knots form when polymers self-entangle, a process enhanced by compaction with important implications in biological and artificial systems involving chain confinement. In particular, new experimental tools are needed to assess the impact of multiple variables influencing knotting probability. Here, we introduce a nanofluidic knot factory for efficient knot formation and detection. Knots are produced during hydrodynamic compression of single DNA molecules against barriers in a nanochannel; subsequent extension of the chain enables direct assessment of the number of independently evolving knots. Knotting probability increases with chain compression as well as with waiting time in the compressed state. Using a free energy derived from scaling arguments, we develop a knot-formation model that can quantify the effect of interactions and the breakdown of Poisson statistics at high compression. Our model suggests that highly compressed knotted states are stabilized by a decreased free energy as knotted contour contributes a lower self-exclusion derived free energy.

Trefoil Knot Hydrodynamic Delocalization on Sheared Ring Polymers

The behavior of unknotted and trefoil-knotted ring polymers under shear flow is here examined by means of mesoscopic simulations. In contrast to most polymers, ring polymers in a hydrodynamic solvent at high shear rates do not get shortened in the vorticity direction. This is a consequence of the backflow produced by the interaction of the sheared solvent with the end-free polymer topology. The extended structures of the ring in the vorticity-flow plane, when they are aligned in a constant velocity plane, favor ring contour fluctuations. This variety of conformations largely suppresses the tank-treading type of rotation with extended conformations in favor of the tumbling type of rotations, where stretched and collapsed conformations alternate. The extension of trefoil knots is also enhanced, so that the knots become delocalized. We anticipate that these effects, which disappear in the absence of hydrodynamic interactions, will have a crucial impact on the rheological properties of concentrated ring solutions, and will also influence the behavior of more complicated systems such as mixtures of polymers with different topologies.

Epigenetic Transitions and Knotted Solitons in Stretched Chromatin

The spreading and regulation of epigenetic marks on chromosomes is crucial to establish and maintain cellular identity. Nonetheless, the dynamic mechanism leading to the establishment and maintenance of tissue-specific, epigenetic pattern is still poorly understood. In this work we propose, and investigate in silico, a possible experimental strategy to illuminate the interplay between 3D chromatin structure and epigenetic dynamics. We consider a set-up where a reconstituted chromatin fibre is stretched at its two ends (e.g., by laser tweezers), while epigenetic enzymes (writers) and chromatin-binding proteins (readers) are flooded into the system. We show that, by tuning the stretching force and the binding affinity of the readers for chromatin, the fibre undergoes a sharp transition between a stretched, epigenetically disordered, state and a crumpled, epigenetically coherent, one. We further investigate the case in which a knot is tied along the chromatin fibre, and find that the knotted segment enhances local epigenetic order, giving rise to "epigenetic solitons" which travel and diffuse along chromatin. Our results point to an intriguing coupling between 3D chromatin topology and epigenetic dynamics, which may be investigated via single molecule experiments.

Simulations of knotting of DNA during genome mapping

Genome mapping involves the confinement of long DNA molecules, in excess of 150 kilobase pairs, in nanochannels near the circa 50 nm persistence length of DNA. The fidelity of the map relies on the assumption that the DNA is linearized by channel confinement, which assumes the absence of knots. We have computed the probability of forming different knot types and the size of these knots for long chains (approximately 164 kilobase pairs) via pruned-enriched Rosenbluth method simulations of a discrete wormlike chain model of DNA in channel sizes ranging from 35 nm to 60 nm. Compared to prior simulations of short DNA in similar confinement, these long molecules exhibit both complex knots, with up to seven crossings, and multiple knots per chain. The knotting probability is a very strong function of channel size, ranging from 0.3% to 60%, and rationalized in the context of Odijk's theory for confined semiflexible chains. Overall, the knotting probability and knot size obtained from these equilibrium measurements are not consistent with experimental measurements of the properties of anomalously bright regions along the DNA backbone during genome mapping experiments. This result suggests that these events in experiments are either knots formed during the processing of the DNA prior to injection into the nanochannel or regions of locally high DNA concentration without a topological constraint. If so, knots during genome mapping are not an intrinsic problem for genome mapping technology.

Entropic Interactions between Two Knots on a Semiflexible Polymer

Two knots on a string can either be separated or intertwined, and may even pass through each other. At the microscopic scale, such transitions may occur spontaneously, driven by thermal fluctuations, and can be associated with a topological free energy barrier. In this manuscript, we study the respective location of a trefoil ( 3 1 ) and a figure-eight ( 4 1 ) knot on a semiflexible polymer, which is parameterized to model dsDNA in physiological conditions. Two cases are considered: first, end monomers are grafted to two confining walls of varying distance. Free energy profiles and transition barriers are then compared to a subset of free chains, which contain exactly one 3 1 and one 4 1 knot. For the latter, we observe a small preference to form an intertwined state, which can be associated with an effective entropic attraction. However, the respective free energy barrier is so small that we expect transition events to occur spontaneously and frequently in polymers and DNA, which are highly knotted for sufficient strain lengths.

Role of Bending Energy and Knot Chirality in Knot Distribution and Their Effective Interaction along Stretched Semiflexible Polymers

Knots appear frequently in semiflexible (bio)polymers, including double-stranded DNA, and their presence can affect the polymer’s physical and functional properties. In particular, it is possible and indeed often the case that multiple knots appear on a single chain, with effects which have only come under scrutiny in the last few years. In this manuscript, we study the interaction of two knots on a stretched semiflexible polymer, expanding some recent results on the topic. Specifically, we consider an idealization of a typical optical tweezers experiment and show how the bending rigidity of the chain—And consequently its persistence length—Influences the distribution of the entanglements; possibly more importantly, we observe and report how the relative chirality of the otherwise identical knots substantially modifies their interaction. We analyze the free energy of the chain and extract the effective interactions between embedded knots, rationalizing some of their pertinent features by means of simple effective models. We believe the salient aspect of the knot–knot interactions emerging from our study will be present in a large number of semiflexible polymers under tension, with important consequences for the characterization and manipulation of these systems—Be they artificial or biologica in origin—And for their technological application.

Dilute Semiflexible Polymers with Attraction: Collapse, Folding and Aggregation

We review the current state on the thermodynamic behavior and structural phases of self- and mutually-attractive dilute semiflexible polymers that undergo temperature-driven transitions. In extreme dilution, polymers may be considered isolated, and this single polymer undergoes a collapse or folding transition depending on the internal structure. This may go as far as to stable knot phases. Adding polymers results in aggregation, where structural motifs again depend on the internal structure. We discuss in detail the effect of semiflexibility on the collapse and aggregation transition and provide perspectives for interesting future investigations.

Enhanced electrostatic force microscopy reveals higher-order DNA looping mediated by the telomeric protein TRF2

Shelterin protein TRF2 modulates telomere structures by promoting dsDNA compaction and T-loop formation. Advancement of our understanding of the mechanism underlying TRF2-mediated DNA compaction requires additional information regarding DNA paths in TRF2-DNA complexes. To uncover the location of DNA inside protein-DNA complexes, we recently developed the Dual-Resonance-frequency-Enhanced Electrostatic force Microscopy (DREEM) imaging technique. DREEM imaging shows that in contrast to chromatin with DNA wrapping around histones, large TRF2-DNA complexes (with volumes larger than TRF2 tetramers) compact DNA inside TRF2 with portions of folded DNA appearing at the edge of these complexes. Supporting coarse-grained molecular dynamics simulations uncover the structural requirement and sequential steps during TRF2-mediated DNA compaction and result in folded DNA structures with protruding DNA loops as seen in DREEM imaging. Revealing DNA paths in TRF2 complexes provides new mechanistic insights into structure-function relationships underlying telomere maintenance pathways.

Pore Translocation of Knotted Polymer Chains: How Friction Depends on Knot Complexity

Knots can affect the capability of polymers to translocate through narrow pores in complex and counterintuitive ways that are still relatively unexplored. We report here on a systematic theoretical and computational investigation of the driven translocation of flexible chains accommodating a large repertoire of knots trapped at the pore entrance. These include composite knots, which are the most common form of spontaneous entanglement in long polymers. Two unexpected results emerge from this study. First, the high force translocation compliance does not decrease systematically with knot complexity. Second, the response of composite knots is so dependent on the order of their factor knots, that their hindrance can even be lower than some of their prime components. We show that the resulting rich and seemingly disparate phenomenology can be captured in a seamless framework based on the mechanism by which the tractive force is propagated along and past the knots. The quantitative scheme can be viably used for predictive purposes and, hence, ought to be useful in applicative contexts, too.

Stretching Response of Knotted and Unknotted Polymer Chains

Recent theoretical and experimental advances have clarified the major effects of knotting on the properties of stretched chains. Yet, how knotted chains respond to weak mechanical stretching and how this behavior differs from the unknotted case are still open questions and we address them here by profiling the complete stretching response of chains of hundreds of monomers and different topology. We find that the ratio of the knotted and unknotted chain extensions varies nonmonotonically with the applied force. This surprising feature is shown to be a signature of the crossover between the well-known high-force stretching regime and the previously uncharacterized low-force one. The observed differences of knotted and unknotted chain response increases with knot complexity and are sufficiently marked that they could be harnessed in single-molecule contexts to infer the presence and complexity of physical knots in micron-long biomolecules.

Origin of metastable knots in single flexible chains

Recent theoretical progress has explained the physics of knotting of semiflexible polymers, yet knotting of flexible polymers is relatively unexplored. We herein develop a new theory for the size distribution of knots on a flexible polymer and the existence of metastable knots. We show the free energy of a flexible molecule in a tube can be mapped to quantitatively reproduce the free energy distribution of a knot on a flexible chain. The size distribution of knots on flexible chains is expected to be universal and might be observed at a macroscopic scale, such as a string of hard balls.