Biophysics of knotting

Activity-driven polymer knotting for macromolecular topology engineering

Macromolecules can gain special properties by adopting knotted conformations, but engineering knotted macromolecules is a challenging task. Here, we unexpectedly find that knots can be efficiently generated in active polymer systems. When one end of an actively reptative polymer is anchored, it undergoes continual self-knotting as a result of intermittent giant conformation fluctuations and the outward reptative motion. Once a knot is formed, it migrates to the anchoring point due to a nonequilibrium ratchet effect. Moreover, when the active polymer is grafted on a passive polymer, it can function as a self-propelling soft needle to either transfer its own knots or directly braid knots on the passive polymer. We further show that these active needles can create intermolecular bridging knots between two passive polymers. Our finding highlights the nonequilibrium effects in modifying the dynamic pathways of polymer systems, which have potential applications in macromolecular topology engineering.

Morphological Entanglement in Living Systems

Many organisms exhibit branching morphologies that twist around each other and become entangled. Entanglement occurs when different objects interlock with each other, creating complex and often irreversible configurations. This physical phenomenon is well studied in nonliving materials, such as granular matter, polymers, and wires, where it has been shown that entanglement is highly sensitive to the geometry of the component parts. However, entanglement is not yet well understood in living systems, despite its presence in many organisms. In fact, recent work has shown that entanglement can evolve rapidly and play a crucial role in the evolution of tough, macroscopic multicellular groups. Here, through a combination of experiments, simulations, and numerical analyses, we show that growth generically facilitates entanglement for a broad range of geometries. We find that experimentally grown entangled branches can be difficult or even impossible to disassemble through translation and rotation of rigid components, suggesting that there are many configurations of branches that growth can access that agitation cannot. We use simulations to show that branching trees readily grow into entangled configurations. In contrast to nongrowing entangled materials, these trees entangle for a broad range of branch geometries. We, thus, propose that entanglement via growth is largely insensitive to the geometry of branched trees but, instead, depends sensitively on timescales, ultimately achieving an entangled state once sufficient growth has occurred. We test this hypothesis in experiments with snowflake yeast, a model system of undifferentiated, branched multicellularity, showing that lengthening the time of growth leads to entanglement and that entanglement via growth can occur for a wide range of geometries. Taken together, our work demonstrates that entanglement is more readily achieved in living systems than in their nonliving counterparts, providing a widely accessible and powerful mechanism for the evolution of novel biological material properties.

Thermal properties of knotted block copolymer rings with charged monomers subjected to short-range interactions

The thermal properties of coarse-grained knotted copolymer rings fluctuating in a highly screening solution are investigated on a simple cubic lattice using the Wang-Landau Monte Carlo algorithm. The rings contain two kinds of monomers A and B with opposite charges that are subjected to short-range interactions. In view of possible applications in medicine and the construction of intelligent materials, it is shown that the behavior of copolymer rings can be tuned by changing both their monomer configuration and topology. We find several phase transitions depending on the monomer distribution. They include the expansion and collapse of the knotted polymer as well as rearrangements leading to metastable states. The temperatures at which these phase transitions are occurring and other features can be tuned by changing the topology of the system. The processes underlying the observed transitions are identified. In knots formed by diblock copolymers, two different classes of behaviors are detected depending on whether there is an excess of monomers of one kind or not. Moreover, we find that the most stable compact states are formed by copolymers in which units of two A monomers are alternated by units of two B monomers. Remarkably, these compact states are in a lamellar phase. The transition from the lamellar to the expanded state produces in the specific heat capacity a narrow and high peak that is centered at temperatures that are much higher than those of the peaks observed in all other monomer distributions.

Diffusion of knots in nanochannel-confined DNA molecules

We used Langevin dynamics simulations without hydrodynamic interactions to probe knot diffusion mechanisms and the time scales governing the evolution and the spontaneous untying of trefoil knots in nanochannel-confined DNA molecules in the extended de Gennes regime. The knot untying follows an "opening up process," wherein the initially tight knot continues growing and fluctuating in size as it moves toward the end of the DNA molecule before its annihilation at the chain end. The mean knot size increases significantly and sub-linearly with increasing chain contour length. The knot diffusion in nanochannel-confined DNA molecules is subdiffusive, with the unknotting time scaling with chain contour length with an exponent of 2.64 ± 0.23 to within a 95% confidence interval. The scaling exponent for the mean unknotting time vs chain contour length, along with visual inspection of the knot conformations, suggests that the knot diffusion mechanism is a combination of self-reptation and knot region breathing for the simulated parameters.

The Story of the Little Blue Box: A Tribute to Siegfried Hünig

The tetracationic cyclophane, cyclobis(paraquat-p-phenylene), also known as the little blue box, constitutes a modular receptor that has facilitated the discovery of many host-guest complexes and mechanically interlocked molecules during the past 35 years. Its versatility in binding small π-donors in its tetracationic state, as well as forming trisradical tricationic complexes with viologen radical cations in its doubly reduced bisradical dicationic state, renders it valuable for the construction of various stimuli-responsive materials. Since the first reports in 1988, the little blue box has been featured in over 500 publications in the literature. All this research activity would not have been possible without the seminal contributions carried out by Siegfried Hünig, who not only pioneered the syntheses of viologen-containing cyclophanes, but also revealed their rich redox chemistry in addition to their ability to undergo intramolecular π-dimerization. This Review describes how his pioneering research led to the design and synthesis of the little blue box, and how this redox-active host evolved into the key component of molecular shuttles, switches, and machines.

Phase transition of DNA knotting in spherical space

Knots have been discovered in various biological systems, such as DNA. The knotting probability of DNA in free space depends non-monotonically on its bending rigidity and has a prominent peak. The current work aims to understand the underlying mechanism of the non-monotonic dependence of DNA knotting probability on bending rigidity. Monte Carlo simulations are performed on a closed DNA molecule confined in spherical space described by a worm-like chain model and a flexible kink model, respectively. The closed DNA's contour length and the spherical space radius both increase knotting probability, but also alter the unimodal dependence of knotting probability on bending rigidity. This is generalized using universal phase diagrams based on the two models. Under the flexible kink model, the total knotting probability of closed DNA is obviously increased at a relatively high excited energy. This supports the expectation that the entropy effect of knot size favours knot formation at a relatively low bending rigidity. In a given spherical space, the increasing contour length of closed DNA described by the worm-like chain model results in a visible shift in the knotting probability distribution. At the same time, the gyration radius of non-trivial closed DNA becomes comparable to that of trivial closed DNA, so that their ratio is not anti-correlated with average knot length. For closed DNA of various contour lengths, the relationship between average knot length and bending rigidity has a universal behaviour: the average knot length decreases to a local minimum at a bending rigidity of ∼5 and then gradually increases to a constant value. The existence of the local minimum is determined by the cut-off distance in repulsive Lennard-Jones potential. The bending rigidity corresponding to the beginning of the constant average knot length is consistent with that at the peak in the knotting distribution. At this point, the knot-size effect balances with the fragment free-energy effect and, at an even greater bending rigidity, knot length breathes around the average knot length value.

Vernier template synthesis of molecular knots

Molecular knots are often prepared using metal helicates to cross the strands. We found that coordinatively mismatching oligodentate ligands and metal ions provides a more effective way to synthesize larger knots using Vernier templating. Strands composed of different numbers of tridentate 2,6-pyridinedicarboxamide groups fold around nine-coordinate lanthanide (III) ions to generate strand-entangled complexes with the lowest common multiple of coordination sites for the ligand strands and metal ions. Ring-closing olefin metathesis then completes the knots. A 3:2 (ditopic strand:metal) Vernier assembly produces +3₁#+3₁ and -3₁#-3₁ granny knots. Vernier complexes of 3:4 (tetratopic strand:metal) stoichiometry selectively form a 378-atom-long trefoil-of-trefoils triskelion knot with 12 alternating strand crossings or, by using opposing stereochemistry at the terminus of the strand, an inverted-core triskelion knot with six alternating and six nonalternating strand crossings.

Channels with Helical Modulation Display Stereospecific Sensitivity for Chiral Superstructures

By means of coarse-grained molecular dynamics simulations, we explore chiral sensitivity of confining spaces modelled as helical channels to chiral superstructures represented by polymer knots. The simulations show that helical channels exhibit stereosensitivity to chiral knots localized on linear chains by effect of external pulling force and also to knots embedded on circular chains. The magnitude of the stereoselective effect is stronger for torus knots, the effect is weaker in the case of twist knots, and amphichiral knots do exhibit no chiral effects. The magnitude of the effect can be tuned by the so-far investigated radius of the helix, the pitch of the helix and the strength of the pulling force. The model is aimed to simulate and address a range of practical situations that may occur in experimental settings such as designing of nanotechnological devices for the detection of topological state of molecules, preparation of new gels with tailor made stereoselective properties, or diffusion of knotted DNA in biological conditions.

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.

DNA Knot Malleability in Single-Digit Nanopores

Knots in long DNA molecules are prevalent in biological systems and serve as a model system for investigating static and dynamic properties of biopolymers. We explore the dynamics of knots in double-stranded DNA in a new regime of nanometer-scale confinement, large forces, and short time scales, using solid-state nanopores. We show that DNA knots undergo isomorphic translocation through a nanopore, retaining their equilibrium morphology by swiftly compressing in a lateral direction to fit the constriction. We observe no evidence of knot tightening or jamming, even for single-digit nanopores. We explain the observations as the malleability of DNA, characterized by sharp buckling of the DNA in nanopores, driven by the transient disruption of base pairing. Our molecular dynamics simulations support the model. These results are relevant not only for the understanding of DNA packing and manipulation in living cells but also for the polymer physics of DNA and the development of nanopore-based sequencing technologies.

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.

Time-dependent knotting of agitated chains

Agitated strings serve as macroscale models of spontaneous knotting, providing valuable insight into knotting dynamics at the microscale while allowing explicit analysis of the resulting knot topologies. We present an experimental setup for confined macroscale knot formation via tumbling along with a software interface to process complex knot data. Our setup allows characterization of knotting probability, knot complexity, and knot formation dynamics for knots with as many as 50 crossings. We find that the probability of knotting saturates below 80% within 100 s of the initiation of tumbling and that this saturation probability does not increase for chains above a critical length, an indication of nonequilibrium knot-formation conditions in our experiment. Despite the saturation in knot formation, we show that longer chains, while being more confined, will always tend to form knots of higher complexity since the free end can access a greater number of loops during tumbling.

Spontaneous knotting of a flexible fiber in chaotic flows

We consider the problem of an inextensible but flexible fiber advected by a steady chaotic flow, and ask the simple question of whether the fiber can spontaneously knot itself. Using a one-dimensional Cosserat model, a simple local viscous drag model and discrete contact forces, we explore the probability of finding knots at any given time when the fiber is interacting with the ABC class of flows. The bending rigidity is shown to have a marginal effect compared to that of increasing the fiber length. Complex knots are formed up to 11 crossings, but some knots are more probable than others. The finite-time Lyapunov exponent of the flow is shown to have a positive effect on the knot probability. Finally, contact forces appear to be crucial since knotted configurations can remain stable for times much longer than the turnover time of the flow, something that is not observed when the fiber can freely cross itself.

Creation of acoustic vortex knots

Knots and links have been conjectured to play a fundamental role in a wide range of scientific fields. Recently, tying isolated vortex knots in the complex optical field has been realized. However, how to construct the acoustic vortex knot is still an unknown problem. Here we propose theoretically and demonstrate experimentally the creation of acoustic vortex knots using metamaterials, with decoupled modulation of transmitted phase and amplitude. Based on the numerical simulation, we find that the knot function can be embedded into the acoustic field by designed metamaterials with only 24 × 24 pixels. Furthermore, using the optimized metamaterials, the acoustic fields with Hopf link and trefoil knot vortex lines have been observed experimentally.

Although knots in complex optical fields have been realized experimentally, the realization of acoustic vortex knots is still problematic. Here, the authors have demonstrated the creation of acoustic vortex knots by embedding the knot function into a propagating acoustic field using a metasurface hologram.

Computational methods in the study of self-entangled proteins: a critical appraisal

The existence of self-entangled proteins, the native structure of which features a complex topology, unveils puzzling, and thus fascinating, aspects of protein biology and evolution. The discovery that a polypeptide chain can encode the capability to self-entangle in an efficient and reproducible way during folding, has raised many questions, regarding the possible function of these knots, their conservation along evolution, and their role in the folding paradigm. Understanding the function and origin of these entanglements would lead to deep implications in protein science, and this has stimulated the scientific community to investigate self-entangled proteins for decades by now. In this endeavour, advanced experimental techniques are more and more supported by computational approaches, that can provide theoretical guidelines for the interpretation of experimental results, and for the effective design of new experiments. In this review we provide an introduction to the computational study of self-entangled proteins, focusing in particular on the methodological developments related to this research field. A comprehensive collection of techniques is gathered, ranging from knot theory algorithms, that allow detection and classification of protein topology, to Monte Carlo or molecular dynamics strategies, that constitute crucial instruments for investigating thermodynamics and kinetics of this class of proteins.

Periodic Motion of Sedimenting Flexible Knots

We study the dynamics of knotted deformable closed chains sedimenting in a viscous fluid. We show experimentally that trefoil and other torus knots often attain a remarkably regular horizontal toroidal structure while sedimenting, with a number of intertwined loops, oscillating periodically around each other. We then recover this motion numerically and find out that it is accompanied by a very slow rotation around the vertical symmetry axis. We analyze the dependence of the characteristic timescales on the chain flexibility and aspect ratio. It is observed in the experiments that this oscillating mode of the dynamics can spontaneously form even when starting from a qualitatively different initial configuration. In numerical simulations, the oscillating modes are usually present as transients or final stages of the evolution, depending on chain aspect ratio and flexibility, and the number of loops.

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

The KymoKnot software package and web server identifies and locates physical knots or proper knots in a series of polymer conformations. It is mainly intended as an analysis tool for trajectories of linear or circular polymers, but it can be used on single instances too, e.g. protein structures in PDB format. A key element of the software package is the so-called minimally interfering chain closure algorithm that is used to detect physical knots in open chains and to locate the knotted region in both open and closed chains. The web server offers a user-friendly graphical interface that identifies the knot type and highlights the knotted region on each frame of the trajectory, which the user can visualize interactively from various viewpoints. The dynamical evolution of the knotted region along the chain contour is presented as a kymograph. All data can be downloaded in text format. The KymoKnot package is licensed under the BSD 3-Clause licence. The server is publicly available at http://kymoknot.sissa.it/kymoknot/interactive.php .

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.

Steady-State and Transient Behavior of Knotted Chains in Extensional Fields

Recently, there has been a push to understand how molecular topology alters the nonequilibrium dynamics of polymer systems. In this paper, we probe how knotted polymers evolve in planar extensional fields using Brownian dynamics simulations and single-molecule experiments. In the first part of the study, we quantify the extension versus strain-rate curves of polymers and find that knots shift these curves to larger strain-rates. These trends can be quantitatively explained by Rouse-like scaling theories. In the second half of the study, we examine the consequences of knot untying on the time-dependent conformations of polymers in these external fields. We find that knot untying creates significant, transient changes in chain extension. If the topology is complex, the chain undergoes a wide range of time-dependent conformations since knot untying proceeds through many different stages. We provide examples of such untying trajectories over time.

The energy cost of polypeptide knot formation and its folding consequences

Knots are natural topologies of chains. Yet, little is known about spontaneous knot formation in a polypeptide chain—an event that can potentially impair its folding—and about the effect of a knot on the stability and folding kinetics of a protein. Here we used optical tweezers to show that the free energy cost to form a trefoil knot in the denatured state of a polypeptide chain of 120 residues is 5.8 ± 1 kcal mol⁻¹. Monte Carlo dynamics of random chains predict this value, indicating that the free energy cost of knot formation is of entropic origin. This cost is predicted to remain above 3 kcal mol⁻¹ for denatured proteins as large as 900 residues. Therefore, we conclude that naturally knotted proteins cannot attain their knot randomly in the unfolded state but must pay the cost of knotting through contacts along their folding landscape.

The effect of knots on protein stability and folding kinetics is not well understood. Here the authors combine optical tweezer experiments and calculations to experimentally determine the energy cost for knot formation, which indicates that knotted proteins evolved specific folding pathways because knot formation in unfolded chains is unfavorable.

Non-monotonic knotting probability and knot length of semiflexible rings: the competing roles of entropy and bending energy

We consider self-avoiding rings of up to 1000 beads and study, by Monte Carlo techniques, how their equilibrium knotting properties depend on the bending rigidity. When the rings are taken from the rigid to fully-flexible limit, their average compactness increases, as expected. However, this progressive compactification is not parallelled by a steady increase of the abundance of knots. In fact the knotting probability, P_k, has a prominent maximum when the persistence length is a few times larger than the bead size. At similar bending rigidities, the knot length has, instead, a minimum. We show that the observed non-monotonicity of P_k arises from the competition between two effects. The first one is the entropic cost of introducing a knot. The second one is the gain in bending energy due to the presence of essential crossings. These, in fact, constrain the knotted region and keep it less bent than average. The two competing effects make knots maximally abundant when the persistence length is 5-10 times larger than the bead size. At such intermediate bending rigidities, knots in the chains of 500 and 1000 beads are 40 times more likely than in the fully-flexible limit.

Direct observation of DNA knots using a solid-state nanopore

Long DNA molecules can self-entangle into knots. Experimental techniques for observing such DNA knots (primarily gel electrophoresis) are limited to bulk methods and circular molecules below 10 kilobase pairs in length. Here, we show that solid-state nanopores can be used to directly observe individual knots in both linear and circular single DNA molecules of arbitrary length. The DNA knots are observed as short spikes in the nanopore current traces of the traversing DNA molecules and their detection is dependent on a sufficiently high measurement resolution, which can be achieved using high-concentration LiCl buffers. We study the percentage of molecules with knots for DNA molecules of up to 166 kilobase pairs in length and find that the knotting occurrence rises with the length of the DNA molecule, consistent with a constant knotting probability per unit length. Our experimental data compare favourably with previous simulation-based predictions for long polymers. From the translocation time of the knot through the nanopore, we estimate that the majority of the DNA knots are tight, with remarkably small sizes below 100 nm. In the case of linear molecules, we also observe that knots are able to slide out on application of high driving forces (voltage).

The elusive quest for RNA knots

Physical entanglement, and particularly knots arise spontaneously in equilibrated polymers that are sufficiently long and densely packed. Biopolymers are no exceptions: knots have long been known to occur in proteins as well as in encapsidated viral DNA. The rapidly growing number of RNA structures has recently made it possible to investigate the incidence of physical knots in this type of biomolecule, too. Strikingly, no knots have been found to date in the known RNA structures. In this Point of View Article we discuss the absence of knots in currently available RNAs and consider the reasons why knots in RNA have not yet been found, despite the expectation that they should exist in Nature. We conclude by singling out a number of RNA sequences that, based on the properties of their predicted secondary structures, are good candidates for knotted RNAs.

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.

Tuning knot abundance in semiflexible chains with crowders of different sizes: a Monte Carlo study of DNA chains

We use stochastic simulation techniques to sample the conformational space of linear semiflexible polymers in a crowded medium and study how the knotting properties depend on the crowder size and concentration. The abundance of physical knots in the chains, which for definiteness we model on 10 kb long DNA filaments, is shown to have a non-monotonic, unimodal dependence on the colloid diameter, dc. The maximum incidence of knots occurs when dc is about equal to half of the gyration radius of the isolated chain. The degree of enhancement of knots grows rapidly with the solution density and can be very conspicuous relative to the case of isolated chains with no crowders. For instance, at 30% volume fraction the relative increase is more than fourfold. This dramatic enhancement is shown to originate from the depletion-induced chain compaction over multiple and concurring length scales. The same effect accounts for the variations of the knot length that accompany the changes in knotting probability. The findings suggest that crowded media could be viably used as a passive physical means for controlling and modulating the incidence and length of knots in DNA and other types of semiflexible polymers.

Translocation dynamics of knotted polymers under a constant or periodic external field

We perform Brownian dynamics simulations to examine how knots alter the dynamics of polymers moving through nanopores under an external field. In the first part of this paper, we study the situation when the field is constant. Here, knots halt translocation above a critical force with jamming occurring at smaller forces for twist topologies compared to non-twist topologies. Slightly below the jamming transition, the polymer's transit times exhibit large fluctuations. This phenomenon is an example of the knot's molecular individualism since the conformation of the knot plays a large role in the chain's subsequent dynamics. In the second part of the paper, we study the motion of the chain when one cycles the field on and off. If the off time is comparable to the knot's relaxation time, one can adjust the swelling of the knot at the pore and hence design strategies to ratchet the polymer in a controllable fashion. We examine how the off time affects the ratcheting dynamics. We also examine how this strategy alters the fluctuations in the polymer's transit time. We find that cycling the force field can reduce fluctuations near the knot's jamming transition, but can enhance the fluctuations at very high forces since knots get trapped in metastable states during the relaxation process. The latter effect appears to be more prominent for non-torus topologies than torus ones. We conclude by discussing the feasibility of this approach to control polymer motion in biotechnology applications such as sequencing.

Tightening slip knots in raw and degummed silk to increase toughness without losing strength

Knots are fascinating topological elements, which can be found in both natural and artificial systems. While in most of the cases, knots cannot be loosened without breaking the strand where they are tightened, herein, attention is focused on slip or running knots, which on the contrary can be unfastened without compromising the structural integrity of their hosting material. Two different topologies are considered, involving opposite unfastening mechanisms, and their influence on the mechanical properties of natural fibers, as silkworm silk raw and degummed single fibers, is investigated and quantified. Slip knots with optimized shape and size result in a significant enhancement of fibers energy dissipation capability, up to 300–400%, without affecting their load bearing capacity.

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.

Untangling the mechanics and topology in the frictional response of long overhand elastic knots

We combine experiments and theory to study the mechanics of overhand knots in slender elastic rods under tension. The equilibrium shape of the knot is governed by an interplay between topology, friction, and bending. We use precision model experiments to quantify the dependence of the mechanical response of the knot as a function of the geometry of the self-contacting region, and for different topologies as measured by their crossing number. An analytical model based on the nonlinear theory of thin elastic rods is then developed to describe how the physical and topological parameters of the knot set the tensile force required for equilibrium. Excellent agreement is found between theory and experiments for overhand knots over a wide range of crossing numbers.

Molecular knots in biology and chemistry

Knots and entanglements are ubiquitous. Beyond their aesthetic appeal, these fascinating topological entities can be either useful or cumbersome. In recent decades, the importance and prevalence of molecular knots have been increasingly recognised by scientists from different disciplines. In this review, we provide an overview on the various molecular knots found in naturally occurring biological systems (DNA, RNA and proteins), and those created by synthetic chemists. We discuss the current knowledge in these fields, including recent developments in experimental and, in some cases, computational studies which are beginning to shed light into the complex interplay between the structure, formation and properties of these topologically intricate molecules.

Stretching self-entangled DNA molecules in elongational fields

We present experiments of self-entangled DNA molecules stretching under a planar elongational field, and their stretching dynamics are compared to identical molecules without entanglements. Self-entangled molecules stretch in a stage-wise fashion, persisting in an "arrested" state for decades of strain prior to rapidly stretching, slowing down the stretching dynamics by an order of magnitude compared to unentangled molecules. Self-entangled molecules are shown to proceed through a transient state where one or two ends of the molecule are protruding from an entangled, knotted core. This phenomenon sharply contrasts with the wide array of transient configurations shown here and by others for stretching polymers without entanglements. The rate at which self-entangled molecules stretch through this transient state is demonstrably slower than unentangled molecules, providing the first direct experimental evidence of a topological friction. These experimental observations are shown to be qualitatively and semi-quantitatively reproduced by a dumbbell model with two fitting parameters, the values of which are reasonable in light of previous experiments of knotted DNA.

Absence of knots in known RNA structures

The ongoing effort to detect and characterize physical entanglement in biopolymers has so far established that knots are present in many globular proteins and also, abound in viral DNA packaged inside bacteriophages. RNA molecules, however, have not yet been systematically screened for the occurrence of physical knots. We have accordingly undertaken the systematic profiling of the several thousand RNA structures present in the Protein Data Bank (PDB). The search identified no more than three deeply knotted RNA molecules. These entries are rRNAs of about 3,000 nt solved by cryo-EM. Their genuine knotted state is, however, doubtful based on the detailed structural comparison with homologs of higher resolution, which are all unknotted. Compared with the case of proteins and viral DNA, the observed incidence of knots in available RNA structures is, therefore, practically negligible. This fact suggests that either evolutionary selection or thermodynamic and kinetic folding mechanisms act toward minimizing the entanglement of RNA to an extent that is unparalleled by other types of biomolecules. A possible general strategy for designing synthetic RNA sequences capable of self-tying in a twist-knot fold is finally proposed.

Knot theory realizations in nematic colloids

Nematic braids are reconfigurable knots and links formed by the disclination loops that entangle colloidal particles dispersed in a nematic liquid crystal. We focus on entangled nematic disclinations in thin twisted nematic layers stabilized by 2D arrays of colloidal particles that can be controlled with laser tweezers. We take the experimentally assembled structures and demonstrate the correspondence of the knot invariants, constructed graphs, and surfaces associated with the disclination loop to the physically observable features specific to the geometry at hand. The nematic nature of the medium adds additional topological parameters to the conventional results of knot theory, which couple with the knot topology and introduce order into the phase diagram of possible structures. The crystalline order allows the simplified construction of the Jones polynomial and medial graphs, and the steps in the construction algorithm are mirrored in the physics of liquid crystals.

Untying Knotted DNA with Elongational Flows

We present Brownian dynamics simulations of initially knotted double-stranded DNA molecules untying in elongational flows. We show that the motions of the knots are governed by a diffusion-convection equation by deriving scalings that collapse the simulation data. When being convected, all knots displace nonaffinely, and their rates of translation along the chain are topologically dictated. We discover that torus knots "corkscrew" when driven by flow, whereas nontorus knots do not. We show that a simple mechanism can explain a coupling between this rotation and the translation of a knot, explaining observed differences in knot translation rates. These types of knots are encountered in nanoscale manipulation of DNA, occur in biology at multiple length scales (DNA to umbilical cords), and are ubiquitous in daily life (e.g., hair). These results may have a broad impact on manipulations of such knots via flows, with applications to genomic sequencing and polymer processing.

Tangling of tethered swimmers: interactions between two nematodes

The tangling of two tethered microswimming worms serving as the ends of "active strings" is investigated experimentally and modeled analytically. C. elegans nematodes of similar size are caught by their tails using micropipettes and left to swim and interact at different separations over long times. The worms are found to tangle in a reproducible and statistically predictable manner, which is modeled based on the relative motion of the worm heads. Our results provide insight into the intricate tangling interactions present in active biological systems.

Knotting and Unknotting Dynamics of DNA Strands in Nanochannels

The self-knotting dynamics of DNA strands confined in nanochannels is studied with Brownian simulations. The model DNA chains are several microns long and placed inside channels that are 50-300 nm wide. This width range covers the transition between different metric scaling regimes and the concomitant drop of DNA knotting probability for channel widths below ∼75 nm. We find that knots typically originate from deep looping and backfoldings of the chain ends. Upon lowering the channel width, backfoldings become shallower and rarer and the lifetime of knots decreases while that of unknots increases. This lifetimes interplay causes the dramatic reduction of knots incidence for increasing confinement. The results can aid the design of nanochannels capable of harnessing the self-knotting dynamics to quench or relax the DNA topological state as desired.

Driving knots on DNA with AC/DC electric fields: topological friction and memory effects

The dynamical properties of entangled polyelectrolytes are investigated theoretically and computationally for a proposed novel micromanipulation setup. Specifically, we investigate the effects of DC and AC electric fields acting longitudinally on knotted DNA chains, modelled as semiflexible chains of charged beads, under mechanical tension. We consider various experimentally accessible values of the field amplitude and frequency as well as several of the simplest knot types. In particular, we consider both torus and twist knots because they are respectively known to be able or unable to slide along macroscopic threads and ropes. Strikingly, this qualitative distinction disappears in this microscopic context because all the considered knot types acquire a systematic drift in the direction of the electric force. Notably, the knot drift velocity and diffusion coefficient in zero field (both measurable also experimentally) can be used to define a characteristic "frictional" lengthscale for the various knot types. This previously unexplored length provides valuable information on the extent of self-interactions in the nominal knotted region. It is finally observed that the motion of a knot can effectively follow the AC field only if the driving period is larger than the knot relaxation time (for which the self-diffusion time provides an upper bound). These results suggest that salient aspects of the intrinsic dynamics of knots in DNA chains could be probed experimentally by means of external, time-dependent electric fields.

The illusive search for the lowest free energy state of globular proteins and RNAs

As a consequence of the one-dimensional storage and transfer of genetic information, DNA→RNA→protein, the process by which globular proteins and RNAs achieve their three-dimensional structure involves folding of a linear chain. The folding process itself could create massive activation barriers that prevent the attainment of many stable protein and RNA structures. We consider several kinds of energy barriers inherent in folding that might serve as kinetic constraints to achieving the lowest energy state. Alternative approaches to forming 3D structure, where a substantial number of weak interactions would be created prior to the formation of all the peptide (or phosphodiester) bonds, might not be subjected to such high barriers. This could lead to unique 3D conformational states, potentially more stable than "native" proteins and RNAs, with new functionalities.

Computational study on the progressive factorization of composite polymer knots into separated prime components

Using Monte Carlo simulations and advanced knot localization methods, we analyze the length and distribution of prime components in composite knots tied on freely jointed rings. For increasing contour length, we observe the progressive factorization of composite knots into separated prime components. However, we observe that a complete factorization, equivalent to the "decorated ring" picture, is not obtained even for rings of contour lengths N ≃ 3 N(0), about tens of times the most probable length of the prime knots tied on the rings. The decorated ring hypothesis has been used in the literature to justify the factorization of composite knot probabilities into the knotting probabilities of their prime components. Following our results, we suggest that such a hypothesis may not be necessary to explain the factorization of the knotting probabilities, at least when polymers excluding volume is not relevant. We rationalize the behavior of the system through a simple one-dimensional model in which prime knots are replaced by slip links randomly placed on a circle, with the only constraint being that the length of the loops has the same distribution as that of the length of the corresponding prime knots.

Topological jamming of spontaneously knotted polyelectrolyte chains driven through a nanopore

The advent of solid state nanodevices allows for interrogating the physicochemical properties of a polyelectrolyte chain by electrophoretically driving it through a nanopore. Salient dynamical aspects of the translocation process have been recently characterized by theoretical and computational studies of model polymer chains free from self-entanglement. However, sufficiently long equilibrated chains are necessarily knotted. The impact of such topological "defects" on the translocation process is largely unexplored, and is addressed in this Letter. By using Brownian dynamics simulations on a coarse-grained polyelectrolyte model we show that knots, despite being trapped at the pore entrance, do not per se cause the translocation process to jam. Rather, knots introduce an effective friction that increases with the applied force, and practically halts the translocation above a threshold force. The predicted dynamical crossover, which is experimentally verifiable, ought to be relevant in applicative contexts, such as DNA nanopore sequencing.