Knotted Structures Research Articles

Knotted artifacts in predicted 3D RNA structures.

Unlike proteins, RNAs deposited in the Protein Data Bank do not contain topological knots. Recently, admittedly, the first trefoil knot and some lasso-type conformations have been found in experimental RNA structures, but these are still exceptional cases. Meanwhile, algorithms predicting 3D RNA models have happened to form knotted structures not so rarely. Interestingly, machine learning-based predictors seem to be more prone to generate knotted RNA folds than traditional methods. A similar situation is observed for the entanglements of structural elements. In this paper, we analyze all models submitted to the CASP15 competition in the 3D RNA structure prediction category. We show what types of topological knots and structure element entanglements appear in the submitted models and highlight what methods are behind the generation of such conformations. We also study the structural aspect of susceptibility to entanglement. We suggest that predictors take care of an evaluation of RNA models to avoid publishing structures with artifacts, such as unusual entanglements, that result from hallucinations of predictive algorithms.

Animating hydrogel knotbots with topology-invoked self-regulation

Steering soft robots in a self-regulated manner remains a grand challenge, which often requires continuous symmetry breaking and recovery steps for persistent motion. Although structural morphology is found significant for robotic functions, geometric topology has rarely been considered and appreciated. Here we demonstrate a series of knotbots, namely hydrogel-based robots with knotted structures, capable of autonomous rolling and spinning/rotating motions. With symmetry broken by external stimuli and restored by self-regulation, the coupling between self-constraint-induced prestress and photothermal strain animates the knotbots continuously. Experiments and simulations reveal that nonequilibrium processes are regulated dynamically and cooperatively by self-constraints, active deformations, and self-shadowing effect of the photo-responsive gel. The active motions enable the knotbots to execute tasks including gear rotation and rod climbing. This work paves the way to devise advanced soft robots with self-regulated sustainable motions by harnessing the topology.

The Arc Unknotting Game

Summary Who doesn’t like games? And when games can involve interesting, visual, mathematical objects like knot diagrams, all the better! In this paper, we propose a new 2-player game on knot diagrams, the Arc Unknotting Game, that uses hot-off-the-press insights about the structure of knots. This game differs from previously studied games on knot diagrams—it’s much more slippery, with available moves appearing and disappearing in perplexing ways throughout the game. Intrigued? Then come play with us!

Connecting literary cultures

What are the assumptions (institutional, individual, cultural, structural, artistic) of encounter andexchange? How can they be measured and tested? And how do they play out in creative writing andliterary communities in the region broadly defined as the Asia-Pacific? This paper argues thatmethodologies enabling an examination of the ethics and power relations inherent to interculturalencounters must be predicated on creative uncertainty; be collaborative; and be testable in the sensethat they allow artists and researchers to ‘meet’ structures of power and ethical knots through evolving,creative-led, iterative, practices. This paper is interested in what might be understood as a pedagogy ofencounter. The emergent methodology holds a number of overlapping principles including ethics as aprocess, holding (prepositional) space, and uncertainty and the not-yet-made.

Compression of a confined semiflexible polymer under direct and oscillating fields.

The folding transition of biopolymers from the coil to compact structures has attracted wide research interest in the past and is well studied in polymer physics. Recent seminal works on DNA in confined devices have shown that these long biopolymers tend to collapse under an external field, which is contrary to the previously reported stretching of the chain. In this work, we capture the compression of a confined semiflexible polymer under direct and oscillating fields using a coarse-grained computer simulation model in the presence of long-range hydrodynamics. In the case of a semiflexible polymer chain, the inhomogeneous hydrodynamic drag from the center to the periphery of the coil couples with the chain bending to cause a swirling movement of the chain segments, leading to structural intertwining and compaction. Contrarily, a flexible chain of the same length lacks such structural deformation and forms a well-established tadpole structure. While bending rigidity profoundly influences the chain's folding favorability, we also found that subject to the direct field, chains in stronger confinements exhibit substantial compaction, contrary to the one in moderate confinements or bulk where such compaction is absent. However, an alternating field within an optimum frequency can effectuate this compression even in moderate or no confinement. This field-induced collapse is a quintessential hydrodynamic phenomenon, resulting in intertwined knotted structures even for shorter chains, unlike other spontaneous knotting experiments where it happens exclusively for longer chains.

Trawl Grid Structure Design and Analysis Using the Finite Element Method

The structure of fishnet knots has been simplified in previous studies to facilitate the construction of numerical equations of the fishnet structure. This leads to errors in the dynamic analysis of the trawl mesh structure with water flow. In this study, the finite element method was used to analyze the interaction of the trawl mesh structure with the solid object in a dynamic explicit environment. At the same time, design variables were optimized through impact assessment and the displacement of grid cells. The results show that the polyamide (PA) material, a 0.4 mm cross-section, and a 25 mm mesh size are the optimal choices. When the displacement speed of the solid body increased, the displacement and collision values of the mesh structure tended to increase gradually along the quadratic curve. Confirmation tests performed on the tensile tester machine showed a good load-carrying capacity of up to 1280 MPa for trawl mesh structures using the PA material. The characteristic curve for the stress of the trawl mesh structure is shown through the higher-order curve.

AlphaFold predicts novel human proteins with knots.

The fact that proteins can have their chain formed in a knot is known for almost 30 years. However, as they are not common, only a fraction of such proteins is available in the Protein Data Bank. It was not possible to assess their importance and versatility up until now because we did not have access to the whole proteome of an organism, let alone a human one. The arrival of efficient machine learning methods for protein structure prediction, such as AlphaFold and RoseTTaFold, changed that. We analyzed all proteins from the human proteome (over 20,000) determined with AlphaFold in search for knots and found them in less than 2% of the structures. Using a variety of methods, including homolog search, clustering, quality assessment and visual inspection, we determined the nature of each of the knotted structures and classified it as either knotted, potentially knotted, or an artifact, and deposited all of them in a database available at: https://knotprot.cent.uw.edu.pl/alphafold. Overall, we found 51 credible knotted proteins (0.2% of human proteome). The set of potentially knotted structures includes a new complex type of a knot not reported in proteins yet. That knot type, denoted 63 in mathematical notation, would necessitate a more complex folding path than any knotted protein characterized to date. This article is protected by copyright. All rights reserved.

Explicit formulae for Chern-Simons invariants of the hyperbolic J(2n,-2m) knot orbifolds

We calculate the Chern-Simons invariants of the hyperbolic double twist knot orbifolds using the Schläfli formula for the generalized Chern-Simons function on the family of cone-manifold structures of double twist knots.

Ligand induced dimerization and topology of a knotted spout methyltransferase.

Understanding the complex knotted topology in protein folding is an important structural and functional indication as increasing families of protein are discovered to contain deeply knotted structure. Methyltransferases in SpoU-TrmD (SPOUT) family catalyze the post-translational methylation of RNA assisted by the cofactor, S-Adenosyl methionine (SAM), and have a deeply folded trefoil knot that associated with the dimer interface and ligand binding domain. Using the “minimalism” of knotted methyltransferase, 1vh0, which only contains α/β SPOUT domain and a 31 trefoil knot, the biological function and structural change have been studied by cofactor binding. The complexity of the trefoil knot and localized frustration bring up the discussion about dimeric enzymatic active states formation. The monomeric apo-state of 1vh0 in solution is indicated by HSQC spectrum and the unfolding curve monitored by CD and fluorescence spectroscopy. In comparison, isothermal titration calorimetry (ITC) and NMR spectroscopy indicate that the holo-state of 1vh0 forms dimer. With the observed allosteric conformational change of the frustrated region that implicates knot formation, we conclude that the cofactor binding leads to allosteric conformational change, which induces dimerization.

Topologically protected vortex knots and links

In 1869, Lord Kelvin found that the way vortices are knotted and linked in an ideal fluid remains unchanged in evolution, and consequently hypothesized atoms to be knotted vortices in a ubiquitous ether, different knotting types corresponding to different types of atoms. Even though Kelvin’s atomic theory turned out incorrect, it inspired several important developments, such as the mathematical theory of knots and the investigation of knotted structures that naturally arise in physics. However, in previous studies, knotted and linked structures have been found to untie via local cut-and-paste events referred to as reconnections. Here, in contrast, we construct knots and links of non-Abelian vortices that are topologically protected in the sense that they cannot be dissolved employing local reconnections and strand crossings. Importantly, the topologically protected links are supported by a variety of physical systems such as dilute Bose-Einstein condensates and liquid crystals. We also propose a classification scheme for topological vortex links, in which two structures are considered equivalent if they differ from each other by a sequence of topologically allowed reconnections and strand crossings, in addition to the typical continuous transformations. Interestingly, this scheme produces a remarkably simple classification.

Dynamics of hot galactic winds launched from spherically-stratified starburst cores

ABSTRACT The analytic galactic wind model derived by Chevalier and Clegg in 1985 (CC85) assumes uniform energy and mass-injection within the starburst galaxy nucleus. However, the structure of nuclear star clusters, bulges, and star-forming knots are non-uniform. We generalize to cases with spherically-symmetric energy/mass injection that scale as r−Δ within the starburst volume R, providing solutions for Δ = 0, 1/2, 1, 3/2, and 2. In marked contrast with the CC85 model (Δ = 0), which predicts zero velocity at the centre, for a singular isothermal sphere profile (Δ = 2), we find that the flow maintains a constant Mach number of $\mathcal {M}=\sqrt{3/5} \simeq 0.77$ throughout the volume. The fast interior flow can be written as $v_{r \lt R} = (\dot{E}_T/3\dot{M}_T)^{1/2} \simeq 0.41 \, v_\infty$, where v∞ is the asymptotic velocity, and $\dot{E}_T$ and $\dot{M}_T$ are the total energy and mass injection rates. For $v_\infty \simeq 2000 \, \mathrm{km \, s^{-1}}$, $v_{r\lt R} \simeq 820 \, \mathrm{km\, s^{-1}}$ throughout the wind-driving region. The temperature and density profiles of the non-uniform models may be important for interpreting spatially-resolved maps of starburst nuclei. We compute velocity resolved spectra to contrast the Δ = 0 (CC85) and Δ = 2 models. Next generation X-ray space telescopes such as XRISM may assess these kinematic predictions.

Model for ring closure in ER tubular network dynamics

Tubular networks of the endoplasmic reticulum (ER) are dynamic structures whose steady-state conformations are maintained by a balance between the persistent generation and vanishing of the network elements. While factors producing the ER tubules and intertubular junctions have been investigated, the mechanisms behind their elimination remained unknown. Here, we addressed the ER ring closure, the process resulting in the tubule and junction removal through constriction of the network unit cells into junctional knots followed by the knot remodeling into regular junctions. We considered the ring closure to be driven by the tension existing in ER membranes. We based our consideration on the notion of Gibbs’ thermodynamic tension and reviewed its relationship to other tension definitions used in the literature. We modeled, computationally, the structures of the junctional knots containing internal nanopores and analyzed their tension dependence. We analyzed the process of the pore sealing through membrane fission resulting in the formation of regular junctions. Considering the hemi-fission as the rate-limiting stage of the fission reaction, we evaluated the membrane tensions guaranteeing the spontaneous character of the pore sealing. We concluded that feasible membrane tensions explain all stages of the ER ring closure.

How Cooperatively Folding Are Homopolymer Molecular Knots?

Detailed thermodynamic analysis of complex systems with multiple stable configurational states allows for insight into the cooperativity of each individual transition. In this work we derive a heat capacity decomposition comprising contributions from each individual configurational state, which together sum to a baseline heat capacity, and contributions from each state-to-state transition. We apply this analysis framework to a series of replica exchange molecular dynamics simulations of linear and 1-1 coarse-grained homo-oligomer models which fold into stable, configurationally well-defined molecular knots, in order to better understand the parameters leading to stable and cooperative folding of these knots. We find that a stiff harmonic backbone bending angle potential is key to achieving knots with specific 3D structures. Tuning the backbone equilibrium angle in small increments yields a variety of knot topologies, including $3_1$, $5_1$, $7_1$, and $8_{19}$ types. Populations of different knotted states as functions of temperature can also be manipulated by tuning backbone torsion stiffness or by adding side chain beads. We find that sharp total heat capacity peaks for the homo-oligomer knots are largely due to a coil-to-globule transition, rather than a cooperative knotting step. However, in some cases the cooperativity of globule-to-knot and coil-to-globule transitions are comparable, suggesting that highly cooperative folding to knotted structures can be achieved by refining the model parameters or adding sequence specificity.

Study of Spin-Crossover in Iron(II) Complexes with 2,6-Bis(4,5-Dimethyl-1H-Imidazol-2-yl)Pyridine and closo-Borate Anions

New iron(II) coordination compounds with 2,6-bis(4,5-dimethyl-1H-imidazol-2-yl)pyridine (L) and closo-borate(2–) anions [FeL2]B10H10⋅2H2O and [FeL2]B12H12⋅H2O have been synthesized. The compounds have been identified and studied by CHN analysis, electron spectroscopy (diffuse reflection spectroscopy), IR, Mössbauer, and EXAFS spectroscopies, X-ray powder diffraction, and static magnetic susceptibility. The structures of the coordination knots of complexes [FeL2]B10H10⋅2H2O and [FeL2]B12H12⋅H2O has been obtained by modeling the EXAFS spectra. The ligand is coordinated by the iron(II) ion in a tridentate-cyclic manner by two nitrogen atoms of imidazole cycles and a nitrogen atom of pyridine to form the FeN6 coordination knot. The study of the temperature dependence of the magnetic susceptibility in the range of 80–500 K has showed that the high-temperature spin-crossover 1А1 ↔ 5Т2 manifests itself in the obtained compounds.

Cell-Free Expression to Probe Co-Translational Insertion of an Alpha Helical Membrane Protein.

The majority of alpha helical membrane proteins fold co-translationally during their synthesis on the ribosome. In contrast, most mechanistic folding studies address refolding of full-length proteins from artificially induced denatured states that are far removed from the natural co-translational process. Cell-free translation of membrane proteins is emerging as a useful tool to address folding during translation by a ribosome. We summarise the benefits of this approach and show how it can be successfully extended to a membrane protein with a complex topology. The bacterial leucine transporter, LeuT can be synthesised and inserted into lipid membranes using a variety of in vitro transcription translation systems. Unlike major facilitator superfamily transporters, where changes in lipids can optimise the amount of correctly inserted protein, LeuT insertion yields are much less dependent on the lipid composition. The presence of a bacterial translocon either in native membrane extracts or in reconstituted membranes also has little influence on the yield of LeuT incorporated into the lipid membrane, except at high reconstitution concentrations. LeuT is considered a paradigm for neurotransmitter transporters and possesses a knotted structure that is characteristic of this transporter family. This work provides a method in which to probe the formation of a protein as the polypeptide chain is being synthesised on a ribosome and inserting into lipids. We show that in comparison with the simpler major facilitator transporter structures, LeuT inserts less efficiently into membranes when synthesised cell-free, suggesting that more of the protein aggregates, likely as a result of the challenging formation of the knotted topology in the membrane.

Self-Localized Liquid Crystal Micro-Droplet Arrays on Chemically Patterned Surfaces

Liquid crystal (LC) micro-droplet arrays are elegant systems that have a range of applications, such as chemical and biological sensing, due to a sensitivity to changes in surface properties and strong optical activity. In this work, we utilize self-assembled monolayers (SAMs) to chemically micro-pattern surfaces with preferred regions for LC occupation. Exploiting discontinuous dewetting, dragging a drop of fluid over the patterned surfaces demonstrates a novel, high-yield method of confining LC in chemically defined regions. The broad applicability of this method is demonstrated by varying the size and LC phase of the droplets. Although the optical textures of the droplets are dictated by topological constraints, the additional SAM interface is shown to lock in inhomogeneous alignment. The surface effects are highly dependent on size, where larger droplets exhibit asymmetric director configurations in nematic droplets and highly knotted structures in cholesteric droplets.

Circuit Topology for Bottom-Up Engineering of Molecular Knots

The art of tying knots is exploited in nature and occurs in multiple applications ranging from being an essential part of scouting programs to engineering molecular knots. Biomolecular knots, such as knotted proteins, bear various cellular functions, and their entanglement is believed to provide them with thermal and kinetic stability. Yet, little is known about the design principles of naturally evolved molecular knots. Intra-chain contacts and chain entanglement contribute to the folding of knotted proteins. Circuit topology, a theory that describes intra-chain contacts, was recently generalized to account for chain entanglement. This generalization is unique to circuit topology and not motivated by other theories. In this conceptual paper, we systematically analyze the circuit topology approach to a description of linear chain entanglement. We utilize a bottom-up approach, i.e., we express entanglement by a set of four fundamental structural units subjected to three (or five) binary topological operations. All knots found in proteins form a well-defined, distinct group which naturally appears if expressed in terms of these basic structural units. We believe that such a detailed, bottom-up understanding of the structure of molecular knots should be beneficial for molecular engineering.

Advancing mathematics by guiding human intuition with AI

The practice of mathematics involves discovering patterns and using these to formulate and prove conjectures, resulting in theorems. Since the 1960s, mathematicians have used computers to assist in the discovery of patterns and formulation of conjectures1, most famously in the Birch and Swinnerton-Dyer conjecture2, a Millennium Prize Problem3. Here we provide examples of new fundamental results in pure mathematics that have been discovered with the assistance of machine learning—demonstrating a method by which machine learning can aid mathematicians in discovering new conjectures and theorems. We propose a process of using machine learning to discover potential patterns and relations between mathematical objects, understanding them with attribution techniques and using these observations to guide intuition and propose conjectures. We outline this machine-learning-guided framework and demonstrate its successful application to current research questions in distinct areas of pure mathematics, in each case showing how it led to meaningful mathematical contributions on important open problems: a new connection between the algebraic and geometric structure of knots, and a candidate algorithm predicted by the combinatorial invariance conjecture for symmetric groups4. Our work may serve as a model for collaboration between the fields of mathematics and artificial intelligence (AI) that can achieve surprising results by leveraging the respective strengths of mathematicians and machine learning.

Strain concentration effects of wood knots under longitudinal tension obtained through digital image correlation

Knots are natural, inevitable structures in wood left by branches. Most of the negative mechanical effects of knots are caused by strain concentration effects and fiber variability. In this study, digital image correlation (DIC) was used to observe the global strain distribution around intergrown knots, encased knots, knotholes, and artificial holes in thin Japanese cedar ( Cryptomeria japonica ) tangential plates of different sizes subjected to tensile load. The results indicated that the strain around knots is more homogeneous than that around holes, and that the influence of wood anatomy is greater than that of the geometric factors of a hole. Knots have naturally optimized structures for branch junctions, altering wood properties, such as grain angle, microfibril angle, and branch collar, to disperse strain and prevent fractures. The fractures caused by four types of knots or holes differed. The specimens with artificial holes were the most consistent with strain concentration theory, but none of the specimens within the other three natural knots or holes exhibited fractures around knots. Although the strain peak of natural knots occurred at knot edges, this was unrelated to the fractures. Finally, the strain characteristics of the knot tissue revealed that the internal structure of intergrown knots was completely connected with surrounding tissues, meaning that they can deform synchronously with the specimen. By contrast, encased knots tissue exhibited an extremely low modulus in the transverse direction of the specimens (tangent axis of wood) because the strain was higher in the knot tissue than in the surrounding wood. • Anatomy influenced more than the geometry of a hole in strain concentration. • Strain around the knots is more homogeneous than the hole. • Fracture causes of knots or holes are different and categorized in four types. • Strain characteristics of the intergrown knot can reveal its internal structure. • Loose knots has higher strain than the vicinity in the transverse direction.

Construction and application of knotted acoustic fields with intensity maxima

Knots have always played an important role in the life sciences because of their complex topology. Some previous investigations have shown that an optical field can be modulated into knot structures, and a knotted trap formed by light beams has also been demonstrated. Very recently, it has also been demonstrated that an acoustic vortex field with phase singularities can be tied into a knotted structure. However, for knotted tweezers, we need to use the relative maximum points of the amplitude distribution to construct the knotted field (although it is still not known how to create such a knotted line acoustic field) which is beneficial for particle trapping into knotted shapes. In this work we propose a method to generate acoustic fields with knotted intensity maxima in three dimensions. Based on the finite element method and angular spectrum theory, we prove that both Hopf link and trefoil knot lines in acoustic fields can be generated by the designed holograms. Furthermore, under particle tracking simulation in the time domain, we demonstrate that the knotted line acoustic fields can be used to capture particles into different topologies in three dimensions.