Trefoil Knot Research Articles

Construction of a molecular prime link by interlocking two trefoil knots

Construction of a molecular prime link by interlocking two trefoil knots

Creation of quantum knots and links driven by minimal surfaces

Using an improved numerical code we investigate the creation and evolution of quantum knots and links as defects of the Gross–Pitaevskii equation. The particular constraints put on quantum hydrodynamics make this an ideal context for application of geometric and topological methods to investigate dynamical properties. Evolutionary processes are classified into three generic scenarios representing (i) direct topological cascade and collapse, (ii) structural and topological cycles, and (iii) inverse topological cascade of complex structures. Several examples and test cases are studied; the head-on collision of quantum vortex rings and the creation of a trefoil knot from initially unlinked, unknotted loops are realized for the first time. Each type of scenario is studied by carrying out a detailed evaluation of fundamental geometric and dynamical properties associated with evolution. Direct topological cascade that governs the decay of complex structures to small-scale vortex rings is identified by writhe measures, while picks of total curvature are found to provide a clear signature of reconnection events. We demonstrate that isophase minimal surfaces spanning knots and links have a privileged role in the decay process by detecting surface energy relaxation of complex structures. Minimal surfaces are shown to be critical markers for energy and prove to be appropriate detectors for the evolution of complex systems.

Character Varieties and Algebraic Surfaces for the Topology of Quantum Computing

It is shown that the representation theory of some finitely presented groups thanks to their SL2(C) character variety is related to algebraic surfaces. We make use of the Enriques–Kodaira classification of algebraic surfaces and related topological tools to make such surfaces explicit. We study the connection of SL2(C) character varieties to topological quantum computing (TQC) as an alternative to the concept of anyons. The Hopf link H, whose character variety is a Del Pezzo surface fH (the trace of the commutator), is the kernel of our view of TQC. Qutrit and two-qubit magic state computing, derived from the trefoil knot in our previous work, may be seen as TQC from the Hopf link. The character variety of some two-generator Bianchi groups, as well as that of the fundamental group for the singular fibers E˜6 and D˜4 contain fH. A surface birationally equivalent to a K3 surface is another compound of their character varieties.

Inside Back Cover: Self‐Assembly of Molecular Trefoil Knots Featuring Pentadecanuclear Homoleptic AuI‐, AuI/AgI‐, or AuI/CuI‐Alkynyl Coordination (Angew. Chem. Int. Ed. 21/2022)

Molecular trefoil knots are usually covalently assembled from rationally designed building blocks with multiple functional groups. In their Research Article (e202200748), Jie-Sheng Huang, Chi-Ming Che, and co-workers report on the use of monoalkynes and a mononuclear AuI compound for the self-assembly of Au−C bonded Au15 trefoil knots featuring non-covalent AuI–AuI interactions. The structures, solution behavior, and photoluminescence properties of the Au15 and Au9M6 (M=Cu/Ag) trefoil knots were studied.

A Trefoil Knot Polymer Chain Translocates through a Funnel-like Channel: A Multi-Particle Collision Dynamics Study.

With combining multi-particle collision dynamics (MPCD) for the solvent and molecular dynamics (MD) for the polymer chains, we have studied the conformation and untying behaviors of a trefoil knot polymer chain translocated through a confined funnel-like channel. For the trefoil knot chain, we found that the untying knot behavior mostly happens during the translocation process, and the translocation behavior of linear chains is also simulated as a comparison. Some characteristics of the trefoil knot chain during translocation process, such as average gyration radius <Rg> and the average end-to-end distances <S> are discussed, and we statistic the scale relations of the translocation time versus the chain length, and that of the chain rigidity. This study may help to understand translocation behaviors of the knotted linear polymer chain in the capillary flow field.

Self‐Assembly of Molecular Trefoil Knots Featuring Pentadecanuclear Homoleptic AuI‐, AuI/AgI‐, or AuI/CuI‐Alkynyl Coordination

AbstractMetal‐free and metal‐containing molecular trefoil knots are fascinating ensembles that are usually covalently assembled, the latter requiring the rational design of di‐ or multidentate/multipodal ligands as connectors. In this work, we describe the self‐assembly of pentadecanuclear AuI trefoil knots [Au15(C≡CR)15] from monoalkynes HC≡CR (R=9,9‐X2‐fluorenyl with X=nBu, n‐hexyl) and [AuI(THT)Cl]. Hetero‐bimetallic counterparts [Au9M6(C≡CR)15] (M=Cu/Ag) were self‐assembled by reactions of [Au15(C≡CR)15] with [Cu(MeCN)4]+/AgNO3 and HC≡CR. The type of pentadecanuclear trefoil knots described herein is characterized by X‐ray crystallography, 2D NMR and HR‐ESI‐MS. [Au9Cu6(C≡CR)15] is relatively stable in hexane; its excited state properties were investigated. DFT calculations revealed that non‐covalent metal–metal and metal–ligand interactions, together with longer alkyl chain‐strengthened inter‐ligand dispersion interactions, govern the stability of the trefoil knot structures.

Self-Assembly of Molecular Trefoil Knots Featuring Pentadecanuclear Homoleptic AuI -, AuI /AgI -, or AuI /CuI -Alkynyl Coordination.

Metal-free and metal-containing molecular trefoil knots are fascinating ensembles that are usually covalently assembled, the latter requiring the rational design of di- or multidentate/multipodal ligands as connectors. In this work, we describe the self-assembly of pentadecanuclear AuI trefoil knots [Au15 (C≡CR)15 ] from monoalkynes HC≡CR (R=9,9-X2 -fluorenyl with X=nBu, n-hexyl) and [AuI (THT)Cl]. Hetero-bimetallic counterparts [Au9 M6 (C≡CR)15 ] (M=Cu/Ag) were self-assembled by reactions of [Au15 (C≡CR)15 ] with [Cu(MeCN)4 ]+ /AgNO3 and HC≡CR. The type of pentadecanuclear trefoil knots described herein is characterized by X-ray crystallography, 2D NMR and HR-ESI-MS. [Au9 Cu6 (C≡CR)15 ] is relatively stable in hexane; its excited state properties were investigated. DFT calculations revealed that non-covalent metal-metal and metal-ligand interactions, together with longer alkyl chain-strengthened inter-ligand dispersion interactions, govern the stability of the trefoil knot structures.

Shortest paths of trefoil knot designs using Rubik’s Snakes

Shortest paths of trefoil knot designs using Rubik’s Snakes

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.

Revealing Topological Barriers against Knot Untying in Thermal and Mechanical Protein Unfolding by Molecular Dynamics Simulations.

The knot is one of the most remarkable topological features identified in an increasing number of proteins with important functions. However, little is known about how the knot is formed during protein folding, and untied or maintained in protein unfolding. By means of all-atom molecular dynamics simulation, here we employ methyltransferase YbeA as the knotted protein model to analyze changes of the knotted conformation coupled with protein unfolding under thermal and mechanical denaturing conditions. Our results show that the trefoil knot in YbeA is occasionally untied via knot loosening rather than sliding under enhanced thermal fluctuations. Through correlating protein unfolding with changes in the knot position and size, several aspects of barriers that jointly suppress knot untying are revealed. In particular, protein unfolding is always prior to knot untying and starts preferentially from separation of two α-helices (α1 and α5), which protect the hydrophobic core consisting of β-sheets (β1–β4) from exposure to water. These β-sheets form a loop through which α5 is threaded to form the knot. Hydrophobic and hydrogen bonding interactions inside the core stabilize the loop against loosening. In addition, residues at N-terminal of α5 define a rigid turning to impede α5 from sliding out of the loop. Site mutations are designed to specifically eliminate these barriers, and easier knot untying is achieved under the same denaturing conditions. These results provide new molecular level insights into the folding/unfolding of knotted proteins.

A Topological Selection of Folding Pathways from Native States of Knotted Proteins

Understanding how knotted proteins fold is a challenging problem in biology. Researchers have proposed several models for their folding pathways, based on theory, simulations and experiments. The geometry of proteins with the same knot type can vary substantially and recent simulations reveal different folding behaviour for deeply and shallow knotted proteins. We analyse proteins forming open-ended trefoil knots by introducing a topologically inspired statistical metric that measures their entanglement. By looking directly at the geometry and topology of their native states, we are able to probe different folding pathways for such proteins. In particular, the folding pathway of shallow knotted carbonic anhydrases involves the creation of a double-looped structure, contrary to what has been observed for other knotted trefoil proteins. We validate this with Molecular Dynamics simulations. By leveraging the geometry and local symmetries of knotted proteins’ native states, we provide the first numerical evidence of a double-loop folding mechanism in trefoil proteins.

Chaos and integrability in -geometry

We review the integrability of the geodesic flow on a threefold admitting one of the three group geometries in Thurston’s sense. We focus on the case. The main examples are the quotients , where is a cofinite Fuchsian group. We show that the corresponding phase space contains two open regions with integrable and chaotic behaviour, with zero and positive topological entropy, respectively. As a concrete example we consider the case of the modular threefold with the modular group . In this case is known to be homeomorphic to the complement of a trefoil knot in a 3-sphere. Ghys proved the remarkable fact that the lift of a periodic geodesic on the modular surface to produces the same isotopy class of knots as that which appears in the chaotic version of the celebrated Lorenz system and was studied in detail by Birman and Williams. We show that these knots are replaced by trefoil knot cables in the integrable limit of the geodesic system on . Bibliography: 60 titles.

Taking no risk is the greatest risk: Coping with the paradoxes of Outcome-based services

Traditional manufacturing firms take more operational responsibility and business risk when offering their customers outcome-based services (OBS). Research on servitization recognizes multiple paradoxical tensions between product- and service business that manufacturers must to accept and cope with. Using an explorative case study approach on three OBS providers, we find that the paradoxes related to this extreme servitization business model are related to control, knowledge and dependency. The found paradoxes are interlinked through an insoluble trefoil knot, which will persist while the OBS relationship continues to exist. Thus, OBS provider firms’ managers cope with the paradoxical tensions using coping strategies we coin commitment, acknowledgement and alliancing. Overall, the current study contributes to the paradox theory – an alternative to traditional organization theories. More specifically, we contribute to paradoxes in servitization literature by elaborating the interplay of paradoxical tensions in an advanced form of servitization, that is, OBS.

A Specific Set of Heterogeneous Native Interactions Yields Efficient Knotting in Protein Folding.

Native interactions are crucial for folding, and non-native interactions appear to be critical for efficiently knotting proteins. Therefore, it is important to understand both their roles in the folding of knotted proteins. It has been proposed that non-native interactions drive the correct order of contact formation, which is essential to avoid backtracking and efficiently self-tie. In this study, we ask if non-native interactions are strictly necessary to tangle a protein or if the correct order of contact formation can be assured by a specific set of native, but otherwise heterogeneous (i.e., having distinct energies), interactions. In order to address this problem, we conducted extensive Monte Carlo simulations of lattice models of protein-like sequences designed to fold into a preselected knotted conformation embedding a trefoil knot. We were able to identify a specific set of heterogeneous native interactions that drives efficient knotting and is able to fold the protein when combined with the remaining native interactions modeled as homogeneous. This specific set of heterogeneous native interactions is strictly enough to efficiently self-tie. A distinctive feature of these native interactions is that they do not backtrack because their energies ensure the correct order of contact formation. Furthermore, they stabilize a knotted intermediate state, which is en route to the native structure. Our results thus show that-at least in the context of the adopted model-non-native interactions are not necessary to knot a protein. However, when they are taken into account in protein energetics, it is possible to find specific, nonlocal non-native interactions that operate as a scaffold that assists the knotting step.

Topologically non-trivial metal-organic assemblies inhibit β2-microglobulin amyloidogenesis

Inhibiting amyloid aggregation through high-turnover dynamic interactions could be an efficient strategy that is already used by small heat-shock proteins in different biological contexts. We report the interactions of three topologically non-trivial, zinc-templated metal-organic assemblies, a [2]catenane, a trefoil knot (TK), and Borromean rings, with two β 2 -microglobulin (β2m) variants responsible for amyloidotic pathologies. Fast exchange and similar patterns of preferred contact surface are observed by NMR, consistent with molecular dynamics simulations. In vitro fibrillation is inhibited by each complex, whereas the zinc-free TK induces protein aggregation and does not inhibit fibrillogenesis. The metal coordination imposes structural rigidity that determines the contact area on the β2m surface depending on the complex dimensions, ensuring in vitro prevention of fibrillogenesis. Administration of TK, the best protein-contacting species, to a disease-model organism, namely a Caenorhabditis elegan s mutant expressing the D76N β2m variant, confirms the bioactivity potential of the knot topology and suggests new developments. Topologically non-trivial (TnT) species may engage labile supramolecular interactions Proteins can be ideal targets of TnT metal-templated structures Protein fibrillogenesis can be inhibited by TnT metal-organic assemblies Rigidity and dimensions of the TnT assemblies modulate interactions with proteins Prakasam et al. report that topologically non-trivial metal-organic species inhibit the in vitro fibrillogenesis of the paradigmatic amyloidogen β 2 -microglobulin and the effects of the D76N mutant expression in a Caenorhabditis elegans disease model. Acting as small-molecule chaperones, the metal-templated assemblies suppress the formation of soluble protofibrillar oligomers by engaging labile supramolecular interactions that interfere with protein-protein pairing.

Negative differential resistance in all-benzene molecule of trefoil knot

Based on first-principles calculations, the electronic transport of a recently synthesized all-benzene molecule of trefoil knot is investigated, by contacted with Au electrodes. Negative differential resistance behavior is observed. Further analysis shows that, it is resulted from the weaken of the states on the molecule modulated by the bias voltage, which suppresses the transmission eigenchannels and transmission peaks. Moreover, such a NDR behavior is also found in knot molecules with more benzene rings, indicating NDR is the intrinsic feature of such kind of molecules and robust to the number of benzene rings. These findings are expected to be extended to many other molecules with similar geometries, and may throw light on the development of future nanodevices.

Lorenz attractors and the modular surface * *TP was supported by the Israel Science Foundation (Grant No. 1504/18).

We define an extension of the geometric Lorenz model, defined on the three sphere. This geometric model has an invariant one dimensional trefoil knot, a union of invariant manifolds of the singularities. It is similar to the invariant trefoil knot arising in the classical Lorenz flow near the classical parameters. We prove that this geometric model is topologically equivalent to the geodesic flow on the modular surface, once compactifying the latter.

Selective Construction of Trefoil knots and a Molecular Borromean Ring Induced by Steric Hindrance of Thioether Ligands.

Two Cp*-RhIII based trefoil knots were obtained in high yield under ambient conditions via the coordination-driven self-assembly of semi-rigid thioether dipyridyl ligand 1,4-bis[(pyridin-4-ylthio)methyl]benzene (L1 ), ligand chloranilic acid (H2 -CA) and 6,11-dihydroxytetracene-5,12-dione (H2 -TtDo) with Cp*RhIII metal corner units, respectively. Furthermore, using the bulkier 4,4'-{[(2,5-dimethyl-1,4-phenylene)bis(methylene)]bis(sulfanediyl)}dipyridine (L2 ) in the place of ligand L1 in the construction process resulted in the formation of a teranuclear metallacycle and a template-free Borromean ring in high yields thanks to significantly altered intermolecular forces between the constituent ligands induced by the sterically-hindering methyl groups of L2 , as demonstrated via a detailed X-ray crystallographic analysis and NMR spectroscopy.

Converging experimental and computational views of the knotting mechanism of a small knotted protein

MJ0366 from Methanocaldococcus jannaschii is the smallest topologically knotted protein known to date. 92 residues in length, MJ0366 ties a trefoil (31) knot by threading its C-terminal helix through a buttonhole formed by the remainder of the secondary structure elements. By generating a library of point mutations at positions pertinent to the knot formation, we systematically evaluated the contributions of individual residues to the folding stability and kinetics of MJ0366. The experimental Φ-values were used as restraints to computationally generate an ensemble of conformations that correspond to the transition state of MJ0366, which revealed several nonnative contacts. The importance of these nonnative contacts in stabilizing the transition state of MJ0366 was confirmed by a second round of mutagenesis, which also established the pivotal role of F15 in stapling the network of hydrophobic interactions around the threading C-terminal helix. Our converging experimental and computational results show that, despite the small size, the transition state of MJ0366 is formed at a very late stage of the folding reaction coordinate, following a polarized pathway. Eventually, the formation of extensive native contacts, as well as a number of nonnative ones, leads to the threading of the C-terminal helix that defines the topological knot.

Spatially Multiplexing of Metasurface for Manipulating the Focused Trefoil and Cinquefoil Vector Light Field.

The trefoil and cinquefoil vector field are of essential significance for fundamental topology properties as the Hopf link and trefoil knots in the light field. The spatially multiplexing metasurfaces were designed with two sets of periodical nanoslits arranged alternately, each had independent geometric spiral phases and metalens phases to produce and focus vortex of the corresponding circular polarized (CP) light. By arranging the orientations of the two slit sets, the two CP vortices of the desired topological charges were obtained, the superposition of the vortices were realized to generate the vector field. With the topological charges of the vortices set to one and two, and three and two, respectively, the focused trefoil and cinquefoil vector light fields were acquired. The work would be important in broadening the applications of metasurface in areas as vector beam generations and topology of light field.