Topological Knot Research Articles

Experimental detection of knotted conformations in denatured proteins

Structures that contain a knot formed by the path of the polypeptide backbone represent some of the most complex topologies observed in proteins. How or why these topological knots arise remains unclear. By developing a method to experimentally trap and detect knots in nonnative polypeptide chains, we find that two knotted methyltransferases, YibK and YbeA, can exist in a trefoil-knot conformation even in their chemically unfolded states. The unique denatured-state topology of these molecules explains their ability to efficiently fold to their native knotted structures in vitro and offers insights into the potential role of knots in proteins. Furthermore, the high prevalence of the denatured-state knots identified here suggests that they are either difficult to untie or that threading of any untied molecules is rapid and spontaneous. The occurrence of such knotted topologies in unfolded polypeptide chains raises the possibility that they could play an important, and as yet unexplored, role in folding and misfolding processes in vivo.

Studying uniform thickness I: Legendrian simple iterated torus knots

We prove that the class of topological knot types that are both Legendrian simple and satisfy the uniform thickness property (UTP) is closed under cabling. An immediate application is that all iterated cabling knot types that begin with negative torus knots are Legendrian simple. We also examine, for arbitrary numbers of iterations, iterated cablings that begin with positive torus knots, and establish the Legendrian simplicity of large classes of these knot types, many of which also satisfy the UTP. In so doing we obtain new necessary conditions for both the failure of the UTP and Legendrian non-simplicity in the class of iterated torus knots, including specific conditions on knot types.

The diffeotopy group of S1×S2 via contact topology

AbstractAs shown by Gluck in 1962, the diffeotopy group of S1×S2 is isomorphic to ℤ2⊕ℤ2 ⊕ℤ2. Here an alternative proof of this result is given, relying on contact topology. We then discuss two applications to contact topology: (i) it is shown that the fundamental group of the space of contact structures on S1×S2, based at the standard tight contact structure, is isomorphic to ℤ; (ii) inspired by previous work of Fraser, an example is given of an integer family of Legendrian knots in S1×S2#S1×S2 (with its standard tight contact structure) that can be distinguished with the help of contact surgery, but not by the classical invariants (topological knot type, Thurston–Bennequin invariant, and rotation number).

Untangling the folding mechanism of the 52‐knotted protein UCH‐L3

Proteins possessing deeply embedded topological knots in their structure add a stimulating new challenge to the already complex protein-folding problem. The most complicated knotted topology observed to date belongs to the human enzyme ubiquitin C-terminal hydrolase UCH-L3, which is an integral part of the ubiquitin-proteasome system. The structure of UCH-L3 contains five distinct crossings of its polypeptide chain, and it adopts a 5(2)-knotted topology, making it a fascinating target for folding studies. Here, we provide the first in depth characterization of the stability and folding of UCH-L3. We show that the protein can unfold and refold reversibly in vitro without the assistance of molecular chaperones, demonstrating that all the information necessary for the protein to find its knotted native structure is encoded in the amino acid sequence, just as with any other globular protein, and that the protein does not enter into any deep kinetic traps. Under equilibrium conditions, the unfolding of UCH-L3 appears to be two-state, however, multiphasic folding and unfolding kinetics are observed and the data are consistent with a folding pathway in which two hyperfluorescent intermediates are formed. In addition, a very slow phase in the folding kinetics is shown to be limited by proline-isomerization events. Overall, the data suggest that a knotted topology, even in its most complex form, does not necessarily limit folding in vitro, however, it does seem to require a complex folding mechanism which includes the formation of several distinct intermediate species.

LEGENDRIAN RIBBONS IN OVERTWISTED CONTACT STRUCTURES

We show that a null-homologous transverse knot K in the complement of an overtwisted disk in a contact 3-manifold is the boundary of a Legendrian ribbon if and only if it possesses a Seifert surface S such that the self-linking number of K with respect to S satisfies sl (K, S) = -χ(S). In particular, every null-homologous topological knot type in an overtwisted contact manifold can be represented by the boundary of a Legendrian ribbon. Finally, we show that a contact structure is tight if and only if every Legendrian ribbon minimizes genus in its relative homology class.

Higher-order Alexander invariants and filtrations of the knot concordance group

We establish certain results for several filtrations of smooth and topological knot concordance groups. First, as regards n-solvable filtration of topological knot concordance group, C, defined by K. Orr, P. Teichner and first author: 0 ⊂... ⊂ F (n,.5) ⊂ F (n) ⊂... ⊂ F (1.5) ⊂ F (1.0) ⊂ F (0.5) ⊂ F (0) ⊂ C, we refine recent nontriviality results of Cochran and Teichner by including information on Alexander modules. These results also extend those of C. Livingston and second author. We exhibit similar structure in closely related symmetric Grope filtration ofC. We also show that Grope filtration of smooth concordance group is nontrivial using examples that cannot be distinguished by Ozsvath-Szabo τ-invariant nor by J. Rasmussen's s-invariant. Our broader contribution is to establish, in the relative case, key homological results whose analogues Cochran-Orr-Teichner established in the absolute case. We say two knots K 0 and K 1 are concordant modulo n-solvability if K o #(-K 1 ) ∈F (n) . Our main result is that, for any knot K whose classical Alexander polynomial has degree greater than 2, and for any positive integer n, there exist infinitely many knots K i that are concordant to K modulo n-solvability, but are all distinct modulo n.5-solvability. Moreover, K i and K share same classical Seifert matrix and Alexander module as well as sharing same higher-order Alexander modules and Seifert presentations up to order n - 1.

GEOMETRICAL AND TOPOLOGICAL MODELING OF SUPERCOILING IN SUPRAMOLECULAR STRUCTURES

Both microscopic and macroscopic worlds are plentiful in knotted and linked structures. But even at the mesoscopic level, works in molecular topology have shown that Nature builds DNA molecules, which form actual knots and links between themselves. Qualitative similarities may be observed between forms of the microscopic world, as well as between those of the macroscopic world; and by extracting the structures, it is possible to study their dynamics thanks to topological methods and techniques. The aim of this paper is to show that knots and links are ubiquitous scale-independent objects carrying a tremendous amount of precious information on the emergence of new forms and structures especially in living matter. In fact, knotting and unknotting are "universal" principles underlying these forms and structures. This study is focused on showing that differential geometry and topological knot's theory can be used, notably (a) to describe three-dimensional structures of DNA and protein-DNA complexes, and (b) to explain how geometrical supercoiling occur in the most fundamental supra-molecular structures and why it is essential for their biological activity.

Erratum: On iterated torus knots and transversal knots

In [3] the author’s Theorems 1.1 and 1.2, combined, implied that iterated torus knots are transversally simple. This result is in error and this erratum pinpoints the error. In [4] a more subtle result is proven resulting in giving a geometric realization of the Honda‐Etnyre transverse (2,3)‐cable of the (2,3)‐torus knot example (appendix joint with H Matsuda). 57M27, 57N16; 57R17, 37F20 In [3] the author’s Theorems 1.1 and 1.2, combined, implied that iterated torus knots are transversally simple. The previous best result in this direction had been the proof, due to John Etnyre, that positive torus knots are transversally simple. After the publication of [3], Etnyre and Honda argued in [2], using very different techniques from those in [3], that among the infinitely many Legendrian knots whose topological knot type is that of the .3;2/‐cable on the .3;2/‐torus knot, there must be one whose transversal pushoff cannot be transversally simple. This lead to a review of the proof in [3] and the discovery of an error.

Structural and evolutionary bioinformatics of the SPOUT superfamily of methyltransferases.

BackgroundSPOUT methyltransferases (MTases) are a large class of S-adenosyl-L-methionine-dependent enzymes that exhibit an unusual alpha/beta fold with a very deep topological knot. In 2001, when no crystal structures were available for any of these proteins, Anantharaman, Koonin, and Aravind identified homology between SpoU and TrmD MTases and defined the SPOUT superfamily. Since then, multiple crystal structures of knotted MTases have been solved and numerous new homologous sequences appeared in the databases. However, no comprehensive comparative analysis of these proteins has been carried out to classify them based on structural and evolutionary criteria and to guide functional predictions.ResultsWe carried out extensive searches of databases of protein structures and sequences to collect all members of previously identified SPOUT MTases, and to identify previously unknown homologs. Based on sequence clustering, characterization of domain architecture, structure predictions and sequence/structure comparisons, we re-defined families within the SPOUT superfamily and predicted putative active sites and biochemical functions for the so far uncharacterized members. We have also delineated the common core of SPOUT MTases and inferred a multiple sequence alignment for the conserved knot region, from which we calculated the phylogenetic tree of the superfamily. We have also studied phylogenetic distribution of different families, and used this information to infer the evolutionary history of the SPOUT superfamily.ConclusionWe present the first phylogenetic tree of the SPOUT superfamily since it was defined, together with a new scheme for its classification, and discussion about conservation of sequence and structure in different families, and their functional implications. We identified four protein families as new members of the SPOUT superfamily. Three of these families are functionally uncharacterized (COG1772, COG1901, and COG4080), and one (COG1756 represented by Nep1p) has been already implicated in RNA metabolism, but its biochemical function has been unknown. Based on the inference of orthologous and paralogous relationships between all SPOUT families we propose that the Last Universal Common Ancestor (LUCA) of all extant organisms contained at least three SPOUT members, ancestors of contemporary RNA MTases that carry out m1G, m3U, and 2'O-ribose methylation, respectively. In this work we also speculate on the origin of the knot and propose possible 'unknotted' ancestors. The results of our analysis provide a comprehensive 'roadmap' for experimental characterization of SPOUT MTases and interpretation of functional studies in the light of sequence-structure relationships.

Unraveling the equilibrium chain exchange kinetics of polymeric micelles using small-angle neutron scattering – architectural and topological effects

In this paper, we present a study of micellar structures formed by poly(styrene)-poly(butadiene) (PS10-PB10; the numbers indicate the molecular weight in kg mol−1) diblock copolymers and PB10-PS20-PB10 triblock copolymers in different n-alkane solvents. Particular emphasis is placed on the dynamic properties of these micelles under equilibrium which are studied using a novel time-resolved small-angle neutron scattering technique. The results show that the structures of the micelles are very similar for both the diblock and triblock copolymers, which allows a direct comparison of the dynamic properties. A novel logarithmic relaxation is found for both the triblock and the diblock micelles which is not consistent with theoretical expectations. However, for the diblock micelles, the relaxation kinetics seem to approach the rate and the single exponential decay predicted by Halperin & Alexander [Macromolecules, (1989), 22, 2403–2412] when the micellar cores are strongly swollen with solvent. For the triblock micelles a logarithmic relaxation is found for all cases as an effect of additional topological knots present even in highly swollen micellar cores. This behavior is assigned to an increased coupling of chain motion within the dense confined core – an effect which seems to vanish in diblock micelles when the core is sufficiently swollen.

The Dimerization of an α/β-Knotted Protein Is Essential for Structure and Function

alpha/beta-Knotted proteins are an extraordinary example of biological self-assembly; they contain a deep topological trefoil knot formed by the backbone polypeptide chain. Evidence suggests that all are dimeric and function as methyltransferases, and the deep knot forms part of the active site. We investigated the significance of the dimeric structure of the alpha/beta-knot protein, YibK, from Haemophilus influenzae by the design and engineering of monomeric versions of the protein, followed by examination of their structural, functional, stability, and kinetic folding properties. Monomeric forms of YibK display similar characteristics to an intermediate species populated during the formation of the wild-type dimer. However, a notable loss in structure involving disruption to the active site, rendering it incapable of cofactor binding, is observed in monomeric YibK. Thus, dimerization is vital for preservation of the native structure and, therefore, activity of the protein.

Long Chain‐Substituted and Triply Functionalized Molecular Knots – Synthesis, Topological Chirality and Monolayer Formation

AbstractNew topological chiral molecular knots (knotanes) 3a–6 of the amide type were substituted with long aliphatic side chains (butyloxy, hexyloxy, octenyloxy, decyloxy) and with p‐bromobenzyloxy and oligoethylene glycol ether units, which resulted in improved solubility and enhanced potential for racemate separation. In consequence, the knotanes with longer substituted chains could be separated even on commercially available noncovalently bound chiral stationary phases. The (p‐bromobenzyloxy)‐knotane 6 and the octenyloxy‐knotane 4 were synthesized with the intention of using their functional sidechains in further reactions, 6 being used in Suzuki‐type reactions, while 4 was consequently allowed to react in the presence of Grubbs catalyst in metathesis reactions. The achieved solubility allowed us to test the abilities of the alkyl chain‐substituted knotanes 3a–c and the (p‐bromobenzyloxy)‐knotane 6 to form Langmuir–Blodgett films. The assemblies were investigated on mica by AFM technique. A key point of focus was the (topological) chirality of the long chain‐substituted knotanes. To address this point, the racemates were enantioseparated by HPLC on various chiral phases, and the obtained pure enantiomers were found to exhibit pronounced Cotton effects in their circular dichroism spectra. The absolute configurations of all new knots were derived by comparison with the unsubstituted knot. (© Wiley‐VCH Verlag GmbH & Co. KGaA, 69451 Weinheim, Germany, 2007)

Challenges and opportunities in polymer theory

Challenges and opportunities in polymer theory

Glueball mass from quantized knot solitons and gauge-invariant gluon mass

We propose an approach which enables one to obtain simultaneously the glueball mass and the gluon mass in the gauge-invariant way to shed new light on the mass gap problem in Yang-Mills theory. First, we point out that the Faddeev (Skyrme--Faddeev-Niemi) model can be induced through the gauge-invariant vacuum condensate of mass dimension two from SU(2) Yang-Mills theory. Second, we obtain the glueball mass spectrum by performing the collective coordinate quantization of the topological knot soliton in the Faddeev model. Third, we demonstrate that a relationship between the glueball mass and the gluon mass is obtained, since the gauge-invariant gluon mass is also induced from the relevant vacuum condensate. Finally, we determine physical values of two parameters in the Faddeev model and give an estimate of the relevant vacuum condensation in Yang-Mills theory. Our results indicate that the Faddeev model can play the role of a low-energy effective theory of the quantum SU(2) Yang-Mills theory.

Manipulation of deformable linear objects using knot invariants to classify the object condition based on image sensor information

Using a topological model and knot theory, we propose a method for describing the condition of a rope. We also propose a recognition method based on the image information obtained from the charge-coupled device cameras to obtain the structure of the rope when manipulated by a robot. This method will help solve the difficulties of robots manipulating deformable objects by providing a theoretical framework of error recovery for deformable object manipulation. We confirm the effectiveness of the methods through experiments

Non-Abrikosov vortex and topological knot in two-gap superconductors

We establish the existence of topologically stable knot in two-gap superconductor whose topology $\pi_3(S^2)$ is fixed by the Chern-Simon index of the electromagnetic potential. We present a helical magnetic vortex solution in Ginzburg-Landau theory of two-gap superconductor which has a non-vanishing condensate at the core, and identify the knot as a twisted magnetic vortex ring made of the helical vortex. We discuss how the knot can be constructed in the recent two-gap $\rm MgB_2$ superconductor.

A volume form on the SU(2)–representation space of knot groups

For a knot K in S^3 we construct according to Casson--or more precisely taking into account Lin and Heusener's further works--a volume form on the SU(2)-representation space of the group of K. We prove that this volume form is a topological knot invariant and explore some of its properties.

Novel Structural Motifs in Clusters of Dipolar Spheres: Knots, Links, and Coils

We present the structures of putative global potential energy minima for clusters bound by the Stockmayer (Lennard-Jones plus point dipole) potential. A rich variety of structures is revealed as the cluster size and dipole strength are varied. Most remarkable are groups of closed-loop structures with the topology of knots and links. Despite the large number of possibilities, energetically optimal structures exhibit only a few such topologies.

Cabling and transverse simplicity

We study Legendrian knots in a cabled knot type. Specifically, given a topological knot type /C, we analyze the Legendrian knots in knot types ob tained from K by cabling, in terms of Legendrian knots in the knot type K. As a corollary of this analysis, we show that the (2,3)-cable of the (2,3)-torus knot is not transversely simple and moreover classify the transverse knots in this knot type. This is the first classification of transverse knots in a non transversely-simple knot type. We also classify Legendrian knots in this knot type and exhibit the first example of a Legendrian knot that does not destabi lize, yet its Thurston-Bennequin invariant is not maximal among Legendrian representatives in its knot type. In this paper we continue the investigation of Legendrian knots in tight contact 3-manifolds using 3-dimensional contact-topological methods. In [EH1], the authors introduced a general framework for analyzing Legendrian knots in tight contact 3-manifolds. There we streamlined the proof of the classification of Legendrian unknots, originally proved by Eliashberg-Eraser in [EF], and gave a complete classification of Legendrian torus knots and figure eight knots. In [EH2], we gave the first structure theorem for Legendrian knots, namely the reduction of the analysis of connected sums of Legendrian knots to that of the prime summands. This yielded a plethora of non-Legendrian-simple knot types. (A topological knot type is Legendrian simple if Legendrian knots in this knot type are determined by their Thurston-Bennequin invariant and rotation number.) Moreover, we exhibited pairs of Legendrian knots in the same topological knot type with the same Thurston-Bennequin and rotation numbers, which required arbitrarily many stabilizations before they became Legendrian isotopic (see [EH2]).

THE PENDULUM WEAVES ALL KNOTS AND LINKS

The behavior of a dissipative chaotic dynamical system is determined by the skeleton of its strange attractor, which consists of an uncountably infinite set of unstable periodic orbits. Each orbit is a topological knot, and the set is an infinite link. The types of knots and links supported by the system may be determined by collapsing the attractor along its local stable manifolds to form a template, a branched two manifold with boundary that supports the same set of knots and links as the original attractor. We show that the strange attractor of a chaotic, vertically forced physical pendulum can be collapsed to a template that supports all knots and links. Thus, one of the simplest and most well known dynamical systems is capable of the most complex behavior possible.