Figure - available from: Scientific Reports
This content is subject to copyright. Terms and conditions apply.
(A) Illustration of some of the knots relevant to the present study and their nomenclature. The chirality is consistent with that in Brasher et al.²¹. The green arrows along the unknot 01 represent the two reconnection sites. The sites shown are equidistant and in direct repeats. A complete table of prime knots with up to 10 crossings and information on how they compare to those in Rolfsen²⁰ can be obtained from the authors upon request. (B) Nomenclature for two component links relevant to the present study. The green arrows represent the reconnection sites, which confer an orientation to each link component. The nomenclature is described in the Supplementary Methods and in Supplementary Fig. S5. For 2-component links with 9 or more crossings we revert to the default Knotplot naming convention. (C) The four possible combinations of chirality and orientation for the 4-crossing torus link. A comparison between the nomenclature used in this paper and that in Rolfsen²⁰ and in works by Kanenobu28,29 is included in Supplementary Fig. S5. Arrows indicate the relative orientations of the sites.
Source publication
In Escherichia coli DNA replication yields interlinked chromosomes. Controlling topological changes associated with replication and returning the newly replicated chromosomes to an unlinked monomeric state is essential to cell survival. In the absence of the topoisomerase topoIV, the site-specific recombination complex XerCD- dif-FtsK can remove re...
Similar publications
In Escherichia coli DNA replication yields interlinked chromosomes. Controlling topological changes associated with replication and returning the newly replicated chromosomes to an unlinked monomeric state is essential to cell survival. In the absence of the topoisomerase topoIV, the site-specific recombination complex XerCD- dif -FtsK can remove r...
Citations
... In [14,16], it is shown that the XerCD-dif-FtsK system unlinks replication DNA catenanes in a stepwise manner. The link types of the replication catenanes are torus link T (2, c). ...
... As an application, we can uniquely characterize the unlinking pathway of the replication catenane of type T (2, c) if we assume that the crossing number goes down at each recombination. In [16], we relaxed the assumption a little and considered the case where cr(L ′ ) = cr(L) for a smoothing on L = T (2, c). Then L ′ satisfies σ(L ′ ) = 2−cr(L ′ ). ...
This paper investigates the relationship between the signature and the crossing number of knots and links. We refine existing theorems and provide a comprehensive classification of links with specific properties, particularly those with signatures that deviate by a fixed amount from their crossing numbers. The main results include the identification of all links for which the sum of the signature and crossing number equals 2, which are shown to be closures of positive 3-braids. Additionally, we explore the implications of these findings in the context of band surgeries and their applications to vortex knots and DNA topology.
... In the real world, where fluids are not ideal, knots decay and can change their topology. The study of the sequences of knots, knot cascades, that can occur during such processes is an active area of research [48,30], see also [38] for a study of knot cascades in DNA. ...
The curves of zero intensity of a complex optical field can form knots and links: optical vortex knots. Both theoretical constructions and experiments have so far been restricted to the very small families of torus knots or lemniscate knots. Here we describe a mathematical construction that presumably allows us to generate optical vortices in the shape of any given knot or link. We support this claim by producing for every knot K in the knot table up to 8 crossings a complex field that satisfies the paraxial wave equation and whose zeros have a connected component in the shape of K. These fields thus describe optical beams in the paraxial regime with knotted optical vortices that go far beyond previously known examples.
... Tangles have had various applications in knot theory and biology. Notably, tangles have been used to model enzyme actions on DNA [19,46,9]. This paper presents an application of tangle calculus to the study of protein knotting. ...
Although rare, an increasing number of proteins have been observed to contain entanglements in their native structures. To gain more insight into the significance of protein knotting, researchers have been investigating protein knot formation using both experimental and theoretical methods. Motivated by the hypothesized folding pathway of -haloacid dehalogenase (DehI) protein, Flapan, He, and Wong proposed a theory of how protein knots form, which includes existing folding pathways described by Taylor and B\"olinger et al. as special cases. In their topological descriptions, two loops in an unknotted open protein chain containing at most two twists each come close together, and one end of the protein eventually passes through the two loops. In this paper, we build on Flapan, He, and Wong's theory where we pay attention to the crossing signs of the threading process and assume that the unknotted protein chain may arrange itself into a more complicated configuration before threading occurs. We then apply tangle calculus, originally developed by Ernst and Sumners to analyze the action of specific proteins on DNA, to give all possible knots or knotoids that may be discovered in the future according to our model and give recipes for engineering specific knots in proteins from simpler pieces. We show why twists knots are the most likely knots to occur in proteins. We use chirality to show that the most likely knots to occur in proteins via Taylor's twisted hairpin model are the knots , , and .
... In turn, this provided strong evidence that the simplification action of specific DNA enzymes is driven by a geometric selection of sites [3]. One further potential use of grid diagrams is to help with the search of band changes and the determination of Gordian distance between knot types [17,25,5,6]. ...
Grid diagrams are a combinatorial version of classical link diagrams, widely used in theoretical, computational and applied knot theory. Motivated by questions from (bio)-physical knot theory, we introduce GridPyM, a Sage compatible Python module that handles grid diagrams. GridPyM focuses on generating and simplifying grids, and on modelling local transformations between them.
... See Figure 1 for examples. Knots and links can be found in many biological, chemical, and physical systems, including closed circular DNA [1][2][3][4][5], proteins [6][7][8], and topological vortices [9]. For references on knot theory and its applications, see [10][11][12][13][14][15], for example. ...
Knots and links are ubiquitous in chemical systems. Their structure can be responsible for
a variety of physical and chemical properties, making them very important in materials development. In this article, we analyze the topological structures of interlocking molecules composed of metal-peptide rings using the concept of polyhedral links. To that end, we discuss the topological classification of alternating polyhedral links.
... Rigorous methods to characterize these changes are necessary. In recent decades, measures from knot theory have been applied to biopolymers [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30] and in particular to proteins to classify their conformations [27][28][29][30][31][32][33][34][35][36]. One of the simplest measures of conformational complexity of proteins dates back to Gauss: the writhe of a curve [37][38][39][40][41][42][43]. ...
The novel coronavirus SARS-CoV-2 infects human cells using a mechanism that involves binding and structural rearrangement of its Spike protein. Understanding protein rearrangement and identifying specific amino acids where mutations affect protein rearrangement has attracted much attention for drug development. In this manuscript, we use a mathematical method to characterize the local topology/geometry of the SARS-CoV-2 Spike protein backbone. Our results show that local conformational changes in the FP, HR1, and CH domains are associated with global conformational changes in the RBD domain. The SARS-CoV-2 variants analyzed in this manuscript (alpha, beta, gamma, delta Mink, G614, N501) show differences in the local conformations of the FP, HR1, and CH domains as well. Finally, most mutations of concern are either in or in the vicinity of high local topological free energy conformations, suggesting that high local topological free energy conformations could be targets for mutations with significant impact of protein function. Namely, the residues 484, 570, 614, 796, and 969, which are present in variants of concern and are targeted as important in protein function, are predicted as such from our model.
... In this representation topological complexity is simply given by the crossing number q ≥ 2N, and topological change by the reduction of q due to a single reconnection event (supposedly taking place in the encircled region of figure 1a-f ). The topological sequence (2.4) was observed in the unlinking pattern of DNA catenanes subject to site-specific enzyme recombinations (Shimokawa et al. 2013;Stolz et al. 2017), while the last stages of the sequence were observed experimentally in the evolution of a vortex trefoil knot in water (Kleckner & Irvine 2013) and in GPE simulations (see, for instance, Kleckner et al. (2016) and ZR17). A more dramatic reduction of topological complexity occurs when multiple reconnections take place concurrently; if n ≤ q simultaneous reconnections occur, we have T ( p, q) → T ( p, q − n) with a drastic reduction of topology. ...
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.
... A smooth knot allows for arbitrary large curvature, which does not model the rigidity of DNA. Recently, lattice knots were used to study Escherichia coli DNA in [6]; vertex distortion may relate to the "unlinking pathway" of chromosomes since it measures the proximity of points weighted by their closeness to being antipodal along the knot. ...
The first two authors introduced vertex distortion and showed that the vertex distortion of the unknot is trivial. It was conjectured that the vertex distortion of a knot is trivial if and only if the knot is trivial. We will use Denne-Sullivan's bound on Gromov distortion to bound the vertex distortion of nontrivial lattice knots. We will then conclude that trivial vertex distortion implies the unknot, proving the conjecture. Additionally, the first conjecture in vertex distortion's debut article is proven and a vertex distortion calculator is given.
... Many physical systems are composed of entangled filamentous structures whose complexity affects their mechanical properties and their function [1][2][3][4][5][6][7][8]. Under some conditions, we can see these filamentous structures like mathematical curves in 3-space whose entanglement we can measure using tools from knot theory [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. A knot (link) is one (or more) simple closed curve(s) in 3-space, and knots (links) are classified using the notion of topological equivalence. ...
In this article, we define Vassiliev measures of complexity for open curves in 3-space. These are related to the coefficients of the enhanced Jones polynomial of open curves in 3-space. These Vassiliev measures are continuous functions of the curve coordinates; as the ends of the curve tend to coincide, they converge to the corresponding Vassiliev invariants of the resulting knot. We focus on the second Vassiliev measure from the enhanced Jones polynomial for closed and open curves in 3-space. For closed curves, this second Vassiliev measure can be computed by a Gauss code diagram and it has an integral formulation, the double alternating self-linking integral. The double alternating self-linking integral is a topological invariant of closed curves and a continuous function of the curve coordinates for open curves in 3-space. For polygonal curves, the double alternating self-linking integral obtains a simpler expression in terms of geometric probabilities.
... Unfortunately, attempts to create reproducible knots by colliding vortex rings have not been successful [29]. Moreover, most of the theoretical studies focus on the dissolving of knots and links [2,4,7,20,[30][31][32][33] or head-on collisions [34,35], where only standard reconnections occur. In this process, the reduced complexity of the topological structures is identified through stepwise reduction, which shows the transfer of helicity of the system evolves only in one direction from knotted (linked) structures to helical coils [4,36]. ...
... As described in Refs. [2,31], the stepwise simplification during DNA untying or vortex knots and/or links dissolution in viscous fluids for an L2a1 linked configuration (a K3-1 knotted configuration) should follow the link-ring-rings pathway (knot-link-ring-rings) with decreasing crossing numbers. ...
The topological transition of vortex lines to vortex rings and hopfions is numerically investigated by the Gross–Pitaevskii equation in three-dimensional trapped Bose–Einstein condensates. The shape of the vortex rings formed by the two vortex lines of the vortex dipole depends strongly on the initial separation of the lines. An approximately perfect vortex ring can be obtained by choosing some suitable values of the separation. The deformation of the formed rings depends on the shape of the rings in turn. Furthermore, we show a feasible approach to generate vortex hopfions by imprinting a vortex line in the center of the generated vortex rings. Specifically, the movement of the vortex rings can excite helical waves along the central vortex line of the hopfion structure if the vortex ring is not perfect.
