Mathematical Knot Research Articles

Tying a true topological protein knot by cyclization

Knotted proteins are fascinating to biophysicists because of their robust ability to fold into intricately defined three-dimensional structures with complex and topologically knotted arrangements. Exploring the biophysical properties of the knotted proteins is of significant interest, as they could offer enhanced chemical, thermal, and mechanostabilities. A true mathematical knot requires a closed path; in contrast, knotted protein structures have open N- and C-termini. To address the question of how a truly knotted protein differs from the naturally occurring counterpart, we enzymatically cyclized a 31 knotted YibK protein from Haemophilus influenza (HiYibK) to investigate the impact of path closure on its structure-function relationship and folding stability. Through the use of a multitude of structural and biophysical tools, including X-ray crystallography, NMR spectroscopy, small angle X-ray scattering, differential scanning calorimetry, and isothermal calorimetry, we showed that the path closure minimally perturbs the native structure and ligand binding of HiYibK. Nevertheless, the cyclization did alter the folding stability and mechanism according to chemical and thermal unfolding analysis. These molecular insights contribute to our fundamental understanding of protein folding and knotting that could have implications in the protein design with higher stabilities.

Knot the Usual Suspects: Finding the Diagrammatic Representations of Physical Knots

In the last few hundred years, mathematicians have been attempting to describe the topological and algebraic properties of mathematical knots. Regarding the study of knots, there exists a disconnect between examining a knot’s mathematical and physical definitions. This is due to the inherent difference in the topology of an open-ended physical knot and a closed mathematical knot. By closing the ends of a physical knot, this paper presents a method to break this discontinuity by establishing a clear relation between physical and mathematical knots. By joining the ends and applying Reidemeister moves, this paper will calculate the equivalent mathematical prime or composite knots for several commonly used physical knots. In the future, it will be possible to study the physical properties of these knots and their potential to expand the field of mathematical knot theory.

Mathematical Knot Sematic Logic:Theory and Example in Surah Al-Kahf

Start your abstract here Knot semantic logic theory is a new idea to explain the symmetry structure of a literature. The methodology used in this research is the combination of Theme-Rheme framework, Paltridge’s classification of thematic progression and symmetry diagram mapping. The results of this research are the establishment of knot semantic logic theory and the application to analyze the symmetrical structure in Surah Al-Kahf. After applying this theory to Surah Al-Kahf resulting that the chapter contains 9 groups form symmetrical structure. The first theme on this Surah is about Believers, non-believers and consequences connected to the last group with the similar theme. The theme of the second group is about self-quarantine mirrors to the second to the last group about lockdown. The third theme is about Patience and its reward related to the seventh group. The test of wealthyis the theme of the fourth group which mirrors to the sixth group talking about provision as a sign. Finally, the fifth group as the central of this chapter states that Quran has provided many lessons and guidance for human beings. Surah Al-Kahf is knot chapter with a mathematical symbol :

The mathematics of tie knots

In 2000, Thomas Fink and Young Mao studied neck ties and, with certain assumptions, found 85 different ways to tie a neck tie. They gave a formal language which describes how a tie is made, giving a sequence of moves for each neck tie. The ends of a neck tie can be joined together, which gives a physical model of a mathematical knot that we call a tie knot. In this paper we classify the knot type of each of Fink and Mao's 85 tie knots. We describe how the unknot, left and right trefoil, twist knots and $(2,p)$ torus knots can be recognized from their sequence of moves. We also view tie knots as a family within the set of all knots. Among other results, we prove that any tie knot is prime and alternating.

A Suggestive Interface for Untangling Mathematical Knots.

In this paper we present a user-friendly sketching-based suggestive interface for untangling mathematical knots with complicated structures. Rather than treating mathematical knots as if they were 3D ropes, our interface is designed to assist the user to interact with knots with the right sequence of mathematically legal moves. Our knot interface allows one to sketch and untangle knots by proposing the Reidemeister moves, and can guide the user to untangle mathematical knots to the fewest possible number of crossings by suggesting the moves needed. The system highlights parts of the knot where the Reidemeister moves are applicable, suggests the possible moves, and constrains the user's drawing to legal moves only. This ongoing suggestion is based on a Reidemeister move analyzer, that reads the evolving knot in its Gauss code and predicts the needed Reidemeister moves towards the fewest possible number of crossings. For our principal test case of mathematical knot diagrams, this for the first time permits us to visualize, analyze, and deform them in a mathematical visual interface. In addition, understanding of a fairly long mathematical deformation sequence in our interface can be aided by visual analysis and comparison over the identified "key moments" where only critical changes occur in the sequence. Our knot interface allows users to track and trace mathematical knot deformation with a significantly reduced number of visual frames containing only the Reidemeister moves being applied. All these combine to allow a much cleaner exploratory interface for us to analyze and study mathematical knots and their dynamics in topological space.

Accelerating visual communication of mathematical knot deformation

Mathematical knots are different from everyday ropes, in that they are infinitely stretchy and flexible when being deformed into their ambient isotopic. For this reason, challenges of visualization and computation arise when communicating mathematical knot’s static and changing structures during its topological deformation. In this paper, we focus on visual and computational methods to facilitate the communication of mathematical knot’ dynamics by simulating the topological deformation and capturing the critical changes during the entire simulation. To improve our visual experience, we design and exploit parallel functional units to accelerate both topological refinements in simulation phase and view selection in presentation phase. To further allow a real-time keyframe-based communication of knot deformation, we propose a fast and adaptive method to extract key moments where only critical changes occur to represent and summarize the long deformation sequence in real-time fashion. We conduct performance study and present the efficacy and efficiency of our methods. .

GISA: using Gauss Integrals to identify rare conformations in protein structures.

The native structure of a protein is important for its function, and therefore methods for exploring protein structures have attracted much research. However, rather few methods are sensitive to topologic-geometric features, the examples being knots, slipknots, lassos, links, and pokes, and with each method aimed only for a specific set of such configurations. We here propose a general method which transforms a structure into a ”fingerprint of topological-geometric values” consisting in a series of real-valued descriptors from mathematical Knot Theory. The extent to which a structure contains unusual configurations can then be judged from this fingerprint. The method is not confined to a particular pre-defined topology or geometry (like a knot or a poke), and so, unlike existing methods, it is general. To achieve this our new algorithm, GISA, as a key novelty produces the descriptors, so called Gauss integrals, not only for the full chains of a protein but for all its sub-chains. This allows fingerprinting on any scale from local to global. The Gauss integrals are known to be effective descriptors of global protein folds. Applying GISA to sets of several thousand high resolution structures, we first show how the most basic Gauss integral, the writhe, enables swift identification of pre-defined geometries such as pokes and links. We then apply GISA with no restrictions on geometry, to show how it allows identifying rare conformations by finding rare invariant values only. In this unrestricted search, pokes and links are still found, but also knotted conformations, as well as more highly entangled configurations not previously described. Thus, an application of the basic scan method in GISA’s tool-box revealed 10 known cases of knots as the top positive writhe cases, while placing at the top of the negative writhe 14 cases in cis-trans isomerases sharing a spatial motif of little secondary structure content, which possibly has gone unnoticed. Possible general applications of GISA are fold classification and structural alignment based on local Gauss integrals. Others include finding errors in protein models and identifying unusual conformations that might be important for protein folding and function. By its broad potential, we believe that GISA will be of general benefit to the structural bioinformatics community. GISA is coded in C and comes as a command line tool. Source and compiled code for GISA plus read-me and examples are publicly available at GitHub (https://github.com).

Knots in Mathematics and Medicine

The analogy between knots in mathematical knot theory and “knots” in medicine is studied for the example of a certain invariant of Legendrian knots. The connection between the “Chern class” and the “Maslov class” is analyzed. A conjecture on the connection between the “Connes-Chern character” and the “Maslov class” is proposed.

A topological characterization of the products arising from site-specific recombination on $$T(2,n) \# C(2,r)$$ DNA substrates

A mathematical knot is a simple closed curve in 3-space. Following work of Buck, Flapan, and others, we model circular DNA as a knot to classify the possible knotted or linked products which can arise as a result of site-specific recombination on DNA substrates of the topological form \(T(2,n) \# C(2,r)\). We show that, given some reasonable biological assumptions, all of the possible products are contained in one of two families.

Ways of Being Strands: Cross-Disciplinary Collaboration Using Craft and Mathematics

This article presents a collaborative project between a textile practitioner-researcher (Nimkulrat) and a textile practitioner-mathematician (Matthews). Mathematical investigation of Nimkulrat's craft knots through mathematical knot diagrams by Matthews revealed knot properties which were indiscernible from the work alone. This approach led to a way of visualizing knot designs using more than one color prior to making. The result of this current research phase illuminates the role of mathematics in making the knotting process explicit. It demonstrates the influence of mathematical analysis on craft practice and the significance of cross-disciplinary collaboration on the development of knotted pattern design.

「結び目の数学」の教材化について : 高等学校の実践を踏まえて

「結び目の数学」の教材化について : 高等学校の実践を踏まえて

The properties of conformal blocks, the AGT hypothesis, and knot polynomials

Various properties of correlators of the two-dimensional conformal field theory are discussed. Specifically, their relation to the partition function of the four-dimensional supersymmetric theory is analyzed. In addition to being of interest in its own right, this relation is of practical importance. For example, it is much easier to calculate the known expressions for the partition function of supersymmetric theory than to calculate directly the expressions for correlators in conformal theory. The examined representation of conformal theory correlators as a matrix model serves the same purpose. The integral form of these correlators allows one to generalize the obtained results for the Virasoro algebra to more complicated cases of the W algebra or the quantum Virasoro algebra. This provides an opportunity to examine more complex configurations in conformal field theory. The three-dimensional Chern–Simons theory is discussed in the second part of the present review. The current interest in this theory stems largely from its relation to the mathematical knot theory (a rather well-developed area of mathematics known since the 17th century). The primary objective of this theory is to develop an algorithm that allows one to distinguish different knots (closed loops in three-dimensional space). The basic way to do this is by constructing the so-called knot invariants.

Historical highlights of non-cyclic knot theory

This paper illustrates some classical non-cyclic milestones in mathematical knot theory.

An Introduction to Learn Mathematical Knot Theory with Knot Polynomial

An Introduction to Learn Mathematical Knot Theory with Knot Polynomial

TOPOLOGICAL ANALYSIS OF MU-TRANSPOSITION

An n-string tangle is a three dimensional ball with n-strings which are properly embedded in the ball. In early 90's, C. Ernst and D. Sumners first used a tangle to describe a DNA-protein complex. In this model, DNA is represented by a string and protein is represented by a ball. Mu is a protein which binds to DNA at three sites and a DNA-Mu complex is called Mu-transpososome. Knowing the DNA topology within Mu-transpososome is very important to understand DNA transposition by Mu protein. In 2002, Pathania et al. determined that the DNA configuration within the Mu transpososome is three branched and five noded [12]. In 2007, Darcy et al. analyzed this by using mathematical tangle and concluded that the three branched and five noded DNA configuration is the only biologically reasonable solution [4]. In this paper, based on the result of Pathania et al. and Darcy et al., the author determines the DNA topology within the DNA-Mu complex after the whole Mu transposition process. Furthermore, a new experiment is designed which can support the Pathania et al.'s result. The result of this new experiment is predicted through mathematical knot thory.

Computing a Knot Invariant as a Constraint Satisfaction Problem

We point out the connection between mathematical knot theory and spin glass/search problem. In particular, we present a statistical mechanical formulation of the problem of computing a knot invariant; p-colorability problem, which provides an algorithm to find the solution. The method also allows one to get some deeper insight into the structural complexity of knots, which is expected to be related with the landscape structure of constraint satisfaction problem.

Pokefind: a novel topological filter for use with protein structure prediction.

Motivation: Our focus has been on detecting topological properties that are rare in real proteins, but occur more frequently in models generated by protein structure prediction methods such as Rosetta. We previously created the Knotfind algorithm, successfully decreasing the frequency of knotted Rosetta models during CASP6. We observed an additional class of knot-like loops that appeared to be equally un-protein-like and yet do not contain a mathematical knot. These topological features are commonly referred to as slip-knots and are caused by the same mechanisms that result in knotted models. Slip-knots are undetectable by the original Knotfind algorithm. We have generalized our algorithm to detect them, and analyzed CASP6 models built using the Rosetta loop modeling method.Results: After analyzing known protein structures in the PDB, we found that slip-knots do occur in certain proteins, but are rare and fall into a small number of specific classes. Our group used this new Pokefind algorithm to distinguish between these rare real slip-knots and the numerous classes of slip-knots that we discovered in Rosetta models and models submitted by the various CASP7 servers. The goal of this work is to improve future models created by protein structure prediction methods. Both algorithms are able to detect un-protein-like features that current metrics such as GDT are unable to identify, so these topological filters can also be used as additional assessment tools.Contact: firas@u.washington.edu

Spontaneous knotting of an agitated string

It is well known that a jostled string tends to become knotted; yet the factors governing the "spontaneous" formation of various knots are unclear. We performed experiments in which a string was tumbled inside a box and found that complex knots often form within seconds. We used mathematical knot theory to analyze the knots. Above a critical string length, the probability P of knotting at first increased sharply with length but then saturated below 100%. This behavior differs from that of mathematical self-avoiding random walks, where P has been proven to approach 100%. Finite agitation time and jamming of the string due to its stiffness result in lower probability, but P approaches 100% with long, flexible strings. We analyzed the knots by calculating their Jones polynomials via computer analysis of digital photos of the string. Remarkably, almost all were identified as prime knots: 120 different types, having minimum crossing numbers up to 11, were observed in 3,415 trials. All prime knots with up to seven crossings were observed. The relative probability of forming a knot decreased exponentially with minimum crossing number and Möbius energy, mathematical measures of knot complexity. Based on the observation that long, stiff strings tend to form a coiled structure when confined, we propose a simple model to describe the knot formation based on random "braid moves" of the string end. Our model can qualitatively account for the observed distribution of knots and dependence on agitation time and string length.

Classification of protein fold classes by knot theory and prediction of folds by neural networks: A combined theoretical and experimental approach

We present different means of classifying protein structure. One is made rigorous by mathematical knot invariants that coincide reasonably well with ordinary graphical fold classification and another classification is by packing analysis. Furthermore when constructing our mathematical fold classifications, we utilize standard neural network methods for predicting protein fold classes from amino acid sequences. We also make an analysis of the redundancy of the structural classifications in relation to function and ligand binding. Finally we advocate the use of combining the measurement of the VA, VCD, Raman, ROA, EA and ECD spectra with the primary sequence as a way to improve both the accuracy and reliability of fold class prediction schemes.

Helicity conservation under Reidemeister moves

We discuss a connection between two fields that appear to have little in common: plasma physics and mathematical knot theory. Plasma physicists are interested in studying helicity conservation in magnetic flux ropes and knot theorists commonly consider “Reidemeister moves,” transformations that preserve a property called “knottedness.” To study the tangling, twisting, and untwisting of magnetic flux ropes, it is helpful to know which topological transformations conserve helicity. Although the second and third types of Reidemeister moves applied to a magnetic flux rope clearly conserve the helicity of the flux rope, the first type of Reidemeister move appears to be in conflict with helicity conservation. We show that all three Reidemeister moves conserve helicity in magnetic flux ropes.