Physical and Numerical Models in Knot Theory - Including Applications to the Life Sciences (original) (raw)
Related papers
Annual Review of Biophysics, 2010
Knots appear in a wide variety of biophysical systems, ranging from biopolymers, such as DNA and proteins, to macroscopic objects, such as umbilical cords and catheters. Although significant advancements have been made in the mathematical theory of knots and some progress has been made in the statistical mechanics of knots in idealized chains, the mechanisms and dynamics of knotting in biophysical systems remain far from fully understood. We report on recent progress in the biophysics of knotting-the formation, characterization, and dynamics of knots in various biophysical contexts.
Subknots in ideal knots, random knots, and knotted proteins
Scientific Reports, 2015
We introduce disk matrices which encode the knotting of all subchains in circular knot configurations. The disk matrices allow us to dissect circular knots into their subknots, i.e. knot types formed by subchains of the global knot. The identification of subknots is based on the study of linear chains in which a knot type is associated to the chain by means of a spatially robust closure protocol. We characterize the sets of observed subknot types in global knots taking energy-minimized shapes such as KnotPlot configurations and ideal geometric configurations. We compare the sets of observed subknots to knot types obtained by changing crossings in the classical prime knot diagrams. Building upon this analysis, we study the sets of subknots in random configurations of corresponding knot types. In many of the knot types we analyzed, the sets of subknots from the ideal geometric configurations are found in each of the hundreds of random configurations of the same global knot type. We also compare the sets of subknots observed in open protein knots with the subknots observed in the ideal configurations of the corresponding knot type. This comparison enables us to explain the specific dispositions of subknots in the analyzed protein knots.
New biologically motivated knot table
Biochemical Society Transactions, 2013
The knot nomenclature in common use, summarized in Rolfsen's knot table [Rolfsen (1990) Knots and Links, American Mathematical Society], was not originally designed to distinguish between mirror images. This ambiguity is particularly inconvenient when studying knotted biopolymers such as DNA and proteins, since their chirality is often significant. In the present article, we propose a biologically meaningful knot table where a representative of a chiral pair is chosen on the basis of its mean writhe. There is numerical evidence that the sign of the mean writhe is invariant for each knot in a chiral pair. We review numerical evidence where, for each knot type K, the mean writhe is taken over a large ensemble of randomly chosen realizations of K. It has also been proposed that a chiral pair can be distinguished by assessing the writhe of a minimal or ideal conformation of the knot. In all cases examined to date, the two methods produce the same results.
Knotting and linking in macromolecules
Reactive and Functional Polymers
In the 1980's, knotting in DNA became a fundamental research dimension in the study of the mechanisms by which enzymes act on it. Later, the first compelling identification of knotting in proteins, in 2000, launched the study of knotting in protein structures and linear macromolecules more generally following on theoretical efforts of the 1960's. While the linking occurring in structures such as DNA, with the articulation of the relationship between linking, twisting, and writhe, and, more directly, linking in Olympic gels has been of interest to geometers, molecular biologists and, polymer physicists since the 1960's, a new mathematical analysis of both global and local facets of knotting and linking is again providing new promising discoveries. Following a discussion of the two topological structures of knotting and linking, we will consider some of their applications, and close with a consideration of new questions that suggest attractive directions for future research.
Typical knots: size, link component count, and writhe
arXiv: Geometric Topology, 2020
We model the typical behavior of knots and links using grid diagrams. Links are ubiquitous in the sciences, and their normal or typical behavior is of significant importance in understanding situations such as the topological state of DNA or the statistical mechanics of ring polymers. We examine three invariants: the expected size of a random knot; the expected number of components of a random link; and the expected writhe of a random knot. We investigate the first two numerically and produce generating functions which codify the observed patterns, we perform an exploratory data analysis for the third. We continue this project in a future work, where we investigate genus and the effects of crossing change on it.
The Knot Theory A Comprehensive Discussion
Knot theory is a fascinating branch of mathematics that delves into the study of loops in three-dimensional space. By examining how these loops can be twisted, entangled, and deformed without cutting or breaking, knot theory provides profound insights into the structure of our universe, ranging from the microscopic strands of DNA to the complex topology of quantum fields. With its roots in 19th-century mathematics and its influence extending to modern physics, biology, and computer science, knot theory bridges pure mathematical abstraction and practical applications. This paper offers a comprehensive exploration of knot theory, starting with its foundational concepts and definitions. It examines the tools mathematicians use to classify knots, including powerful invariants like the Jones polynomial and knot groups. The paper also highlights advanced topics, such as the relationship between braids and knots and the role of knot theory in higher dimensions. Finally, it sheds light on the practical relevance of knot theory, showcasing its applications in understanding DNA replication, quantum computing, and fluid dynamics. By weaving together intuition, rigorous mathematics, and real-world applications, this paper aims to provide both an introduction to knot theory and a glimpse into the open problems that continue to challenge and inspire mathematicians. Whether you're a mathematician, a scientist, or simply a curious mind, this journey through the tangled world of knots promises to be as intriguing as the knots themselves.
Much Ado About Knotting: On the Physics of Knots
From a physicist's point of view knots are interesting for two reasons: they are stable, meaning that they possess certain invariants, and there is a large variety of them, meaning that they could account for different phenomena. We give examples of applications of knot theory in classical physics, namely the magnetic helicity integral and its topological interpretation, and how knot invariants arise from the 2D solvable Ice-type model. We also discuss knot theory in modern physics where it has found use in particle physics to describe the energy levels of 'glueballs' in the form of knotted and linked tubes, and we introduce a quantum money scheme which draws its security from Alexander polynomials. We conclude that after being out of fashion in the physical community for almost a century, knot theory has convincingly re-entered physics, to stay this time around.
2014
A mathematical knot is simply a closed curve in three-space. Classifying open knots, or knots that have not been closed, is a relatively unexplored area of knot theory. In this note, we report on our study of open random walks of varying length, creating a collection of open knots. Following the strategy of Millett, Dobay and Stasiak, an open knot is closed by connecting its two open endpoints to a third point, lying on a large sphere that encloses the random walk deeply within its interior. The resulting polygonal knot can be analyzed and its knot type determined, up to the indeterminacy of standard knot invariants, using the HOMFLY polynomial. With many closure points uniformly distributed on the large sphere, a statistical distribution of knot types is created for each open knot. We use this method to continue the exploration of the knottedness of linear random walks and apply it also to the study of several protein chains. One new feature of this work is the use of an Eckert IV ...
DNA Knots: Theory and Experiments
Progress of Theoretical Physics Supplement, 2011
Cellular DNA is a long, thread-like molecule with remarkably complex topology. Enzymes that manipulate the geometry and topology of cellular DNA perform many vital cellular processes (including segregation of daughter chromosomes, gene regulation, DNA repair, and generation of antibody diversity). Some enzymes pass DNA through itself via enzyme-bridged transient breaks in the DNA; other enzymes break the DNA apart and reconnect it to different ends. In the topological approach to enzymology, circular DNA is incubated with an enzyme, producing an enzyme signature in the form of DNA knots and links. By observing the changes in DNA geometry (supercoiling) and topology (knotting and linking) due to enzyme action, the enzyme binding and mechanism can often be characterized. This paper will discuss some personal research history, and the tangle model for the analysis of site-specific recombination experiments on circular DNA.
A chapter in physical mathematics: theory of knots in the sciences
2000
A systematic study of knots was begunin the second half of the 19th century by Tait and his followers. They were motivated by Kelvin's theory of atoms modelled on knotted vortex tubes of ether. It was expected that physical and chemical properties of various atoms could beexpressed in terms of properties of knots such as the knot invariants. Even though Kelvin's theory did not work, the theory of knots grew as a sub eld of combinatorial topology. Recently new invariants of knots have been discovered and they have led to the solution of long standing problems in knot theory. Surprising connections between the theory of knots and statistical mechanics, quantum groups and quantum eld theory are emerging. We give a geometric formulation of some of these invariants using ideas from topological quantum eld theory. We also discuss some recent connections and application of knot theory to problems in Physics, Chemistry and Biology. It is interesting to note that as we stand on the threshold of the new millenium, di cult questions arising in the sciences continue to serve as a driving force for the development of new mathematical tools needed to understand and answer them.