Kenneth Millett | University of California, Santa Barbara (original) (raw)
Papers by Kenneth Millett
Employing the MDS analysis, one defines the knot type of open chains thereby providing a basic to... more Employing the MDS analysis, one defines the knot type of open chains thereby providing a basic tool in the study of fine –grained knotting structure in proteins (or other open chains) as well as classical topological knots (for example tight knots). The resulting matrix of data arising from the entire collection of sub chains is represented as a triangular or circular knotting fingerprint. From the knotting regions one derives an oriented graph from whose characteristic structure one can derive characteristics of the complexity of the knotting and the knotting pathways associated to the given protein structure or the specific tight knot type. This is joint research with Henrich, Hyde, Rawdon, and Stasiak.
Lecture Notes in Mathematics, 1985
This paper grew out of an effort to decide if R 3 can be foliated by circles. Although the result... more This paper grew out of an effort to decide if R 3 can be foliated by circles. Although the results are, as yet, insufficient to provide an answer to the question they do appear to be of independent interest and do allow us to focus upon some very specific problems concerning the extent to which embedded (or immersed) surfaces in a 3 manifold, foliated by (topologically)
Lecture Notes in Mathematics, 1975
Without Abstract
Nucleic acids research, Jun 22, 2016
Freshly replicated DNA molecules initially form multiply interlinked right-handed catenanes. In b... more Freshly replicated DNA molecules initially form multiply interlinked right-handed catenanes. In bacteria, these catenated molecules become supercoiled by DNA gyrase before they undergo a complete decatenation by topoisomerase IV (Topo IV). Topo IV is also involved in the unknotting of supercoiled DNA molecules. Using Metropolis Monte Carlo simulations, we investigate the shapes of supercoiled DNA molecules that are either knotted or catenated. We are especially interested in understanding how Topo IV can unknot right-handed knots and decatenate right-handed catenanes without acting on right-handed plectonemes in negatively supercoiled DNA molecules. To this end, we investigate how the topological consequences of intersegmental passages depend on the geometry of the DNA-DNA juxtapositions at which these passages occur. We observe that there are interesting differences between the geometries of DNA-DNA juxtapositions in the interwound portions and in the knotted or catenated portions ...
Knots in Hellas '98, 2000
Series on Knots and Everything, 2005
Lecture Notes in Mathematics, 1975
ABSTRACT Without Abstract
Journal of Knot Theory and Its Ramifications, 2010
The probability that a random walk or polygon in the 3-space or in the simple cubic lattice conta... more The probability that a random walk or polygon in the 3-space or in the simple cubic lattice contains a knot goes to one at the length goes to infinity. Here, we prove that this is also true for slipknots consisting of unknotted portions, called the slipknot, that contain a smaller knotted portion, called the ephemeral knot. As is the case with knots, we prove that any topological knot type occurs as the ephemeral knotted portion of a slipknot.
Biophysical Journal, 2012
Knot fingerprints provide a fine-grained resolution of the local knotting structure of tight knot... more Knot fingerprints provide a fine-grained resolution of the local knotting structure of tight knots. From this fine structure and an analysis of the associated planar graph, one can define a measure of knot complexity using the number of independent unknotting pathways from the global knot type to the short arc unknot. A specialization of the Cheeger constant provides a measure of constraint on these independent unknotting pathways. Furthermore, the structure of the knot fingerprint supports a comparison of the tight knot pathways to the unconstrained unknotting pathways of comparable length.
For a polygonal knot K, it is shown that a tube of radius R(K), the polygonal thickness radius, i... more For a polygonal knot K, it is shown that a tube of radius R(K), the polygonal thickness radius, is an embedded torus. Given a thick configuration K, perturbations of size r<R(K) define satellite structures, or local knotting. We explore knotting within these tubes both theoretically and numerically. We provide bounds on perturbation radii for which we can see small trefoil and figure-eight summands and use Monte Carlo simulations to approximate the relative probabilities of these structures as a function of the number of edges.
We present two descriptions of the the local scaling and shape of ideal rings, primarily featurin... more We present two descriptions of the the local scaling and shape of ideal rings, primarily featuring subsegments. Our focus will be the squared radius of gyration of subsegments and the squared internal end to end distance, defined to be the average squared distance between vertices k edges apart. We calculate the exact averages of these values over the space of all such ideal rings, not just a calculation of the order of these averages, and compare these to the equivalent values in open chains. This comparison will show that the structure of ideal rings is similar to that of ideal chains for only exceedingly short lengths. These results will be corroborated by numerical experiments. They will be used to analyze the convergence of our generation method and the effect of knotting on these characteristics of shape.
Random walks and polygons are used to model polymers. In this paper we consider the extension of ... more Random walks and polygons are used to model polymers. In this paper we consider the extension of writhe, self-linking number and linking number to open chains. We then study the average writhe, self-linking and linking number of random walks and polygons over the space of configurations as a function of their length. We show that the mean squared linking number, the mean squared writhe and the mean squared self-linking number of oriented uniform random walks or polygons of length n, in a convex confined space, are of the form O(n^2). Moreover, for a fixed simple closed curve in a convex confined space, we prove that the mean absolute value of the linking number between this curve and a uniform random walk or polygon of n edges is of the form O(√(n)). Our numerical studies confirm those results. They also indicate that the mean absolute linking number between any two oriented uniform random walks or polygons, of n edges each, is of the form O(n). Equilateral random walks and polygons...
A mathematical knot is simply a closed curve in three-space. Classifying open knots, or knots tha... more 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 ...
This paper contains selected topics from four lectures given at the Abdus Salam International Cen... more This paper contains selected topics from four lectures given at the Abdus Salam International Centre for Theoretical Physics in May 2009. We introduce the study of the influence of knotting and linking on the spatial characteristics of linear and ring polymer chains with examples of scientific interest. We describe a few basic concepts of the geometry and topology of knots and measures of the spatial shape of open and closed polymer chains. We then present some fundamental mathematical results concerning them. Next we discuss random sampling methods of collections of open and closed chains that are employed to provide estimates of the spatial properties of the chains. Finally, we discuss implementations of the sampling algorithms, survey consequences of theoretical and experimental results, and discuss some interesting problems deserving further research.
Let M denote a compact topological 3-manifold, without boundary, foliated by topological circles ... more Let M denote a compact topological 3-manifold, without boundary, foliated by topological circles in the sense of Seifert's gefaserter Rdume[8]. This will be called a Seifert structure, or Seifert fibration on M; the leaves of the foliation are called (regular or exceptional) fibres. Our main result is the following theorem, reminiscent
Springer Proceedings in Mathematics & Statistics
With a focus on one-dimensional periodic boundary systems, we describe the application of extensi... more With a focus on one-dimensional periodic boundary systems, we describe the application of extensions of the Gauss linking number of closed rings to open chains and, then, to systems of such chains via the periodic linking and periodic self-linking of chains. These lead to the periodic linking matrix and its associated eigenvalues providing measures of entanglement that can be applied to complex systems. We describe the general one-dimensional case and applications to one-dimensional Olympic gels and to tubular filamental structures.
Employing the MDS analysis, one defines the knot type of open chains thereby providing a basic to... more Employing the MDS analysis, one defines the knot type of open chains thereby providing a basic tool in the study of fine –grained knotting structure in proteins (or other open chains) as well as classical topological knots (for example tight knots). The resulting matrix of data arising from the entire collection of sub chains is represented as a triangular or circular knotting fingerprint. From the knotting regions one derives an oriented graph from whose characteristic structure one can derive characteristics of the complexity of the knotting and the knotting pathways associated to the given protein structure or the specific tight knot type. This is joint research with Henrich, Hyde, Rawdon, and Stasiak.
Lecture Notes in Mathematics, 1985
This paper grew out of an effort to decide if R 3 can be foliated by circles. Although the result... more This paper grew out of an effort to decide if R 3 can be foliated by circles. Although the results are, as yet, insufficient to provide an answer to the question they do appear to be of independent interest and do allow us to focus upon some very specific problems concerning the extent to which embedded (or immersed) surfaces in a 3 manifold, foliated by (topologically)
Lecture Notes in Mathematics, 1975
Without Abstract
Nucleic acids research, Jun 22, 2016
Freshly replicated DNA molecules initially form multiply interlinked right-handed catenanes. In b... more Freshly replicated DNA molecules initially form multiply interlinked right-handed catenanes. In bacteria, these catenated molecules become supercoiled by DNA gyrase before they undergo a complete decatenation by topoisomerase IV (Topo IV). Topo IV is also involved in the unknotting of supercoiled DNA molecules. Using Metropolis Monte Carlo simulations, we investigate the shapes of supercoiled DNA molecules that are either knotted or catenated. We are especially interested in understanding how Topo IV can unknot right-handed knots and decatenate right-handed catenanes without acting on right-handed plectonemes in negatively supercoiled DNA molecules. To this end, we investigate how the topological consequences of intersegmental passages depend on the geometry of the DNA-DNA juxtapositions at which these passages occur. We observe that there are interesting differences between the geometries of DNA-DNA juxtapositions in the interwound portions and in the knotted or catenated portions ...
Knots in Hellas '98, 2000
Series on Knots and Everything, 2005
Lecture Notes in Mathematics, 1975
ABSTRACT Without Abstract
Journal of Knot Theory and Its Ramifications, 2010
The probability that a random walk or polygon in the 3-space or in the simple cubic lattice conta... more The probability that a random walk or polygon in the 3-space or in the simple cubic lattice contains a knot goes to one at the length goes to infinity. Here, we prove that this is also true for slipknots consisting of unknotted portions, called the slipknot, that contain a smaller knotted portion, called the ephemeral knot. As is the case with knots, we prove that any topological knot type occurs as the ephemeral knotted portion of a slipknot.
Biophysical Journal, 2012
Knot fingerprints provide a fine-grained resolution of the local knotting structure of tight knot... more Knot fingerprints provide a fine-grained resolution of the local knotting structure of tight knots. From this fine structure and an analysis of the associated planar graph, one can define a measure of knot complexity using the number of independent unknotting pathways from the global knot type to the short arc unknot. A specialization of the Cheeger constant provides a measure of constraint on these independent unknotting pathways. Furthermore, the structure of the knot fingerprint supports a comparison of the tight knot pathways to the unconstrained unknotting pathways of comparable length.
For a polygonal knot K, it is shown that a tube of radius R(K), the polygonal thickness radius, i... more For a polygonal knot K, it is shown that a tube of radius R(K), the polygonal thickness radius, is an embedded torus. Given a thick configuration K, perturbations of size r<R(K) define satellite structures, or local knotting. We explore knotting within these tubes both theoretically and numerically. We provide bounds on perturbation radii for which we can see small trefoil and figure-eight summands and use Monte Carlo simulations to approximate the relative probabilities of these structures as a function of the number of edges.
We present two descriptions of the the local scaling and shape of ideal rings, primarily featurin... more We present two descriptions of the the local scaling and shape of ideal rings, primarily featuring subsegments. Our focus will be the squared radius of gyration of subsegments and the squared internal end to end distance, defined to be the average squared distance between vertices k edges apart. We calculate the exact averages of these values over the space of all such ideal rings, not just a calculation of the order of these averages, and compare these to the equivalent values in open chains. This comparison will show that the structure of ideal rings is similar to that of ideal chains for only exceedingly short lengths. These results will be corroborated by numerical experiments. They will be used to analyze the convergence of our generation method and the effect of knotting on these characteristics of shape.
Random walks and polygons are used to model polymers. In this paper we consider the extension of ... more Random walks and polygons are used to model polymers. In this paper we consider the extension of writhe, self-linking number and linking number to open chains. We then study the average writhe, self-linking and linking number of random walks and polygons over the space of configurations as a function of their length. We show that the mean squared linking number, the mean squared writhe and the mean squared self-linking number of oriented uniform random walks or polygons of length n, in a convex confined space, are of the form O(n^2). Moreover, for a fixed simple closed curve in a convex confined space, we prove that the mean absolute value of the linking number between this curve and a uniform random walk or polygon of n edges is of the form O(√(n)). Our numerical studies confirm those results. They also indicate that the mean absolute linking number between any two oriented uniform random walks or polygons, of n edges each, is of the form O(n). Equilateral random walks and polygons...
A mathematical knot is simply a closed curve in three-space. Classifying open knots, or knots tha... more 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 ...
This paper contains selected topics from four lectures given at the Abdus Salam International Cen... more This paper contains selected topics from four lectures given at the Abdus Salam International Centre for Theoretical Physics in May 2009. We introduce the study of the influence of knotting and linking on the spatial characteristics of linear and ring polymer chains with examples of scientific interest. We describe a few basic concepts of the geometry and topology of knots and measures of the spatial shape of open and closed polymer chains. We then present some fundamental mathematical results concerning them. Next we discuss random sampling methods of collections of open and closed chains that are employed to provide estimates of the spatial properties of the chains. Finally, we discuss implementations of the sampling algorithms, survey consequences of theoretical and experimental results, and discuss some interesting problems deserving further research.
Let M denote a compact topological 3-manifold, without boundary, foliated by topological circles ... more Let M denote a compact topological 3-manifold, without boundary, foliated by topological circles in the sense of Seifert's gefaserter Rdume[8]. This will be called a Seifert structure, or Seifert fibration on M; the leaves of the foliation are called (regular or exceptional) fibres. Our main result is the following theorem, reminiscent
Springer Proceedings in Mathematics & Statistics
With a focus on one-dimensional periodic boundary systems, we describe the application of extensi... more With a focus on one-dimensional periodic boundary systems, we describe the application of extensions of the Gauss linking number of closed rings to open chains and, then, to systems of such chains via the periodic linking and periodic self-linking of chains. These lead to the periodic linking matrix and its associated eigenvalues providing measures of entanglement that can be applied to complex systems. We describe the general one-dimensional case and applications to one-dimensional Olympic gels and to tubular filamental structures.