A long-standing problem in knot theory concerns the additivity of crossing numbers of links under the connected sum operation. It is conjectured that if A and B are links, then Cr(A#B)=Cr(A)+Cr(B), but so far this has been proved only for certain classes of links. One such class is the zero-deficiency links, which includes some alternating links and some non-alternating, such as torus knots. In this paper the known realm of zero-deficiency links is expanded to include some cases of links represented by alternating closed braids. It is shown that for a link L represented by a reduced, alternating, k-string closed braid diagram D having at most three sequences of consecutive crossings between each pair of adjacent strings, the braid index of L is k. This result makes use of a well-known property of the HOMFLY polynomial, which provides a lower bound for the braid index of a link. It is then seen that the deficiency of L is zero. It seems likely that this result can be extended to more complex alternating closed braids.

Topological methods have been mostly used to study the action of enzymes and properties of naked DNA molecules. However topology can also be used to study chromosome organization. In this talk I will present three problems in which topology can be used to study complex organization of DNA.

First I will present the problem of DNA knotting in bacteriophages. Understanding the organization of the genome in bacteriophages is important because bacteriophages are good models for DNA organization in some animal viruses (such as herpex viruses) and in DNA lipo-complexes used in gene therapy. Our approach is based on work by Liu, Calendar, Wang and colleagues that showed that DNA extracted from bacteriophage P4 is knotted. We have investigated these knots and shown that they are informative of the organization of the genome inside the capsid. I will present models that have been derived from these knots as well as the mathematical problems that this biological problem has generated.



Second I will discuss the problem of chromosome intermingling of chromosome territories in the eukaryotic cell. During the G0/G1 phase of the cell cycle the eukaryotic genome is organized into chromosome territories. The positions of these territories as well as the structure along their surface are believed to play a major role in the formation of recurrent aberrations found in some genetic diseases and in some cancers. I will present some of our results on the topological implications of the Interchromosomal Network Model proposed by Branco and Pombo. In particular I will introduce our estimation of the linking probability of two neighboring chromosome territories assuming that chromatin fibers follow random trajectories.



Third I will present some new and unpublished results on the linking of mitochondrial DNA in trypanosomes. Trypanosomatid parasites, trypanosoma and lishmania, are the cause of disease and death in many third world countries. One of the most unusual features of these organisms is the 3 dimensional organization of their mitochondrial DNA into maxi and minicircles. Minicircles are confined into a small volume and are interlocked forming a huge network. Some initial models for the organization of this network were proposed by Cozzarelli and Englund. Here we discuss some of the possible pathways for the formation and maintenance of this network as well as the mathematical results that derive from this problem.

This work is in collaboration with: Y. Diao (UNC Charlotte), R. Scharein (SFSU), R. Kaplan (SFSU) and M. Vazquez (SFSU).

A tangle consists of strings properly embedded within a 3-dimensional ball. Solutions of tangle equations have proven quite useful when applied to recombinases. Recombinases are enzymes that cut DNA strands and interchange the ends, changing the topology of the DNA. The recombinase action will be mathematically modeled by replacing the zero tangle with the tangle t/w, resulting in a new DNA product. If we model experiments involving two topologically different substrates and/or two topologically different products, we have a corresponding system of four tangle equations. Given a1, a2, b1, b2, z1, z2, v1, and v2, we are solving the following system of four tangle equations for t/w:

N(j1/p1 + 0/1) = N(a1/b1) N(j1/p1 + t/w) = N(z1/v1) N(j2/p2 + 0/1) = N(a2/b2) N(j2/p2 + t/w) = N(z2/v2).

Joint work with Matthew R. Kesinger^*, Massa J. Shoura^*, Alexandre Vetcher^*, and Stephen D. Levene^*†.

Biological processes such as DNA recombination, replication, and gene expression involve specific interactions between one or more DNA-binding proteins and multiple protein-binding sites along a single DNA molecule. Such interactions lead to the formation of a topologically closed DNA loop between protein-recognition sites, whose energetics depends on the structure and the flexibility of the intervening DNA, the degree of supercoiling, and the binding of additional proteins such as HU and Fis in bacterial systems or histones and HMG proteins in the case of eukaryotic cells. We present here a novel technique for incorporating multiple modifications such as covalently attached fluorescent probes to multiple defined sites within covalently closed DNA molecules. Applications of this technology include the use of two- and three-color FRET to investigate effects of DNA supercoiling on lac-repressor DNA interactions both in vitro and in vivo.

Departments of Molecular and Cell Biology^* and Physics^† University of Texas at Dallas Richardson, TX 75080

Single Particle Reconstructions are commonly used to elucidate structures of different types of small particles with precision approaching that of x-ray techniques. Such reconstructions assume a homogeneity of data or a small heterogeneous collection of homogeneous subgroups. Current models of the packing of small capsid viruses like bacteriophages suggest a spooling type model with some non-uniformity resulting in small numbers of knots. Three dimensional reconstructions have been used to justify different models but such assumptions assuming uniform homogeneity of a data set disregard slight variations which could be present. We present a look at the single particle reconstruction process as a whole as well as a view of two different sets of data. We ask, "what happens if single particle reconstruction is used to reconstruct a single model from a data set in which each particle is very similar but no two are exactly alike?".

Joint work with Thomas LeHew and Joe Eichholz.

We are creating a webpage which will allow users to create tables of links, knots and their invariants. Our plan is to provide a platform which visualizes the information about knots and links on a table that will satisfy most types of users. The best platform for this plan is a webpage. The webpage would be separated into three main components. First, the actually html document that the user will see and interact with to properly generate the specific table of knots/links that the user would like to see. Second, a series of scripts will be set up to take in the input provided to retrieve the information needed from the database and then output the information in an easy to read table for the user to see. Last, a component specific to our project is the desire to allow others to contribute to the table. Livingston and Cha’s KnotInfo is an outstanding webpage for creating knot tables. However, for our project, we put more emphasis on links and their relevant invariants. It will also handle composition of links, orientation, and mirror images.

Proteins are linear chains of amino acids. Proteins are composed of secondary structure units called alpha helices and beta sheets, which are energetically stable, and random coils, which are not. Many diseases are caused by protein folding disorders. Local knots in proteins are much rarer than is expected for a long polymer. As of 2006, only thirty-nine proteins out of 9,553 proteins with determined structures contain local knots. Eighteen of those thirty-nine proteins contain shallow knots which can deform to the unknot with the removal of five to ten residues from the N- and C-termini, which are the ends of the protein. The most complicated knot in proteins with known structures is the 5_2 knot found in ubiquitin hydrolase (pdb code 1xd3).

The purpose of this project is to engineer knots into proteins with known structures that currently do not contain knots. The archive for all current structures of proteins and nucleic acids is the Protein Data Bank (PDB). For the purpose of this project, the central carbon atom will signify the amino acid. We will use a program called KnotPlot to graph the coordinates of each alpha carbon and join the two termini. From here, we will perform crossing changes in random coils on the chain and determine whether this creates a local knot that is not the unknot.

Since most proteins are linear strands and not closed loops, it is not generally possible to talk about mathematical knots in proteins. We will start with choosing a method to close the gap between the N- and C-termini. As a positive control, we will start by using proteins with local knots to see if the algorithm works. We will then perform crossing changes caused by changing random coils. We will create a protocol to do these functions using KnotPlot, and then write a program to do this automatically so that we can discover all places that can be knotted.

Joint work with Stephen D. Levene^*†.

The thermodynamic properties of a semi-flexible linear polymer, DNA for example, are examined using discrete wormlike chains (dWLCs) as a model. Monte Carlo ensembles of dWLCs were generated to investigate the effect of excluded volume on the configurational entropy of the chain, SC, which is calculated based on the Schlitter approximation. We examined the dependence of absolute and relative entropies on the cylinder diameter and also practical aspects of this approach such as fluctuations in computed SC values as a function of ensemble size. Future applications include estimating the free energy of DNA looping in complex nucleoprotein assemblies.

Departments of Molecular and Cell Biology^* and Physics^† University of Texas at Dallas Richardson, TX 75080

Joint work with Bruce MacTaggart.

Our educational lesson plans focus on elementary properties of knots and are meant to be a mini-unit in basic knot theory for high school students. There are five total lessons in our introductory unit with a summative assessment on the fifth day. The lessons address knot notation, basic definitions, knot equivalence, and knot arithmetic. There are also various activities with hands-on manipulatives for modeling knots and activities involving the program KnotPlot. We believe that since knot theory is a relatively new field of both mathematics and biology it is important to generate interest with younger mathematics students.

Two links are equivalent if, roughly speaking, one can be physically deformed into the other. However, we have a choice as to what information we are keeping track of. For example, if we label the components of a link we could ask whether or not the components can be permuted. A labeling of components could arise naturally in application, for example if the components are different polymers. The poster describes a method of recording all of the symmetry information of links as a certain group. We also preview an upcoming paper in which the symmetry groups for prime links through 8 crossings are computed and discuss future directions including the tabulation of composite links.

Tangle analysis, a branch of mathematical knot theory, in conjunction with difference topology experiments has become a powerful emerging approach for the analysis of complex nucleoprotein assemblies containing DNA loops. A tangle consists of strings properly embedded in a 3-dimensional ball. The protein complex can be thought of as a 3D ball while the DNA segments bound by the protein complex can be thought of as strings embedded within the ball. At present, tangle analysis can only provide information about 2-dimensional diagrams representing the topology of DNA bound within a protein complex. Many DNA geometries can be consistent with a particular topological solution, however, limiting the value of tangle analysis in deducing biological mechanism and function. In addition, many problems of interest do not yield unique tangle solutions. Information about the relative energies of geometric solutions is badly needed to evaluate the plausibility of a particular mathematical solution both physically and biologically. We will demonstrate preliminary software for determining likely DNA geometries consistent with protein-bound DNA topologies.

Our group studies topological properties of DNA molecules in solution. We consider highly compacted models of knotted DNA, such as DNA extracted from P4 phages. Circular DNA molecules are modeled as self-avoiding polygons (SAPs) in three-dimensional space. Using different Monte Carlo algorithms, we sample the space of knotted SAPs and study knotting probabilities. To better understand how DNA knotting is affected in confined environments, we generate knotted configurations confined inside small spheres. Writhe is a geometric invariant that measures the entanglement complexity of a given configuration. A comparison of the writhe of confined versus free knots suggests that the sign of the average writhe is invariant for each chiral knot type under varying polygonal lengths on the simple cubic lattice and in R3. We propose that the sign of the average space writhe is a robust measure of knot chirality.

We present provisional data on the minimal step number of cubic lattice knots confined to a thin slab. In particular, we investigate thin slabs of thickness 1, 2 and 3. For most knot types, several ergodicity classes are found, often with dramatically different minimal step numbers. We discuss the number of distinct minimal step embeddings found within each class. We show that in the case of the 1-slab, arbitrarily high step number representatives for each knot type may be found that are irreducible within the 1-slab. Finally, we examine recurring patterns across the entire database of minimal step knots, both in thin slabs and for the unconstrained case.

In a thermally agitated environment, randomly generated polygons are used to model the conformations of fluctuating polymer chains. To characterize the shapes of these polygons, we created 3D density plots of the vertex distributions of families of random 6 edge polygons. The distributions give a measure of the shapes of the polygons, and our symmetry-breaking alignment procedure is not only able to reveal their average bulk shape, but also distinguish between different knot topologies and chirality. We looked at the family of 6 edge polygons, separating them by knot type, and we also looked at 6 edge open chains.

Given a polygonal knot we present an efficient construction of polygonal cables of the knot. The construction is applied to polygonal unknots to obtain results about stick numbers of torus knots. In particular, we show that (2,q) torus links can be constructed with about two-thirds q sticks. This is used to show that for q greater than 14, minimal stick representatives of (2,q) torus links are "supercoiled". Finally we show that for 2p < q < 3p the stick number of (p,q) torus links is 4p.

Like knots, tabulating tangles is done by crossing number. Tangles are similar to knots, but contain strings whose endpoints are "nailed down" on the boundary of a 3-dimensional ball. The crossing number is the minimal number of crossings needed to draw the diagram of a knot (tangle). We will discuss some basic concepts common between knots and tangles, how to code this and issues that have arisen. Finally, we will introduce why we are interested in tabulating tangles: math biology.

The long, rich history of topology in mathematics has proven extremely useful for the study of DNA. DNA, the genetic blueprint for life, undergoes tremendous flux as it is packaged, replicated, segregated, transcribed, recombined and repaired. Extremely long and skinny, DNA is prone to entanglement. Every time it is copied, the two resulting "daughter chromosomes" are entangled. And nearly all organisms maintain duplex DNA in a slightly underwound state. Linking number (Lk), the major descriptor for DNA apart from base pair sequence, defines the three forms of DNA topology, which are known to biologists as knots, catenanes, and supercoils. Changes in Lk have dramatic effects on biological processes.

In this talk I will provide an overview of DNA topology and the biological ramifications of topology, including exciting new developments in the application to medicine.

The following authors have contributed to the work:

Jonathan M. Fogg₁, Daniel J. Catanese, Jr.₁, Donald Schrock, II₁, Richard W. Deibler_1,2,3, Jennifer K. Mann_1,4, De Witt L. Sumners₄, Brian E. Gilbert₁, Youli Zu₅, Nianxi Zhao₅.

₁Departments of Molecular Virology & Microbiology, Biochemistry and Molecular Biology, and Pharmacology, Baylor College of Medicine, Houston, TX 77025

₂Interdepartmental Program in Cell and Molecular Biology, Baylor College of Medicine, Houston, TX 77030

₃Department of Systems Biology, Harvard Medical School, Boston, MA 02115

₄Department of Mathematics, Florida State University, Tallahassee, FL 32306

₅Department of Pathology, The Methodist Hospital Research Institute, Houston, TX 77030 USA

We consider $mathcal{P}_n$, the space of equilateral, n-sided polygons embedded in $mathbb{R}^3$. There are several descriptions of the global shape of a polygon $P in mathcal{P}_n$, including the convex hull volume, miniball radius, asphericity and radius of gyration. We sought a description of shape that was sensitive to both local and global behavior, and to look at average trends over both the whole population of $mathcal{P}_n$, as well as finding the average over sub populations of a specific knot type.

We developed two such descriptions. For a given $P in mathcal{P}_n$, we find the average of the squared distance between vertex $i$ and $i+k$ for all $1 leq i leq n$. We call this the Average Squared End to End Distance of length $k$ of $P$.

Similarly, we find the squared radius of gyration for all sub-segments of $P$ of length $k$, and we call the average of these values the Average Squared Radius of Gyration of length $k$ of $P$.

We determine the theoretical averages of these values, taken over all of $mathcal{P}_n$, in terms of $n$ and $k$. In addition, we examine specific examples of embedded polygons, to determine the effect of knotting of these descriptions of shape.