K. Hinson

Sampling Large Random Knots in a Confined Space

DNA knots formed under extreme conditions of condensation, as in bacteriophage P4, are difficult to analyze experimentally and theoretically. In this paper, we propose to use the uniform random polygon model as a supplementary method to the existing methods for generating random knots in confinement. The uniform random polygon model allows us to sample knots with large crossing numbers and also to generate large diagrammatically prime knot diagrams. We show numerically that uniform random polygons sample knots with large minimum crossing numbers and certain complicated knot invariants (as those observed experimentally). We do this in terms of the knot determinants or colorings. Our numerical results suggest that the average determinant of a uniform random polygon of n vertices grows faster than . We also investigate the complexity of prime knot diagrams. We show rigorously that the probability that a randomly selected 2D uniform random polygon of n vertices is almost diagrammatically prime goes to 1 as n goes to infinity. Furthermore, the average number of crossings in such a diagram is at the order of O(n2). Therefore, the two-dimensional uniform random polygons offer an effective way in sampling large (prime) knots, which can be useful in various applications.

The growth of minicircle networks on regular lattices

The mitochondrial DNA of trypanosomes is organized into a network of topologically linked minicircles. In order to investigate how key topological properties of the network change with minicircle density, the authors introduced, in an earlier study, a mathematical model in which randomly oriented minicircles were placed on the vertices of the simple square lattice. Using this model, the authors rigorously showed that when the density of minicircles increases, percolation clusters form. For higher densities, these percolation clusters are the backbones for networks of minicircles that saturate the entire lattice. An important relevant question is whether these findings are generally true. That is, whether these results are independent of the choice of the lattices on which the model is based. In this paper, we study two additional lattices (namely the honeycomb and the triangular lattices). These regular lattices are selected because they have been proposed for trypanosomes before and after replication. We compare our findings with our earlier results on the square lattice and show that the mathematical statements derived for the square lattice can be extended to these other lattices qualitatively. This finding suggests the universality of these properties. Furthermore, we performed a numerical study which provided data that are consistent with our theoretical analysis, and showed that the effect of the choice of lattices on the key network topological characteristics is rather small.

The effects of density on the topological structure of the mitochondrial DNA from trypanosomes

Trypanosomatida parasites, such as trypanosoma and lishmania, are the cause of deadly diseases in many third world countries. A distinctive feature of these organisms is the three dimensional organization of their mitochondrial DNA into maxi and minicircles. In some of these organisms minicircles are confined into a small disk volume and are topologically linked, forming a gigantic linked network. The origins of such a network as well as of its topological properties are mostly unknown. In this paper we quantify the effects of the confinement on the topology of such a minicircle network. We introduce a simple mathematical model in which a collection of randomly oriented minicircles are spread over a rectangular grid. We present analytical and computational results showing that a finite positive critical percolation density exists, that the probability of formation of a highly linked network increases exponentially fast when minicircles are confined, and that the mean minicircle valence (the number of minicircles that a particular minicircle is linked to) increases linearly with density. When these results are interpreted in the context of the mitochondrial DNA of the trypanosome they suggest that confinement plays a key role on the formation of the linked network. This hypothesis is supported by the agreement of our simulations with experimental results that show that the valence grows linearly with density. Our model predicts the existence of a percolation density and that the distribution of minicircle valences is more heterogeneous than initially thought.

The mean squared writhe of alternating random knot diagrams

The writhe of a knot diagram is a simple geometric measure of the complexity of the knot diagram. It plays an important role not only in knot theory itself, but also in various applications of knot theory to fields such as molecular biology and polymer physics. The mean squared writhe of any sample of knot diagrams with n crossings is n when for each diagram at each crossing one of the two strands is chosen as the overpass at random with probability one-half. However, such a diagram is usually not minimal. If we restrict ourselves to a minimal knot diagram, then the choice of which strand is the over- or under-strand at each crossing is no longer independent of the neighboring crossings and a larger mean squared writhe is expected for minimal diagrams. This paper explores the effect on the correlation between the mean squared writhe and the diagrams imposed by the condition that diagrams are minimal by studying the writhe of classes of reduced, alternating knot diagrams. We demonstrate that the behavior of the mean squared writhe heavily depends on the underlying space of diagram templates. In particular this is true when the sample space contains only diagrams of a special structure. When the sample space is large enough to contain not only diagrams of a special type, then the mean squared writhe for n crossing diagrams tends to grow linearly with n, but at a faster rate than n, indicating an intrinsic property of alternating knot diagrams. Studying the mean squared writhe of alternating random knot diagrams also provides some insight into the properties of the diagram generating methods used, which is an important area of study in the applications of random knot theory.

Invariants of Composite Networks Arising as a Tensor Product

We generalize Brylawski’s formula of the Tutte polynomial of a tensor product of matroids to colored connected graphs, matroids, and disconnected graphs. Unlike the non-colored tensor product where all edges have to be replaced by the same graph, our colored generalization of the tensor product operation allows individual edge replacement. The colored Tutte polynomials we compute exists by the results of Bollobás and Riordan. The proof depends on finding the correct generalization of the two components of the pointed Tutte polynomial, first studied by Brylawski and Oxley, and on careful enumeration of the connected components in a tensor product. Our results make the calculation of certain invariants of many composite networks easier, provided that the invariants are obtained from the colored Tutte polynomials via substitution and the composite networks are represented as tensor products of colored graphs. In particular, our method can be used to calculate (with relative ease) the expected number of connected components after an accident hits a composite network in which some major links are identical subnetworks in themselves.

The effect of volume exclusion on the formation of DNA minicircle networks: implications to kinetoplast DNA

Kinetoplast DNA (kDNA) is the mitochondrial of DNA of disease causing organisms such as Trypanosoma Brucei (T. Brucei) and Trypanosoma Cruzi (T. Cruzi). In most organisms, KDNA is made of thousands of small circular DNA molecules that are highly condensed and topologically linked forming a gigantic planar network. In our previous work we have developed mathematical and computational models to test the confinement hypothesis, that is that the formation of kDNA minicircle networks is a product of the high DNA condensation achieved in the mitochondrion of these organisms. In these studies we studied three parameters that characterize the growth of the network topology upon confinement: the critical percolation density, the mean saturation density and the mean valence (i.e. the number of mini circles topologically linked to any chosen minicircle). Experimental results on insect-infecting organisms showed that the mean valence is equal to three, forming a structure similar to those found in medieval chain-mails. These same studies hypothesized that this value of the mean valence was driven by the DNA excluded volume. Here we extend our previous work on kDNA by characterizing the effects of DNA excluded volume on the three descriptive parameters. Using computer simulations of polymer swelling we found that (1) in agreement with previous studies the linking probability of two minicircles does not decrease linearly with the distance between the two minicircles, (2) the mean valence grows linearly with the density of minicircles and decreases with the thickness of the excluded volume, (3) the critical percolation and mean saturation densities grow linearly with the thickness of the excluded volume. Our results therefore suggest that the swelling of the DNA molecule, due to electrostatic interactions, has relatively mild implications on the overall topology of the network. Our results also validate our topological descriptors since they appear to reflect the changes in the physical properties of the polymeric chains and at the same time remain faithful to their description of kDNA.

A Tutte-style proof of Brylawski's tensor product formula

We provide a new proof of Brylawskis formula for the Tutte polynomial of the tensor product of two matroids. Our proof involves extending Tuttes formula, expressing the Tutte polynomial using a calculus of activities, to all polynomials involved in Brylawskis formula. The approach presented here may be used to show a signed generalization of Brylawskis formula, which may be used to compute the Jones polynomial of some large non-alternating knots. Our proof inspires an extension of Brylawskis formula to the case when the pointed element is a factor.