A Transfer Matrix Approach to Studying the Entanglement Complexity of Self-Avoiding Polygons in Lattice Tubes
Self-avoiding polygons (SAPs) are a well-established useful model of ring polymers and they have also proved useful for addressing DNA topology questions. Motivated by exploring the effects of confinement on DNA topology, in this thesis, SAPs are confined to a tubular sublattice of the simple cubic lattice. Transfer matrix methods are applied to examine the entanglement complexity of SAPs in lattice tubes. Transfer matrices are generated for small tube sizes, and exact enumeration of knotting distributions are obtained for small SAP sizes. Also, a novel sampling procedure that utilizes the generated transfer matrices is implemented to obtain independent uniformly distributed random samples of large SAPs in tubes. Using these randomly generated polygons, asymptotic growth rates for the number of fixed knot-type SAPs are estimated, and evidence is provided to support a conjectured asymptotic form for the growth of the number of fixed knot-type polygons of a given size. In particular, the evidence supports that the entropic critical exponent goes up by one with each knot factor. Additionally, a system consisting of two SAPs (called a 2SAP) in a tube is also studied to explore linking. New transfer matrices are generated for 2SAPs in small tube sizes, and exact enumeration of linking distributions are obtained for small 2SAP sizes. A sampling procedure similar to that developed for SAPs is implemented by using the 2SAP transfer matrices to obtain independent uniform samples of large 2SAPs in tubes. An asymptotic form for the number of fixed link-type 2SAPs is conjectured with some supporting evidence from the sampled 2SAPs. All the evidence obtained supports the conclusion that the knotted parts in long polymers confined to tubular environments occur in a relatively localized manner. This is supported by the entropic critical exponent results, and by preliminary evidence that average spans of knot factor patterns are not growing significantly with polygon size. Similar evidence is obtained for the knotted parts in 2SAPs. The SAP study has also revealed further characteristics of knotting in tubes. For example, when the cross-sectional area of tubes are equal, evidence indicates that knotting is more likely in more symmetrical tubes as opposed to flatter tubes. Additionally, two types of knot pattern modes have been observed and strong evidence is provided that the so-called non-local mode is dominant for small tube sizes. These two modes have also been observed in non-equilibrium simulations and in DNA nanopore experiments. The evidence for the characteristics of the linked part of 2SAPs in a tube is less conclusive but its study has opened up numerous interesting questions for further study. In summary, the novelty of the contributions in the thesis include both computational and polymer modelling contributions. Computationally: transfer matrices, Monte Carlo methods, and a novel approach for knot identification for knots in tubes are developed and extended to larger tube sizes than ever before. Polymer modelling: strong numerical evidence supporting knot localization for polymers in tubes and the first evidence regarding characterising linking for polymers in tubes are obtained.
Polygon, knotting, linking, lattice, tube, transfer, matrix
Doctor of Philosophy (Ph.D.)
Mathematics and Statistics