Link transition probabilities after a local strand exchange in a self-avoiding polygon model of enzyme-DNA interactions

Abstract

A new Local Strand Exchange (LSE) model is presented in this thesis as a first-step towards modelling recombinase-DNA interactions. The model uses self-avoiding polygons (SAPs) on the simple cubic lattice to represent DNA configurations. The new LSE model is related to the Local Strand Passage (LSP) model that was developed by Szafron and Soteros to model topoisomerase-DNA interactions. The thesis begins with a review of the biological background for the topoisomerase and recombinase enzymes, followed by a review of current approaches to model the interactions of these enzymes with DNA. A review is then presented for the LSP model of a type II topoisomerase-DNA interaction, where a fixed structure, called Θ, is used as the interaction location. Following this, the new LSE model is introduced for modelling a site-specific recombinase-DNA interaction, where the Θ structure is again used as the interaction location. A recombination action is modelled with two different structures to take into account the two biologically relevant types of site-specific recombination. Specifically: 1) Direct repeat recombination sites are modelled with the direct-repeat-to-link strand exchange (DLE) structure; 2) Inverted repeat recombination sites are modelled with the inverted-repeat-to-knot strand exchange (IKE) structure. The LSE model is studied using composite Markov chain Monte Carlo data that was previously generated to study the LSP model; the LSE study involves estimating link transition probabilities, i.e. the probabilities of going from knot type K to link type L after a strand exchange via DLE or IKE. Strong numerical evidence is provided that the link transition probabilities from the LSE model have asymptotic (as polygons get larger) properties consistent with conjectures based on polymer scaling theory. For example, it is shown that as polygon lengths increase, it becomes less likely that the fixed Θ structure will interact with any pre-existing knot. This means that after a strand exchange, typically, the original knot remains intact and a secondary knot or link can be formed at the Θ structure. In the DLE case, a link is created with one component of the link still containing the original knot, and in the IKE case, a composite knot containing the original knot can be created. Lastly, for a subset of all the sampled polygons, numerical results that are biologically relevant are presented. Specifically, when the Θ structure is in the middle of a polygon (i.e. each of the two walks connected to Θ have the same length), then the most probable outcome is that the complexity of the knot will be reduced. In the DLE case, this is consistent with the mathematical and biological unlinking pathway observed in recombinase experiments. In the IKE case, the knot is more likely to become simpler than more complicated, reflecting recombinases’s unknotting potential. This evidence establishes the new LSE model as a useful model for studying recombinase-DNA interactions, and worthy of future research.