# Modelling DNA Knotting Using Interacting Lattice Self-Avoiding Polygon Models

## Date

2022-09-06

## Authors

## Journal Title

## Journal ISSN

## Volume Title

## Publisher

## ORCID

0000-0002-2852-8136

## Type

Thesis

## Degree Level

Doctoral

## Abstract

The work presented here considers simple cubic lattice self-avoiding polygon (SAP) models for studying the average conformational properties of randomly cyclized relaxed DNA in a salt solution. The work involves novel studies of two classes of SAP models, strand-passage models and knot probability models, each of which are related to different DNA knotting experiments. \\
\noindent The first part of the thesis focuses on strand-passage models. These models are intended to better understand the favourable DNA unknotting action of type-II topoisomerases, which are enzymes that perform segment passages (a.k.a. strand passages) between two segments of double stranded DNA. A DNA experiment performed by Rybenkov \textit{et al.} (1997) showed this enzyme action reduces the fraction of DNA knots to values well below those observed at equilibrium. Studies of various models of this DNA-enzyme interaction have demonstrated that the amount of knot reduction depends significantly on the local geometry at the strand passage site and on DNA supercoiling. Amongst these, studies of the Local Strand Passage (LSP) model, a lattice SAP model of ring polymers in a good solvent, established that the amount of knot reduction can depend on the crossing-sign in addition to the local geometry at the strand passage site. Here we explore whether the conclusions of the LSP model studies are affected by varying solvent conditions. Building from the LSP model, the work in this thesis models the DNA-enzyme complex using $\Theta$-SAPs which are SAPs containing a fixed structure $\Theta$ representing where the enzyme has attached to two nearby polygon segments in preparation for a strand passage action. A short-range nearest neighbour (contact) interaction and a long-range screened coulomb (Yukawa) potential are incorporated in order to model interactions between negatively charged DNA segments and ions in a salt solution. The resulting model is referred to as the Interacting Local Strand Passage (ILSP) model. Samples of $\Theta$-SAPs from the ILSP model are generated via the Interacting-$\Theta$-BFACF Algorithm (a Markov Chain Monte Carlo algorithm) and then used to estimate the probabilities of transitioning from one knot type to another due to a crossing-sign dependent strand passage at $\Theta$. By conditioning the strand-passage action to be dependent on various geometric factors (\textit{e.g.} the local geometry near the strand passage site or the space writhe of the SAP), we also estimate knot comparison factors in order to measure the relative unknotting ability of geometry-restricted strand passage actions. In comparison to the LSP model results, we similarly find that the relative unknotting ability of a strand passage is strongly dependent on the opening angle of the juxtaposed segments at the strand passage site; however, this relative unknotting ability decreases as model salt concentration increases. We also find that the relative unknotting ability of a crossing-sign dependent strand passage is higher if the supercoiling of the $\Theta$-SAP matches the crossing sign at $\Theta$. \\
\noindent The second part of this work focuses on knot probability models. Experiments involving random cyclization of relaxed DNA by Shaw and Wang (1993) and Rybenkov \textit{et al.} (1993) resulted in DNA knot probability data for varying salt concentrations and a small range of DNA lengths. Starting with the worm-like chain model of Rybenkov \textit{et al.} (1993), DNA modellers have used these experiments to tune their model parameters. This was first done for a lattice SAP model by Tesi \textit{et al.} (1994). Here, we build on the Tesi \textit{et al.} model towards finding a model that better fits the knot probability experiment data. We consider two different lattice SAP models with differing interaction energies. The ﬁrst model, referred to here as the \textit{Contact/Yukawa model}, considers the same short-range nearest neighbour interaction and long-range screened coulomb potential as used in the ILSP model and as in the Tesi \textit{et al.} model. The second model, referred to here as the \textit{Bending/Yukawa model}, replaces the nearest neighbour interaction in the Contact/Yukawa model with another short-range interaction based on a polymer’s bending rigidity. Parameters in the Contact/Yukawa and Bending/Yukawa models are optimized to the Shaw and Wang experimental knotting probabilities using the Simultaneous Perturbation Stochastic Approximation (SPSA) method. Implementation of the SPSA method for these models involves simulations from the Interacting-Pivot Monte Carlo algorithm and involves non-trivial model-dependent adjustments. The resulting fits from the optimized Contact/Yukawa and Bending/Yukawa models are of similar quality; however estimates of persistence length from these two models shows that the ﬁtted Bending/Yukawa model gives a persistence length which is much closer to the reported persistence length of DNA. Using the optimized parameters from the ﬁtted Bending/Yukawa model, we then study how knotting probabilities in this model vary with the number of edges in the SAP (\textit{i.e.} different lengths of DNA). In particular, estimates of these knotting probabilities agree well with results obtained from a discrete worm-like chain model used by Virnau and Rieger (2016). We conclude that the Bending/Yukawa SAP model is a suitable SAP model for modelling relaxed DNA in solution. This points to important future work which includes modifying the model to study supercoiled DNA and also incorporating its features into SAP strand passage models.

## Description

## Keywords

self-avoiding polygons, SAPs, MCMC, Markov Chain Monte Carlo, Stochastic Approximation, BFACF Algorithm, Pivot Algorithm, Schmirler, DNA, modelling

## Citation

## Degree

Doctor of Philosophy (Ph.D.)

## Department

Mathematics and Statistics

## Program

Mathematics