# Strand Passage And Knotting Probabilities In An Interacting Self-Avoiding Polygon Model

## Date

2012-09-19

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## Degree Level

Masters

## Abstract

The work presented in this thesis develops a new model for local strand passage in a ring polymer in a dilute salt solution. This model, called the Interacting Local Strand Passage (ILSP) model, models ring polymers via Theta-SAPs, which are self-avoiding polygons (SAPs) in the simple cubic lattice that contain a fixed structure Theta. This fixed structure represents two segments of the self-avoiding polygon being brought ''close'' together for the purpose of performing a strand passage. Theta-SAPs were first studied in the Local Strand Passage (LSP) model developed by Szafron (2000, 2009), where each Theta-SAP is considered equally likely in order to model good solvent conditions. In the ILSP model, each Theta-SAP has a modified Yukawa potential energy which contains an attractive term as well as a screened Coulomb potential that accounts for the effect of salt in the model. The energy function used in this model was first proposed by Tesi et al. (1994) for studying self-avoiding polygons in the simple cubic lattice.
The ILSP model is studied in this thesis using the Interacting Theta-BFACF (I-Theta-BFACF) Algorithm, an algorithm which is developed in this thesis and is proven to be ergodic on the set of all Theta-SAPs of a particular knot type and connection class. The I-Theta-BFACF algorithm was created by modifying the Theta-BFACF algorithm developed by Szafron (2000, 2009) to include energy-based Metropolis sampling. This modification allows one to sample Theta-SAPs of a particular knot type and connection class based on a priori chosen solvent conditions.
Multiple simulations (each consisting of 40 billion time steps) of composite Markov Chain Monte Carlo implementations of the I-Theta-BFACF algorithm are performed on unknotted connection class II Theta-SAPs over a wide range of salt concentrations. The data from these simulations is used to estimate, as a function of polygon length, the probability of an unknotted Theta-SAP remaining an unknot after a strand passage, as well as the probability of it becoming a positive trefoil knot. The results strongly suggest that as the length of a Theta-SAP goes to infinity, the probability of the Theta-SAP becoming knotted after a strand passage increases as the salt concentration in the model increases. These results serve as a first step for studying how the knot reduction factor (studied by Liu et al. (2006) and Szafron and Soteros (2011)) of a ring polymer varies in differing solvent conditions. The goal of this future research is to find solvent conditions and a local geometry of the strand passage site that yields a knot reduction factor comparable to the research of Rybenkov et al. (1997), which shows an 80-fold reduction of knotting after type II topoisomerase enzymes act on DNA.

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Mathematics, Statistics, Topoisomerase, MCMC, Markov Chain Monte Carlo, Metropolis Sampling, DNA, Self Avoiding Polygons

## Citation

## Degree

Master of Science (M.Sc.)

## Department

Mathematics and Statistics

## Program

Mathematics