Self-Avoiding Walks (SAWs), Entanglement and Biomolecules

The Self-Avoiding Walk (SAW) on a lattice are often used to study properties of polymers in good solvents such as entanglement, knotting (ring polymers), and statistical mechanical properties of polymers. Recently it has been used to explain the increased probability of phage DNA being knotted when compared to DNA in found in unconstrained environments.

We propose to examine different aspects of SAWs on the square (2D) and cubic (3D) lattice using a dynamical Monte Carlo (MC) method in known as the pivot algorithm. Initially we only consider linear (or open) polymers and look at the entangledness of a (lattice) polymer using the writhe of a curve. A first goal is to study the relationship between the writhe and extension of a polymer/SAW.

Several questions arise naturally in the course of this project including (but not restricted to): the statistical quality of data obtained by MC sampling (autocorrelation times), how to implement geometrical (writhe) and topological (alexander polynomial) numerically, closing the curve we can start to ask questions about random (lattice) knots, optimizing the implementation e.g. using hash-coding ...