Some applications of topology and statistical physics to polymer physics and molecular biology

This work involving topology and statistical physics consists of three separate parts. In the first part, we study knots in proteins and protein-like polymers on a lattice. We verify the Gaussian behaviour of subchains or short segments of the compact conformations on a lattice. We use knot invariants to measure the probabilities of obtaining certain types of knots in lattices and in proteins. We verify that there are few non-trivial knots in proteins, and we examine the behaviour of the segments of proteins to gain some insight. In the second part, we use a model of a particle diffusing under a sawtooth potential to compute the average time, or mean first passage time, for single-stranded DNA to escape after being threaded through a narrow pore. The sawtooth potential emphasizes the asymmetry and interaction of the DNA bases with the pore. The first passage time and its dependence on the length N of the ssDNA are considered in the context of classical diffusion and in the context of subdiffusion. In the third part, using the simple model of an ideal gas inside a cavity, we explore the issue of the practical evaluation of free energy differences via the Jarzynski identity. The Jarzynski identity connects the average exponential of the work over the non-equilibrium trajectories to the free energy difference between initial and final states. We derive an exact expression for the distribution of work done by the gas as the piston expands or contracts the cavity and we verify the Jarzynski relation for this model.