UC Berkeley

Calculations based on the Wright-Fisher process and its limits are the primary tools of population genetics theory. Decades of theoretical work have elucidated much about the properties of the neutral Wright-Fisher model, in which different mutations have the exact same Darwinian fitness. The model becomes significantly more complicated when the action of natural selection is taken into account, and we are only just beginning to understand the details of evolution subject to natural selection. In this thesis, I take a path integral approach to understanding the Wright-Fisher diffusion with selection, in contrast to the typical approach, using partial differential equations. This allows me to use powerful machinery from quantum physics and mathematical finance to come up with novel ways to perform difficult calculations. The work here is composed of three parts. First, I develop a rejection sampling approach to obtaining Wright-Fisher diffusion paths when the allele frequency trajectory is conditioned to start and end at certain points (such paths are called bridges). Next, I use perturbation theory to calculate the transition densities of the Wright-Fisher process with genic selection, assuming weak selection. Finally, I implemented a Markov chain Monte Carlo approach to estimating selection coefficients from allele frequency time series that makes use of aspects of both of the previous chapters.