Ornstein Uhlenbeck- Jump Models of Evolution

Abstract

In this thesis we explore models of continuous trait evolution which allow for sudden changes in trait values. Starting with the Ornstein Uhlenbeck process, we introduce two new models based on combinations of the Ornstein Uhlenbeck process and a jump process.

Chapter 1 introduces the biological background. We summarise the evolutionary processes relevant to our models, introduce terminology associated with phylogenetic trees, and describe the experiment to which we will eventually apply our models.

Chapter 2 discusses models of continuous trait evolution. We first review a number of models found in the literature, then describe the versions of the Ornstein Uhlenbeck process which are the focus of this work. For each of our models we define the transition densities needed to calculate the likelihood of data generated under the model on a phylogeny.

Chapter 3 describes three methods for approximating the likelihood under one of our new models. The first of these uses weighted sums of Hermite polynomials to approximate partial likelihoods and makes use of nice properties of the Hermite polynomials to calculate an approximate likelihood based on these partial approximations. The second method is based on numerical quadrature in which partial likelihoods are approximated by interpolatory polynomials. The third method approximates a weighted sum of Gaussians by a smaller weighted sum of Gaussians. The accuracy and convergence properties of all three approximations are evaluated numerically.

Chapter 4 attempts to infer model parameters from data. We discuss how to incorporate sampling of individuals and multiple traits into our model. We take a Bayesian approach to parameter estimation and sample from the marginal posterior distribution of each parameter using MCMC. We investigate our ability to correctly estimate model parameters on simulated data and find that only some parameters are estimated well. Finally, we estimate model parameters from data on protein expression levels sampled from white clover (Trifolium repens) under artificial selection.

Chapter 5 summarises the main conclusions and discusses outstanding questions and further work to be done.