Finite Population Mutation-Selection Models with Recombination

The objective of the research is to extend existing mutation-selection models with recombination to incorporate the stochastic effects induced by a finite population. Mutation-selection models describe the spread of mutations through a population, as new mutations appear, and old deleterious mutations are eliminated by natural selection. An important feature of those models is recombination, that is the process by which chromosomes from the two parents are cut and pasted together to create new chromosomes.

This has the effect of spreading mutations more reliably across the population. The mutation-selection models enable us to understand how the various biological forces acting on a species interact with one another to produce the diversity and complexity we observe. It is also a useful framework to study the question of the evolution of aging.

It is often convenient to represent the population in such models as being infinite and continuous. As a consequence, those models do not exhibit a biological phenomenon known as genetic drift, which is the variation in the genetic makeup of the population that is not due to selective pressures, but simply to the inherent randomness of mating and reproduction. Genetic drift can have a significant impact on predictions made by those models.

To build a more accurate model, we will replace the assumption of an infinite continuous population with that of a large but finite population. Instead of following a deterministic trajectory, the model thus obtained will be a random process. We will focus on the number of mutations per individual and how this quantity will fluctuate over a long period of time, and in particular how this differs from the deterministic model.

For the mathematical analysis of the model, we will use tools from probability theory, such as stochastic calculus and Markov chain theory. If necessary, we may approximate the true finite population model with a continuous model which exhibits similar genetic drift. This continuous model is likely to take the form of a system of stochastic differential equations.

Solutions of this system should converge after a long period of time to a stationary distribution. We will characterize this distribution as best as possible and compare it with the repartition of mutations in the deterministic model. This project falls within the EPSRC mathematical sciences research area.