# Minerva Foundation Lectures, Columbia University, September 2007

## Some mathematical problems from population genetics

Alison Etheridge ( etheridg@stats.ox.ac.uk )

Page last updated September 12, 2007. Some typos in the slides for the 5th lecture have been corrected.

• Lecture 1: Classical models.
We introduce some terminology and review some established models from mathematical population genetics. This includes Wright-Fisher, Moran and stepping stone models for the (forwards in time) evolution of gene frequencies and Kingman's coalescent and its extensions for modelling the genealogical relationship (found by tracing backwards in time) between individuals in a sample from a population.

• Lecture 2: Recombination.
In sexually reproducing organisms such as our own, in which chromosomes are carried in pairs, each individual will inherit one chromosome of each pair from their mother and one from their father. But their chromosomes are not faithful copies of parental chromosomes. One reason for this is recombination. We describe the action of recombination and explain the resulting mathematical complications in our models. Backwards in time analytic results are hard to find. Here we consider the simpler problem of the descent of a block of genome forwards in time. Our model, based on branching processes, predicts the probability of survival of any genetic material from a single block of genome $t$ generations in the future.

• Lecture 3 :Selection.
If a selectively advantageous mutation appears in a population, then with some probability it will increase in frequency until everyone in the population carries it. We then say that a selective sweep has occurred. How can we detect selective sweeps in data? The key is an effect known as genetic hitchhiking which we discuss here. We then turn to another form of selection, balancing selection, which contrives to maintain different forms of the same gene at non-trivial frequencies in the population. Examination of the effect on gene frequencies at a neutral locus on the same chromosome has important ramifications for the classical models introduced in the first lecture.

• Lecture 4: Spatial Models.
An important conclusion from the work on selection described in lecture 3 is that we cannot disentangle the effects of demography and genetics. In this lecture we describe some of the challenges of demographic modelling and investigate which features of a demographic model must be understood if we are to feed demographic information into our genetic models.

• Lecture 5: Muller's ratchet and the rate of adaptation.
The evolutionary force of recombination is lacking in an asexually reproducing population. As a consequence, the population can suffer an irreversible accumulation of deleterious mutations, a phenomenon known as Muller's ratchet. Since recombination can overcome this effect, it is sometimes used as an explanation for the evolution of sex. But other forces can overcome the ratchet too; for example the presence of some beneficial mutations. A mathematical model for the ratchet was formulated by Haigh (1978), but in spite of the apparent simplicity of the formulation the model has proved to be remarkably resistant to analytic study. We investigate variants of Haigh's model in two settings: first when there are only deleterious mutations and second when a small proportion of mutations are beneficial. In the latter case we discover that large enough populations increase in mean fitness and we establish a lower bound on this `rate of adaptation'.

• Probability Seminar, Sept 14: Drift, draft and structure: mathematical models of evolution.
One of the outstanding successes of mathematical population genetics is Kingman's coalescent. This process provides a simple and elegant description of the genealogical trees relating a sample of neutral genes from a {\em panmictic} population, that is, one in which every individual is equally likely to mate with every other and all individuals experience the same conditions. But real populations are not like this. They are structured by spatial location and genetic type. Spurred on by the recent flood of DNA sequence data, an enormous industry has developed that seeks to extend Kingman's coalescent to incorporate things like varying population size, natural selection and spatial and genetic structure. But a satisfactory approach to populations evolving in a spatial continuum has proved elusive. In this talk we describe the effects of some of these biologically important phenomena on the genealogical trees that relate individuals in a sample from the population before describing a new approach (joint work with Nick Barton) to modelling the evolution of populations distributed in a spatial continuum.

• Annotated reference list.