M2 Lectures, Orsay, February/March 2009

Some mathematical models from population genetics

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

Page last updated March 19th, 2009.

Any student requiring validation for this course should contact me. Corrected notes are being added below as they are typed. NOTES FOR LECTURE 4 WILL BE ADDED LATER AS THEY ARE BEING PRODUCED BY STUDENTS AS PART OF THEIR VALIDATION. Pictures will only be added much later and the pagination will change, so the contents in the first section is only a guide.

  • Part 1a: Notes for lecture 1.
    This section of the notes introduces some classicl models of theoretical population genetics, the Wright-Fisher model and the Cannings model. We show that for large populations the genealogy of a sample from such a population is approximately governed by Kingman's coalescent. We also investigate how allele frequencies change over long timescales in large populations.

  • Part 1b: Notes for lecture 2.
    In the second lecture we talked about adding mutations and inferring genealogies from data. We also proved some elementary properties of the Kingman coalescent and commented briefly on the relationship between genealogies and pedigrees. Finally we introduced the Moran model. Pictures have not yet been added and the final version of the notes will probably include more in sections 1.4 and 1.6.

  • Part 1c Notes for lecture 3.
    In this lecture we used the duality between the Moran model and the Kingman coalescent to invesitigate expectations in the site frequency spectrum of a sample under the infinitely many sites mutation model. We then showed that for very large populations, over large timescales, changes in allele frequency for the Moran model match those for the Wright-Fisher model. In the limit of infinite population size we obtain a one-dimensional diffusion process as an approximate description of the dynamics of allele frequencies. The second half of the lecture presents basic facts about one-dimensional diffusions, in particular providing a very brief introduction to the theory of speed and scale. (The notes include a description of the Donnelly-Kurtz lookdown process and a discussion of the Green's function for a one-dimensional diffusion even though we did not have time to cover these topics in lectures.)

    Part 1d. Notes for lecture 4.
    After a brief introduction to stationary measures and reversibility arguments for our one-dimensional diffusions, in this lecture we move on to discuss the case when our population is subdivided into more than two types. In this context we discuss the Dirichlet, Poisson-Dirichlet and GEM distributions. Our study of neutral models is rounded off with Ewens Sampling Formula.

  • Part 2 Notes for lecture 5 (by Eric Luçon). Additional material will be posted later.
    In this lecture we give a brief introduction to the mathematics of natural selection. We begin with one-dimensional diffusions before looking at the effect of selection on genealogies. Our discussion begins with the ancestral selection graph of Krone and Neuhauser and then moves on to an alternative approach in which the sample is thought of as embedded in a subdivided population. We then investigate what happens in the strong selection limit and describe Gillespie's pseudohitchhiking model and its coalescent dual. Finally we describe the family of Lambda-coalescents.

  • Annotated reference list. Not yet available, sorry!