Some Mathematical Models from Population Genetics
École d'Été de Probabilités de Saint-Flour XXXIX-2009
Published on 5. January 2011
VIII, 119 pages
978-3-642-16632-7 (ISBN)
Description
This work reflects sixteen hours of lectures delivered by the author at the 2009 St Flour summer school in probability. It provides a rapid introduction to a range of mathematical models that have their origins in theoretical population genetics. The models fall into two classes: forwards in time models for the evolution of frequencies of different genetic types in a population; and backwards in time (coalescent) models that trace out the genealogical relationships between individuals in a sample from the population. Some, like the classical Wright-Fisher model, date right back to the origins of the subject. Others, like the multiple merger coalescents or the spatial Lambda-Fleming-Viot process are much more recent. All share a rich mathematical structure. Biological terms are explained, the models are carefully motivated and tools for their study are presented systematically.
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Alison Etheridge
Some Mathematical Models from Population Genetics
École d'Été de Probabilités de Saint-Flour XXXIX-2009
Book
01/2011
Springer
€42.79
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Content
- Some Mathematical Models from Population Genetics
- Preface
- Contents
- 1 Introduction
- 2 Mutation and Random Genetic Drift
- 2.1 The Wright-Fisher Model and the Kingman Coalescent
- 2.2 The Cannings Model
- 2.3 Selfing
- 2.4 Adding Mutations
- 2.5 Inferring Genealogies From Data
- 2.6 Some Properties of Kingman's Coalescent
- 2.7 Genealogies and Pedigrees
- 2.8 The Moran Model
- 2.9 The Site Frequency Spectrum
- 2.10 The Lookdown Process
- 2.11 A More Simplistic Limit
- 3 One Dimensional Diffusions
- 3.1 Diffusions
- 3.2 Convergence to Diffusions
- 3.3 Speed and Scale
- 3.4 Hitting Probabilities and Feller's Boundary Classification
- 3.5 Green's Functions
- 3.6 Stationary Distributions and Reversibility
- 4 More than Two Types
- 4.1 Multi-Dimensional Diffusion Models
- 4.2 The Poisson-Dirichlet and GEM Distributions
- 4.3 Ewens Sampling Formula
- 5 Selection
- 5.1 Genetic Diversity
- 5.2 Wright-Fisher Model with Selection
- 5.3 Selection in a Diploid Population
- 5.4 The Ancestral Selection Graph
- 5.5 Adding Structure to the Coalescent
- 5.6 Selective Sweeps
- 5.7 Coalescents with Multiple Mergers
- 6 Spatial Structure
- 6.1 Subdivided Populations and the Structured Coalescent
- 6.2 Duality
- 6.3 Collapse of Structure
- 6.4 Evolution in a Spatial Continuum and the Pain in the Torus
- 6.5 The Spatial -Fleming-Viot Process
- 6.6 More General Models
- References
- List of Participants
- Index
