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A coalescent dual process for a Wright-Fisher diffusion with recombination and its application to haplotype partitioning
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Griffiths, Robert C., Jenkins, Paul and Lessard, Sabin (2016) A coalescent dual process for a Wright-Fisher diffusion with recombination and its application to haplotype partitioning. Theoretical Population Biology, 112 . pp. 126-138. doi:10.1016/j.tpb.2016.08.007 ISSN 0040-5809.
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Official URL: https://doi.org/10.1016/j.tpb.2016.08.007
Abstract
Duality plays an important role in population genetics. It can relate results from forwards-in-time models of allele frequency evolution with those of backwards-in-time genealogical models; a well known example is the duality between the Wright–Fisher diffusion for genetic drift and its genealogical counterpart, the coalescent. There have been a number of articles extending this relationship to include other evolutionary processes such as mutation and selection, but little has been explored for models also incorporating crossover recombination. Here, we derive from first principles a new genealogical process which is dual to a Wright–Fisher diffusion model of drift, mutation, and recombination. The process is reminiscent of the ancestral recombination graph , a widely-used multilocus genealogical model, but here ancestral lineages are typed and transition rates are regarded as being conditioned on an observed configuration at the leaves of the genealogy. Our approach is based on expressing a putative duality relationship between two models via their infinitesimal generators, and then seeking an appropriate test function to ensure the validity of the duality equation. This approach is quite general, and we use it to find dualities for several important variants, including both a discrete L-locus model of a gene and a continuous model in which mutation and recombination events are scattered along the gene according to continuous distributions. As an application of our results, we derive a series expansion for the transition function of the diffusion. Finally, we study in further detail the case in which mutation is absent. Then the dual process describes the dispersal of ancestral genetic material across the ancestors of a sample. The stationary distribution of this process is of particular interest; we show how duality relates this distribution to haplotype fixation probabilities. We develop an efficient method for computing such probabilities in multilocus models.
Item Type: | Journal Article | ||||||||
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Alternative Title: | |||||||||
Subjects: | Q Science > QH Natural history > QH426 Genetics | ||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Computer Science Faculty of Science, Engineering and Medicine > Science > Statistics |
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Library of Congress Subject Headings (LCSH): | Population genetics, Population genetics--Mathematical models, Mutation (Biology), Stochastic processes, Gene mapping | ||||||||
Journal or Publication Title: | Theoretical Population Biology | ||||||||
Publisher: | Academic Press | ||||||||
ISSN: | 0040-5809 | ||||||||
Official Date: | December 2016 | ||||||||
Dates: |
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Volume: | 112 | ||||||||
Page Range: | pp. 126-138 | ||||||||
DOI: | 10.1016/j.tpb.2016.08.007 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||
Date of first compliant deposit: | 26 August 2016 | ||||||||
Date of first compliant Open Access: | 1 September 2017 |
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