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The Common Ancestor Process for a Wright-Fisher Diffusion

  
@article{EJP418,
	author = {Jesse Taylor},
	title = {The Common Ancestor Process for a Wright-Fisher Diffusion},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {12},
	year = {2007},
	keywords = {Common-ancestor process; diffusion process; structured coalescent; substitution rates; selection; genetic drift},
	abstract = {Rates of molecular evolution along phylogenetic trees are influenced by mutation,  selection and genetic drift.  Provided that the branches of the tree correspond to lineages belonging  to genetically isolated populations (e.g., multi-species phylogenies), the interplay between these  three processes can be described by analyzing the process of substitutions to the common ancestor of each population.  We characterize this process for a class of diffusion models from population genetics theory using the structured coalescent process introduced by Kaplan et al. (1988) and formalized in Barton et al. (2004).  For two-allele models, this approach allows both the stationary distribution of  the type of the common ancestor and the generator of the common ancestor process to be determined  by solving a one-dimensional boundary value problem.  In the case of a Wright-Fisher diffusion with  genic selection, this solution can be found in closed form, and we show that our results complement  those obtained by Fearnhead (2002) using the ancestral selection graph.  We also observe that  approximations which neglect recurrent mutation can significantly underestimate the exact substitution  rates when selection is strong.  Furthermore, although we are unable to find closed-form expressions  for models with frequency-dependent selection, we can still solve the corresponding boundary  value problem numerically and then use this solution to calculate the substitution rates to the  common ancestor.  We illustrate this approach by studying the effect of dominance on the  common ancestor process in a diploid population.  Finally, we show that the theory can be formally  extended to diffusion models with more than two genetic backgrounds, but that it leads to systems  of singular partial differential equations which we have been unable to solve.},
	pages = {no. 28, 808-847},
	issn = {1083-6489},
	doi = {10.1214/EJP.v12-418},    
        url = {http://ejp.ejpecp.org/article/view/418}}