The Annals of Applied Probability

Coalescence in a random background

N. H. Barton, A. M. Etheridge, and A. K. Sturm

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Abstract

We consider a single genetic locus which carries two alleles, labelled P and Q. This locus experiences selection and mutation. It is linked to a second neutral locus with recombination rate r. If r=0, this reduces to the study of a single selected locus. Assuming a Moran model for the population dynamics, we pass to a diffusion approximation and, assuming that the allele frequencies at the selected locus have reached stationarity, establish the joint generating function for the genealogy of a sample from the population and the frequency of the P allele. In essence this is the joint generating function for a coalescent and the random background in which it evolves. We use this to characterize, for the diffusion approximation, the probability of identity in state at the neutral locus of a sample of two individuals (whose type at the selected locus is known) as solutions to a system of ordinary differential equations. The only subtlety is to find the boundary conditions for this system. Finally, numerical examples are presented that illustrate the accuracy and predictions of the diffusion approximation. In particular, a comparison is made between this approach and one in which the frequencies at the selected locus are estimated by their value in the absence of fluctuations and a classical structured coalescent model is used.

Article information

Source
Ann. Appl. Probab. Volume 14, Number 2 (2004), 754-785.

Dates
First available in Project Euclid: 23 April 2004

Permanent link to this document
http://projecteuclid.org/euclid.aoap/1082737110

Digital Object Identifier
doi:10.1214/105051604000000099

Mathematical Reviews number (MathSciNet)
MR2052901

Zentralblatt MATH identifier
02100753

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J85: Applications of branching processes [See also 92Dxx] 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Coalescent selection recombination identity by descent random environment

Citation

Barton, N. H.; Etheridge, A. M.; Sturm, A. K. Coalescence in a random background. Ann. Appl. Probab. 14 (2004), no. 2, 754--785. doi:10.1214/105051604000000099. http://projecteuclid.org/euclid.aoap/1082737110.


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