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Solving the Fisher-Wright and coalescence problems with a discrete Markov chain analysis

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Samuel R. Buss  and
Peter Clote
Samuel R. Buss*
Affiliation:
University of California, San Diego
Peter Clote*
Affiliation:
Boston College
*
∗ Postal address: Department of Mathematics, University of California, San Diego, La Jolla, CA 92093, USA. Email address: sbuss@ucsd.edu
∗∗ Postal address: Department of Biology, Boston College, Chestnut Hill, MA 02467, USA. Email address: clote@bc.edu
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Abstract

We develop a new, self-contained proof that the expected number of generations required for gene allele fixation or extinction in a population of size n is O(n) under general assumptions. The proof relies on a discrete Markov chain analysis. We further develop an algorithm to compute expected fixation or extinction time to any desired precision. Our proofs establish O(nH(p)) as the expected time for gene allele fixation or extinction for the Fisher-Wright problem, where the gene occurs with initial frequency p and H(p) is the entropy function. Under a weaker hypothesis on the variance, the expected time is O(n(p(1-p))1/2) for fixation or extinction. Thus, the expected-time bound of O(n) for fixation or extinction holds in a wide range of situations. In the multi-allele case, the expected time for allele fixation or extinction in a population of size n with n distinct alleles is shown to be O(n). From this, a new proof is given of a coalescence theorem about the mean time to the most recent common ancestor (MRCA), which applies to a broad range of reproduction models satisfying our mean and weak variation conditions.

Keywords

Fisher-Wright modeldiffusion equationpopulation geneticsmitochondrial EveMarkov chainmean stopping timecoalescencemartingale

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) 60J22: Computational methods in Markov chains 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 4 , December 2004 , pp. 1175 - 1197
Copyright
Copyright © Applied Probability Trust 2004 

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Footnotes

∗

Supported in part by NSF Grant DMS-9803515 and DMS-0100589

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