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Approximate Sampling Formulae for General Finite-Alleles Models of Mutation

Part of: Approximations and expansions Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  04 January 2016

Anand Bhaskar ,
John A. Kamm  and
Yun S. Song
Anand Bhaskar*
Affiliation:
University of California, Berkeley
John A. Kamm*
Affiliation:
University of California, Berkeley
Yun S. Song*
Affiliation:
University of California, Berkeley
*
∗ Postal address: Computer Science Division, University of California, Berkeley, CA 94720, USA.
∗∗ Postal address: Department of Statistics, University of California, Berkeley, CA 94720, USA.
∗∗∗ Postal address: Computer Science Division and Department of Statistics, University of California, Berkeley, CA 94720, USA. Email address: yss@stat.berkeley.edu
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Abstract

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Many applications in genetic analyses utilize sampling distributions, which describe the probability of observing a sample of DNA sequences randomly drawn from a population. In the one-locus case with special models of mutation, such as the infinite-alleles model or the finite-alleles parent-independent mutation model, closed-form sampling distributions under the coalescent have been known for many decades. However, no exact formula is currently known for more general models of mutation that are of biological interest. In this paper, models with finitely-many alleles are considered, and an urn construction related to the coalescent is used to derive approximate closed-form sampling formulae for an arbitrary irreducible recurrent mutation model or for a reversible recurrent mutation model, depending on whether the number of distinct observed allele types is at most three or four, respectively. It is demonstrated empirically that the formulae derived here are highly accurate when the per-base mutation rate is low, which holds for many biological organisms.

Keywords

Sampling probabilitycoalescent theoryurn modelmartingale

MSC classification

Primary: 92D15: Problems related to evolution
Secondary: 65C50: Other computational problems in probability 92D10: Genetics 41A58: Series expansions (e.g. Taylor, Lidstone series, but not Fourier series)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 2 , June 2012 , pp. 408 - 428
Copyright
© Applied Probability Trust 

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