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Duality Between the Two-Locus Wright–Fisher Diffusion Model and the Ancestral Process with Recombination

Part of: Special processes Markov processes

Published online by Cambridge University Press:  30 January 2018

Shuhei Mano
Shuhei Mano*
Affiliation:
The Institute of Statistical Mathematics
*
∗ Postal address: The Institute of Statistical Mathematics and The Japan Science and Technology Agency, 10-3 Midori-cho, Tachikawa, Tokyo 190-8562, Japan. Email address: [email protected]
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Abstract

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Known results on the moments of the distribution generated by the two-locus Wright–Fisher diffusion model, and the duality between the diffusion process and the ancestral process with recombination are briefly summarized. A numerical method for computing moments using a Markov chain Monte Carlo simulation and a method to compute closed-form expressions of the moments are presented. By applying the duality argument, the properties of the ancestral recombination graph are studied in terms of the moments.

Keywords

dualitydiffusion processancestral graphrecombinationpopulation genetics

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 92D15: Problems related to evolution 92D25: Population dynamics (general)
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 1 , March 2013 , pp. 256 - 271
Copyright
Copyright © Applied Probability Trust 2013 

References

Bailey, W. N. (1935). Generalized Hypergeometric Series. Cambridge University Press.Google Scholar
Crow, J. F. and Kimura, M. (1971). An Introduction to Population Genetics Theory. Harper and Low, New York.Google Scholar
Erdély, A. (ed.) (1953). Higher Transcendental Functions, Vol. I. McGraw-Hill, New York.Google Scholar
Ethier, S. N. (1979). A limit theorem for two-locus diffusion models in population genetics. J. Appl. Prob. 16, 402–408.CrossRefGoogle Scholar
Ethier, S. N. and Griffiths, R. C. (1990). The neutral two-locus model as a measure-valued diffusion. Adv. Appl. Prob. 22, 773–786.CrossRefGoogle Scholar
Ethier, S. N. and Griffiths, R. C. (1990). On the two-locus sampling distribution. J. Math. Biol. 29, 131–159.CrossRefGoogle Scholar
Griffiths, R. C. (1991). The two-locus ancestral graph. In Selected Proceedings of the Sheffield Symposium on Applied Probability (Sheffield, 1989; IMS Lecture Notes Monogr. 18), eds Basawa, I. V. and Taylor, R. L., Institute of Mathematical Statistics, Hayward, CA, pp. 100–117.CrossRefGoogle Scholar
Griffiths, R. C. and Tavaré, S. (1994). Sampling theory for neutral alleles in a varying environment. Phil. Trans. R. Soc. London B 344, 403–410.Google Scholar
Hudson, R. R. and Kaplan, N. L. (1985). Statistical properties of the number of recombination events in the history of a sample of DNA sequences. Genetics 111, 147–164.CrossRefGoogle ScholarPubMed
Kimura, M. (1955). Solution of a process of random genetic drift with a continuous model. Proc. Nat. Acad. Sci. USA 41, 144–150.CrossRefGoogle ScholarPubMed
Kimura, M. (1955). Stochastic process and distribution of gene frequencies under natural selection. Cold Spring Harbor Symp. Quantitative Biol. 20, 33–53.CrossRefGoogle ScholarPubMed
Kingman, J. F. C. (1982). The coalescent. Stoch. Process. Appl. 13, 235–248.CrossRefGoogle Scholar
Krone, S. M. and Neuhauser, C. (1997). Ancestral process with selection. Theoret. Pop. Biol. 51, 210–237.CrossRefGoogle ScholarPubMed
Liggett, T. M. (1985). Interacting Particle Systems. Springer, Berlin.CrossRefGoogle Scholar
Littler, R. A. (1972). Multidimensional stochastic models in genetics. Doctoral Thesis, Monash University.Google Scholar
Malécot, G. (1948). Les Mathematiques de l'Hérédité. Masson et Cie, Paris.Google Scholar
Mano, S. (2005). Random genetic drift and gamete frequency. Genetics 171, 2043–2050.CrossRefGoogle ScholarPubMed
Mano, S. (2009). Duality, ancestral and diffusion processes in models with selection. Theoret. Pop. Biol. 75, 164–175.CrossRefGoogle ScholarPubMed
Mano, S. (2013). Ancestral graph with bias in gene conversion. J. Appl. Prob. 50, 239–255.CrossRefGoogle Scholar
Ohta, T. and Kimura, M. (1969). Linkage disequilibrium due to random genetic drift. Genet. Res. 13, 47–55.CrossRefGoogle Scholar
Shiga, T. (1981). Diffusion processes in population genetics. J. Math. Kyoto Univ. 21, 133–151.Google Scholar
Tavaré, S. (2004). Ancestral inference in population genetics. In Lectures on Probability Theory and Statistics (Lecture Notes Math. 1837), ed. Picard, J., Springer, Berlin, pp. 1–188.Google Scholar