Journal of Applied Probability

Duality between the two-locus Wright–Fisher diffusion model and the ancestral process with recombination

Shuhei Mano

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Abstract

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.

Article information

Source
J. Appl. Probab., Volume 50, Number 1 (2013), 256-271.

Dates
First available in Project Euclid: 20 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.jap/1363784437

Digital Object Identifier
doi:10.1239/jap/1363784437

Mathematical Reviews number (MathSciNet)
MR3076785

Zentralblatt MATH identifier
1302.92075

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 92D15: Problems related to evolution 92D25: Population dynamics (general)

Keywords
duality diffusion process ancestral graph recombination population genetics

Citation

Mano, Shuhei. Duality between the two-locus Wright–Fisher diffusion model and the ancestral process with recombination. J. Appl. Probab. 50 (2013), no. 1, 256--271. doi:10.1239/jap/1363784437. https://projecteuclid.org/euclid.jap/1363784437


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References

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