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Convergence to the structured coalescent process

Part of: Limit theorems

Published online by Cambridge University Press:  21 June 2016

Ryouta Kozakai ,
Akinobu Shimizu  and
Morihiro Notohara
Ryouta Kozakai*
Affiliation:
Nagoya City University
Akinobu Shimizu*
Affiliation:
Nagoya City University
Morihiro Notohara*
Affiliation:
Nagoya City University
*
* Postal address: Graduate School of Natural Sciences, Nagoya City University, Mizuho, Nagoya, 467-8501, Japan.
* Postal address: Graduate School of Natural Sciences, Nagoya City University, Mizuho, Nagoya, 467-8501, Japan.
* Postal address: Graduate School of Natural Sciences, Nagoya City University, Mizuho, Nagoya, 467-8501, Japan.
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Abstract

The coalescent was introduced by Kingman (1982a), (1982b) and Tajima (1983) as a continuous-time Markov chain model describing the genealogical relationship among sampled genes from a panmictic population of a species. The random mating in a population is a strict condition and the genealogical structure of the population has a strong influence on the genetic variability and the evolution of the species. In this paper, starting from a discrete-time Markov chain model, we show the weak convergence to a continuous-time Markov chain, called the structured coalescent model, describing the genealogy of the sampled genes from whole population by means of passing the limit of the population size. Herbots (1997) proved the weak convergence to the structured coalescent on the condition of conservative migration and Wright–Fisher-type reproduction. We will give the proof on the condition of general migration rates and exchangeable reproduction.

Keywords

Structured coalescentnonconservative migrationCannings' reproductionweak convergence

MSC classification

Primary: 92D25: Population dynamics (general) 92D10: Genetics
Secondary: 60F17: Functional limit theorems; invariance principles
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 2 , June 2016 , pp. 502 - 517
Copyright
Copyright © Applied Probability Trust 2016 

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References

Bahlo, M. and Griffiths, R. C. (2000).Inference from gene trees in a subdivided population.Theoret. Pop. Biol. 57, 79–95.CrossRefGoogle Scholar
Bahlo, M. and Griffiths, R. C. (2001).Coalescence times for two genes from a subdivided population.J. Math. Biol. 43, 397–410.Google Scholar
Cannings, C. (1974).The latent roots of certain Markov chains arising in genetics: a new approach. I. Haploid models.Adv. Appl. Prob. 6, 260–290.CrossRefGoogle Scholar
Ethier, S. N. and Kurtz, T. G. (1986).Markov Processes: Characterization and Convergence.John Wiley, New York.Google Scholar
Herbots, H. M. (1994).Stochastic models in population genetics: genealogy and genetic differentiation in structured populations. Doctoral thesis. University of London.Google Scholar
Herbots, H. M. (1997).The structured coalescent. In Progress in Population Genetics and Human Evolution (IMA Vol. Math. Appl.87), Springer, New York, pp.231–255.Google Scholar
Hössjer, O. (2011).Coalescence theory for a general class of structured populations with fast migration.Adv. Appl. Prob. 43, 1027–1047.Google Scholar
Kaj, I., Krone, S. M. and Lascoux, M. (2001).Coalescent theory for seed bank models.J. Appl. Prob. 38, 285–300.Google Scholar
Kingman, J. F. C. (1982a).On the genealogy of large populations. In Essays in Statistical Science (J. Appl. Prob. Spec. Vol 19A), Applied Probability Trust, Sheffield, pp.27–43.Google Scholar
Kingman, J. F. C. (1982b).The coalescent.Stoch. Process. Appl. 13, 235–248.CrossRefGoogle Scholar
Möhle, M. (1998).A convergence theorem for Markov chains arising in population genetics and the coalescent with selfing.Adv. Appl. Prob. 30, 493–512.Google Scholar
Nath, H. B. and Griffiths, R. C. (1993).The coalescent in two colonies with symmetric migration.J. Math. Biol. 31, 841–851.Google Scholar
Nordborg, M. and Krone, S. M. (2002).Separation of timescales and convergence to the coalescent in structured populations. In Modern Developments In Theoretical Population Genetics, Oxford University Press, pp.194–232.Google Scholar
Notohara, M. (1990).The coalescent and the genealogical process in geographically structured population.J. Math. Biol. 29, 59–75.CrossRefGoogle ScholarPubMed
Notohara, M. (1993).The strong-migration limit for the genealogical process in geographically structured populations.J. Math. Biol. 31, 115–122.Google Scholar
Notohara, M. (1997).The number of segregating sites in a sample of DNA sequences from a geographically structured population.J. Math. Biol. 36, 188–200.Google Scholar
Notohara, M. (2001).The structured coalescent process with weak migration.J. Appl. Prob. 38, 1–17.Google Scholar
Notohara, M. (2010).An application of the central limit theorem to coalescence times in the structured coalescent model with strong migration.J. Math. Biol. 61, 695–714.CrossRefGoogle ScholarPubMed
Pollak, E. (2011).Coalescent theory for age-structured random mating populations with two sexes.Math. Biosci. 233, 126–134.Google Scholar
Sagitov, S. and Jagers, P. (2005).The coalescent effective size of age-structured populations.Ann. Appl. Prob. 15, 1778–1797.Google Scholar
Sampson, K. Y. (2006).Structured coalescent with nonconservative migration.J. Appl. Prob. 43, 351–362.Google Scholar
Shiga, T. (1980a).An interacting system in population genetics.J. Math. Kyoto Univ. 20, 213–242.Google Scholar
Shiga, T. (1980b).An interacting system in population genetics. II.J. Math. Kyoto Univ. 20, 723–733.Google Scholar
Tajima, F. (1983).Evolutionary relationship of DNA sequences in finite populations.Genetics 105, 437–460.Google Scholar
Takahata, N. (1988).The coalescent in two partially isolated diffusion populations.Genetical Res. 52, 213–222.CrossRefGoogle ScholarPubMed
Wakeley, J. (1998).Segregating sites in Wright's island model.Theoret. Pop. Biol. 53, 166–174.Google Scholar
Wakeley, J. (2009).Coalescent Theory: An Introduction.Roberts, Greenwood Village, CO.Google Scholar
Wilkinson-Herbots, H. M. (1998).Genealogy and subpopulation differentiation under various models of population structure.J. Math. Biol. 37, 535–585.Google Scholar