Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-25T04:01:42.840Z Has data issue: false hasContentIssue false
  • English
  • Français

Recurrence Equations for the Probability Distribution of Sample Configurations in Exact Population Genetics Models

Part of: Combinatorial probability

Published online by Cambridge University Press:  14 July 2016

Sabin Lessard
Sabin Lessard*
Affiliation:
Université de Montréal
*
∗Postal address: Département de Mathématiques et de Statistique, Université de Montréal, Montréal, Québec H3C 3J7, Canada. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Recurrence equations for the number of types and the frequency of each type in a random sample drawn from a finite population undergoing discrete, nonoverlapping generations and reproducing according to the Cannings exchangeable model are deduced under the assumption of a mutation scheme with infinitely many types. The case of overlapping generations in discrete time is also considered. The equations are developed for the Wright-Fisher model and the Moran model, and extended to the case of the limit coalescent with nonrecurrent mutation as the population size goes to ∞ and the mutation rate to 0. Computations of the total variation distance for the distribution of the number of types in the sample suggest that the exact Moran model provides a better approximation for the sampling formula under the exact Wright-Fisher model than the Ewens sampling formula in the limit of the Kingman coalescent with nonrecurrent mutation. On the other hand, this model seems to provide a good approximation for a Λ-coalescent with nonrecurrent mutation as long as the probability of multiple mergers and the mutation rate are small enough.

Keywords

Ewens sampling formulacoalescent theoryCannings modelWright-Fisher modelMoran modelexchangeable population genetics modelΛ-coalescent

MSC classification

Secondary: 60C05: Combinatorial probability 92D15: Problems related to evolution
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 3 , September 2010 , pp. 732 - 751
Copyright
Copyright © Applied Probability Trust 2010 

References

Abramowitz, M. and Stegun, I. A. (eds) (1965). Handbook of Mathematical Functions. Dover, New York.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.Google Scholar
Dong, R., Gnedin, A. and Pitman, J. (2007). Exchangeable partitions derived from Markovian coalescents. Ann. Appl. Prob. 17, 1172–1201.Google Scholar
Eldon, B. and Wakeley, J. (2006). Coalescent processes when the distribution of offspring number among individuals is highly skewed. Genetics 172, 2621–2633.CrossRefGoogle ScholarPubMed
Ewens, W. J. (1972). The sampling theory of selectively neutral alleles. Theoret. Pop. Biol. 3, 87–112.Google Scholar
Ewens, W. J. (1990). Population genetics theory—the past and the future. In Mathematical and Statistical Developments of Evolutionary Theory, ed. Lessard, S., Kluwer, Dordrecht, pp. 177–227.CrossRefGoogle Scholar
Ewens, W. J. (2004). Mathematical Population Genetics. I. 2nd edn. Springer, New York.Google Scholar
Fisher, R. A. (1930). The Genetical Theory of Natural Selection. Clarendon Press, Oxford.Google Scholar
Freund, F. and Möhle, M. (2009). On the number of allelic types for samples taken from exchangeable coalescents with mutation. Adv. Appl. Prob. 41, 1082–1101.Google Scholar
Fu, Y. X. (2006). Exact coalescent for the Wright–Fisher model. Theoret. Pop. Biol. 69, 385–394.Google Scholar
Griffiths, R. C. and Lessard, S. (2005). Ewens' sampling formula and related formulae: combinatorial proofs, extensions to variable population size and applications to ages of alleles. Theoret. Pop. Biol. 68, 167–177.Google Scholar
Hoppe, F. M. (1984). Pólya-like urns and the Ewens' sampling formula. J. Math. Biol. 20, 91–94.Google Scholar
Hoppe, F. M. (1987). The sampling theory of neutral alleles and an urn model in population genetics. J. Math. Biol. 25, 123–159.Google Scholar
Joyce, P. and Tavaré, S. (1987). Cycles, permutations and the structures of the Yule process with immigration. Stoch. Process. Appl. 25, 309–314.CrossRefGoogle Scholar
Karlin, S. and McGregor, J. (1972). Addendum to a paper of W. Ewens. Theoret. Pop. Biol. 3, 113–116.CrossRefGoogle ScholarPubMed
Kingman, J. F. C. (1982). The coalescent. Stoch. Process. Appl. 13, 235–248.Google Scholar
Lessard, S. (2007). An exact sampling formula for the Wright–Fisher model and a solution to a conjecture about the finite-island model. Genetics 177, 1249–1254.Google Scholar
Lessard, S. (2009). Diffusion approximations for one-locus multi-allele kin selection, mutation and random drift in group-structured populations: a unifying approach to selection models in population genetics. J. Math. Biol. 59, 659–696.Google Scholar
Möhle, M. (2000). Total variation distances and rates of convergence for ancestral coalescent processes in exchangeable population models. Adv. Appl. Prob. 32, 983–993.Google Scholar
Möhle, M. (2004). The time back to the most recent common ancestor in exchangeable population models. Adv. Appl. Prob. 36, 78–97.Google Scholar
Möhle, M. (2006). On sampling distributions for coalescent processes with simultaneous multiple collisions. Bernoulli 12, 35–53.Google Scholar
Möhle, M. and Sagitov, S. (2001). A classification of coalescent processes for haploid exchangeable population models. Ann. Prob. 29, 1547–1562.Google Scholar
Moran, P. A. P. (1958). Random processes in genetics. Proc. Camb. Phil. Soc. 54, 60–71.Google Scholar
Moran, P. A. P. (1959). The theory of some genetical effects of population subdivision. Austral. J. Biol. Sci. 12, 109–116.Google Scholar
Moran, P. A. P. (1962). The Statistical Processes of Evolutionary Theory. Clarendon Press, Oxford.Google Scholar
Pitman, J. (1999). Coalescents with multiple collisions. Ann. Prob. 27, 1870–1902.CrossRefGoogle Scholar
Sagitov, S. (1999). The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Prob. 36, 1116–1125.Google Scholar
Tavaré, S. (1989). The genealogy of the birth, death, and immigration process. In Mathematical Evolutionary Theory, ed. Feldman, M. W., Princeton University Press, pp. 41–56.Google Scholar
Trajstman, A. C. (1974). On a conjecture of G. A. Watterson. Adv. Appl. Prob. 6, 489–493.Google Scholar
Wakeley, J. (2003). Polymorphism and divergence for island-model species. Genetics 163, 411–420.Google Scholar
Wakeley, J. and Takahashi, T. (2004). The many-demes limit for selection and drift in a subdivided population. Theoret. Pop. Biol. 66, 83–91.Google Scholar
Wright, S. (1931). Evolution in Mendelian populations. Genetics 16, 97–159.Google Scholar