Recurrence equations for the probability distribution of sample configurations in exact population genetics models

Journal of Applied Probability

Volumn 47, Issue 3, 2010, Pages 732-751

  Lessard, Sabin ^a  

^a UNIVERSITÉ DE MONTRÉAL   (Canada)

Author keywords

coalescent; Cannings model; Coalescent theory; Ewens sampling formula; Exchangeable population genetics model; Moran model; Wright Fisher model

Indexed keywords

EID: 80053971752     PISSN: 00219002     EISSN: None     Source Type: Journal    
DOI: 10.1239/jap/1285335406     Document Type: Article

References (32)

1. Abramowitz, M. and Stegun, I. A. (eds) (1965). Handbook of Mathematical Functions. Dover, NewYork.
2. 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.
3. Dong, R., Gnedin, A. and Pitman, J. (2007). Exchangeable partitions derived from Markovian coalescents. Ann. Appl. Prob. 17, 1172-1201.
4. Eldon, B. andWakeley, J. (2006). Coalescent processes when the distribution of offspring number among individuals is highly skewed. Genetics 172, 2621-2633.
5. Ewens, W. J. (1972). The sampling theory of selectively neutral alleles. Theoret. Pop. Biol. 3, 87-112.
6. Ewens,W. J. (1990). Population genetics theory-the past and the future. In Mathematical and Statistical Developments of Evolutionary Theory, ed. S. Lessard, Kluwer, Dordrecht, pp. 177-227.
7. Ewens, W. J. (2004). Mathematical Population Genetics. I. 2nd edn. Springer, NewYork.
8. Fisher, R. A. (1930). The Genetical Theory of Natural Selection. Clarendon Press, Oxford.
9. 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.
10. Fu, Y. X. (2006). Exact coalescent for the Wright-Fisher model. Theoret. Pop. Biol. 69, 385-394.
11. 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. (Pubitemid 41415589)
12. Hoppe, F. M. (1984). Pólya-like urns and the Ewens' sampling formula. J. Math. Biol. 20, 91-94. (Pubitemid 14030507)
13. Hoppe, F. M. (1987). The sampling theory of neutral alleles and an urn model in population genetics. J. Math. Biol. 25, 123-159.
14. Joyce, P. and Tavaré, S. (1987). Cycles, permutations and the structures of theYule process with immigration. Stoch. Process. Appl. 25, 309-314.
15. Karlin, S. and McGregor, J. (1972). Addendum to a paper ofW. Ewens. Theoret. Pop. Biol. 3, 113-116.
16. Kingman, J. F. C. (1982). The coalescent. Stoch. Process. Appl. 13, 235-248.
17. 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. (Pubitemid 350005672)
18. 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.
19. Möhle, M. (2000).Total variation distances and rates of convergence for ancestral coalescent processes in exchangeable population models. Adv. Appl. Prob. 32, 983-993. (Pubitemid 32869493)
20. Möhle, M. (2004). The time back to the most recent common ancestor in exchangeable population models. Adv. Appl. Prob. 36, 78-97.
21. Möhle, M. (2006). On sampling distributions for coalescent processes with simultaneous multiple collisions. Bernoulli 12, 35-53.
22. Möhle, M. and Sagitov, S. (2001). A classification of coalescent processes for haploid exchangeable population models. Ann. Prob. 29, 1547-1562. (Pubitemid 33406000)
23. Moran, P. A. P. (1958). Random processes in genetics. Proc. Camb. Phil. Soc. 54, 60-71.
24. Moran, P. A. P. (1959). The theory of some genetical effects of population subdivision. Austral. J. Biol. Sci. 12, 109-116.
25. Moran, P. A. P. (1962). The Statistical Processes of Evolutionary Theory. Clarendon Press, Oxford.
26. Pitman, J. (1999). Coalescents with multiple collisions. Ann. Prob. 27, 1870-1902.
27. Sagitov, S. (1999). The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Prob. 36, 1116-1125.
28. Tavaré, S. (1989). The genealogy of the birth, death, and immigration process. In Mathematical Evolutionary Theory, ed. M.W. Feldman, Princeton University Press, pp. 41-56.
29. Trajstman, A. C. (1974). On a conjecture of G. A.Watterson. Adv. Appl. Prob. 6, 489-493.
30. Wakeley, J. (2003). Polymorphism and divergence for island-model species. Genetics 163, 411-420. (Pubitemid 36222683)
31. Wakeley, J. and Takahashi, T. (2004). The many-demes limit for selection and drift in a subdivided population. Theoret. Pop. Biol. 66, 83-91. (Pubitemid 39081533)
32. Wright, S. (1931). Evolution in Mendelian populations. Genetics 16, 97-159.