×

Genealogical-tree probabilities in the infinitely-many-site model. (English) Zbl 0716.92012

Summary: This paper considers the distribution of the genealogical tree of a sample of genes in the infinitely-many-site model where the relative age ordering of the mutations (nodes in the tree) is known. A computer implementation of a recursion for the probability of such trees is discussed when the nodes are age-labeled, or not.

MSC:

92D10 Genetics and epigenetics
60J85 Applications of branching processes
92-04 Software, source code, etc. for problems pertaining to biology
05C05 Trees
05C90 Applications of graph theory
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
Full Text: DOI

References:

[1] Beder, A.: Allelic frequencies given the sample’s common ancestral type. Theor. Popul. Biol. 33, 126–137 (1988) · Zbl 0657.92007 · doi:10.1016/0040-5809(88)90009-3
[2] Donnelly, P., Tavaré, S.: The ages of alleles and a coalescent. Adv. Appl. Probab. 18, 1–19 (1986) · Zbl 0588.92013 · doi:10.2307/1427237
[3] Donnelly, P., Tavaré, S.: The population genealogy of the infinitely-many neutral alleles model. J. Math. Biol. 25, 381–391 (1987) · Zbl 0636.92008 · doi:10.1007/BF00277163
[4] Ethier, S. N.: The infmitely-many-neutral-alleles diffusion model with ages. Adv. Appl. Probab., in press (1990) · Zbl 0699.92012
[5] Ethier, S. N., Griffiths, R. C.: The infinitely-many-sites model as a measure-valued diffusion. Ann. Probab. 15, 515–545 (1987) · Zbl 0634.92007 · doi:10.1214/aop/1176992157
[6] Ewens, W. J.: The sampling theory of selectively neutral alleles. Theor. Popul. Biol. 3, 87–112 (1972) · Zbl 0245.92009 · doi:10.1016/0040-5809(72)90035-4
[7] Griffiths, R. C.: Family trees and DNA sequences. In: Francis, I. S., Manly, B. F., Lam, F. C. (eds.). Proceedings of the Pacific Statistical Congress. Amsterdam New York: Elsevier 1986
[8] Griffiths, R. C.: Counting genealogical trees. J. Math. Biol. 25, 423–431 (1987) · Zbl 0652.92011 · doi:10.1007/BF00277166
[9] Hoppe, F. M.: Polyra-like urns and the Ewens sampling formula. J. Math. Biol. 20, 91–94 (1984) · Zbl 0547.92009 · doi:10.1007/BF00275863
[10] Hoppe, F. M.: The sampling theory of neutral alleles and an urn model in population genetics. J. Math. Biol. 25, 123–157 (1987) · Zbl 0636.92007 · doi:10.1007/BF00276386
[11] Kingman, J. F. C.: The coalescent. Stochastic Processes Appl. 13, 235–248 (1982) · Zbl 0491.60076 · doi:10.1016/0304-4149(82)90011-4
[12] Nijenhuis, A., Wilf, H. S.: Combinatorial algorithms, 2nd edn. New York: Academic Press 1978 · Zbl 0476.68047
[13] Tavaré, S.: Line-of-descent and genealogical processes, and their applications in population genetics models. Theor. Popul. Biol. 26, 119–164 (1984) · Zbl 0555.92011 · doi:10.1016/0040-5809(84)90027-3
[14] Tavaré, S.: The birth process with immigration, and the genealogical structure of large populations. J. Math. Biol. 25, 161–171 (1987) · Zbl 0625.92010 · doi:10.1007/BF00276387
[15] Watterson, G. A.: On the number of segergating sites in genetical models without recombination. Theor. Popul. Biol. 7, 256–276 (1975) · Zbl 0294.92011 · doi:10.1016/0040-5809(75)90020-9
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.