Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-25T16:34:32.732Z Has data issue: false hasContentIssue false

Do Eve's Alleles Live On?

Published online by Cambridge University Press:  14 April 2009

G. A. Watterson*
Affiliation:
Department of Mathematics, Monash University, Victoria 3168, Australia
P. Donnelly
Affiliation:
School of Mathematical Sciences, Queen Mary and Westfield College, London, E1 4NS, U.K.
*
*Corresponding author.
Rights & Permissions [Opens in a new window]

Summary

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.

Consider a random sample of genes at a locus, drawn from a population evolving according to the infinitely many, neutral, alleles model. The sample will have a most recent common ancestor gene, which we shall call ‘Eve’. The probability distribution, for the number of genes of oldest allelic type in a sample, is known and has a neat form. Rather less is known about the distribution for the number of genes in the sample which are of the same allelic type as Eve possessed. If the latter number is positive, then these genes are automatically of the oldest type in the sample. But Eve may have no non-mutant descendants in the sample; then, the oldest allele will be a mutant arising in a line of descent after Eve. The paper studies the number of non-mutant descendants from Eve, its distribution and moments. It seems that there may be few neat results. In large samples, the proportion of genes of Eve's type has an approximate β-like density, together with a discrete probability atom at zero, if the mutation rate parameter is low. Extinction of the allele of even the population's common ancestor is possible, but not certain, and bounds are obtained for its probability. Some comments are made about the applications and implications of the results for human mitochondrial DNA.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1992

References

Abramowitz, M. & Stegun, I. A. (1972). Handbook of Mathematical Functions. New York: Dover.Google Scholar
Beder, B. (1988). Allelic frequencies given the sample's common ancestral type. Theoretical Population Biology 33, 126137.CrossRefGoogle ScholarPubMed
Cann, R. L., Stoneking, M. & Wilson, A. C. (1987) Mitochondrial DNA and human evolution. Nature 325, 3136.CrossRefGoogle ScholarPubMed
Cavalli-Sforza, L. L. (1991). Genes, peoples and languages. Scientific American (November), 7278.Google Scholar
Donnelly, P. (1986). Partition structures, Polya urns, the Ewens sampling formula, and the ages of alleles. Theoretical Population Biology 30, 271288.CrossRefGoogle ScholarPubMed
Feller, W. (1968). An Introduction to Probability Theory and its Applications, vol. 1, 3rd edn.New York: J. Wiley.Google Scholar
Griffiths, R. C. (1986). Family trees and DNA sequences. In Proceedings of the Pacific Statistical Congress (ed. Francis, I. S., Manly, B. F. J. and Lam, F. C.), pp. 225227. Elsevier Science Publishers.Google Scholar
Griffiths, R. C. (1989). Genealogical-tree probabilities in the infinitely-many-sites model. Journal of Mathematical Biology 27, 667680.CrossRefGoogle Scholar
Hey, J. (1991). A multi-dimensional coalescent process applied to multiallelic selection models and migration models. Theoretical Population Biology 39, 3048.CrossRefGoogle ScholarPubMed
Hoppe, F. M. (1987). The sampling theory of neutral alleles and an urn model in population genetics. Journal of Mathematical Biology 25, 123159.CrossRefGoogle Scholar
Kelly, F. P. (1979). Reversibility and Stochastic Networks. New York: J. Wiley.Google Scholar
Kingman, J. F. C. (1982). The coalescent. Stochastic Processes and their Applications 13, 235248.CrossRefGoogle Scholar
Notohara, M. (1990). The coalescent and the genealogical process in geographically structured population. Journal of Mathematical Biology 29, 5975.CrossRefGoogle ScholarPubMed
Saunders, I. W., Tavare, S. & Watterson, G. A. (1984). On the genealogy of nested subsamples from a haploid population. Advances in Applied Probability 16, 471491.CrossRefGoogle Scholar
Shimizu, A. (1987). Stationary distribution of a diffusion process taking values in probability distributions on the partitions. In Stochastic Models in Biology (ed. Kimura, M., Kallianpur, G. and Hida, T.). Berlin/New York: Springer-Verlag.Google Scholar
Takahata, N. (1986). An attempt to estimate the effective size of the ancestral species common to two extant species from which homologous genes are sequenced. Genetical Research, Cambridge 48, 187190.CrossRefGoogle ScholarPubMed
Takahata, N. (1988). The coalescent in two partially isolated diffusion populations. Genetical Research, Cambridge 52, 213222.CrossRefGoogle ScholarPubMed
Takahata, N. & Slatkin, M. (1990). Genealogy of neutral genes in two spatially isolated populations. Theoretical Population Biology 38, 331350.CrossRefGoogle Scholar
Tavaré, S. (1984). Line-of-descent and genealogical processes and their applications in population genetics. Theoretical Population Biology 26, 119164.CrossRefGoogle ScholarPubMed
Wainscoat, J. (1987). Out of the garden of Eden. Nature 325, 13.CrossRefGoogle ScholarPubMed
Watterson, G. A. (1984). Lines of descent and the coalescent. Theoretical Population Biology 26, 7792.CrossRefGoogle Scholar
Watterson, G. A. (1989). Allele frequencies in multigene families. I. Diffusion equation approach. Theoretical Population Biology 35, 142160.CrossRefGoogle ScholarPubMed