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Recent common ancestors of all present-day individuals

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Joseph T. Chang
Joseph T. Chang*
Affiliation:
Yale University
*
∗ Postal address: Statistics Department, Yale University, Box 208290, New Haven, CT 06520, USA. Email address: [email protected]
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Abstract

Previous study of the time to a common ancestor of all present-day individuals has focused on models in which each individual has just one parent in the previous generation. For example, ‘mitochondrial Eve’ is the most recent common ancestor (MRCA) when ancestry is defined only through maternal lines. In the standard Wright-Fisher model with population size n, the expected number of generations to the MRCA is about 2n, and the standard deviation of this time is also of order n. Here we study a two-parent analog of the Wright-Fisher model that defines ancestry using both parents. In this model, if the population size n is large, the number of generations, 𝒯n, back to a MRCA has a distribution that is concentrated around lgn (where lg denotes base-2 logarithm), in the sense that the ratio 𝒯n(lgn) converges in probability to 1 as n→∞. Also, continuing to trace back further into the past, at about 1.77 lgn generations before the present, all partial ancestry of the current population ends, in the following sense: with high probability for large n, in each generation at least 1.77lgn generations before the present, all individuals who have any descendants among the present-day individuals are actually ancestors of all present-day individuals.

Keywords

CoalescentWright-Fisher modelGalton-Watson processgenealogical modelspopulation genetics

MSC classification

Primary: 92D25: Population dynamics (general)
Secondary: 60J85: Applications of branching processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 31 , Issue 4 , December 1999 , pp. 1002 - 1026
Copyright
Copyright © Applied Probability Trust 1999 

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References

Athreya, K. B. and Ney, P. (1972). Branching Processes. Springer, New York.CrossRefGoogle Scholar
Ayala, F. J. (1995). The myth of Eve: molecular biology and human origins. Science 270, 1930–1936.Google Scholar
Bernstein, S. (1946). The Theory of Probabilities. Gastehizdat, Moscow.Google Scholar
Cann, R. L., Stoneking, M. and Wilson, A. C. (1987). Mitochondrial DNA and human evolution. Nature 325, 31–36.Google Scholar
Donnelly, P. (1986). Genealogical approach to variable-population-size models in population genetics. J. Appl. Prob. 23, 283–296.Google Scholar
Donnelly, P. and Tavaré, S. (eds.) (1997). Progress in Population Genetics and Human Evolution. Springer, New York.Google Scholar
Dorit, R. L., Akashi, H. and Gilbert, W. (1995). Absence of polymorphism at the ZFY locus on the human Y chromosome. Science 268, 1183–1185.Google Scholar
Griffiths, R. and Marjoram, P. (1997). An ancestral recombination graph. In Progress in Population Genetics and Human Evolution, eds. Tavaré, S. and Donnelly, P., Springer, New York, pp. 257–270.Google Scholar
Hudson, R. R. (1983). Properties of a neutral allele model with intragenic recombination. Theoret. Popn. Biol. 23, 183–201.Google Scholar
Hudson, R. R. (1990). Gene genealogies and the coalescent process. Oxford Surveys in Evolutionary Biology 7, 1–44.Google Scholar
Kämmerle, K., (1989). Looking forwards and backwards in a bisexual model. J. Appl. Prob. 27, 880–885.Google Scholar
Kämmerle, K., (1991). The extinction probability of descendants in bisexual models of fixed population size. J. Appl. Prob. 28, 489–502.Google Scholar
Kingman, J. F. C. (1982a). Exchangeability and the evolution of large populations. In Exchangeability in Probability and Statistics, eds. Koch, G. and Spizzichino, F., North-Holland, New York, pp. 97–112.Google Scholar
Kingman, J. F. C. (1982b). On the genealogy of large populations. In Essays in Statistical Science, eds. Gani, J. and Hannan, E. J. (J. Appl. Prob. 19A,). Applied Probability Trust, Sheffield, UK, pp. 27–43.Google Scholar
Möhle, M., (1994). Forward and backward processes in bisexual models with fixed population sizes. J. Appl. Prob. 31, 309–332.CrossRefGoogle Scholar
Möhle, M., (1997). Fixation in bisexual models with variable population sizes. J. Appl. Prob. 34, 436–448.Google Scholar
Pääbo, S., (1995). The Y chromosome and the origin of all of us (men). Science 268, 1141–1142.Google Scholar
Vigilant, L., Stoneking, M., Harpending, H., Hawkes, K. and Wilson, A. C. (1991). African populations and the evolution of human mitochondrial DNA. Science 253, 1503–1507.Google Scholar