# zbMATH — the first resource for mathematics

Quasi-equilibrium theory for the distribution of rare alleles in a subdivided population: Justification and implications. (English) Zbl 0974.92021
Summary: This paper examines a quasi-equilibrium theory of rare alleles for subdivided populations that follow an island-model version of the Wright-Fisher model of evolution. All mutations are assumed to create new alleles. We present four results:
(1) conditions for the theory to apply are formally established using properties of the moments of the binomial distribution; (2) approximations currently in the literature can be replaced with exact results that are in better agreement with our simulations; (3) a modified maximum likelihood estimator of migration rate exhibits the same good performance on island-model data or on data simulated from the multinomial mixed with the Dirichlet distribution, and (4) a connection between the rare-allele method and the Ewens Sampling Formula [W.J. Ewens, Theor. Popul. Biol. 3, 87-112 (1972; Zbl 0245.92009)]; Mathematical population genetics. (1979; Zbl 0422.92011)] for the infinite-allele mutation model is made. This introduces a new and simpler proof for the expected number of alleles implied by the Ewens Sampling Formula.

##### MSC:
 92D15 Problems related to evolution 62P10 Applications of statistics to biology and medical sciences; meta analysis
Full Text:
##### References:
 [1] Barton, N.; Slatkin, M., A quasi-equilibrium theory of the distribution of rare alleles in a subdivided population, Heredity, 56, 409-415, (1986) [2] Burr, T., Estimating and modeling gene flow for a spatially distributed species, (1992), Florida State University Tallahassee [3] Chung, K.L., A course in probability theory, (1974), Academic Press Orlando · Zbl 0159.45701 [4] Cockerham, C.C.; Weir, B.S., Correlations, descent measures: drift with migration and mutation, Proc. nat. acad. sci. USA, 84, 8512-8514, (1987) · Zbl 0626.92011 [5] Cockerham, C.C.; Weir, B.S., Estimation of gene-flow from F-statistics, Evolution, 47, 855-863, (1993) [6] Crow, J.F.; Kimura, M., An introduction to population genetics theory, (1970), Harper & Row New York · Zbl 0246.92003 [7] Ewens, W.J., The sampling theory of selectively neutral alleles, Theor. popul. biol., 3, 87-112, (1972) · Zbl 0245.92009 [8] Ewens, W.J., Mathematical population genetics, (1979), Springer-Verlag Berlin · Zbl 0422.92011 [9] Kimura, M.; Crow, J.F., The number of alleles that can be maintained in a finite population, Genetics, 49, 725-738, (1964) [10] Kleinman, J., Proportions with extraneous variance: single and independent samples, J. amer. stat. assoc., 68, 46-54, (1973) [11] Mosimann, J., On the compound multinomial distribution, the multivariate β-distribution, and correlations among proportions, Biometrika, 49, 65-82, (1962) · Zbl 0105.12502 [12] Rannala, B., The sampling theory of neutral alleles in an island population of fluctuating size, Theor. popul. biol., 50, 91-104, (1996) · Zbl 0856.92013 [13] Rannala, B.; Hartigan, J.A., Estimating gene flow in island populations, Genet. res., 67, 147-158, (1996) [14] Slatkin, M., Estimating levels of gene flow in natural populations, Genetics, 99, 323-335, (1981) [15] Slatkin, M., Rare alleles as indicators of gene flow, Evolution, 39, 53-65, (1985) [16] Slatkin, M.; Barton, N., A comparison of three indirect methods for estimating average levels of gene flow, Evolution, 43, 1349-1368, (1989) [17] Slatkin, M.; Takahata, N., The average frequency of private alleles in a partially isolated population, Theor. popul. biol., 28, 314-331, (1985) · Zbl 0573.92014 [18] Watterson, G.A., The stationary distribution of the infinitely-many neutral alleles model, J. app. prob., 13, 639-651, (1976) · Zbl 0356.92012 [19] Wright, S., Evolution in Mendelian populations, Genetics, 16, 97-159, (1931)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.