Stationary Allele Frequency Distributions

Abstract

Forces that determine the allele frequencies in natural populations include genetic drift, natural selection, migration and mutation. A balance of opposing forces can, in some cases, cause allele frequencies to approach a stationary distribution over time. The form of this distribution is not influenced by initial allele frequencies, but instead is determined by the relative magnitudes of different evolutionary forces. Statistical distributions are presented for the stationary allele frequencies under several simple population genetic models (including the k alleles symmetrical mutation model and the Wright island model of migration). In addition, the sampling distribution of allele counts under these models are described. The latter is useful when using genetic marker data to estimate population parameters. A population is at genetic equilibrium if evolutionary forces have persisted long enough for a population to have reached the stationary distribution, this is not often the case in nature.

Key Concepts:

  • If evolutionary forces such as natural selection, genetic drift and migration remain constant allele frequencies may reach to a stationary distribution.

  • Because evolutionary forces are often changing on a relatively short time‐scale the allele frequencies in many populations will not be at equilibrium.

  • Statistics based on parameters of stationary distributions may provide a useful summary of the genetic variation in populations that are not at equilibrium.

  • Simulation methods can be used to study allele frequency distributions for more complex models.

  • Modern approaches focus on nonstationary sampling distributions derived using the coalescent process model.

Keywords: probability distribution; Fisher‐Wright model; genetic drift; migration; mutation; sampling distribution

References

Buzbas EO and Joyce P (2009) Maximum likelihood estimates under K‐allele models can be numerically unstable. Annals of Applied Statistics 3: 1147–1162.

Fisher RA (1930) The Genetical Theory of Natural Selection. Oxford, UK: Oxford University Press.

Johnson NL, Kotz S and Balakrishnan N (1995) Continuous Univariate Distributions, vol. 2. New York: Wiley.

Johnson NL, Kotz S and Balakrishnan N (1997) Discrete Multivariate Distributions. New York: Wiley.

Kimura M (1955) Solution of a process of random genetic drift with a continuous model. Proceedings of the National Academy of Sciences of the USA 41: 144–150.

Kotz S, Balakrishnan N and Johnson NL (2000) Continuous Multivariate Distributions, vol. 1. New York: Wiley.

Rannala B and Hartigan JA (1996) Estimating gene flow in island populations. Genetical Research 67: 147–158.

Wright S (1931) Evolution in Mendelian populations. Genetics 16: 97–159.

Wright S (1969) Evolution and the genetics of populations. In: The Theory of Gene Frequencies, vol. 2. Chicago, IL: University of Chicago Press.

Further Reading

Ewens WJ (1979) Mathematical Population Genetics. New York: Springer.

Fearnhead P (2006) The stationary distribution of allele frequencies when selection acts at unlinked loci. Theoretical Population Biology 70: 376–386.

Gale JS (1990) Theoretical Population Genetics. London, UK: Unwin Hyman.

Medhi J (1994) Stochastic Processes, 2nd edn. New Delhi, India: Wiley Eastern.

Rice JA (1995) Mathematical Statistics and Data Analysis, 2nd edn. Belmont, CA: Duxbury Press.

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Rannala, Bruce(Apr 2013) Stationary Allele Frequency Distributions. In: eLS. John Wiley & Sons Ltd, Chichester. http://www.els.net [doi: 10.1002/9780470015902.a0005465.pub3]