Fixation Probabilities and Times


The fixation probability of an allele is the probability that it will eventually be the ancestor of all the alleles within a population at that locus. Population genetics theory has demonstrated that the probability of fixation is approximately proportional to the selection coefficient of a weak beneficial mutation, because such mutations are susceptible to stochastic loss while being rare despite their advantage. The time to fixation is the number of generations that it takes for an allele to progress from its initial frequency to fixation. This time is inversely proportional to the selection coefficient of a beneficial allele. Interestingly, though deleterious alleles are much less likely to fix, the time that they take to do so is, on average, the same as for a beneficial allele with the same magnitude of selective effect.

Key Concepts:

  • The fixation probability of an allele is the probability that it will eventually be the ancestor of all the alleles within a population at that locus.

  • Even beneficial mutations may not fix within a population.

  • The fixation probability of a beneficial allele is approximately proportional to its selection coefficient.

  • Deleterious mutations are unlikely to fix, but they can fix if their selective disadvantage is sufficiently weak and the population size sufficiently small.

  • The time for a new allele to fix within a population is inversely proportional to the magnitude of selection for both beneficial and deleterious alleles.

Keywords: branching process; diffusion; fixation; probability of fixation; time to fixation

Figure 1.

Mean time to fixation for additive alleles in a diploid population (from eqn ). The average time to fixation, conditioned on the fact that the allele does fix, is a decreasing function of |s|. As the strength of selection increases, both beneficial and deleterious alleles, if they fix, will fix faster. Alleles will also fix faster, on average, in populations of smaller effective size. Note that the mean time to fixation for neutral alleles (s=0) is 4Ne (for Ne=10 000, this time is 40 000 generations).



Alexander HK and Wahl LM (2008) Fixation probabilities depend on life history: fecundity, generation time and survival in a burst‐death model. Evolution 62: 1600–1609.

Barton NH (1993) The probability of fixation of a favoured allele in a subdivided population. Genetical Research 62: 149–158.

Barton NH (1995) Linkage and the limits to natural selection. Genetics 140: 821–841.

Caballero A and Hill WG (1992) Effects of partial inbreeding on fixation rates and variation of mutant genes. Genetics 131: 493–507.

Crow JF and Kimura M (1970) An Introduction to Population Genetics Theory. New York, NY: Harper & Row.

Desai MM and Fisher DS (2007) Beneficial mutation‐selection balance and the effect of linkage on positive selection. Genetics 176: 1759–1798.

Ewens WJ (1967) The probability of survival of a new mutant in a fluctuating environment. Heredity 22: 438–443.

Ewens WJ (1979) Mathematical Population Genetics. Berlin: Springer.

Gerrish PJ and Lenski RE (1998) The fate of competing beneficial mutations in an asexual population. Genetica 102/103: 127–144.

Haldane JBS (1927) A mathematical theory of natural and artificial selection, part V: selection and mutation. Proceedings of the Cambridge Philosophical Society 23: 838–844.

Hill WG and Robertson A (1966) The effect of linkage on the limits to artificial selection. Genetical Research 8: 269–294.

Johnson T and Gerrish PJ (2002) The fixation probability of a beneficial allele in a population dividing by binary fission. Genetica 115: 283–287.

Kimura M (1957) Some problems of stochastic processes in genetics. Annals of Mathematical Statistics 28: 882–901.

Kimura M (1962) On the probability of fixation of mutant genes in a population. Genetics 47: 713–719.

Kimura M (1983) The Neutral Theory of Molecular Evolution. Cambridge, UK: Cambridge University Press.

Kimura M and Ohta T (1969) The average number of generations until fixation of a mutant gene in a finite population. Genetics 61: 763–771.

Lynch M, Conery J and Bürger R (1995) Mutational meltdowns in sexual populations. Evolution 49: 1067–1080.

Martin G and Lenormand T (2008) The distribution of beneficial and fixed mutation fitness effects close to an optimum. Genetics 179: 907–916.

Maruyama T and Kimura M (1974) A note on the speed of gene‐frequency changes in reverse directions in a finite population. Evolution 28: 161–163.

Neher RA, Shraiman BI and Fisher DS (2010) Rate of adaptation in large sexual populations. Genetics 184: 467–481.

Orr HA and Coyne JA (1992) The genetics of adaptation: a reassessment. American Naturalist 140: 725–742.

Orr HA and Turelli M (2001) The evolution of postzygotic isolation: accumulating Dobzhansky–Muller incompatibilities. Evolution 55: 1085–1094.

Otto SP and Whitlock MC (1997) The probability of fixation in populations of changing size. Genetics 146: 723–733.

Peischl S and Kirkpatrick M (2012) Establishment of new mutations in changing environments. Genetics 191: 895–906.

Poon A and Otto SP (2000) Compensating for our load of mutations: freezing the meltdown of small populations. Evolution 54: 1467–1479.

Schluter D (2009) Evidence for ecological speciation and its alternative. Science 323: 737–741.

Uecker H and Hermisson J (2011) On the fixation process of a beneficial mutation in a variable environment. Genetics 188: 915–930.

Wahl LM and Gerrish PJ (2001) Fixation probability in populations with periodic bottlenecks. Evolution 55: 2606–2610.

Waxman D (2011) A unified treatment of the probability of fixation when population size and the strength of selection change over time. Genetics 188: 907–913.

Weissman DB and Barton NH (2012) Limits to the rate of adaptive substitution in sexual populations. PLoS Genetics 8: e1002740.

Whitlock MC (2003) Fixation probability and time in subdivided populations. Genetics 164: 767–779.

Further Reading

Fisher RA (1922) On the dominance ratio. Proceedings of the Royal Society of Edinburgh 42: 321–341.

Fisher RA (1930) The distribution of gene ratios for rare mutations. Proceedings of the Royal Society of Edinburgh 50: 204–219.

Gifford DR, de Visser JAG and Wahl LM (2013) Model and test in a fungus of the probability that beneficial mutations survive drift. Biology Letters 9: 20120310.

Li WH (1997) Molecular Evolution. Sunderland, MA: Sinauer Associates.

Nurminsky DI (2001) Genes in sweeping competition. Cellular and Molecular Life Sciences 58: 125–134.

Orr HA (1998) The population genetics of adaptation: the distribution of factors fixed during adaptive evolution. Evolution 52: 935–949.

Patwa Z and Wahl LM (2008) The fixation probability of beneficial mutations. Journal of the Royal Society Interface 5: 1279–1289.

Whitlock MC (2000) Fixation of new alleles and the extinction of small populations: drift load, beneficial alleles, and sexual selection. Evolution 54: 1855–1861.

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

Zeyl C, Mizesko M and de Visser J (2001) Mutational meltdown in laboratory yeast populations. Evolution 55: 909–917.

Contact Editor close
Submit a note to the editor about this article by filling in the form below.

* Required Field

How to Cite close
Otto, Sarah P, and Whitlock, Michael C(Jun 2013) Fixation Probabilities and Times. In: eLS. John Wiley & Sons Ltd, Chichester. [doi: 10.1002/9780470015902.a0005464.pub3]