Fixation Probabilities and Times

Abstract

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).

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