Fixation Probabilities andTimesSarah P Otto, University of British Columbia, Vancouver, British Columbia, Canada

Michael C Whitlock, University of British Columbia, Vancouver, British Columbia, Canada

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 approxi-

mately proportional to the selection coefficient of a weak

beneficial mutation, because such mutations are suscep-

tible to stochastic loss while being rare despite their

advantage. The time to fixation is the number of gener-

ations 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. Inter-

estingly, 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.

Introduction

Within a population, any allele must ultimately persist orbe lost. If, sometime in the future, all members of thepopulation are descendants of the allele, then that allele issaid to have fixedwithin the population. If, sometime in thefuture, no members of the population are descendants ofthe allele, then that allele is said to have been lost. Usingmathematical models, the probability of fixation and thetime until fixation have been calculated under a range ofassumptions. These two quantities are related to manyimportant processes in evolutionary biology. Evolution bynatural selection relies on the greater fixation probabilityof beneficial alleles than deleterious alleles. The rate ofadaptation depends on the fixation probabilities and timesfor alleles that increase fitness. The relative efficacy ofselection and random genetic drift can be determined froma comparison of the fixation probabilities of selected andneutral alleles. The extinction risk of small populationsdepends on the fixation rates of beneficial and deleterious

mutations. At a molecular level, the rate of substitutionwithin the deoxyribonucleic acid (DNA) sequencesdepends on the rate at which new mutations appear withina population and on their fixation probability. Thus, anunderstanding of fixation probabilities and times providesan important basis for a clear understanding of evolutionitself. See also: Fitness and Selection; Genetic Drift inHuman Populations

Probability of Fixation

Let us consider the fate (fixation or loss) of an allele (A) thatarises within a population; A may represent a nucleotidemutation, an insertion or deletion, or any other geneticchange. The probability of fixation (C) of allele A dependson several factors, the most important of which is whetherthe allele has a positive, negative or neutral effect on fitness.If the average fitness of individuals who carry one copy ofAis (1+s) times greater than the average fitness of indi-viduals who carry no copies of A, then the fitness effect ofallele A is measured by s, the selection coefficient. See also:Mutational Change in EvolutionWhen s is zero, the allele is said to be selectively neutral.

For a neutral allele, the probability of fixation is equal to p,its initial allele frequency. If one considers a time point longin the future, all members of the population must descendfrom one and only one allele currently present becausenopolymorphism can last for ever in a finite population.Aslong as theA alleles always have the same average fitness asthe non-A alleles, the chance that an A allele will be thelucky ancestor of all descendants far in the future is simplyequal to the chance that an allele chosen at random fromthe initial population is the A allele, which is p. If, initially,there is one copy of the A allele in a population of censussize N and ploidy level c (c=1 for haploids, c=2 for dip-loids, etc.), p=1/(cN) and hence the fixation probability fora single-copy neutral allele,CNeutral, is 1/(cN). As cN is thetotal number of alleles at that locus in the population as awhole, each one has an equal chance of being the lucky onethat will eventually become the ancestor of the entirepopulation in the absence of selection.When allele A is beneficial (s40), it is more likely to

persist over time because selection acts to increase its fre-quency.Haldane (1927) derived the fixation probability fora single beneficial allele that appears within a population

Advanced article

Article Contents

. Introduction

. Probability of Fixation

. Time to Fixation

. Implications

Online posting date: 13th June 2013

eLS subject area: Evolution & Diversity of Life

How to cite:Otto, Sarah P; and Whitlock, Michael C (June 2013) FixationProbabilities and Times. In: eLS. John Wiley & Sons, Ltd: Chichester.

DOI: 10.1002/9780470015902.a0005464.pub3

eLS & 2013, John Wiley & Sons, Ltd. www.els.net 1

using a branching process model. Haldane assumed thatthe total number of alleles within a population (cN) is verylarge and constant (so that the average number of offspringper parent equals 1). Furthermore, he assumed that theprobability that a parent carrying allele A would have joffspring that also bear theA allele would follow a Poissonprobability distribution with mean (1+s); call this prob-ability distribution Pr[j]. Consequently, forA eventually tobe lost, which occurs with probability 12C, each of the joffspring copies of theA allele must also be lost eventually,which occurs with probability (12C)j under the assump-tion that the fate of each offspring allele is independent ofthe others. This insight allowed Haldane to write down anequation that the probability of fixation must obey:

1�C¼X1j¼ 0

Pr½j� 1�Cð Þj

¼X1j¼ 0

e�ð1þsÞ1þ sð Þj

j!1�Cð Þj ¼ e�ð1þsÞC

½1�

The above equation can be solved numerically for the fix-ation probability, C, but if the selection is weak, a Taylorseries expansion can be used to show that

C � 2s ½2�

This remarkable result proves that selection is notomnipotent. Selection cannot ensure the fixation of everybeneficial allele, because there is always the chance that aparent may fail to pass on the allele; for example, a newallele that confers a 1% fitness benefit (s=0.01) is expectedto fix only 1.97% of the time using eqn [1] (or � 2% of thetime using eqn [2]).Kimura (1957, 1962) generalised this result to include

deleterious alleles (s50), arbitrary population size andarbitrary initial allele frequency. By using a diffusionanalysis, Kimura found that the probability that anA allele

would eventually fix is

C � 1� expð�2cNespÞ1� expð�2cNesÞ

½3�

In this equation, Ne is the ‘variance effective populationsize’, which measures the degree of genetic drift experi-enced by a population. Equation [3] is derived assumingthat allele A has the same effect on fitness (s) regardless ofits genetic background. For diploids, this implies that alleleA is additive (codominant), but dominance may alsobe incorporated (see Crow and Kimura, 1970, p. 427).See also: Diffusion Theory; Effective Population SizeEquation [3] is extremely powerful. Although the dif-

fusion approximation technically holds only for weakselection and a large population size, in practice, it isapproximately correct even for very small populations andlarge selection coefficients (Table 1). Equation [3] alsodescribes the probability that a deleterious allele will driftto fixation despite selection against it; for example, a newallele that causes a 1% selective disadvantage will fix withprobability 0.00038 within a diploid population of size 100but will essentially never fix (C� 8� 10220) in a diploidpopulation of size 1000 (putting c=2, p=1/(2N), Ne=Nand s=20.01 into eqn [3]). By taking the limit of eqn [3]when s=0, we can also regain the result that CNeutral=p.When Ne is large and s is small but positive, the fixationprobability (3) for an allele that is introduced in a singlecopy (p=1/(cN)) approaches

C � 2sðNe=NÞ ½4�

Thus, factors that increase the extent of random geneticdrift (i.e. decreaseNe relative toN) will decrease the chancethat a beneficial allele will become established within apopulation. Such factors include a skewed sex ratio and ahigh variance in reproductive success.The above results assume apopulation that is constant in

size, homogeneous over space, randomly mating and

Table 1 Probability of fixation (C) for an allele initially present in one copy in a diploid population (c5 2; Ne5N)

Population size (N) Selection (s) CObserved CBranching process CDiffusion

10 0.001 0.051 0.002 0.051

0.01 0.06 0.0197 0.0601

0.1 0.1787 0.1761 0.1847

1 0.7902 0.7968 0.8647

100 0.001 0.0061 0.002 0.0061

0.01 0.0201 0.0197 0.0202

0.1 0.176 0.1761 0.1813

1 0.7958 (0.0004) 0.7968 0.8647

1000 0.001 0.0021 (0.00005) 0.002 0.002

0.01 0.0193 (0.0001) 0.0197 0.0198

0.1 0.1760 (0.0004) 0.1761 0.1813

1 0.7975 (0.0004) 0.7968 0.8647

The ‘observed’ probability of fixation was obtained from an exact matrix calculation, when practical, or by simulation with 106 replicates(followed by standard errors in parentheses).C based on a branching process (eqn [1]) is accurate for strong selection and/or large populations(Ns4� 1). C based on diffusion analyses (eqn [3]) is accurate except with very strong selection (s4� 0.1).

Fixation Probabilities and Times

eLS & 2013, John Wiley & Sons, Ltd. www.els.net2

unaffected by selection at other loci, but the models havebeen extended in a number of important ways, whichmake the results more applicable to natural populations.Fixation probability can be significantly affected bypopulation growth and decline (Ewens, 1967; Otto andWhitlock, 1997; Wahl and Gerrish, 2001; Uecker andHermisson, 2011), spatial population structure (Barton,1993; Whitlock, 2003), inbreeding (Caballero and Hill,1992), life history details (Johnson and Gerrish, 2002;Alexander and Wahl, 2008), varying selection over time(Uecker andHermisson, 2011;Waxman, 2011; Peischl andKirkpatrick, 2012) and selection at linked loci (Hilland Robertson, 1966; Barton, 1995; Neher et al., 2010;Weissman and Barton, 2012), especially in asexual organ-isms (Gerrish and Lenski, 1998; Desai and Fisher, 2007).

Time to Fixation

Even when an allele ultimately rises to fixation within apopulation, it may take many generations, especially whenselection is weak or absent. For a neutral allele (s=0) ini-tially present in one copy, the average time to fixation isapproximately 2cNe generations, conditional on the allelefixing, a result first derived by Kimura and Ohta (1969).Note that this result is approximately the same as thatobtained from coalescence theory for the number of gen-erations, looking back in time, until all members of a cur-rent population trace back to a single allele (i.e. the timeuntil the first common ancestor). Either looking forwardsor backwards in time, it takes approximately 2cNe gener-ations for a neutral allele to spread through drift from asingle copy to fixation. See also: Coalescent TheorySelection causes beneficial alleles to rise more rapidly to

fixation. For an additive allele initially present in one copy,the average time to fixation satisfies

T �Z 1

1=ðcNeÞ

e2cNesx � 1� �

e2cNesð1�xÞ � 1� �

sx 1� xð Þ e2cNes � 1ð Þ dx ½5�

(Ewens, 1979, p. 151,where results for arbitrary dominancemay also be found). For very weak selection (s51/(cNe)),eqn [5] approaches 2cNe generations, the fixation timefor a neutral allele. Conversely, for very strong selection(s440, such that the ‘21’ terms can be dropped toapproximate eqn [5] and replacing the upper limit of inte-gration with 12(1/(cNe))), the fixation time approaches

T � 2 ln cNe � 1ð Þs

½6�

Equation [6] also equals the time for a beneficial allele torise from frequency 1/(cNe) to 12(1/(cNe)) in a model thatignores drift (see Crow and Kimura (1970), eqn (5.3.13)).Thus, when selection is weak relative to the force of drift,

the fixation time for a beneficial allele is of the order of thepopulation size, whereas when selection is very strong thefixation time is much shorter and is inversely proportionalto the strength of selection (Figure 1). What is much more

surprising is that the time to fixation, conditional on fix-ation, is the same for deleterious and beneficial alleles.Although deleterious alleles are much less likely to fix,when they do rise to fixation, the time that it takes is alsogiven by eqn [5] (Maruyama and Kimura, 1974). This isbecause drift must, by chance, occur more rapidly if it is tooffset stronger selection against a deleterious allele.

Implications

The probability of fixation has many interesting impli-cations in evolutionary biology. At a molecular level, therate of nucleotide substitution will equal the rate ofappearance of new alleles within a population times theirprobability of fixation. For a neutral site with a per gen-erationmutation rate of m per site (or mcN per population),the substitution rate will equal (mcN)� 1/(cN) or simply m.This calculation predicts that the rate of substitution ofneutral alleles should depend only on themutation rate andnot on population size, a fact that has been used to explainwhy the rate of substitution is often similar in differentevolutionary lineages (the so-called ‘molecular clock’hypothesis; Kimura, 1983). Higher (lower) substitutionrates would be observed if the site was under positive(negative) selection, as can be calculated using eqn [3].See also: Molecular Clocks; Mutation Rate; NucleotideSubstitution: Rate ofIf the rate of appearance of beneficial alleles bymutation

limits the rate of evolution, then the probability that thesenew alleles fix is critical to the evolutionary advance of aspecies. Furthermore, isolated populations may accumu-late different mutations, potentially leading to geneticincompatibilities that initiate the speciation process (Orrand Turelli, 2001; Schluter, 2009). It is not yet known towhat extent waiting for new mutations limits the rate of

Ne= 100

Ne= 1000

Ne=10000

2000

4000

6000

8000

10 000

0

Tim

e to

fixa

tion

(gen

erat

ions

)

–0.04 –0.02 0 0.02 0.04

Selection coefficient (s)

Figure 1 Mean time to fixation for additive alleles in a diploid population

(from eqn [5]). 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).

Fixation Probabilities and Times

eLS & 2013, John Wiley & Sons, Ltd. www.els.net 3

evolution, but potentially this is a large problem, especiallyfor structural mutations such as insertions, deletions andrearrangements. Furthermore, because alleles are morelikely to fix if they have a large advantage, the alleles thatdo in fact fix in a population are much more likely to havea large effect than would be expected by the mutationaldistribution. This may explain in part why the genetic basisof many selected traits seems dominated by alleles of largeeffect (Orr and Coyne, 1992).The fact that deleterious alleles can fix via genetic drift

allows the possibility that the mean fitness of a populationcan decrease over evolutionary time. In small populations,the risk of decreasing fitness can be significant and maycause a ‘mutational meltdown’, where extinction resultsfrom the accumulation of deleterious mutations (Lynchet al., 1995). Furthermore, because mutations appearrarely in small populations, the rate of fixation of beneficialmutationsmaynot be high enough for small populations toadapt to and persist in a changing environment. Thus,because populations with a small effective size fix moredeleterious and fewer beneficial alleles over time than alarger population, they are doubly endangered by theeffects of genetic drift on fixation rates.See also: CoalescentTheory; Diffusion TheoryIt should, however, be remembered that the distribution

of selective effects is not constant and depends on the extentto which an organism has previously adapted to a par-ticular environment (e.g. Martin and Lenormand, 2008).As populations become less fit, for example, during amutational meltdown, beneficial mutations may becomemore plentiful and/or more advantageous, which can haltthe meltdown (Poon and Otto, 2000).

References

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.

EwensWJ (1967) The probability of survival of a newmutant 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 arti-

ficial 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 gen-

etics. Annals of Mathematical Statistics 28: 882–901.

KimuraM(1962)On the probability of fixation ofmutant genes in

a population. Genetics 47: 713–719.

Kimura M (1983) The Neutral Theory of Molecular Evolution.

Cambridge, UK: Cambridge University Press.

KimuraM andOhta T (1969) The average number of generations

until fixation of a mutant gene in a finite population. Genetics

61: 763–771.

LynchM,Conery J andBurgerR (1995)Mutationalmeltdowns 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.

NeherRA, Shraiman BI and FisherDS (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 isol-

ation: 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 muta-

tions 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. Evo-

lution 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 popu-

lations with periodic bottlenecks. Evolution 55: 2606–2610.

Waxman D (2011) A unified treatment of the probability of fix-

ation 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.

WhitlockMC (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.

Fixation Probabilities and Times

eLS & 2013, John Wiley & Sons, Ltd. www.els.net4

FisherRA(1930)Thedistributionofgene ratios for raremutations.

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 dis-

tribution 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 inMendelian populations.Genetics 16:

97–159.

ZeylC,MizeskoManddeVisser J (2001)Mutationalmeltdown in

laboratory yeast populations. Evolution 55: 909–917.

Fixation Probabilities and Times

eLS & 2013, John Wiley & Sons, Ltd. www.els.net 5