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Selection in a Subdivided Population with Local Extinction and Recolonization

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Selection in a Subdivided Population with Local Extinction and Recolonization Copyright 2003 by the Genetics Society of America Selection in a Subdivided Population With Local Extinction and Recolonization Joshua L. Cherry1 Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, Massachusetts 02138 Manuscript received December 13, 2002 Accepted for publication March 4, 2003 ABSTRACT In a subdivided population, local extinction and subsequent recolonization affect the fate of alleles. Of particular interest is the interaction of this force with natural selection. The effect of selection can be weakened by this additional source of stochastic change in allele frequency. The behavior of a selected allele in such a population is shown to be equivalent to that of an allele with a different selection coefficient in an unstructured population with a different size. This equivalence allows use of established results for panmictic populations to predict such quantities as fixation probabilities and mean times to fixation. The magnitude of the quantity Nese, which determines fixation probability, is decreased by extinction and recolonization. Thus deleterious alleles are more likely to fix, and advantageous alleles less likely to do so, in the presence of extinction and recolonization. Computer simulations confirm that the theoretical predictions of both fixation probabilities and mean times to fixation are good approximations. HE consequences of population subdivision for evol- tion of an infinite population size. These results provide Tution depend on the nature of gene flow between not only fixation probabilities but also a complete de- subpopulations. Gene flow might be restricted to ordi- scription of the trajectory of the frequency of a selected nary migration, but might also include extinction of allele. This description is equivalent to that of an un- subpopulations followed by recolonization. Extinction structured population with a different size and a differ- and recolonization affect not only the amount of neutral ent selection coefficient. variation maintained in the population, but also the effi- cacy of natural selection compared to stochastic change in allele frequency. In the absence of extinction and recolonization, sub- MODEL AND RESULTS division increases the effective population size. None- To obtain results for a finite island model of subdivi- theless, fixation probabilities of alleles subject to genic sion, I first consider a model with a single sink popula- selection are unaffected by subdivision under fairly gen- tion and a source population with constant allele fre- eral conditions (Maruyama 1970, 1974). These facts quency. The results translate to a quasi-equilibrium for can in some cases be reconciled with the aid of the subpopulations in a finite island model. notion of effective selection coefficient (se; Cherry and Consider a haploid population that receives migrants Wakeley 2003). from a source population and is also subject to extinc- Extinction and recolonization lower effective popula- tion and subsequent recolonization by a single individ- tion size (Slatkin 1977; Maruyama and Kimura 1980; ual from that source population. Suppose that a locus Whitlock and Barton 1997). When sufficiently strong has two allelic forms, a and A, and that there is no they can reduce effective size below the actual popula- further mutation. Denote by x the frequency of the A tion size. Extinction and recolonization can affect the allele in the sink population, and let x be its (constant) probability of fixation of alleles subject to selection frequency in the source population. Let m be the rate (Barton 1993). of migration and let ␭ be the probability of extinction Barton (1993) has derived expressions for the fixa- and recolonization in any generation. Assume for the tion probability of a favored allele initially present in a moment that there is no selection. single copy in an infinite population with extinction If there were no extinction and recolonization, the and recolonization. Here I present results that cover equilibrium distribution of the allele frequency x would deleterious as well as advantageous alleles, apply to any be well approximated by a ␤-distribution (Wright 1931). initial allele frequency, and do not require the assump- This follows from a diffusion approximation. The case with extinction and recolonization may not be amenable to a diffusion approach because extinction/recoloniza- 1Address for correspondence: National Center for Biotechnology Infor- tion events drastically change the allele frequency in the mation, National Library of Medicine, National Institutes of Health, 8600 Rockville Pike, Bldg. 45, Bethesda, MD 20894. course of a generation. However, we need not be con- E-mail: [email protected] cerned with the exact distribution for the present pur- Genetics 164: 789–795 ( June 2003) 790 J. L. Cherry (N Ϫ 1)(2m Ϫ m2) ␭ poses. Because we are interested only in the mean and 1 Ϫ F ϭ ΂1 Ϫ ΃ ST Ϫ Ϫ Ϫ 2 Ϫ Ϫ␭ Ϫ Ϫ 2 variance of the change in allele frequency in a genera- N (N 1)(1 m) 1 (1 )(1 1/N )(1 m) tion, the main concern is to find the expected value of 2Nm Ϸ , x(1 Ϫ x), which is half the expected heterozygosity. The 2Nm ϩ N␭ϩ1 mean change in allele frequency due to selection is where the approximate equality holds for small m, large approximately proportional to this quantity. The com- ␭ ponent of the variance of this change that is due to N, and small . The first factor in the exact form would give 1 Ϫ F if there were no extinction and recoloniza- ordinary drift is also approximately proportional to this ST tion and is approximately equal to 2Nm/(2Nm ϩ 1), a quantity. The additional variance due to extinction/ familiar approximation for 1 Ϫ F for an island model recolonization events can be calculated separately and ST with ordinary migration (Wright 1940; Dobzhansky combined with this. and Wright 1941). The second factor represents the It is convenient to think in terms of the “virtual hetero- additional loss of heterozygosity due to extinction/re- zygosity” H, the expected value of 2x(1 Ϫ x). This quan- colonization events; for ␭ϭ0 it equals one, and it is tity can be interpreted as the probability that two copies less than one for ␭Ͼ0. The approximate equality is of the locus, drawn uniformly and independently from equivalent to a case of the expression for F given by the population with replacement, are in different allelic ST Whitlock and Barton (1997, Equation 21), with their states. We can write a recursion for H that depends on k equal to 1⁄ . This approximation makes it clear that the mean allele frequency. For the two copies of the 2 extinction and recolonization decreases heterozygosity gene to be in different allelic states, it is necessary that and that the expression for 1 Ϫ F is close to 2Nm/ there has not just been an extinction/recolonization ST (2Nm ϩ 1) when ␭ϭ0. Note that 1 Ϫ F is independent event (probability 1 Ϫ␭) and also that the same copy ST of x, which might not have been obvious from the start. of the locus has not been sampled twice (probability The sink population described above serves as a 1 Ϫ 1/N). If these criteria are met there are three model for a subpopulation in a finite island model. In possibilities to consider. If neither sampled allele is a this model D demes (“islands”), each consisting of N migrant, the probability that the alleles are different is individuals, exchange migrants among themselves and simply the value of H in the previous generation. If also serve as sources for recolonization. Let x now refer exactly one is a migrant, the probability depends on the to the frequency of the A allele in the population as a value of E[x] in the previous generation and is equal Ϫ ϩ Ϫ whole (the mean of the within-deme allele frequencies). to E[x](1 x) (1 E[x])x. If both are migrants, this In any generation, the population as a whole in effect probability is that of picking two different alleles from Ϫ serves as a source population for any particular deme, the source population, 2x(1 x). Putting this all to- with an allele frequency x. In the finite island model, gether gives in contrast to the source-sink model, the value of x ϭ Ϫ␭ Ϫ changes over time. However, when the number of Htϩ1 (1 )(1 1/N) demes is large, x changes only slowly compared to the ϫ Ϫ 2 ϩ Ϫ Ϫ ϩ Ϫ [(1 m) Ht 2m(1 m)(E[x](1 x) (1 E[x])x) rate of equilibration of within-deme allele frequencies, unless selection is very strong (see below), and so a ϩ 2 Ϫ m 2x(1 x)] , (1) quasi-equilibrium is attained (Wright 1931; Dobzhan- sky and Wright 1941). This is so because the stochastic where Ht is the heterozygosity at time t and E[x] refers to the expectation in the previous generation. At equi- change in the population as a whole is the mean of librium E[x] ϭ x. The equilibrium condition for H is the D independent within-deme stochastic changes. The therefore quasi-equilibrium distribution of within-deme allele fre- quency approaches the equilibrium distribution that H ϭ (1 Ϫ␭)(1 Ϫ 1/N)[(1 Ϫ m)2H ϩ (2m Ϫ m2)2x(1 Ϫ x)]. holds for the sink population of a source-sink model. (2) Thus FST, the fractional loss of heterozygosity due to subdivision, is given approximately by Equation 4. This Solution for H gives expression will allow derivation of a diffusion approxi- (1 Ϫ␭)(1 Ϫ 1/N )(2m Ϫ m2) mation for the behavior of an allele in the population H ϭ 2x(1 Ϫ x) 1 Ϫ (1 Ϫ␭)(1 Ϫ 1/N )(1 Ϫ m)2 as a whole when the number of demes is large.
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