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 Nϩ1 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. The above treatment of the quasi-equilibrium ne- (N Ϫ 1)(2m Ϫ m2) ϭ 1 Ϫ 2x(1 Ϫ x). N Ϫ (N Ϫ 1)(1 Ϫ m)2 1 Ϫ (1 Ϫ)(1 Ϫ 1/N )(1 Ϫ m)2 glected selection. Selection raises two issues. First, if (3) selection is very strong, x may change so rapidly that a quasi-equilibrium is not approached. Second, selection
The quantity FST, which is later interpreted as the frac- will alter the (quasi-) equilibrium distribution of allele tional loss of heterozygosity due to subdivision, is de- frequency, even in the source-sink model, where a true Ϫ ϵ Ϫ Ϫ ϭ Ϫ fined by 1 FST E[x(1 x)]/x(1 x) H/2x(1 equilibrium is reached. If selection is weak compared x). From Equation 3 it follows that to stochastic change in allele frequency within a subpop- Selection With Extinction and Recolonization 791 ulation, it has negligible effect on the (quasi-) equilib- components, one corresponding to binomial sampling rium distribution of allele frequency. The mean change and another corresponding to extinction and recoloni- due to selection is proportional to s, and the variance zation. Conditional on no extinction, the second moment N) x(1 Ϫ/1)ف due to ordinary drift is proportional to 1/N. Thus a about E[xЈ] is (1/N)E[xЈ](1 Ϫ E[xЈ]) or sufficient condition for directional change to be weaker x). Conditional on extinction and recolonization, xЈ is than stochastic change within a deme is 1 with probability x and 0 with probability 1 Ϫ x, so the second moment about E[xЈ]isx(1 Ϫ E[xЈ])2 ϩ (1 Ϫ |Ns| Ӷ 1, (5) x)E[xЈ]2 Ϸ x(1 Ϫ x)2 ϩ (1 Ϫ x)x2. Thus the variance in which is conservative because it does not take into ac- xЈ as a function of x is given by count the stochastic change due to extinction and recol- 1 Var(xЈ) Ϸ (1 Ϫ) x(1 Ϫ x) ϩ[x(1 Ϫ x)2 ϩ (1 Ϫ x)x2]. onization. This condition also guarantees that the rate N of change of x is small enough that a quasi-equilibrium (7) is approached. The per-generation change in x due to selection is at most about sx(1 Ϫ x) (this would be the Because 1 ϪϷ 1, this simplifies to change if all demes had the same allele frequency). The 1 Var(xЈ) Ϸ x(1 Ϫ x) ϩ[x(1 Ϫ x)2 ϩ (1 Ϫ x)x2]. (8) maximum value of this change is therefore s/4. As is N evident from Equation 1, heterozygosity in a deme de- cays toward its equilibrium on a timescale of N genera- The expected value of this expression follows readily Ϫ tions or faster (migration and extinction/recoloniza- from our expressions for E[x] and E[x(1 x)]. Using ϭ Ϫ ϭ Ϫ Ϫ tion hasten this decay). Thus condition (5) is also E[x] x and E[x(1 x)] (1 FST)x(1 x), we ob- sufficient for the quasi-equilibrium to be approached. tain Because stochastic change in the whole population is a E[Var(⌬x)] ϭ E[Var(xЈ)] much weaker force than stochastic change in a subpopu- lation, selection may be sufficiently weak in the sense Ϸ 1 Ϫ ϩ Ϫ Ϫ Ϫ Ά (1 FST) [2 (1 FST)]·x(1 x). that |Ns| Ӷ 1 and yet strongly affect the trajectory of N allele frequency in the population as a whole. I assume (9) that this condition holds in all that follows. Therefore V⌬x, the variance in the change in population- Let xi be the allele frequency in the ith deme. The wide allele frequency x, is given by mean change in allele frequency in this deme due to sx (1 Ϫ x ) in one generation. The mean 1ف selection is i i V⌬ ϭ ͚Var(⌬x ) Ϫ x 2 i in the entire population is the mean of the sxi(1 xi) D i Ϫ ϭ Ϫ Ϫ ف over all i,or sE[xi(1 xi)] (1 FST)sx(1 x). This Ϸ 1 ⌬ mean has the same form as that in a panmictic popula- E[Var( xi)] Ϫ D tion, but with s replaced by (1 FST)s. Thus the effective selection coefficient se is given by Ϸ 1 1 Ϫ ϩ Ϫ Ϫ Ϫ Ά (1 FST) [2 (1 FST)]·x(1 x). (10) ϭ Ϫ D N se (1 FST)s This variance is proportional to x(1 Ϫ x), as it is in a (N Ϫ 1)(2m Ϫ m2) ϭ 1 Ϫ s N Ϫ (N Ϫ 1)(1 Ϫ m)2 1 Ϫ (1 Ϫ)(1 Ϫ 1/N)(1 Ϫ m)2 panmictic population. The definition of variance effec- ϭ Ϫ tive population size is given by V⌬x (1/Ne)x(1 x). 2Nm Ϸ s . Therefore 2Nm ϩ Nϩ1 (6) ND ⌬ Ϸ An analogous treatment of the variance in x will Ne Ϫ ϩ Ϫ Ϫ , (11) (1 FST) N [2 (1 FST)] yield the effective population size Ne. This variance is approximated by a simple function of the expected vari- with FST given by Equation 4. Ne can be larger or smaller ance of within-deme change in allele frequency. To than the actual population size ND, depending on the obtain this quantity I first derive an expression for this parameters N, m, and . variance conditional on within-deme allele frequency These results show that a selected allele in a subdi- and then take the mean across the quasi-equilibrium vided population with extinction and recolonization be- distribution of allele frequency. haves much like an allele with a different selection coef- Suppose that x is the allele frequency within a deme ficient in a panmictic population with a different size. in one generation and xЈ is the frequency in the next. This follows from the fact that both the mean and the The variance of ⌬x, the change in allele frequency, is variance of the change in allele frequency are approxi- equal to the variance of xЈ, the second moment of xЈ mately proportional to x(1 Ϫ x), as they are in a panmic- about E[xЈ]. Because the mean change in allele fre- tic population (expressions for this mean and variance quency in a single generation is small for s, m, Ӷ1, completely determine the diffusion approximation). E[xЈ] Ϸ x. The distribution of xЈ is a mixture of two The parameters of the equivalent panmictic population, 792 J. L. Cherry se and Ne, are given by Equations 6 and 11. In the pres- Predictions of fixation probabilities and fixation times ence of extinction and recolonization, the value of Nese, follow from a combination of the theory presented here which determines fixation probability, is different from with classical diffusion results. Substitution of Ne and se its value in a panmictic population. The magnitude of for N and s in a familiar expression for fixation probabil- this product is decreased by extinction and recoloniza- ity (Wright 1931) gives this probability as tion. Specifically, Ϫ 1 Ϫ e 2Nese x0 Ϫ Ϫ , Ϸ (1 FST) 1 Ϫ e 2Nese Nese Ϫ ϩ Ϫ Ϫ NDs. (12) (1 FST) N [2 (1 FST)] where x0 is the initial allele frequency in the population and se and Ne are given by Equations 6 and 11. Similarly, use of an expression given by Kimura and Ohta (1969, COMPUTER SIMULATIONS Equation 17), with se substituted for s and adjustments To test the approximations used above, I ran com- made for haploidy, gives predictions of mean times to puter simulations and compared the results to theoreti- fixation. cal predictions. In these simulations the state of the Tables 1 and 2 compare such theoretical predictions Ϫ population is represented by an array of D integers, to simulation results for s ϭ 10 3, N ϭ 100, D ϭ 100, each corresponding to a deme. Each integer indicates and various values of m and . Table 1 shows predicted the number of copies of allele A in the deme and hence and observed probabilities of fixation of an allele ini- ranges from 0 to N. Each generation the new value for tially present as a single copy, relative to the neutral each deme is determined as follows. With probability fixation probability. All of the predictions are close to the deme undergoes extinction and recolonization, the observed quantities (largest deviation is 7%). Table after which the new number of A alleles is N with proba- 2 shows predicted and observed mean times to fixation. bility x and 0 with probability 1 Ϫ x. With probability The predictions are very close to the observations, dif- Ϫ 1 Ϫthe deme does not go extinct and the new number fering by at most 2%. Simulations with s ϭ 3 ϫ 10 4 or Ϫ is chosen from a binomial distribution. The index pa- s ϭ 10 4 also demonstrate a good agreement between rameter n of this binomial (number of “trials”) is equal prediction and observation (data not shown). to N. The probability parameter p (probability of “suc- Tables 3 and 4 show relative fixation probabilities for cess”) is determined by the current allele frequency in selectively disfavored alleles. Table 3 gives results for s ϭ Ϫ ϫ Ϫ4 the deme xi, the population-wide mean allele frequency 3 10 . The predictions are all very close to the x, the migration rate m, and the selection coefficient s. observations (within 3%). Table 4 shows results for a ϭ Ϫ ϩ ϭϪ Ϫ3 Let p˜ (1 m)xi mx. This would be the expected more strongly disfavored allele (s 10 ), for which allele frequency in the ith deme in the next generation fixation probabilities would be minuscule without ex- if there were no selection. Therefore p ϭ (1 ϩ s)p˜/ tinction and recolonization. For some sets of parame- (1 ϩ sp˜). ters, no fixations were observed in the simulations. This Figure 1 compares the distribution of allele frequen- is consistent with the very low predicted values of fixa- cies among many independent simulation runs to theo- tion probability. For the other cases, the agreement retical predictions at two time points. In the simulations between theory and observation is good (maximum de- N ϭ 100, D ϭ 100, s ϭ 3 ϫ 10Ϫ4, m ϭ 0.001, ϭ0.001, viation is 15%). Without extinction and recolonization, and the initial allele frequency was 1/2 (each deme or without population structure altogether, the relative ϫ Ϫ8 initially contained 50 A and 50 a alleles). The predic- fixation probability would be 4 10 . Thus the theory tions shown were obtained by iteration of the transition captures the more than seven orders of magnitude matrix for a Wright-Fisher population of 100 individuals. change in fixation probability due to extinction and The selection coefficient for this population was chosen recolonization. Predictions of fixation times are also such that the product of the population size and the good for both values of selection coefficient (maximum deviation 3%), as are predictions for s ϭϪ10Ϫ4 (data selection coefficient was equal to Nese, with Ne and se given by Equations 6 and 11. Time was scaled to account not shown). for the difference in effective size between the small Tables 5 and 6 show results for a smaller number of ϭ Wright-Fisher population and the subdivided popula- demes (D 30). Even with so few demes, predictions tion: the number of generations in the Wright-Fisher of fixation probabilities are close to the simulation re- population was smaller than that in the simulations by sults (all within 6%). Fixation times (not shown) are also in excellent agreement with predictions. a factor of Ne/100. The plots demonstrate excellent agreement between the predictions and the simulation results, confirming that the trajectory of allele frequency DISCUSSION in the structured population is similar to that in a pan- mictic population with parameters given by Equations Local extinction and recolonization affect the fates of 6 and 11. alleles in subdivided populations. The results presented Selection With Extinction and Recolonization 793 0.001, and the . The histograms 5 Ϫ ϭ ibution after 10,087 10 election coefficient of ϫ 0.001, 4.57 ϭ ϭ m e s , 4 Ϫ 10 ϫ 3 ϭ s 100, ϭ D 100, ϭ N 29,668, about half of what it would be with no extinction and recolonization, and ϭ e N 1.—Predicted and observed distributions of allele frequencies at two times. In all simulations Figure (bars) represent the results0.01357. of (A) 500,000 The simulation distributiongenerations. runs. after The The prediction 5044 predictions is generations. (“stair” based The plots) on prediction are 34 is based generations based of on on the a 17 Wright-Fisher Wright-Fisher generations population. population of of the size small 100, Wright-Fisher with population. a (B) s The distr initial allele frequency was 1/2. For these parameters 794 J. L. Cherry
TABLE 1 TABLE 3 Predicted and observed relative fixation probabilities for Predicted and observed relative fixation probabilities for s ϭ 10Ϫ3, D ϭ 100, and N ϭ 100 s ϭϪ3 ϫ 10Ϫ4, D ϭ 100, and N ϭ 100
m 0.001 0.003 0.01 m 0.001 0.003 0.01 0.001 9.04/9.53 3.90/4.21 1.51/1.61 0.001 0.193/0.181 0.535/0.532 0.872/0.858 0.01 16.5/16.6 11.8/12.2 5.00/5.11 0.01 0.0352/0.0363 0.106/0.105 0.434/0.427 Fixation probabilities are given relative to that of a selec- Fixation probabilities are given relative to that of a selec- tively neutral allele for various values of migration rate (m) tively neutral allele. Numbers to the left of slashes are theoreti- and extinction probability (). Numbers to the left of slashes cal predictions, and numbers to the right are observed values. are theoretical predictions, and numbers to the right are ob- In all cases the allele was initially present in a single copy. served values (simulation results). In all cases the allele was initially present in a single copy.
ton (1993) for a favored allele in an infinite population, but applies much more generally. here describe the trajectory of the frequency of an allele Computer simulations confirm that these theoretical under selection in the presence of this force when the results make good predictions. Fixation probabilities number of demes is large and a deme is recolonized by observed in simulations are close to theoretical predic- a single haploid individual. The behavior of the subdi- tions. Predicted mean times to fixation also agree well vided population was shown to be equivalent to that of a with simulation results. certain panmictic population. The size of this equivalent The results presented here are based on the assump-
panmictic population (Ne) is different from the actual tion that an empty deme is recolonized by a single hap-
size of the subdivided population. Ne can be larger or loid founder. Results can be quite different when recolo- smaller than this actual size. The selection coefficient nization involves more than one founding allele. For
in the equivalent panmictic population (se) is different example, FST is raised by extinction and recolonization
from the actual selection coefficient. se is always smaller with just one founding allele, but can be lowered by in magnitude than the actual selection coefficient and extinction and recolonization when there are multiple
has the same sign. Expressions for se and Ne are given founders (Takahata 1994; Whitlock and Barton by Equations 6 and 11. These allow application of estab- 1997). Thus with multiple founders |se| can be raised lished results for panmictic populations to subdivided rather than lowered by extinction and recolonization, populations with extinction and recolonization. although |se| can never be larger than |s|. Barton (1993) noted an apparent discrepancy be- The product Nese determines the probability of fixation of an allele. Extinction and recolonization reduces the tween the Ne that applies to fixation probabilities of magnitude of this quantity. Thus selection plays less of a selected alleles and Ne that is relevant to maintenance role in determining the fate of an allele in the presence of neutral variation. However, the value of Ne given for of extinction and recolonization. This reflects the fact that fixation probabilities was based on the assumption that extinction and recolonization are additional stochastic fixation probability depends on Nes. The distinction be- forces that can overwhelm the directional effects of selec- tween s and se, and hence between Nes and Nese, resolves tion. This result is consistent with those obtained by Bar-
TABLE 4
TABLE 2 Predicted and observed relative fixation probabilities for s ϭϪ10Ϫ3, D ϭ 100, and N ϭ 100 Predicted and observed fixation times for s ϭ 10Ϫ3, D ϭ 100, and N ϭ 100
m 0.001 0.003 0.01
ϫ Ϫ3 m 0.001 0.003 0.01 0.001 1.07 10 / 0.0859/0.0763 0.618/0.582 0.97 ϫ 10Ϫ3 0.001 34,798/34,106 24,513/24,367 9,894/9,813 0.01 1.11 ϫ 10Ϫ6/0 8.76 ϫ 10Ϫ5/0 0.0350/0.0304 0.01 10,337/10,178 9,804/9,729 7,820/7,837 Fixation probabilities are given relative to that of a selec- Numbers to the left of slashes are theoretical predictions, and tively neutral allele. Numbers to the left of slashes are theoreti- numbers to the right are observed values (simulation results). cal predictions, and numbers to the right are observed values. In all cases the allele was initially present in a single copy. In all cases the allele was initially present in a single copy. Selection With Extinction and Recolonization 795
TABLE 5 TABLE 6 Predicted and observed relative fixation probabilities for Predicted and observed relative fixation probabilities for s ϭ 10Ϫ3, D ϭ 30, and N ϭ 100 s ϭϪ10Ϫ3, D ϭ 30, and N ϭ 100
m 0.001 0.003 0.01 m 0.001 0.003 0.01 0.001 2.91/3.05 1.68/1.79 1.14/1.19 0.001 0.193/0.183 0.535/0.507 0.872/0.824 0.01 4.99/5.05 3.65/3.74 1.92/1.98 0.01 0.0352/0.0355 0.106/0.101 0.434/0.419 Fixation probabilities are given relative to that of a selec- Fixation probabilities are given relative to that of a selec- tively neutral allele. Numbers to the left of slashes are theoreti- tively neutral allele. Numbers to the left of slashes are theoreti- cal predictions, and numbers to the right are observed values. cal predictions, and numbers to the right are observed values. In all cases the allele was initially present in a single copy. In all cases the allele was initially present in a single copy.
Ϫ the apparent discrepancy: the ratio of se to s is 1 FST, I thank Christina Muirhead and Jon Wilkins for comments on the which approximately equals the ratio of the two values manuscript. This work was supported by National Science Foundation of Ne given by Barton (1993) for the model analyzed grant DEB-9815367 to John Wakeley. here. Thus the concept of effective selection coefficient
allows the use of a single value of Ne to describe both the behavior of neutral variation and the probability of LITERATURE CITED fixation of an allele under selection. Barton, N. H., 1993 The probability of fixation of a favoured allele The results presented here have implications for the in a subdivided population. Genet. Res. 62: 149–157. Bulmer, M., 1991 The selection-mutation-drift theory of synony- interpretation of population-genetic data. Attempts mous codon usage. Genetics 129: 897–907. have been made to estimate selection coefficients from Cherry, J. L., and J. Wakeley, 2003 A diffusion approximation for molecular data. For example, Bulmer (1991) and selection and drift in a subdivided population. Genetics 163: 421–428. Hartl et al. (1994) estimated the selective cost of a Dobzhansky, T., and S. Wright, 1941 Genetics of natural popula- nonoptimal codon in Escherichia coli as on the order of tions. V. Relations between mutation rate and accumulation of 10Ϫ9–10Ϫ8. What such studies actually estimate is the lethals in populations of Drosophila pseudoobscura. Genetics 26: 23–51. effective selection coefficient se. This quantity is of inter- Hartl, D. L., E. N. Moriyama and S. A. Sawyer, 1994 Selection est because it dictates the population-genetic behavior intensity for codon bias. Genetics 138: 227–234. of, in this example, alleles differing by synonymous Kimura, M., and T. Ohta, 1969 The average number of generations until fixation of a mutant gene in a finite population. Genetics changes. However, if one is interested in the physiology 61: 763–771. of bacteria and the magnitude of the decrease in growth Maruyama, T., 1970 On the fixation probability of mutant genes rate due to nonoptimal codons, the quantity of interest in a subdivided population. Genet. Res. 15: 221–225. Maruyama, T., 1974 A simple proof that certain quantities are inde- is s. Just as Ne can differ by orders of magnitude from pendent of the geographical structure of population. Theor. the actual population size, se might be radically different Popul. Biol. 5: 148–154. from s. The difference between se and s might explain Maruyama, T., and M. Kimura, 1980 Genetic variability and effec- tive population size when local extinction and recolonization the discrepancy noted by Bulmer (1991) between esti- of subpopulations are frequent. Proc. Natl. Acad. Sci. USA 77: mates of “selection coefficients” and predictions based 6710–6714. on physiological considerations. Slatkin, M., 1977 Gene flow and genetic drift in a species subject to frequent local extinctions. Theor. Popul. Biol. 12: 253–262. Knowledge of the actual population size could be Takahata, N., 1994 Repeated failures that led to the eventual suc- used to obtain estimates of s if there were no extinction cess in human evolution. Mol. Biol. Evol. 11: 803–805. and recolonization, since Nese is unaffected by that type Whitlock, M. C., and N. H. Barton, 1997 The effective size of a of population structure. In the presence of extinction subdivided population. Genetics 146: 427–441. Wright, S., 1931 Evolution in Mendelian populations. Genetics 16: and recolonization the actual population size is of no 97–159. help, since Nese is in this case altered by population Wright, S., 1940 Breeding structure of populations in relation to speciation. Am. Nat. 74: 232–248. structure. However, knowledge of FST could be used to relate se to s via Equation 6. Communicating editor: N. Takahata