## Abstract

To test whether quantitative traits are under directional or homogenizing selection, it is common practice to compare population differentiation estimates at molecular markers (*F*_{ST}) and quantitative traits (*Q*_{ST}). If the trait is neutral and its determinism is additive, then theory predicts that *Q*_{ST} = *F*_{ST}, while *Q*_{ST} > *F*_{ST} is predicted under directional selection for different local optima, and *Q*_{ST} < *F*_{ST} is predicted under homogenizing selection. However, nonadditive effects can alter these predictions. Here, we investigate the influence of dominance on the relation between *Q*_{ST} and *F*_{ST} for neutral traits. Using analytical results and computer simulations, we show that dominance generally deflates *Q*_{ST} relative to *F*_{ST}. Under inbreeding, the effect of dominance vanishes, and we show that for selfing species, a better estimate of *Q*_{ST} is obtained from selfed families than from half-sib families. We also compare several sampling designs and find that it is always best to sample many populations (>20) with few families (five) rather than few populations with many families. Provided that estimates of *Q*_{ST} are derived from individuals originating from many populations, we conclude that the pattern *Q*_{ST} > *F*_{ST}, and hence the inference of directional selection for different local optima, is robust to the effect of nonadditive gene actions.

UNDERSTANDING the evolutionary forces that shape ecologically important traits among populations of the same species is one of the central themes of evolutionary biology research (Merila and Crnokrak 2001). These forces are first selection, which can homogenize phenotypes across populations or on the contrary make them diverge because of different local optima, a phenomenon called local adaptation. But the other microevolutionary forces also affect quantitative traits. These forces are classically mutation, and particularly migration and random genetic drift. In the absence of selection, these last three forces are the only ones acting on traits at least partly genetically determined (Lande 1992; Whitlock 1999; Hendry 2002). The same forces affect patterns of variation at molecular markers, hence supporting the idea of comparing statistics obtained from molecular markers and from quantitative traits. Lande (1992) and Whitlock (1999) showed that for a neutral trait with a strictly additive determinism, differentiation estimated from quantitative traits should be equal to that estimated from molecular markers. Spitze (1993), using results obtained by Wright (1951), derived a statistic for quantitative traits equivalent to Wright's (1969) *F*_{ST}, which he called *Q*_{ST}. Under strict neutrality and additivity, *Q*_{ST} = *F*_{ST}. Different local optima in different populations lead to *Q*_{ST} > *F*_{ST}, while selection for the same optimum across populations of the same species lead to *Q*_{ST} < *F*_{ST} (Crnorkrak and Merila 2002; McKay and Latta 2002). These predictions for the relation between *Q*_{ST} and *F*_{ST} were confirmed, using computer simulations by Le Corre and Kremer (2003) for both random-mating and highly selfing situations.

Merila and Crnokrak (2001) and McKay and Latta (2002) have recently reviewed the empirical literature on comparisons between differentiation estimates obtained from quantitative traits and molecular markers. The general pattern that emerges from these reviews is that quantitative traits are on average more differentiated than molecular markers despite showing a very large variability.

While these reviews seem to confirm the ubiquitousness of local adaptation, the conclusions are based on the assumption that the quantitative traits have a purely additive determinism. Several authors have pointed out that it is crucial to investigate how *Q*_{ST} would behave in the presence of dominance and epistasis at quantitative traits (Whitlock 1999; Le Corre and Kremer 2003). Lynch *et al.* (1999) suggested that epistasis would drive *Q*_{ST} upward. Whitlock (1999) demonstrated that additive-by-additive epistasis would on the contrary drive *Q*_{ST} downward and suggests that dominance could affect *Q*_{ST} in either way. Merila and Crnokrak (2001) and Yang *et al.* (1996) have also pointed out that inbreeding could affect the relation between *Q*_{ST} and *F*_{ST}.

Lopez-Fanjul *et al.* (2003) investigated the effect of dominance and epistasis, using a two loci, two alleles model. They concluded that with dominance, *Q*_{ST} < *F*_{ST} for low to moderate frequency of the recessive alleles, and *Q*_{ST} > *F*_{ST} otherwise. Epistasis diminished *Q*_{ST} relative to *F*_{ST}, unless the recessive alleles are very frequent. They therefore concluded that the comparison between *Q*_{ST} and *F*_{ST} should be restricted to purely additive traits. This would certainly be a strong limitation of this approach, as the genetic determinism of quantitative traits is seldom understood. They arrived at these conclusions by looking at the effect on allelic frequencies of a one-generation bottleneck of size *N* = 2. However, several authors (Robertson 1952; Willis and Orr 1993; Cheverud and Routman 1996; Naciri-Graven and Goudet 2003; Barton and Turelli 2004) showed that bottlenecks affect strongly the additive variance within lines. It is thus difficult to conclude whether the pattern observed by Lopez-Fanjul *et al.* (2003) is general or specific to the situation where bottlenecks have occurred in the very recent past.

A second issue touched upon in Lopez-Fanjul *et al.* (2003) concerns the large errors in the estimation of *F*_{ST} and particularly of *Q*_{ST}. This point was already noted in the early 1980s by Rogers and Harpending (1983, p. 985), who pointed out that “one polygenic character contains as much information about population relationships as one single-locus marker.” As comparisons between *Q*_{ST} and *F*_{ST} depend critically on the variance of these statistics, it seems worthwhile to investigate which sampling scheme minimizes the variance of these estimators. Sampling design issues have been addressed for *F*_{ST} (Pons and Chaouche 1995; Pons and Petit 1995), and O'Hara and Merila (2005) have recently investigated the statistical properties of *Q*_{ST}.

The goal of this article is to characterize the effects of dominance and inbreeding on *Q*_{ST} in the absence of selection. We first obtain analytical results for the expression of *Q*_{ST} for a biallelic trait and identify situations in which *Q*_{ST} is expected to be larger than *F*_{ST}. As the analytical results are limited to biallelic loci, we use computer simulations to explore the effect that dominance and inbreeding have on *Q*_{ST}, using estimators of *Q*_{ST} based (i) on allele frequency and (ii) on covariance among relatives obtained from classical crossing designs in common garden experiments. We also explore how the variance of *Q*_{ST} is affected by the experimental design.

## METHODS AND RESULTS

The quantities needed to obtain the expression for *F*_{ST} and *Q*_{ST} are the gene diversity within populations *H*_{S}, the overall *H*_{T}, the variance among populations *V*_{B}, and the additive variance within populations *V*_{AW}.

With these quantities, *F*_{ST} is defined as (Hartl and Clark 1997), while *Q*_{ST} is defined as(1)(Bonnin *et al.* 1996), where *V*_{B} is the among population component of variance for the trait, and *V*_{AW} is the additive genetic variance within populations. The factor 2 associated with *V*_{AW} is due to the fact that for quantitative traits genotypes are compared, while genes are compared when computing *F*_{ST} (Lynch and Spitze 1994).

Consider a locus with two alleles, *A* and *B*, with respective frequencies *p _{i}* and

*q*= 1 −

_{i}*p*in population

_{i}*i*. We use the notation of Falconer (Falconer and MacKay 1996) for genotypic value. Under regular inbreeding, genotypic values and frequencies of the different genotypes are given in Table 1.

Gene diversity within population *H*_{S} depends only on allelic frequencies. It writes as

Overall diversity *H*_{T} writes aswhere is the average frequency of the recessive allele *A*.

*F*_{ST} is defined as(2)

The variance among populations of trait means, *V*_{B} is defined aswhere *M _{i}*, the mean trait value in population

*i*can be written (

*q*−

_{i}*p*)

_{i}*a*+ 2

*p*(1 −

_{i}q_{i}*f*)

*d*.

After replacement and simplifications, *V*_{B} becomes(3)While under pure additivity, *V*_{B} is proportional to (and therefore to the first and second moments of allele frequencies), in the presence of dominance *V*_{B} becomes a complex function of higher moments of allele frequencies. The effect of dominance depends on allelic frequencies and gene diversity. When the recessive allele is frequent (), the covariance term is negative and *V*_{B} increases compared to the case without dominance. When the recessive allele is rare (), *V*_{B} increases provided that , where β(*p*, *H*_{S}) is the slope of the regression of the frequency of the recessive allele on *H*_{S}.

Finally, we seek within-population additive variance. For *n _{l}* loci, additive variance is quantified as (Lynch and Walsh 1998), where

*e*represents the average excess of allele

_{ij}*i*at locus

*j*and α

_{ij}is the average effect of allele

*i*at locus

*j*. For one locus, following Templeton (1987), we obtain(4)

Expression for the additive variance within population *i* is thenwhich, after replacement and simplifications gives

For a number *n* of populations, the expression becomes(5)

From this expression we see that dominance decreases the additive variance within populations when the recessive allele is rare (*p* < 0.5), while it increases it when the recessive allele is frequent. This is easily understood since when the recessive allele is rare, it will be found mainly in heterozygotes that do not differ much in phenotype from the dominant homozygote.

The expression for *Q*_{ST} is obtained by replacing *V*_{AW} and *V*_{B} in Equation 1.

From Equations 3 and 5, we see that inbreeding diminishes the contribution of dominance to both *V*_{B} and *V*_{AW}. Thus, as inbreeding increases, the effect of dominance on *Q*_{ST} diminishes, and, unless , dominance will have little effect on *Q*_{ST} under strong inbreeding.

The expression of *Q*_{ST} for specific cases is listed below:

No dominance, ∀

*f*: In the absence of dominance (*d*= 0),*V*_{AW}reduces to

while

*V*_{B}takes the expression

and

*Q*_{ST}becomes

Overdominance (no additivity), ∀

*f*: When*a*= 0, expressions for*V*_{B}and*V*_{AW}become

and

and

*Q*_{ST}becomes

This is clearly very different from

*F*_{ST}(Equation 2).*f*= 0: The expression for*Q*_{ST}does not simplify greatly when*f*= 0, as it remains a function of*a*,*d*,*H*_{T},*H*_{S}, and*p*:

*f*= 1: When*f*= 1, since the dominance term*d*comes as a product with (1 −*f*) in*V*_{B}and*V*_{AW}, it disappears altogether from their expressions and therefore also from that of*Q*_{ST}, as expected. Thus, .

As we have seen, the expression for *Q*_{ST} in the presence of dominance and inbreeding is not simple. To gain a better understanding of its effect, we start with a two populations system and first show contour plots for *F*_{ST}, *Q*_{ST}, and the difference (*Q*_{ST} − *F*_{ST}) for a trait encoded by one locus and two alleles as a function of the frequency of the recessive allele in two populations.

For a purely additive determinism, *F*_{ST} = *Q*_{ST} and the difference *Q*_{ST} − *F*_{ST} is therefore equal to zero for all points of the allele frequency space.

Figure 1 shows the contour plots for the case *a* = 1, *d* = 1, and *f* = 0. *F*_{ST} (Figure 1A) is null when the frequency is the same in the two populations and increases as allele frequencies diverge between the two populations, to reach a maximum of one when one allele is fixed in the first population and the other allele is fixed in the second.

The contour plot for *Q*_{ST} is shown in Figure 1B. When the allele frequencies are the same in the two populations, *Q*_{ST} is null, as expected. The difference in allele frequencies when the recessive allele is rare in the two populations brings less changes in *Q*_{ST} than in *F*_{ST}. The effect of a difference in allele frequency increases as the recessive allele increases in frequency in both populations. When looking at the difference *Q*_{ST} − *F*_{ST} (Figure 1C) we observed that the difference is negative in the bottom left (when the recessive allele is rare in both populations), while it is positive in the top right (when the recessive allele is frequent). The negative area is larger than the positive one, and this is confirmed by integrating over the surface of allele frequencies: averaged over the allele frequency space, *Q*_{ST} = 0.162 while *F*_{ST} = 0.186. Thus, the expected difference between *Q*_{ST} and *F*_{ST} when there is dominance is negative.

Figure 2 shows a contour plot of the difference between *Q*_{ST} and *F*_{ST} for different levels of dominance and inbreeding. Figure 2, A and B, represents the case *a* = 1, *d* = 1 seen in Figure 1. *f* = 0 for Figure 2A and this is therefore the same as for Figure 1C. In Figure 2B, *f* = 0.8, and we see that the difference between *Q*_{ST} and *F*_{ST} vanishes. It is 10-fold less than that with no inbreeding. And the mean value for *Q*_{ST} is now 0.185, very close to the average *F*_{ST} (= 0.186).

Figure 2, C and D, represents the case of strict overdominance (*a* = 0; *d* = 1). With strict overdominance and *f* = 0, a large area covering the secondary diagonal is negative (Figure 2C), and a much smaller portion of the contour plot has positive values for the difference. When integrating over the surface, the average *Q*_{ST} is 0.09. Therefore, with strict overdominance, *Q*_{ST} is on average much less than *F*_{ST}. With *f* = 0.8 (Figure 2D), the area where the difference between *Q*_{ST} and *F*_{ST} is negative reduces drastically while that where it is positive increases. And indeed, average *Q*_{ST} when *f* = 0.8 is 0.186, as is average *F*_{ST}. High inbreeding therefore cancels out the effect of overdominance on *Q*_{ST}.

#### Computer simulations:

With many loci and alleles, analytical results become intractable when there is dominance and inbreeding. We therefore used computer simulations to generate data under different levels of population structure, inbreeding, and trait determinism. Two types of simulations were used, one based on allele frequencies and the other on individuals.

##### Allelic frequencies:

First we drew allelic frequencies from Dirichlet distributions (Kingman 1977). Overall allelic frequencies [*p*] is a vector with its element obtained from a Dirichlet distribution of parameter 1, equivalent to a uniform distribution. To obtain frequencies at each locus in the different populations, a Dirichlet distribution with parameter 2*Nm*[*p*], where *Nm* is the number of migrants between populations, was used (Beaumont 2005).

From these allele frequencies, *F*_{ST} was obtained classically as , where . Similarly, *H*_{T} was estimated as , where *p _{jk}* is the realized overall allele frequency (

*i.e.*, it is not the original theoretical frequency [

*p*] from the Dirichlet distribution). To account for bias due to the number of samples, (Nei 1987) was used instead of

*H*

_{T}in the expression of

*F*

_{ST}.

Traits were encoded by 10 10-allelic loci. Trait values were simulated by assigning to each allele at each locus an additive value drawn from a normal distribution. Similarly, to obtain a dominance effect of each genotype, a value drawn from a normal distribution was assigned to the dominance deviation of the genotype. The sum of the two additive effects and the dominance deviation gives the genotypic value of this genotype at the locus considered. Genotypic values for multilocus genotypes are obtained by summing the contributions of the individual loci, since we assume no epistasis. We also used exponential distributions instead of normal ones to draw additive value and dominance deviations, but this did not alter the results.

Once all these values are assigned, the within-population additive variance is estimated as (Lynch and Walsh 1998), with expressions for *e _{ij}* and α

_{ij}given above (Equation 4); and the among-population variance of trait mean is simply that.

*Q*

_{ST}estimated this way is denoted

*Q*

_{ST}

^{p}, where p stands for parent. Note that

*Q*

_{ST}

^{p}cannot be estimated in experimental situations, as one would need allelic frequencies for all alleles contributing to the trait and the genotypic values for all genotypes.

Figure 3 shows the relation between *F*_{ST} and *Q*_{ST}^{p} for pure additivity (Figure 3, A and B), dominance (Figure 3, C and D), and superdominance (Figure 3, E and F) under no inbreeding (Figure 3, A, C, and E) or strong inbreeding (*f* = 0.8; Figure 3, B, D, and F).

Under strict additivity, *Q*_{ST}^{p} = *F*_{ST}, as expected, and independently of the inbreeding coefficient. For traits with dominance and under random mating (Figure 3C), *Q*_{ST} becomes less than *F*_{ST} on average, and this tendency increases as populations get more structured. But this effect disappears in inbred populations (Figure 3D). With strict overdominance and under random mating (Figure 3E), the pattern observed with dominance is enhanced, and *Q*_{ST}^{p} is much lower than *F*_{ST}, the more so for very structured populations. With selfing and strict overdominance (Figure 3F), *Q*_{ST}^{p} is larger than *F*_{ST} for weakly structured populations, and smaller for those that are strongly structured.

The results presented in Figure 3 are for simulations with 10 populations. The effect of the number of populations involved is presented in Figure 4, where only the situation with purely additive traits under random mating is presented. Figure 4 shows very clearly that the variance among replicates is huge with 2 populations (Figure 4A), still large with 5 populations (Figure 4B), and much smaller with 50 populations (Figure 4D).

##### Individual-based model:

The results just presented are for a theoretical situation where both allele frequencies at all loci involved in the trait and genotypic values for all genotypes are known, but this is never the case. When experimentalists estimate *Q*_{ST}, they use covariance among relatives to estimate additive variance rather than the allele frequencies at the loci underlying the trait, since these loci are generally unknown. To mimic this real situation we used an individual-based model with the following features:

We used Easypop (Balloux 2001) to generate genotypes from an individual-based finite island model, where islands exchange migrants at a fixed rate *m*, constant among populations and across generations. For all the simulations, population size *N* was fixed at 50 hermaphroditic individuals, and the number of populations *n* was also fixed at 50. For each individual, 100 loci with 20 allelic states each were simulated. The mutation rate was fixed at 0.001 and followed a *K*-allele model. Simulations were run for 500 generations, at which point *F*_{ST} had reached its equilibrium value for all levels of migrations. These were fixed at 0.002, 0.005, 0.01, 0.02, 0.05, 0.1, and 0.2, corresponding to *Nm* values of 0.1, 0.25, 0.5, 1, 2.5, 5, and 10, respectively. Two selfing rates were used, either 0 or 0.8. Genotypes of the last generation were stored for further processing.

Traits were encoded as in the simulations on allelic frequencies, by assigning additive values drawn from a normal distribution to each allele at each locus and dominance deviation to each genotype.

To estimate *F*_{ST} we used the genotypes at the trait loci of the parents and calculated *F*_{ST} using the method of Weir and Cockerham (1984) implemented in Fstat (Goudet 1995). To estimate *Q*_{ST}^{o} (where o stands for offspring), we proceeded as follows: A number of parents were chosen from each population of the island model. From these parents, a number of half-sib families (one male mated to 10 different females) were established, with 10 offspring each (1 offspring per female). We used either the full data set (10 individuals from 50 families from 50 populations, a total of 25,000 individuals) or subsamples of 1000 offspring from each data set, using the different sampling scenarios presented in Table 2.

These experimental sampling designs are similar to those used in several studies estimating *Q*_{ST}^{o}, with total number of individuals at 1000 (*e.g.*, Spitze 1993; Lynch *et al.* 1999; Palo *et al.* 2003; Morgan *et al.* 2005). Often, however, the total number of individuals used to infer *Q*_{ST} is <1000 (*e.g.*, Bonnin *et al.* 1996; Petit *et al.* 2001; Steinger *et al.* 2002).

A classical nested ANOVA was used to estimate the different variance components. *V*_{AW} is estimated as four time the among-family component of variance *V*_{fam}, while *V*_{B} is simply the among-population variance component.

Figure 5 shows the relation between *F*_{ST} and *Q*_{ST}^{o} for additive (Figure 5, A and B), dominant (Figure 5, C and D), and superdominant (Figure 5, E and F) traits, for random mating (Figure 5, A, C, and E) and selfing (Figure 5, B, D, and F). For Figure 5, A–F, estimation of *Q*_{ST}^{o} is based on 50 families of 10 half-sibs from 50 populations. Under strict additivity (Figure 5, A and B), *Q*_{ST}^{o} = *F*_{ST}, and there is no difference between random mating and selfing populations. The observed pattern is congruent with that obtained for *Q*_{ST}^{p} (Figure 3, A and B).

For traits with dominance (Figure 5, C and D), *Q*_{ST}^{o} ≤ *F*_{ST}, and the difference increases as population structure increases. This is observed both for random mating and for selfing populations, and there seems to be very little difference in *Q*_{ST}^{o} between the two mating systems.

Finally, Figure 5, E and F, shows the effect of overdominance. Here, the pattern observed with dominance (*Q*_{ST}^{o} < *F*_{ST}, and the difference increasing with population structure) is amplified.

#### Effect of the crossing design:

We have seen that with allele frequency-based estimates of *Q*_{ST}, the effect of dominance on *Q*_{ST}^{p} disappears as selfing increases (Figure 3D), but when *Q*_{ST} is estimated with a half-sib design, the pattern *Q*_{ST}^{o} < *F*_{ST} remains (Figure 5D). However, the half-sib design is likely to unduly inflate additive variance estimates for a species that commonly selfs, and for strongly selfing species experimentalists often use selfed progeny to estimate the different genetic variance components and hence *Q*_{ST} (*e.g.*, Bonnin *et al.* 1996; Steinger *et al.* 2002). With a selfed progeny design, additive variance cannot be singled out from dominance variance, but for a high selfer dominance variance is not naturally expressed as homozygous genotypes are transmitted intact to the next generation. With selfed progeny, *Q*_{ST} is estimated as , which amounts to assuming complete selfing (see Equation 1).

Using individual-based simulations with a high selfing rate (*s* = 0.889 *f* = 0.8), we compared estimates of *Q*_{ST} obtained from either half-sibs or selfed progenies. Figure 6 shows the results. Figure 6, A, C, and E, was obtained using a classical half-sib design, while Figure 6, B, D, and F, was obtained using a selfing design. With pure additivity (Figure 6, A and B), the two crossing designs give equivalent results. With dominance, estimates of *Q*_{ST} obtained from a selfed design (Figure 6D) are slightly less than but closer to *F*_{ST} than those obtained from a half-sib design (Figure 6C). This is particularly true for populations that are strongly structured. Hence, when the species under scrutiny is mainly selfing, estimates of *Q*_{ST} obtained from selfed progeny are less influenced by dominance, and thus preferable, to estimates obtained from a half-sib design. Another advantage of the selfing design is that it does not require a precise assessment of the inbreeding coefficient of the population (a prerequisite for the half-sib design).

Last, Figure 6, E and F, shows the effect of the crossing design on traits that are purely overdominant. Figure 6E shows that under strict overdominance *Q*_{ST} is even less than under dominance when *Q*_{ST} is estimated from a half-sib design in a species with high selfing. If selfed progeny are used instead of half-sibs, *Q*_{ST} does not differ from zero whatever the level of population structure. This is expected, as under overdominance homozygote genotypes all have the same trait value. Selfed progeny from highly selfed parents are strongly homozygous in all populations, and therefore trait means are also identical among populations. Note that this is an artifact of having strict overdominance (*i.e*., the complete absence of additive effects).

#### Sampling strategies:

The effect of different sampling strategies on estimation of is shown in Figure 7. The box-plot representation gives a fair idea of the distribution of under these four sampling strategies. The worst scenario is strategy A (50 families of 10 individuals in two populations), which shows the largest variance for all levels of population differentiation and a negative bias that increases in magnitude as differentiation increases. Strategies C (20 populations) and D (40 populations) are best overall, with strategies C being slightly better for low structure and strategies D better for high structure. For scenarios C and D, the interquartile range is very similar to that under exhaustive sampling (Figure 7, labeled T).

## DISCUSSION

While dominance can theoretically either increase or decrease *Q*_{ST} relative to its expectation under strict additivity, we have shown that, on average, dominance decreases the value of *Q*_{ST}, and the more differentiated the populations are, the stronger the effect. Thus, we conclude that dominance is unlikely to cause the pattern *Q*_{ST} > *F*_{ST}. Since this pattern was also shown to be unlikely under epistasis (Whitlock 1999; Lopez-Fanjul *et al.* 2003), we argue that when *Q*_{ST} > *F*_{ST} it is a good indicator of the presence of directional selection for different local optima. *Q*_{ST} < *F*_{ST}, on the other hand, could be the result of several factors other than homogenizing selection.

These results contrast with those obtained by Lopez-Fanjul *et al.* (2003), who found that the effect of dominance would more often increase *Q*_{ST} relative to *F*_{ST} rather than the reverse. There may be several reasons for this. First, Lopez-Fanjul *et al.* (2003) focused on populations that just underwent a severe bottleneck, and bottlenecks are known to alter (increase or decrease) the genetic variance both within and between lines (Robertson 1952; Willis and Orr 1993; Cheverud and Routman 1996; Naciri-Graven and Goudet 2003; Barton and Turelli 2004). Second, to compare *F*_{ST} and *Q*_{ST}, they did not use the allele frequencies at the loci coding for the trait as we did here, but used the expectation of the inbreeding coefficient among recently bottlenecked populations. Third, they looked at the effect of a one-generation bottleneck of two individuals, which would correspond to an *F*_{ST} of 0.25 on average, and did not investigate (as we did here) a range of population structure. Note that we also find situations in which *Q*_{ST} > *F*_{ST}, when the recessive allele is very common in most populations. But this situation is unlikely to be found frequently in nature, as recessive alleles are often deleterious, thus under the action of purifying selection, and therefore at low frequencies (Naciri-Graven and Goudet 2003; Barton and Turelli 2004).

In general, empirical studies tend to find *Q*_{ST} > *F*_{ST} (see reviews in Merila and Crnokrak 2001 and McKay and Latta 2002). Can we conclude from this pattern that directional selection is in action? Our results show that nonadditive gene actions are not the likely culprits, but other biases could produce this pattern.

Hendry (2002) pointed out that mutation is also likely to alter the relation between *Q*_{ST} and *F*_{ST}. The deflating effect of large mutation rate on *F*_{ST} is well known (Hedrick 1999; Balloux *et al.* 2000). If *Q*_{ST} is less sensitive to mutation than *F*_{ST}, or if the mutation rate at trait loci is smaller than at marker loci, then the mutation rate in itself can create the pattern expected under the action of directional selection. On the other hand, the mutation rate on a quantitative trait coded by several loci might be larger than that of one molecular marker. Thus, we concur with Hendry (2002): The potential effect of mutation rate on the relation between *F*_{ST} and *Q*_{ST} deserves a thorough investigation.

The large variance in *Q*_{ST} estimates based on few populations (and in many field situations, only a few populations are available) limits the statistical power of a test of the relation *F*_{ST} = *Q*_{ST}. O'Hara and Merila (2005) recently investigated the statistical properties of estimators of *Q*_{ST} and came to the same conclusion: With <12 populations, the variance in *Q*_{ST} is huge. We showed that this is particularly the case for very differentiated populations (*F*_{ST} > 0.2). With less differentiation, the problem seems less acute. We also showed that with few populations estimates of *Q*_{ST} seem to be biased downward. Thus, the pattern *Q*_{ST} > *F*_{ST} is unlikely due to a statistical artifact. Note that these arguments are only verbal, and more work is clearly necessary to refine the statistical tools available. In particular, our investigation of the effects of the sampling strategies needs to be pursued: We modeled traits that are purely genetically determined, and the effect of environmental variance on the precision of *Q*_{ST} estimates is not known, but is likely to inflate the variance of *Q*_{ST}.

It would be interesting to investigate the behavior of *Q*_{ST} under the joint action of selection and dominance. If the effect of dominance is just to hide recessive deleterious alleles, then recessive alleles will be rare and we saw that this is a situation where *Q*_{ST} < *F*_{ST}. Hence, the presence of deleterious recessive alleles should tend to make *Q*_{ST} even smaller than *F*_{ST}. If both purifying selection and directional selection for different optima are occurring, the deflating effect of dominance on *Q*_{ST} might cancel out the enhancing effect of directional selection. Hence, the pattern *Q*_{ST} = *F*_{ST} might reflect the joint action of these two selective forces rather than the absence of selection. A similar investigation on the effect of epistasis is also necessary.

#### Conclusions:

Despite these caveats and shortcomings, we have clearly shown that the pattern *Q*_{ST} > *F*_{ST} is unlikely for neutral traits with nonadditive gene action. Importantly, we have also shown that estimates of *Q*_{ST} are reliable only if based on many sampled populations. Providing that this is the case, the comparison between *Q*_{ST} and *F*_{ST} will remain useful in documenting the presence (or not) of local adaptation.

## Acknowledgments

We thank Guillaume Evanno, Benoît Facon, Juha Merilä, Bob O'Hara, and an anonymous reviewer for useful comments on previous versions of this manuscript. J.G. was supported by grant 31-108194/1 from the Swiss National Science Foundation.

## Footnotes

Communicating editor: M. Veuille

- Received September 12, 2005.
- Accepted November 8, 2005.

- Copyright © 2006 by the Genetics Society of America