Testing for Spatially Divergent Selection: Comparing QST to FST

Q_ST is a standardized measure of the genetic differentiation of a quantitative trait among populations. The distribution of Q_ST''s for neutral traits can be predicted from the F_ST for neutral marker loci. To test for the neutral differentiation of a quantitative trait among populations, it is necessary to ask whether the Q_ST of that trait is in the tail of the probability distribution of neutral traits. This neutral distribution can be estimated using the Lewontin–Krakauer distribution and the F_ST from a relatively small number of marker loci. We develop a simulation method to test whether the Q_ST of a given trait is consistent with the null hypothesis of selective neutrality over space. The method is most powerful with small mean F_ST, strong selection, and a large number (>10) of measured populations. The power and type I error rate of the new method are far superior to the traditional method of comparing Q_ST and F_ST.IN 1993, Spitze (1993) and Prout and Barker (1993) introduced Q_ST, a quantitative genetic analog of Wright''s F_ST. Just as F_ST gives a standardized measure of the genetic differentiation among populations for a genetic locus, Q_ST measures the amount of genetic variance among populations relative to the total genetic variance. In the years since, Q_ST has been frequently used to test for the effects of spatially divergent (or less commonly, spatially uniform) selection (see reviews in Lynch et al. 1999; Merilä and Crnokrak 2001; McKay and Latta 2002; Howe et al. 2003; Leinonen et al. 2008; Whitlock 2008). In principle, the average Q_ST of a neutral additive quantitative trait is expected to be equal to the mean value of F_ST for neutral genetic loci. F_ST can be readily measured on commonly available genetic markers, and Q_ST can be measured as well with an appropriate breeding design in a common-garden setting. As a result, Q_ST promises to be an index of the effect of selection on the quantitative trait. If Q_ST is higher than F_ST, then this is taken as evidence of spatially divergent selection on the trait. If Q_ST is much smaller than F_ST, then this has been taken as evidence of spatially uniform stabilizing selection, which makes the trait diverge less than expected by chance.The comparison with F_ST is essential to rule out genetic drift as an alternative mechanism for phenotypic divergence among populations. Because finite populations may diverge genetically in the absence of selection, divergence must be greater than expected by drift alone if we are to conclusively demonstrate that divergent selection has played a role in genetic differentiation among populations. Therefore it has become common practice to use F_ST of putatively neutral markers as a control for the effects of genetic drift and to compare observed Q_ST values for traits to these neutral F_ST values.These comparisons follow two separate methods, to address related but distinct questions. First, many studies of quantitative genetic differentiation measure the Q_ST of many traits and the F_ST of many loci, followed by a comparison of the mean Q_ST to the mean F_ST. Such a comparison may judge whether the conditions are suitable in that species for local adaptation, that is, whether selective differences between populations are large enough relative to gene flow to allow adaptive differentiation (Whitlock 2008). We do not consider this sort of comparison in this article.The other type of comparison asks whether the Q_ST of a single trait is greater than expected by drift, as measured by F_ST. This type of comparison is most common, but it is statistically difficult. Unfortunately, as emphasized in a recent review by Whitlock (2008), there is great variation in the expected F_ST among neutral loci and among the Q_ST of different neutral traits (see Figure 1). The majority of this variation results from evolutionary differences between loci and not sampling error in the observations. Rogers and Harpending (1983) imply that the distribution of Q_ST of a single neutral trait should be approximately equivalent to that for F_ST of a single neutral locus, and this has been confirmed by simulation for traits determined by additive loci compared to biallelic marker loci (Whitlock 2008). The two distributions are similar, but there is great heterogeneity among traits or loci. As a result, to show that selection is acting on a trait, it is necessary to show that the value of Q_ST has a low probability of being observed given the distribution of neutral Q_ST.Open in a separate windowFigure 1.—The distribution of F_ST for neutral loci and Q_ST for neutral quantitative traits. The histograms show the results of simulations of a set of 10 local populations each of 100 individuals, connected by 5% migration following island model assumptions. The solid line shows the distribution predicted by the Lewontin–Krakauer distribution. The distribution of Q_ST for neutral traits is very similar to the distribution of F_ST for single neutral loci, as can be seen by their mutual good fit to the Lewontin–Krakauer distribution (Figure modified from Whitlock 2008).Comparing Q_ST to the distribution inferred from F_ST is difficult for two reasons. First, typical data sets rarely include enough loci to directly infer the distribution of F_ST without extra inferential steps. In our approach, we use the distribution of Q_ST predicted from the mean F_ST and the χ² distribution by Lewontin and Krakauer (1973) to bridge this gap. Whitlock (2008) has shown that this distribution is appropriate for nearly all realistic situations for traits determined by additive genetic effects. Second, Q_ST for a trait is rarely measured with high precision, so the position of a given estimated Q_ST value in the distribution cannot be known without error.To test the null hypothesis that the spatial distribution of a particular trait is not affected by selection, we wish to compare the observed of that trait (marked with a hat to indicate it is an estimate) to the distribution of Q_ST expected for neutral traits. Unfortunately, calculating the distribution of Q_ST for neutral traits is not straightforward, because the estimate of Q_ST for a particular trait is variable for several reasons. The estimate of Q_ST is subject to measurement error, caused by the finite samples of families and individuals in the quantitative genetic experiment. These cause error in the estimate of the additive genetic variance within populations (V_A,within) and the genetic variance among populations (V_G,among), which translate into error of the estimate of Q_ST. In addition, there is another source of variation in Q_ST among neutral traits, caused by the idiosyncrasies of the evolutionary process in each local population in the study. The true value of Q_ST for the set of populations being studied can vary tremendously around its expectation, even for neutral traits, because by chance a finite set of populations may drift in a similar direction (Whitlock 2008). As a result, measurements of Q_ST can vary because of both statistical and evolutionary variation.Fortunately, these two sources of variation are fairly well understood individually. The sampling error for the estimates of the variance components can be estimated from standard approaches, and this variation can be well approximated using information from the mean squares of the analysis of the breeding experiment (O''Hara and Merilä 2005). The variation in neutral Q_ST that results from heterogeneity of evolutionary history can be approximated by the Lewontin–Krakauer distribution (Lewontin and Krakauer 1973), if information is available on the mean Q_ST of neutral traits (Whitlock 2008). This approximation does not depend on the demographic details of the populations in question (Whitlock 2008), but the effects of deviations from assumptions of additive gene effect have not yet been tested. The mean of the distribution of values of Q_ST for neutral traits is usually not known, but fortunately the mean of the distribution of F_ST of neutral loci is expected to be approximately equal to the mean Q_ST of neutral traits (Spitze 1993), and this does not depend on demographic details (Whitlock 1999). Therefore the mean F_ST measured from a series of genetic markers thought to be selectively neutral can be combined with the Lewontin–Krakauer distribution to predict the distribution of true neutral Q_ST across the range of possible evolutionary trajectories.Given that the mean value of of neutral traits is expected to equal the mean F_ST of neutral markers under certain assumptions (discussed later), we will use as a test statistic and compare the observed quantity to the zero value proposed by the null hypothesis. We will use a traditional hypothesis testing approach, which means that we need to specify the sampling distribution of under the assumption of neutrality. Traditionally, the sampling distribution of is inferred from the data on the trait itself, for example, using bootstrapping to infer the sampling distribution. This is appropriate when calculating a confidence interval for Q_ST but is a biased measure of the sampling variance of neutral Q_ST. The variance of the sampling distribution of varies with its expected value; larger values of true Q_ST have more variable sampling distributions than traits with smaller true Q_ST. This association between Q_ST and its sampling error is quite strong, as shown in Figure 2. As a result, if the sampling properties of neutral are inferred from a trait with high Q_ST, the estimate of the variance of the null distribution will be too high, and the hypothesis test comparing to F_ST will be conservative. On the other hand, if a low Q_ST is used to estimate the variance of the null distribution, the estimated error will be too small, and the test will reject true null hypotheses too often.Open in a separate windowFigure 2.—The width of the estimated sampling distribution of varies with mean Q_ST. The solid line shows the sampling distribution of Q_ST when the true mean Q_ST value is 0.05. The dotted line shows the sampling distribution that would be estimated for Q_ST from a trait that by chance was at the first percentile of this distribution, and the dashed line shows the sampling distribution that would be inferred from a value taken at the 99th percentile. If the Q_ST of a trait differs from the expectation by chance, then the width of the sampling distribution will also be estimated with substantial error. In particular, the error variance of is overestimated with Q_ST estimates that are too high and underestimated for small Q_ST values.We address this problem by using F_ST from putatively neutral maker loci in combination with estimates of the additive genetic variance within populations to predict the sampling variance that would be expected for the Q_ST of a neutral trait. We show that the power and type I error rate of this test are greatly superior to traditional methods.