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# Decoupling of differentiation between traits and their underlying genes in response to divergent selection

## Abstract

We dissected the relationship between genetic differentiation (QST) for a trait and its underlying genes (GSTq, differentiation for a quantitative locus) in an evolutionary context, with the aim of identifying the conditions in which these two measurements are decoupled. We used two parameters (θB and θW) scaling the contributions of inter- and intrapopulation allelic covariation between genes controlling the trait of interest. We monitored the changes in θB and θW, QST and GSTq over successive generations of divergent and stabilizing selection, in simulations for an outcrossing species with extensive gene flow. The dynamics of these parameters are characterized by two phases. Initially, during the earliest generations, differentiation of the trait increases very rapidly and the principal and immediate driver of QST is θB. During subsequent generations, GSTq increases steadily and makes an equal contribution to QST. These results show that selection first captures beneficial allelic associations at different loci at different populations, and then targets changes in allelic frequencies. The same patterns are observed when environmental change modifies divergent selection, as shown by the very rapid response of θB to the changes of selection regimes. We compare our results with previous experimental findings and consider their relevance to the detection of molecular signatures of natural selection.

## Introduction

The detection of genomic imprints of natural selection has become a major area of research in ecological genetics (Nielsen, 2005; Wright and Gaut, 2006). One of the approaches used is based on the detection of genes displaying higher levels of population differentiation than would be expected under neutral evolution (the so-called ‘outlier’ loci; Storz, 2005; Holderegger et al., 2008). In most cases, the rationale of ‘outlier’ detection is based on the existence of a high degree of phenotypic differentiation of the trait, as observed in common garden experiments, suggesting similar levels of differentiation at the genomic level. For example, in forest trees, in which extensive common gardens have been established, known as provenance tests, substantial population differentiation has been observed for almost all phenotypic traits assessed (see Wright (1976) and Morgenstern (1996) for reviews on North American species, and König (2005) for a review on European species). Only a few studies have investigated the differentiation of traits and their underlying genes, and these studies have concluded that there is considerable discrepancy between these two measurements (Hall et al., 2007; Luquez et al., 2007; Namroud et al., 2008). Theoretical predictions supported by simulations have indeed indicated that there may be large discrepancies between phenotypic differentiation (QST) and differentiation of the genes controlling the trait (GSTq, for quantitative loci, as suggested by Santure and Wang (2009), to avoid confusion with neutral GST or FST). Latta (1998, 2004) and McKay and Latta (2002) have shown that this discrepancy stems from the intergenic disequilibria accumulated through selection, which decouples differentiation at the trait and gene levels. In a previous study, we investigated the contribution of intergenic disequilibria to overall phenotypic differentiation, theoretically, in more detail (Le Corre and Kremer, 2003). We confirmed that high levels of differentiation could be observed for a trait, with only low levels of differentiation for the genes controlling the trait, and reviewed various evolutionary scenarios under which this discrepancy would be maintained or increased. In extreme cases, such as outcrossing species with large populations and high gene flow undergoing divergent selection, as in trees, there might be no signature at all at individual loci, despite a high degree of phenotypic differentiation. Recent developments along these lines have considered the contribution of dominance to the decoupling of QST and GSTq, (Goudet and Büchi, 2006; Goudet and Martin, 2007; Santure and Wang, 2009). Under divergent selection, the contrast between these two measurements of differentiation increases with dominance (Santure and Wang, 2009). We therefore consider that the additive model used in this study provides the lower limits of decoupling between QST and GSTq in more complex situations, including gene interactions.

We explore here the progressive decoupling of QST and GSTq under different evolutionary scenarios. Our previous study focused on the equilibrium values of QST and GSTq (Le Corre and Kremer, 2003). Here, we address their transient dynamics over time and consider different genetic architectures of the trait undergoing selection. Interest in the speed of adaptive differentiation in heterogeneous landscapes has increased recently (Kawecki, 2008; Lopez et al., 2008; Bjorklund et al., 2009). Evolutionary responses are often inferred from the divergence or differentiation of adaptive traits observed in a set of populations undergoing divergent selection (Hendry et al., 2001; Crispo, 2008; Rasanen and Hendry, 2008) or from the molecular divergence of genes of adaptive relevance (Kapralov and Filatov, 2006; Nosil et al., 2009). Furthermore there is an increasing concern about the rate of response of natural populations to ongoing climate change (Aitken et al., 2008). It is therefore timely to reconsider the dynamics of differentiation in an evolutionary context. Our main objective is to monitor changes of QST and GSTq and their decoupling in silico by considering different divergent selection regimes, including nonequilibrium situations. We used two parameters (θB and θW) scaling the contributions of between- and within-population disequilibria to QST. We highlight the way in which divergent selection generates a positive θB, which becomes in most circumstances the main driver of QST. We focus on the particular case of species with large populations and high levels of gene flow, such as trees, in which high levels of trait differentiation are maintained in the presence of high levels of gene flow (Savolainen et al., 2007). We aimed to identify the situations in which the traces of natural selection acting on the genes controlling the trait might be blurred by intergenic disequilibria, thereby limiting the power to detect ‘outlier’ loci.

## Materials and methods

### Decoupling of QST and GSTq

We focus here on the comparative monitoring of population differentiation of a diploid organism for a phenotypic trait and the underlying genes contributing to the trait, assuming that all these genes are known. The trait is assessed in different populations subject to selection and exchanging genes through gene flow. At any time during the evolutionary process, differentiation of the trait and of the underlying genes can be assessed. In a previous study, we derived a relationship between differentiation for a quantitative trait (QST) and the mean differentiation of the n loci controlling the trait (GSTq; Le Corre and Kremer, 2003) under the following assumptions:

• Diallelic loci for which the two alleles have symmetric additive effects (+α/2 and −α/2).

• Equal contributions of the various loci to the additive value of the trait.

• Additive effects of the alleles at each locus, with no consideration of dominance or epistasis.

where

σWi2 is the within-population variance of additive effects at locus i contributing to the trait. At the single-locus level, σWi2 is also called genic variance. The genic variance is assessed in all populations, and in what follows σWi2 is the average value across populations. σBi2 is the variance of population effects of locus i.

CovWi,j and CovBi,j are the within- and between-population covariances of the additive and population effects.

θW and θB are the relative contributions of the genetic covariances between loci to the additive and between-population variances of the trait (VW and VB, respectively), with respect to the single-locus variances (σWi2and σBi2).

VW is the additive variance within populations.

VB is the between-population variance.

According to (1), QST is a monotone increasing function of GSTq and θB, and a monotone decreasing function of θW, reflecting the contrasting contributions of the within- and between-population disequilibria to QST. Interpopulation disequilibria between loci tend to increase phenotypic differentiation between populations, whereas intrapopulation disequilibria decrease differentiation.

Relationship (1) holds even when gene flow and selection are not at equilibrium. It is an ‘instantaneous’ relationship describing the architecture of QST in terms of single-locus components (GSTq) and multilocus components (θB and θW) at any time through an evolutionary scenario. This relationship also holds for all types and strengths of selection within or between populations and contributing to the divergence between populations.

We can now consider the difference between trait differentiation and gene differentiation by rewriting (1) in a different way:

From (4) and (5), we can conclude that:

There is no simple relationship between QST and GSTq in more complex situations, with unequal contributions of loci and multiallelism. However, we used simulations to determine whether relationship (1) was applied in such situations (Le Corre and Kremer, 2003; Figure 3). As expected, the observed differentiation of the trait (QST) in the case of multiallelism and an unequal contribution of loci were larger than that predicted by Equation (1), but the difference was moderate when QST was below 0.40. These comparisons suggest that the decoupling of QST and GSTq would be even more pronounced in these more realistic situations. Hence, relationship (1) is a rather conservative figure, corresponding to the minimum observable discrepancy between QST and GSTq. Similarly, when dominance is taken into account, the difference between the two measurements of differentiation increases with divergent selection and higher levels of dominance (Santure and Wang, 2009). Furthermore, inequalities (6) were also maintained when multiple alleles and unequal contributions of loci were considered in simulations (Le Corre and Kremer, 2003, Figure 4). From the inequalities in (6), we can conclude that the comparison of disequilibria between and within populations can be used to predict the level of decoupling of GSTq and QST.

We further dissected θB and θW taking into account the genetic architecture of the trait, that is, the number of loci contributing to the trait. Within- and between-population genic variances can be expressed in terms of diversity and differentiation parameters in the case of biallelic loci having symmetrical additive effects (Le Corre and Kremer, 2003):

where Hsi is the mean within-population diversity for locus i, Fi the mean within-population fixation index for locus i, and DSTi is the differentiation among populations for locus i (Nei, 1987 DSTi=HTiHSi, HTi being the total diversity in the metapopulation).

Covariances between allelic effects of loci i and j can be decomposed in terms of disequilibria and allelic effects (Weir, 1990; Hamilton and Cole, 2004) as:

where ΔWi,j is the mean composite genotypic disequilibrium (within population) between loci i and j according to Weir, (ΔAB in Weir's notation), and Cov(pi, pj) is the covariance between frequencies pi (allele of positive effect at locus i) and pj (allele of positive effect at locus j) calculated across all populations.

If we further assume that the αi and Fi values are the same for all loci, and taking HS as the mean intrapopulation gene diversity over all loci, then θW becomes:

Similarly assuming that DST is the mean differentiation across all loci and that allelic effects are the same across loci, then θB becomes:

where n is the number of loci, and ΔW and ΔB are the mean values of ΔWi,j and ΔBi,j over all pairs of loci. As a result, θB and θW are now written in terms of disequilibria and diversity parameters, which can be estimated in natural populations. Expressions (11) and (12) show that θB and θW are dependent on the number of loci contributing to the trait. The decoupling of QST and GSTq can now be expressed as a function of the number of loci by replacing in (4) θW and θB by their expressions given by (11) and (12)

where x=2(n−1) and B = ΔB/DST and W = ΔW/(1+F)HS It can be shown that (13) is an increasing function of x when B>W (or when θB>θW) and a decreasing function of x when B<W (or when θB<θW).

### Simulations

We monitored the changes in θB, θW, GSTq and QST under different evolutionary scenarios, assessing the decoupling of GSTq and QST over successive generations. Individual-based simulations were carried out with large populations and extensive gene flow, subject to divergent selection. Details of the METAPOP simulation model are provided in Le Corre and Kremer (2003). We considered a phenotypic trait controlled by 5, 10 or 30 unlinked loci, each with 20 allelic states. The mutation rate was set at 10−5 per locus and followed a k-allele model. The genetic value of the trait was the sum of allelic effects over all loci. The additive value at each allele was drawn at random from a Gaussian distribution. The environmental contribution to the phenotypic value of the trait was considered as a Gaussian variable of mean 0 and variance 1. The variance of the distribution of allelic effects was adjusted, so that the initial genetic variance within populations was equal to 5 (implying that the trait's heritability was initially 0.833). These starting values were selected, so that they match with estimates of heritabilities in extant progeny tests (Kremer, 1994); other initial values were tested as well, but the evolution in θB, θW, GSTq and QST showed similar patterns. Throughout the simulations, covariance components of θB and θW were numerically calculated as differences between genetic variances (VB and VW, respectively) and genic variances (σBi2 and σWi2 , respectively) according to Equation (3). Hence, they comprise all sources of allelic associations, for example, between different loci arising on a same gamete (linkage disequilibrium stricto sensu) or of different gametic origins (because of Hardy–Weinberg disequilibrium).

We considered 25 populations, each of size 500, with equal gene flow between all pairs of populations (island model with Nm=10). The 25 populations were displayed on a 5 × 5 grid system with one-dimensional environmental variation, such that the optimal phenotypic value ZOPT(k) for the trait in population k followed a cline of variation across the metapopulation. These settings (initial genetic variance and environmental variance within populations, mutation rate, number of populations, population size and migration rate) were kept constant across all simulations.

We focused on the effects of different selection pressures on population differentiation. We considered Gaussian stabilizing selection within each population k toward the local optimal phenotypic value ZOPT(k). The fitness W(Z) of an individual with phenotype Z located in population k is given by:

The strength of within-population selection was scaled by ω2, the selection intensity. We considered two cases with contrasting reported values of ω2 (Roff, 2002): low selection intensity (ω2=50) and high selection intensity (ω2=5).

Divergent selection was introduced into this setting by the linear gradient of ZOPT values. On the 5 × 5 grid system, this resulted in the assignment of –x,x/2, 0, x and x/2 as ZOPT values for the five populations in a single row. The level of divergent selection was scaled by VarZOPT, the variance of ZOPT values between populations (VarZOPT=x2/2). Again, we considered two contrasting cases: low levels of divergent selection (VarZOPT=1) and moderate divergent selection (VarZOPT=5). In total, we investigated four different selection scenarios combining different selection regimes (Table 1a). At the starting point of the simulations (generation 0), we assumed that the 25 populations were at mutation–migration–drift equilibrium (for Nm=10 and GSTq=2.4%). Hence, some population differentiation for the phenotypic trait existed at time zero, purely due to the neutral differentiation of allelic frequencies (QST=2.4%), but θB and θW were zero at the starting point. The number of alleles initially present at each locus in the metapopulation was fixed at six, according to Ewens’ approximate sampling formula (Ewens, 1972). Overall allelic frequencies at each locus were drawn form a Dirichlet distribution with parameter 1. Allelic frequencies for the different populations were then drawn from Dirichlet distributions with parameters equal to the overall frequency scaled by 4 Nm (Beaumont and Balding, 2004). For each scenario, we considered different genetic architectures of the trait undergoing selection, by imposing that the trait was controlled by 5, 10 or 30 loci. A total of 50 replicates (for each combination scenario—number of loci) were run to account for stochastic variation of the various parameters assessed. These simulations were run up to 3000 generations in order to compare equilibrium values, but their dynamics were monitored in detail during the very first 100 generations.

We further investigated additional scenarios aiming at monitoring the decoupling of GSTq and QST under more realistic situations, where selection takes place while some sort of selection was already acting. This was done by subdividing each scenario into two successive phases: phase A from generation 0 to 100, followed by phase B from generation 101 to 200 (Table 1b) Differences in phases A and B consisted essentially in different levels of divergent selection (uniform, moderate divergent and strong divergent). The rationale for subdividing each scenario into two successive phases was twofold:

1. 1)

to evaluate the impact of different initial situations on the dynamics of the parameters (θB, θW, GSTq and QST ) by considering nonequilibrium situations as initial states and

2. 2)

to monitor how differentiation of traits and genes respond to sudden changes in selection regimes (either increasing levels or decreasing levels of divergent selection).

## Results

### Genetic architecture and decoupling of GSTq and QST

Asymptotic values observed for GSTq and QST after 3000 generations confirm theoretical predictions that the decoupling between GSTq and QST is inflated by differences between θB and θW (Figure 1). The largest decoupling was observed under strong divergent and stabilizing selection, whereas low divergent and stabilizing selection contributed to moderate decoupling. One of the most striking findings of the simulations was the contrasting response of θW, θB and GSTq to variation of the number of loci controlling the trait (Table 2 and Figure 1). By contrast, the genetic parameters VW, VB and QST remained constant (Supplementary Table 1), except under relaxed selection regimes (scenario 3), where higher levels of VW and VB were maintained when the number of loci was larger. As θW, θB and GSTq are the only components of QST (Equation (1)), there must be some trade-off between θW, θB and GSTq to maintain QST constant. As predicted by theory (Equations (11) and (12)), θW and θB increase with the number of loci (Table 2). Interestingly, the increase of θB as a function of the number of loci fits to the prediction in the simplified diallelic locus case (assuming ΔB and DST constant). GSTq decreased with increasing number of loci (Figure 1 and Supplementary Table 1). As predicted by (13), the level of the decoupling (QST/GSTq) increased as the number of loci increased (when θB>θW). A high degree of decoupling of QST and GSTq was therefore expected when a large number of genes contributed to the trait, as illustrated in Figure 1.

These observations can be extended to the case in which the number of loci increases indefinitely (infinitesimal model), resulting to very low levels of GSTq, and θB becoming the key component of QST. A basic assumption of the infinitesimal model is indeed that the allelic frequencies of quantitative loci do not change across generations and that all the between-population variance stems from the intergenic disequilibrium (θB; Bulmer (1980)).

Figure 1 also illustrates the variation of QST and GSTq values between stochastic repetitions of a given evolutionary scenario. As the number of genes contributing to the trait decreases, the variation of GSTq increases over the 50 stochastic repetitions. QST is much less variable than GSTq across stochastic repetitions. Given the high degree of heterogeneity of GSTq values, there may be considerable variation of the decoupling of GSTq and QST for a given setting of VarZOPT and ω2. However, this variation occurs mostly in situations of small numbers of loci and a high intensity of stabilizing selection (Figure 1).

### Dynamics of genetic variances and covariances

We first monitored the dynamics of θW, θB and corresponding variances under selection regimes following an initial mutation–migration–drift equilibrium (scenarios 1–4), considering a trait controlled by 10 loci (Figure 2). In the first few generations (<10) following the induction of selection, the intergenic linkage disequilibrium component θW decreases rapidly. This decrease becomes stronger as the strength of stabilizing selection increases (Figures 2e and g). These trends follow the predictions of Bulmer (1974, 1980), who showed that negative covariances are generated by directional selection in a single population. Our results suggest a similar trend for a subdivided population. Selection induces negative covariance at each generation. However, this covariance is halved by recombination at each generation. Further, the negative covariance component added in each generation decreases steadily because stabilizing selection erodes the genetic variance itself (Figure 3). After reaching its minimum, θW begins to increase, possibly even approaching positive values when divergent selection is strong (Figures 2c vs 2a; Figures 2g vs 2e; Table 2a). We suspect that the construction of a positive θW results from an interaction between gene flow and divergent selection. As first observed by Nei and Li (1973), when divergent selection operates in a subdivided population such that allelic frequencies differ between subpopulations, migration produces permanent linkage disequilibrium in each subpopulation. Kirkpatrick et al. (2002; formula 32) showed that the disequilibrium component contributed by migration in each population is a function of the product, across loci, of the differences in allele frequencies between the local population and the migrant pool, and increases with increasing migration rate. As strong divergent selection favors different allelic associations in different populations, migration continually assembles different allelic combinations, thus creating positive disequilibrium within populations. Indeed, our simulations show that divergent selection has a positive effect on θW due to the restoration of positive allelic covariances (Figures 2c and g and Table 2a), as predicted by Nei and Li (1973) and Kirkpatrick et al. (2002).

θB increases very rapidly during the first 5 to 10 generations, particularly under highly divergent selection (Figures 2d and h). θB builds up to compensate for the lag between the variance of optimal values VarZOPT and the existing between-population variance (VB). At the very beginning (generation 0), the only source of VB is the neutral subdivision of the populations induced by gene flow:

where σ02 is the initial genetic variance in the whole metapopulation and ${G}_{{\text{ST}}_{0}}$ is the neutral differentiation. Thus, in the case of a species with low migration, the between-population variance that exists before the onset of divergent selection may be much lower than the variance of optimal values.

Thereafter, θB is maintained at the same level, or decreases slowly under highly divergent selection (Figure 2h). Thus, simulation results show that changes in genetic covariances both within and between populations occur in the first few generations of selection.

Shortly after θB reaches its peak value, the genetic variance between populations, VB, also attains its maximum value (Figure 3), decreasing steadily thereafter. The rate of decrease is stronger when stabilizing selection is weak. It is difficult to detect these dynamics over short time scales (Figure 3), but they are obvious when comparing the asymptotic values of VB (Supplementary Table 1) with the values of this parameter over the first 100 generations (Figure 3). As alleles are purged from populations, intergenic allelic associations are limited (decay of θB; Figures 2f and h) and VB depends principally on the difference in allele frequency between populations. As can be seen on Figure 3, VB does not reach VarZOPT (the variance of optimal values within populations according to stabilizing selection). The maximum value of the VB/VarZOPT ratio was a function of the strength of stabilizing selection and did not vary with the number of loci: the mean value obtained was 0.67 when ω2=50 and 0.94 when ω2=5. Under extensive gene flow (Nm=10), VarZOPT can be reached only if stabilizing selection is strong while under weak selection, the final VB value is less than half of VarZOPT (Supplementary Table 1). No changes to the transient dynamics were observed when the number of loci was changed to 5 or 30 (data not shown). The only change was the overall higher values of θW and θB when the number of loci increased.

The general trends of variation of θB were maintained when the starting conditions of simulations were modified. Indeed, regardless whether populations were at mutation–migration–drift equilibrium, undergoing uniform selection or moderate divergent selection, the responses of θB to the induction of strong divergent selection was a very rapid and steep increase (Figures 4a and b). The steepness of the response is mainly dependent on the strength of the ongoing stabilizing selection, and not on the initial conditions. However, the range of increase of θB was larger when uniform selection was the starting point (θB negative at generation 0), because the lag between the imposed variance of optimal values (VarZOPT) and the between-population variance (VB) at generation 0 was much larger than under other starting conditions. θB is the immediate response to compensate the lag. We also monitored the changes in θB when the number of loci was 5 and 30. There was no change in the dynamics, but the values after 100 generations differed according to the theoretical predictions (Equation (12)): they ranged from 1.76 to 2.11 with 5 loci, from 3.17 to 4.15 with 10 loci and from 7.74 to 8.36 with 30 loci.

### Dynamics of GSTq and QST

As for of θW and θB, we monitored first the variation of QST and GSTq under scenarios 1–4 following an initial mutation–migration–drift equilibrium (Figure 5). There are striking differences between the variation of QST and GSTq on the one hand and θW and θB on the other. θW and θB reach minimum and maximum values during the first few generations, whereas QST and, especially GSTq, increases more steadily. The main driver of GSTq appears to be the strength of stabilizing selection, which opposes the effect of gene flow, as described above.

Throughout the selection scenario, GSTq remained lower than QST, as a straightforward consequence of θW being lower than θB (inequalities 6). QST increased very rapidly in the first few generations, with a steeper slope under strong stabilizing selection. Subsequently, the dynamics of QST followed that of GSTq, with a steady increase when stabilizing selection was strong (Figures 5c and d), and a smoother or flat course when stabilizing selection was weak (Figures 5a and b). The decoupling of QST and GSTq was not changed when initial conditions were either uniform selection or moderate divergent selection (Figures 5b, d and 6a, b). Whether QST is 0 or already higher than the neutral expectation at generation 0, the time trends during the first 100 generations are similar. The same conclusions hold also for GSTq. Particularly, the very rapid increase of QST during the early generations is created by the steep variation of θB, being the main driver of QST.

The patterns depicted in Figures 2, 3, 4, 5 and 6 indicate that the variation of population differentiation induced by selection follows two phases. A detailed dissection of the early phase (comparison of Figures 3 and 5 with Figure 2) suggested that the main driver of the rapid increase in both VB and QST was the variation of θB, rather than that of GSTq. Interestingly, θB may reach even higher values when the trait is controlled by a larger number of loci (data not shown, but see asymptotic values in Table 2b), enhancing the capture of intergenic disequilibria, as the larger number of loci provides a larger number of opportunities for favorable allelic associations. The dynamics during this early phase is highly analogous to that of the genetic variance of a trait undergoing selection within a single population (Bulmer, 1974, 1980). Indeed, Bulmer indicated that allele frequencies probably remain constant over the first few generations of selection, whereas the genetic variance of the trait decreases, due mostly to negative intergenic linkage disequilibrium. In our multiple-population context, the Bulmer effect may be thus generalized as follows: the short-term response to selection is characterized by a decrease in within-population variance mediated by negative values of θW in each population on one the hand, and an increase in between-population variance mediated by a rapid increase in positive θB values on the other hand. This early phase is characterized by the rapid building up of the covariance of allelic effects, but allelic frequencies within populations change very slowly. The increase in covariance is followed by an increase in between-population variance, VB, which peaks at the end of the first phase. As this point, populations are locally adapted (that is, the between-population variance is closest to the optimal variance). The duration of the early phase varies from a little over 10 to 30 generations when local stabilizing selection is strong, and from 50 to 80 generations when local stabilizing selection is weak .In the second phase, there is a slower steady increase in QST driven by GSTq, with no further increase in θB. θB remains larger than θW, but differences in allelic frequencies gradually increase, contributing to the steady increase in QST.

### Changes of selection regimes and decoupling of GSTq and QST

An interesting extension of the comparative in silico monitoring of GSTq and QST is their response to environmental changes, which can be provoked by changing ZOPT values assigned to the different populations. To do so, we compared two symmetric scenarios: divergent selection to uniform selection (scenarios 5 and 6) versus uniform to divergent selection (scenarios 9 and 10; Figures 7 and 8). The rapidity of the response of θB after generation 100 was very similar, whether the level of divergent selection was increasing or decreasing, resulting in almost a symmetric pattern (Figure 7). The same conclusions can be drawn for QST, whereas the changes of selection regimes have only minor impact on GSTq (Figure 8). The same simulations were conducted under moderate divergent selection during phase B (scenario 11 and 12) and by changing the number of loci to 5 and 30, but the dynamics of GSTq and QST following the changing of selection regimes were very similar to those observed in Figure 8 (data not shown).

## Discussion

### Evidence for the decoupling of differentiation between traits and genes

Our results support earlier findings suggesting that substantial decoupling may occur between QST and GSTq (Latta, 1998, 2004; McKay and Latta, 2002). We explored in greater detail the evolutionary factors enhancing decoupling and showed that strong divergent selection and strong selection intensity within populations were the predominant drivers of the lag between QST and GSTq. Furthermore, this discrepancy increases as the number of loci involved in the trait increases. The conclusions drawn from theoretical analyses can be compared with experimental estimates of QST and GSTq. The genes involved in quantitative traits have not yet been fully identified, but candidate genes have been identified for various traits of adaptive significance, not only in model species. A few recent studies have compared QST with GSTq estimates for candidate genes with GST for neutral markers. The traits of interest investigated were resistance to drought stress in Pinus pinaster (Eveno et al., 2008), bud set in P. sylvestris (Pyhäjärvi et al., 2008) and Picea abies (Heuertz et al., 2006), phenological traits (bud burst, length of growing season and leaf abscission) and growth traits (height growth and diameter) in Populus tremula (Hall et al., 2007 and Luquez et al., 2007), bud burst in Quercus petraea (Derory et al., 2010) and growth, phenology and wood characters in Picea glauca (Namroud et al., 2008). In each case, the level of differentiation of the candidate genes was similar to that of neutral markers, whereas QST values were much higher. The selection of candidate genes may not have been stringent enough to identify the only genes causally involved in expression of the trait of interest, but the concordant results obtained in the various studies carried out support our theoretical predictions that differentiation at individual genes may be limited, even for traits displaying high levels of differentiation in common garden experiments. We also suspect that this may be the case in general for trees and other species with large populations, high levels of gene flow and strong divergent selection.

### Transient dynamics of population differentiation

Over and above the simple decoupling of QST and GSTq, our results highlight the striking differences in the dynamics of these two differentiation measurements, once selection has been established. The differences observed in the variation of θW and θB, and of QST and GSTq over successive generations (illustrated in Figures 3, 4, 5, 6, 7 and 8) highlight the differences in trait and gene divergence as a result of selection. The dynamics can clearly be separated into two phases. Very early in the first phase, θW and θB follow a bell-shaped curve, reaching a maximum for θB (or a minimum for θW) after <20 generations. The increase in θB is followed by an increase in between-population variance (VB), which peaks at 10–30 generations when local stabilizing selection is strong, and at 50–80 generations when local stabilizing selection is weak. During this period of increase, QST increases very rapidly, whereas GSTq displays a much slower monotone increase. The second phase starts after VB has reached its maximum value, and is characterized by a steady increase in GSTq. During this second phase, QST increases more slowly, driven mostly by GSTq.

Slight variations of these general patterns were observed for the different selection scenarios, but the overall trends remained the same. In more biological terms, these patterns suggest two important conclusions: (1) that divergent selection first captures beneficial allelic associations at different loci in different populations and then targets changes in allelic frequencies and (2) that allelic associations promote rapid genetic divergence between populations more efficiently than changes in allelic frequencies.

In terms of evolution, this has the immediate consequence of very rapid differentiation at the trait level, once selection is established. The rapid response of QST results from the building up of covariances between allelic effects within and between populations. θW decays rapidly because of recombination, whereas θB becomes the main driver of QST during the first phase. In their review, Reznick and Ghalambor (2001) identified two principal contexts in which rapid evolution has been reported: new environments recently colonized by new populations and metapopulation structures in heterogeneous environments. In both scenarios, there are recent and new selection pressures, resulting in evolution during these early stages. These conditions match our scenario, in which populations rapidly differentiate. Most authors have tended to conclude that rapid evolution results from drastic demographic changes associated with the changes to the environment. However, our results suggest that there may also be a rapid accumulation of beneficial allelic associations. Several examples supporting rapid differentiation of traits have been reported in European tree species. Norway spruce displays the most recent wave of migration in Europe, spreading throughout southern Scandinavia over the last 2000 years by natural means or through human-mediated dispersion (Bradshaw and Lindbladh, 2005). Common garden experiments established with populations originating from these regions have shown extensive population differentiation for all traits assessed (Hannerz and Westin, 2000; Danusevicius and Gabrilavicius, 2001). Over more recent time scales, the introduction of northern red oak (Quercus rubra) into Europe over the last two centuries has been followed by the rapid divergence of the source population from the natural distribution and the introduced populations (Daubree and Kremer, 1993).

Our results have peculiar relevance in the context of climate change. Although we investigated responses to sudden changes of divergent selection in our simulations, we showed that allelic associations were immediately modified. Whether the transition was from moderate to strong divergent selection or from uniform to strong, the response elapsed during less than 10–20 generations. The lag of the response of θB was slightly lower under the former than the latter transition. We suspect that under more gradual changes of environmental conditions, allelic associations would be changed during a shorter period. Evolutionary response to fill the lag between new optimal values induced by climate change and extant values of populations has triggered theoretical research on adaptation to climate change (Lande and Shannon, 1996; Burger and Krall, 2004). Our results suggest that the multilocus structure of complex traits and intergenic disequilibria should be considered in predictive models of adaptive divergence.

### Detection of selection signatures at the molecular level

We showed that the decoupling of QST and GSTq is generated by the discrepancy between allelic covariances both within and between populations. At the molecular level, the signature of selection is therefore more visible on multilocus than on monolocus structures of diversity. At worst, during the early phase of selection, there may be no visible trace of selection at single loci, whereas strong correlations between allelic effects are expected. However, these conclusions must be considered with caution, as our analysis at the monolocus level is limited to the mean differentiation across all loci involved in the trait. We previously showed that GSTq values may differ between loci and that heterogeneity increases with gene flow and low stabilizing selection (Le Corre and Kremer, 2003). Under such circumstances, the detection of outlier loci by genome scans would be successful, but only a small subset of genes would be identified. However, selection imprints can be detected at the molecular level by another method, based on the correlation of allelic effects between populations (θB). It is difficult to estimate allelic effects, but the correlation of allelic frequencies may be used as a surrogate for θB. Correlations between allelic frequencies can be estimated from gene diversity surveys in natural populations, such as GSTq. In this respect, an appropriate initial approach would be to estimate the correlation between outlier loci detected on genome scans. Alternatively, multilocus measurements of population differentiation taking into account disequilibrium between alleles could also be used to detect multilocus structures (Kremer et al., 1997). However, these approaches may be tainted by confounding effects, generating covariances of allelic frequencies, such as demographic or historical effects. One highly predictable confounding effect may result from isolation by distance, and will affect all loci including those involved in the trait of interest. Divergent selection may follow continuous geographic gradients, resulting in an increase in VarZOPT along the gradient. Covariances of allelic effects may therefore be generated independently by two different mechanisms. The second constraint on explorations of θB or its surrogates is the very high degree of variation observed between repeats of the same scenario, suggesting the existence of a very high associated stochastic evolutionary variance (Figure 1).

### Concluding remarks

In contrast to our earlier comparison between QST and GSTq this review is purposely limited to outcrossing species that exhibit large population sizes and extensive gene flow such as forest trees. Indeed, the built up of substantial population differentiation for fitness-related traits that is observed in provenance tests (common garden experiments) in the context of large pollen flow has not been elucidated so far. Here, we show that the pace at which allelic associations are generated can be responsible for the level of divergence observed in extant populations. We suspect that the building up of allelic associations is enhanced by the very large genetic diversity residing within populations, which is sustained by pollen flow. These results need, however, to be confirmed or refined in a more general context beyond the single trait approach and by considering explicitly the contribution of seed versus pollen to gene flow.

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## Acknowledgements

This research was supported by the European Commission through the directorate General Research within the fifth and sixth framework programmes (Research project TREEESNIPS (QLK3-CT-2002-01973) and Network of Excellence EVOLTREE (FP6 No. 016322)). We are grateful to two anonymous reviewers for their very helpful comments and suggestions.

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Correspondence to A Kremer.

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Kremer, A., Le Corre, V. Decoupling of differentiation between traits and their underlying genes in response to divergent selection. Heredity 108, 375–385 (2012). https://doi.org/10.1038/hdy.2011.81

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• DOI: https://doi.org/10.1038/hdy.2011.81

### Keywords

• differentiation
• Q ST
• G ST
• intergenic disequilibria

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