Genetically correlated traits are known to respond to indirect selection pressures caused by directional selection on other traits. It is however unclear how local adaptation in populations diverging along some phenotypic traits but not others is affected by the joint action of gene flow and genetic correlations among traits. This simulation study shows that although gene flow is a potent constraining mechanism of population adaptive divergence, it may induce phenotypic divergence in traits under homogeneous selection among habitats if they are genetically correlated with traits under divergent selection. This correlated phenotypic divergence is a nonmonotonous function of migration and increases with mutational correlation among traits. It also increases with the number of divergently selected traits provided their genetic autonomy relative to the uniformly selected trait is reduced by specific patterns of genetic covariances: populations with lower effective trait dimensionality are more likely to generate very large correlated divergence. The correlated divergence is likely to be picked up by QSTFST analysis of population genetic differentiation and be erroneously ascribed to adaptive divergence under divergent selection. This study emphasizes the necessity to understand the interaction between selection and the genetic basis of adaptation in a multivariate rather than univariate context.

Natural populations inhabiting heterogeneous environments often show local adaptation at even fine geographic scales. Although divergent natural selection among habitats is expected to be the driving force (e.g., Schluter 2000), the process of adaptation to local conditions is also expected to be genetically constrained by intrinsic, developmental, or functional factors (Arnold 1992; Blows and Hoffmann 2005) or extrinsic factors such as gene flow (Lenormand 2002; Kawecki and Ebert 2004). Local adaptation in geographically isolated populations has commonly been considered as a univariate process resulting from a tension between gene flow and natural selection in a single phenotypic trait (e.g., Bulmer 1972; Phillips 1996; Lythgoe 1997; Tufto 2000; Lopez et al. 2008; Yeaman and Guillaume 2009). Genetic constraints arising from developmental or functional relationships among traits must however be considered as a multivariate process where adaptation proceeds within a multitrait space and the evolution of single traits results from direct and indirect (correlated) selection pressures (e.g., Lande 1979; Arnold et al. 2001; Blows 2007; Blows and Walsh 2009). The interaction between gene flow and the genetic basis of local adaptation should thus also be modeled in a multivariate context to better understand how species adapt in multitrait space in the presence of gene flow. It is not clear yet how univariate predictions from migration–selection balance theory are affected by multivariate selection and constraints.

In multivariate trait space, intrinsic constraints are caused by the presence of genetic correlations among phenotypic traits. An adaptive constraint, over short term evolution, is manifested by a discrepancy between the realized and predicted response to selection of a trait based on its heritability and the strength of selection acting on it alone, ignoring correlations with other traits (Agrawal and Stinchcombe 2009). The genetic determinants of trait correlations are the pleiotropic effects of genes coding for a common set of traits and linkage disequilibria among those genes. Genetic correlations, or covariances among traits cause a reduction of a trait's capacity to respond to direct selection independently from other traits; a part of its genetic variation is conditioned on that of the other traits and a set of genetically integrated traits then has a tendency to collectively respond to selection. Consequently, the response to selection of a population cannot be predicted on the basis of the genetic variance of individual traits alone. Instead, it is given by the sum of the direct and indirect (correlated) responses to selection for each trait (Lande 1979). Although the presence of genetic covariances/correlations among traits is usually interpreted as a sign of potentially strong constraints on adaptation, their effect depend on the alignment between the direction selection takes in trait space and the multidimensional pattern of genetic variation of the traits (as represented by the eigenvectors of the G-matrix; see Arnold et al. 2001; Walsh and Blows 2009; Agrawal and Stinchcombe 2009). In other terms, it depends on the concordance between ecological and genetical dimensions. In a case of mismatch, the adaptive path of a population under direct selection will depart from its optimal path given by the selection gradients acting on the traits (i.e., direction of maximum population fitness increase; Lande 1979). Therefore, the phenotypic divergence we might observe between populations in different habitats may not always parallel the actual habitat ecological divergence. Some traits under homogeneous selection may still diverge, entrained by selection on correlated traits, which results in populations potentially more maladapted than in the absence of correlations.

How do gene flow and genetic correlations jointly affect local adaptation? Gene flow causes the mean phenotypic value of a population to be biased toward the mean phenotype of immigrant individuals from divergent habitats. A response to selection of the population will result from this departure from optimality and may induce a correlated response to selection in traits under homogeneous selection between habitats whenever they co-vary genetically with the divergent trait(s). The interaction of gene flow and genetic correlations may thus cause the population mean phenotype at equilibrium to be diverted away from the line of phenotypic divergence connecting the population local optima along which populations diverge (see Fig. 1). With constant gene flow, this bias and the selection gradient that maintains it will persist at migration–selection balance and modify the level of local adaptation of ecologically diverging populations.

Figure 1.

A schematic representation of the emergence of a correlated phenotypic divergence (dcorr) when two quantitative traits (T1 and T2) are genetically correlated. The two population phenotypic optima (θa and θb) differ for the optimal value of T1 but not T2 (x-axis). The mixing of individuals through migration first leads to a change in the population mean phenotype on T1, shown as zm. The subsequent change due to selection after mixing is denoted zms. This response to selection vector is biased toward gmax, the first eigenvector (or PC) of the G-matrix thus leading to a correlated response to selection on T2 and to the emergence of phenotypic divergence on that dimension. The G-matrix is represented by an ellipse, with axes corresponding to the first and second eigenvectors whose size corresponds to their respective eigenvalues.

It is hard to know from current empirical and theoretical studies to what extent the interaction of gene flow and genetic correlations affect local adaptation. Some insight may be gained from quantitative genetics modeling approaches (e.g., Hendry et al. 2001), which will be presented below. On the empirical side, there is unfortunately a marked disconnect between studies of the constraining effects of genetic correlations on phenotypic divergence and studies of local adaptation in the presence of gene flow such that the importance of the interaction of both factors in natural systems remains unknown. The first type of studies asks whether genetic constraints are strong enough to bias the population response to selection away from the selection gradient. The intent is to test whether diversification among populations within a species or among closely related species is genetically constrained. There is evidence that some species show constraining effects of genetic correlations on diversification (e.g., Mitchell-Olds 1996; Schluter 1996; Bégin and Roff 2003; Chenoweth et al. 2010) although this is not always the case (e.g., McGuigan et al. 2005; Eroukhmanoff and Svensson 2008; Berner et al. 2010). These studies however do not directly ask how the response to selection (the observed phenotypic divergence) aligns with the selection gradients (the ecological divergence). Instead, they ask how the phenotypic divergence aligns with the directions in phenotypic space that bear the highest proportion of genetic variance (i.e., the “genetic lines of least resistance”, see e.g., Schluter 1996). Yet, the alignment of the phenotypic and genetic lines does not contribute evidence for the direct action of genetic constraints as it can be confounded by the alternative mechanisms of correlational selection (Arnold 1992; Schluter 1996) and migration (Guillaume and Whitlock 2007). To date, these alternatives have been poorly investigated in empirical or natural systems (but see Berner et al. 2008, for the effects of selection on phenotypic covariances).

Studies of local adaptation in the presence of gene flow usually focus on the two alternative questions of how does gene flow constrain adaptation, and how does adaptation constrain gene flow between divergent habitats? Although multiple ecologically relevant traits are often used in these studies, their underlying genetic architecture, and in particular their genetic correlations, is seldom considered (but see King and Lawson 1995; Hendry et al. 2001). The traits studied are usually picked because they are thought to be under divergent selection and would thus not help us understand whether phenotypic divergence could be a byproduct of genetic covariances inducing a correlated response to divergent selection. Evidence for the adaptive origin of phenotypic divergence is shown by finding ecological and functional correlates to the observed divergence (e.g., host–plant use and predation in Timema cristinae, Nosil and Crespi (2004, 2006); prey size and water regime in Gasterosteus aculeatus, Moore et al. (2007); Berner et al. (2008); or seed hardness and abundance in Geospiza sp.; Schluter and Grant (1984); Grant and Grant (2002)) thus implying that populations diverge in more than a single dimension of the species' ecological niche. Finding phenotypic divergence not linked to ecological divergence may hint at the presence of genetic constraints. Unfortunately, such correlated divergence will obviously also correlate with the ecological factors causing the divergence in possibly unobserved traits such that disentangling both direct and indirect divergence would be practically impossible when using correlational approaches. The lack of evidence present in the literature is thus not that surprising.

Another type of approach uses a mix of molecular and quantitative genetics methods for the parallel measurement of population genetic differentiation at neutral markers (FST) and phenotypic traits (QST) leading to the classical QSTFST comparative approach (Spitze 1993; Merila and Crnokrak 2001; Whitlock 2008). This approach aims at finding traits that are likely under heterogeneous selection among populations by asking whether their index of (additive) genetic differentiation (QST) is significantly greater than what would be expected under neutrality as measured by FST. Traits with significantly greater values of QST than FST can be suspected to be under geographical diversification although neutral traits correlated to traits under divergent selection are also expected to have a high QST (Whitlock 2008). The extent of the effect of genetic correlations is however not known in this case and whether the interplay between gene flow, divergent stabilizing selection, and genetic constraints would lead to detectable patterns of correlated phenotypic divergence remains to be demonstrated.

The aim of this article is to understand the effects of the interplay between gene flow and genetic correlations on local adaptation at mutation-selection balance. Theoretical predictions of the amount of multivariate phenotypic divergence at equilibrium between two populations exchanging migrants are presented using an existing quantitative genetics model (Hendry et al. 2001). The predictions are then tested with individual-based computer simulations. I specifically seek to find the genetic conditions that maximize the amount of correlated phenotypic divergence of traits experiencing homogeneous selection among diverging habitats. Under those conditions, detection of large nonadaptive correlated phenotypic divergence is then possible using, for instance a QSTFST approach.



Figure 1 shows a cartoon of the evolutionary scenario modeled. It considers two populations entering in secondary contact after having evolved in allopatry toward their respective local optimum such that the phenotypic traits are strictly under stabilizing selection and do not show any correlated response to selection. For illustration, the two populations first diverge along one phenotypic dimension (trait 1 or T1) but not along a second one (trait 2 or T2) which experiences stabilizing selection for the same optimum value in both populations. Trait 1 will be said under heterogeneous or divergent selection whereas T2 is under homogeneous or uniform selection among habitats. Because of the presence of a genetic correlation between T1 and T2, the population response to selection after migration is diverted away from its expected path along the line of phenotypic divergence between populations. This in turn generates a correlated phenotypic divergence along T2 (dcorr in Fig. 1).

Quantitative predictions of equilibrium phenotypic divergence among populations in multivariate trait space can be made by using a model developed by Hendry et al. (2001). This model is based on classical quantitative genetic theory and is similar in essence to other models based on the same assumptions of Gaussian distribution of the phenotypes (see Tufto 2000; Tufto 2010; Lopez et al. 2008). Such models have sometimes been labeled as Gaussian Approximation Models (or GAM hereafter). Under these assumptions, Hendry et al. (2001) have shown that the phenotypic divergence between two populations in a set of quantitative traits at migration–selection balance while exchanging migrants at rate m, is a function of the additive genetic covariance matrix G and phenotypic covariance matrix P of the traits and of the selection surface matrix inline image and is expressed as


(see eq. 7 in Hendry et al., 2001) where D* is the equilibrium phenotypic divergence vector and Dθ is a vector of divergence in phenotypic optima among populations. Under this model, a trait submitted to heterogeneous selection diverges less when its genetic correlation with another trait under homogeneous stabilizing selection increases; its level of maladaptation is increased (Fig. 2A). This constraint can be substantial at intermediate migration and strong genetic correlation. The correlated response to selection of the trait under homogenous selection is a nonmonotonic function of gene flow (Fig. 2B). At low migration, it increases with gene flow as the selection builds up with maladaptation on trait 1 (selection is a function of the departure from the optimum). At some intermediate value of m, the correlated response starts decreasing as gene flow becomes strong enough to prevent the genetic differentiation of the two populations (Fig. 2B). Greater strength of stabilizing selection decreases the effects of gene flow on trait 1 and shifts the peak of the correlated response toward higher migration rates.

Figure 2.

Predicted phenotypic divergences from the GAM (eq. 1) (Hendry et al. 2001) for, (A) the trait under divergent selection between two populations (T1), and (B) the trait experiencing homogeneous selection between populations (T2). Predictions are shown for two values of selection ω2= 5 (solid lines) and 50 (dashed lines) (ω21122), and three values of the genetic covariance G12: 0 (thick lines in panel A), 0.5 (intermediate lines in panel A and lower lines in panel B), and 0.9 (lower lines in panel A and upper lines in panel B). Other parameters of the GAM are Dθ={5, 0}, G11=G22= 1, and P=G.

We can also gain some general insights into the response to selection of correlated traits over a generation by inspecting the multivariate breeder's equation (Lande 1979)


where the population mean phenotype vector inline image is a function of the genetic covariance matrix G and the selection gradients vector β. The G-matrix describes the additive genetic variance of each trait along its diagonal and the additive genetic covariances between each pair of traits off the diagonal. The selection gradients correspond to the change in mean fitness due to a small change in average trait values and represent the direction of maximum increase of mean population fitness on the adaptive landscape. From (2) we can see that the correlated response to selection of a trait under stabilizing selection (i.e., β2= 0) but correlated to a trait under direct selection is proportional to the genetic covariance between both traits:


with β1 the selection gradient acting on T1, for instance. This selection gradient is proportional to the phenotypic distance of the population mean to its local optimum and to the strength of selection (Via and Lande 1985; Hendry et al. 2001). It however changes as the second trait evolves due to the indirect selection pressure represented by (3). Likewise, as soon as T2 responds to a change in T1 and departs from its optimum value, a selection gradient on T2, β2, comes into existence and works to bring T2 back to its local optimum value θ2. This, in turn causes a correlated response to selection on T1.

A major limitation of the quantitative genetics approach behind the GAM comes from the assumption of constant Gaussian genetic and phenotypic variances. This comes from the fact that it only predicts the population change in mean phenotype over a single generation for a “given” amount of genetic covariances. The interaction of selection and migration (as well as mutation, drift, and recombination) on variances is not taken explicitly into account. Additionally, departure from (multi-) normality of the phenotypic values caused by the evolutionary processes is ignored (see e.g., Yeaman and Guillaume 2009, for effects on response to selection of nonnormality (skew) generated by gene flow on quantitative traits). This framework can thus not be used to predict the full dynamics of the selection–migration–mutation–drift balance of the system. To overcome this limitation, authors have turned to numerical methods (e.g., Turelli and Barton 1994; Tufto 2000, 2010, under Gaussian assumptions) or individual-based simulations (e.g., Lopez et al. 2008; Yeaman and Guillaume 2009, relaxing Gaussian assumptions). I will here use an extension of the individual-based simulation model developed by Yeaman and Guillaume (2009).


The simulation model is implemented in the individual-based simulation platform Nemo (Guillaume and Rougemont 2006). Individuals can have an arbitrary number of phenotypic traits with set pairwise genetic correlations. The traits have a common genetic basis of n= 100 fully pleiotropic, diploid loci. An allele i (i∈[1, 2]) at locus jj∈[1, n]) has effect aijk on each of the K traits modeled. The genetic make-up of the traits is purely additive; the phenotypic value of a trait is obtained by adding up the allelic effects for that trait at all loci: inline image. The mutation model follows a continuum-of-alleles model where each new mutational effect is added to the existing allelic value at the mutated locus and mutations occur at rate 2μ per locus. The mutation effect size vector is randomly drawn from a multivariate normal distribution with covariance matrix M. The variance, or diagonal terms of M, α2kk, are the effect size of the alleles for a given trait k. The genetic correlation among the trait-specific effects of a given mutation is set by the covariance (off-diagonal) terms of M. The genetic correlation is denoted rμ and may vary among traits. This term corresponds to the input level of genetic correlation due to mutations into the population and may be different from the genetic correlations measured after migration and selection (i.e., in the G-matrix).

Individuals are spread over two populations of same size (N= 1000) inhabiting two different environments with trait optimum vectors Θ1 and Θ2 for populations 1 and 2. The simulations always considered the second trait (or T2) as the trait under homogeneous selection between habitats and habitat divergence on the other trait(s). Migration between populations occurs at rate m. New offspring are created by drawing the parental gametes from either population with probability depending on m (inline image; inline image). Individuals born in a population are first subject to viability selection depending on their traits' value and the local optimum. Selection is modeled as a multivariate Gaussian surface around the population optimum with covariance matrix Ω. The survival probability of a newborn with phenotype vector z is given by its fitness value in population i: inline image. Surviving individuals are then part of the pool of breeding individuals for the next generation. Offspring are produced until each population is filled to its carrying capacity N, thus leading to a regime of soft selection in the system. Population fitness is measured during viability selection.

Simulation settings

The simulations are first run with stabilizing selection on all the traits, without migration to mimic allopatric divergence of the populations, and for 120,000 generation to ensure stability of the genetic variances and covariances within populations at selection-mutation-drift-recombination balance. Migration is then added using this set of simulations as source populations and evolution of the phenotypic divergence is monitored over 120,000 generations. The number of traits under heterogeneous selection is 1, 2, and 4 and the habitat divergence for each trait is Δ= 5 phenotypic units. A minimum fitness threshold is set to 0.1 to avoid situations where hybrid individuals have a near-null fitness caused by large phenotypic divergence and/or strong stabilizing selection (see Table 1). This mimics an ecological scenario where immigrants and hybrid individuals still have the possibility to survive in a different environment although the mathematical description of the fitness landscape would not allow them to do so. The threshold chosen here is close to the minimum estimate for immigrant survival of 8.7% reported by Nosil et al. (2005) in a review of fitness of locally adapted individuals when transplanted to a foreign environment (estimates range: 0.087–0.99). The genetic settings ensure that the amount of mutational variance entering the population is Vm= 2nμα2= 0.001 with n= 100, μ= 10−4, and α22kk= 0.05, for all k. The selection settings are such that the Ω matrix is diagonal, with no correlational selection. The strength of stabilizing selection is given by ω2, the variance terms in Ω, initially equivalent for all traits. Selection varies from ω2= 2.5 to 50, that is from strong to relatively weak stabilizing selection (i.e., ω2 is inversely proportional to the strength of selection). Each set of parameter values is replicated 50 times (or 100 times for some cases).

Table 1.  Fitness of hybrid individuals (and migrants in parentheses) as a function of habitat divergence along trait T1 (Δ) and the strength of selection in both habitats (ω2). The minimum value of 10% survival is set whenever the actual individual fitness is lower than this threshold, as indicated with the asterisks below.
2.50.29 (0.1*)0.1* (0.1*)0.1* (0.1*)
5 0.54 (0.1*) 0.25 (0.1*) 0.1* (0.1*)
100.73 (0.29)0.49 (0.1*)0.29 (0.1*)
25 0.88 (0.61) 0.76 (0.32) 0.61 (0.14)
500.94 (0.78)0.87 (0.57)0.78 (0.37)


The total correlated response to selection of trait T2 depends additively on the number of traits under heterogeneous selection it is correlated to is given by the sum of each correlated response to selection on other traits under divergent selection:


This sum may increase in absolute value if genetic covariances and selection gradients have same sign. The first case modeled considers two correlated traits, T1 and T2, for which the mutational correlation is set to rμ= 0, 0.5, and 0.9. This case is mostly used as a baseline for comparison with theoretical predictions and with trait divergence in higher dimensions. To further illustrate the additive effect of indirect selection on the population divergence of T2, two cases are simulated: one with all traits equally correlated to each other (with the same rμ values as above), and another with genetic correlations set randomly. The M-matrix in the first case has all its off-diagonal elements equal, whereas in the second case, random M-matrices were generated by drawing from a Wishart distribution (Wishart 1928; Anderson 2003). Using the Wishart distribution insures that the random matrices are positive (a requisite for the covariance matrix of a multivariate normal distribution) and allows us to generate matrices with reduced rank. The rank of a covariance matrix is its number of eigenvalues that are different from zero. A zero eigenvalue and its corresponding eigenvector thus represent an “absolute” genetic constraint; a linear combination of the traits that has no genetic variance along it and where the response to selection is null. A further constraint was imposed by selecting those matrices that have their first eigenvector more correlated to T2. The criterion used is a loading of T2 on the first eigenvector greater than 0.71, which translates into an angle of less than 45 between T2 and the first eigenvector of M. This controls for variation in the orientation in the phenotypic space of the distribution of the major part of the genetic variance. Finally, to keep all traits on the same scale and be able to compare results with two-trait matrices, the random matrices were scaled to have constant genetic variances of α2= 0.05 prior to applying the previous criterion. Overall, the total habitat divergence increases with the number of traits under divergent selection, from Δtot= 5 for one, to Δtot= 7.07 for two, and Δtot= 10 for four traits under divergent selection for the same habitat divergence per trait (inline image with Δk= 5). Divergence on T1 for the bivariate case also varies from Δ= 5 to 10. Note here, that the strength of stabilizing selection is not a function of the number of traits modeled as the individual fitness is only a function of the distance to the optimum and of the curvature of the fitness surface, which is kept isotropic (i.e., all diagonal elements of Ω are zero).


G-matrix angle and projection

The correlated divergence should depend on the orientation of the G-matrix relative to the line of phenotypic divergence between habitats (hereafter named LoD) which is measured by the angle θ between gmax and the LoD (see Fig. 1). gmax is the first eigenvector of the G-matrix (or first principal component) and represents the direction in phenotypic space that has the highest amount of genetic variance and thus the greatest response to selection (see e.g., Schluter 1996). The angle θ is computed from the dot product of the two vectors


where ∥L∥ is the norm of the LoD vector and the absolute value of the cosine of the angle is used.

The genetic constraints acting on the system can also be measured by the amount of genetic variance available to the population to respond to selection along the LoD. This is done by projecting the G-matrix onto the LoD, which is given by


where′ means transpose. The projection is further normalized by the total amount of genetic variance contained in G, given by the sum of its eigenvalues.


A similar way of measuring the constraints acting on population divergence is to measure the autonomy of the traits under divergent selection (as defined by Hansen and Houle 2008). The autonomy is the fraction of “unconditional” genetic variation available for a trait to evolve independently of correlated traits under stabilizing selection and relates to traits under direct selection. The autonomy is defined by Hansen and Houle (2008) as


where y is the trait under direct selection, x is a set of traits under stabilizing selection, and R2 is a the multiple correlation coefficient of y with x (see Hansen and Houle 2008, p. 1205). Because, in the multidimensional case, we have one trait under stabilizing selection and many traits under divergence selection, the autonomy of an index trait is computed relative to trait T2. That index trait is the total phenotypic divergence in the subspace of the traits under divergent selection and is computed as the norm of the vector composed of all traits under divergent selection


with Tk the additive genetic value of trait k. The autonomy of TI then reduces to 1 minus the square of the genetic correlation between TI and T2: a(TI|T2) = 1 −ρ(TI, T2)2.



The amount of phenotypic divergence of trait T2 (the correlated divergence) is a nonmonotonic function of migration; as expected it increases with migration up to a point where selection is overwhelmed by gene flow and population divergence collapses (Figs. 3 and 4). Correlated divergence is increased as the strength of selection decreases and is maximum at ω2= 50 and m= 0.05 or 0.1 for most cases explored here. Increased genetic correlation of T2 with the other trait(s) inflates the amount of correlated divergence (Figs. 3 and 4). Specifically, habitat divergence in multiple dimensions leads to quite large divergences in T2, as expected from equation (4), especially for the cases with M-matrices of reduced rank. For these cases of constrained matrices, the correlated divergence can reach 4 phenotypic units (Fig. 5) whereas it remains below 1 unit for the unconstrained cases, even with rμ= 0.9 and more than one correlated trait under divergent selection (Fig. 4). There is however a large variation of the amount of correlated divergence for the constrained cases depending on the amount of genetic covariances of the traits under divergent selection with trait T2: the sum of covariances strongly correlates with the amount of correlated divergence (Fig. 5), as expected from (4). These covariances are those observed in the G-matrix after selection and migration. Matrices of reduced rank have higher median covariance than full-rank matrices and are thus more likely to generate very large correlated divergences (Fig. 5).

Figure 3.

Correlated phenotypic divergence of trait T2 caused by genetic correlations between traits T1 and T2 as a function of the migration rate and the strength of stabilizing selection on both traits (ω2). Results are shown for three values of the mutational correlation rμ: 0 (filled circles), 0.5 (triangles), and 0.9 (plus). Gray lines without points are the expected divergence values obtained from the GAM using the realized genetic variances (realized G-matrices measured from the distribution of individual breeding values at each given generation). Smaller values of ω2 mean stronger stabilizing selection. Habitat divergence is Δ= 5 on T1. Results are averages over 100 replicates.

Figure 4.

Correlated phenotypic divergence of trait T2 in the multidimensional habitat divergence model. The total number of quantitative traits simulated are three (left column) and five (right column). Graphs in (A) and (B) show results for matrices with all genetic covariances equal among traits. Graphs in (C) and (D) show results for random M-matrices of varying rank (see text for explanations). In (A) and (B), gray lines show results obtained for populations with two traits and habitat divergence of Δ= 5 (dotted lines) and Δ= 10 (dashed lines). Two mutational correlations were used; rμ= 0.5 (triangles) and rμ= 0.9 (crosses). In (C) and (D), vertical bars represent two times SE obtained for 50 different random matrices while error bars are not displayed in (A) and (B) as they would be covered by the symbols. Symbols for the different matrix ranks or mutational correlations are displayed on the graphs. Note the change of scale on the y-axis between the first and second rows. Parameter values are: ω2= 50 and Δ= 5 for all divergent traits.

Figure 5.

Correlated phenotypic divergence in populations with five quantitative traits and M-matrices of full rank in (A), rank 4 in (B), and rank 3 in (C). Results are shown for 50 random M-matrices and are based on the observed G-matrices recorded at the last generation of each simulation. The first column shows the sum of the observed covariances of T2 with the other traits, the second column shows the angle (in degrees) between gmax and the line of habitat divergence (LoD), the third column shows the amount of genetic variation available to evolve along the LoD given by the projection of the G-matrix onto the LoD, and the last column shows the genetic autonomy of the traits under divergent selection relative to T2 (see Methods for definitions). The strength of stabilizing selection is ω2= 50 and the migration rate is m= 0.01.

As suggested by our cartoon figure (Fig. 1) and previous findings (e.g., Schluter 1996), the correlated divergence should depend on the angle between gmax and the line of phenotypic divergence between habitats (the LoD). Our results confirm that prediction: the correlated divergence strongly correlates with the angle (see Fig. 5 for the five traits case). Similarly, the projection of G onto the LoD gives a measure of the amount of genetic variation available to the population to respond to selection along that phenotypic dimension. This projection correlates negatively with the correlated divergence (see Fig. 5). As expected, the angle and the projection of G and the LoD are also negatively correlated. Finally, the genetic autonomy of the divergently selected traits (see Methods) also correlates negatively with the correlated divergence (Fig. 5).

Altogether, these measures of genetic constraints show that when traits are strongly genetically associated with T2, as measured by the sum of their covariances with T2, most available genetic variation is diverted away from the axis of habitat phenotypic divergence, as measured by the relationship between G and the LoD. This results in a strongly constrained total population divergence, as reflected by the low autonomy of the index trait, and a large response on the correlated trait under uniform selection. Exceptions arise when random matrices of rank 3 have low sum of covariances on T2 in addition to a specific distribution of the genetic variance that leads to almost no genetic variation on the LoD and an orientation of gmax almost orthogonal to this axis (angle ≥60°, see Fig. 5C). In these cases, both the correlated divergence and the total divergence among population is small.


The level of maladaptation caused by migration and genetic correlations is measured by the decrease of mean population fitness compared to that of populations at equilibrium without migration or correlation among traits. This leads to two measures: the migration and correlational loads. The migration load is the decrease of mean population fitness caused by migration compared to that of a population without migration, and is measured as the difference in mean fitness of the two populations. It is less than 20% on average for even very strong stabilizing selection (ω2= 2.5) and large migration (m= 0.1) in the bivariate case (habitat divergence in a single trait) but can reach up to 60% or more for the multivariate case with strong genetic constraints (i.e., correlated divergence above 2 phenotypic units for habitat divergence in four traits with matrices of rank 3, see Fig. S4) with an average load of about 40% (for same cases). The correlational load is given by the difference in migration load between populations with rμ= 0 and rμ≠ 0 for a given migration rate. This load is usually not large for the bivariate case. An additional fitness decrease of about 0.5% on average is caused when increasing rμ from 0 to 0.9 at strong stabilizing selection (ω2= 2.5) and strong migration (m≥ 0.05). The correlational load for highly constrained systems can however be more than three times that of an unconstrained one, increasing the migration load from about 20% to above 60% for cases with four traits under divergent selection and matrices of rank 3 (Fig. S4, see also Fig. S3 for three traits cases).

The total migration load is not solely due to maladaptation of the trait under uniform selection but also to that of the traits under divergent selection. The total phenotypic divergence of the two populations indeed decreases as the genetic correlation increases among traits (Fig. 6). Total divergence is the phenotypic distance between the two population means. It is well predicted by the same measures of genetic constraint as for the correlated divergence above, although correlations are of opposite sign. The best measures are the angle between gmax and LoD, and the projection of G onto LoD which represents the amount of genetic variance available to the population to respond along that axis (Fig. 6).

Figure 6.

Total phenotypic divergence between the two population with five quantitative traits and M-matrices of full rank in (A), rank 4 in (B), and rank 3 in (C). The total divergence is the Euclidian distance between the two population means. Results are shown for 50 random M-matrices and are based on the observed G-matrices recorded at the last generation of each simulation. Parameters and metrics as for Figure 5.

Finally, another source of constraint on phenotypic divergence appears when selection on one trait is raised while keeping it constant at ω2= 50 on the other one, as shown for the bivariate case. This results in an overall decrease of the direct or the correlated divergences as selection becomes stronger on one trait (Fig. 7). This effect is driven by the concomitant decrease of genetic variance on T1 as selection on T2 increases, and vice versa, and is a consequence of the complete pleiotropy of the underlying loci, over and above the correlational effects of the alleles present at those loci.

Figure 7.

Correlated divergence (A) and divergence of T1 (B) in populations with two quantitative traits. In (A), the equilibrium divergence of T2 is a function of ω211, the strength of stabilizing selection on T1, for m= 0.1 and ω222= 50. Similarly in (B) for T1 as a function of ω222. Habitat divergence is Δ= 5 on T1 and mutational correlation is: rμ= 0 (filled circles), 0.5 (triangles), and 0.9 (plus).


There is usually a large mismatch between the GAM predictions (see eq. 1) and the simulation results, shown for the two-trait case (Figs. 3, S5, and S6). The GAM tends to overestimate the amount of correlated divergence at low migration rates, especially so when selection is strong (ω2≤ 10) and habitat divergence is large (Δ≥ 5). In particular, the position of the peak of maximum divergence is badly predicted by the GAM (see Figs. 3 and S6), which predicts a much greater effect of gene flow on adaptation than is observed (the same is true for the trait under divergent selection, see Fig. S5). Moreover, the GAM predicts no dependence of the migration rate at which the peak of maximum divergence appears on the habitat divergence although simulations show a clear dependence on Δ (Fig. S6).

A recent study by Yeaman and Guillaume (2009) showed that departure from normality of the distribution of breeding values of quantitative traits lead to a mismatch between the GAM predictions and simulation results at migration–selection balance based on genetic architectures different from a diallelic model. A similar analysis was performed (see Supporting information) and additional simulations under the diallelic mutation model match the GAM predictions almost perfectly, especially for the correlated divergence under weak stabilizing selection and large migration rates. In particular, the peak of maximum divergence is accurately predicted under theses assumptions (see Figs. S7 and S8). This analysis shows that departure from normality of the additive values of the traits, as measured by the genetic skew (see Yeaman and Guillaume 2009) explains well the discrepancy between the analytical predictions and the simulation results (Figs. S9 and S10).


The QST value of trait T2 can be very high, much higher than FST when m≥ 0.01 (Table 2). The median QST/FST ratio is much greater when the rank of M is three versus five and ranges from 10 to about 60 (see Table 2). As recently shown by Whitlock and Guillaume (2009), QST can be more easily shown to be greater than FST when the QST/FST ratio is large (>2) and the mean FST is small, although this obviously also depends on the quality of the estimates. The minimum amount of observed divergence necessary to get a QSTFST ratio of 2 or more is in fact quite small and reaches about 0.6, 0.2, and 0.07 phenotypic units for m= 0.01, 0.05, and 0.1, respectively (with ω2= 50, Δ= 5). Those threshold values would also be reached under stronger stabilizing selection or smaller habitat divergence, as well as with habitat divergence in a single trait when migration is high (m≥ 0.05) and genetic correlations are large (rμ≥ 0.5). These results suggest that correlated divergence can generate false positives when testing for selection using FSTQST comparisons.

Table 2.  Q STFST values in large populations of 10 demes of 500 individuals. The habitat divergence along each of the four traits under divergent selection is Δ= 5 and selection is ω2= 50 for all traits. Each second deme is assigned to a similar habitat. Individuals migrate following the island model with rate m given below. The FST is estimated on 50 unlinked, biallelic neutral loci with mutation rate μ= 10−5 using statistics from Weir and Cockerham (1984). The median of the correlated phenotypic divergence on T2 (dcor) and QST values, and the average FST are presented for M matrices of rank 5 and 3. Median values of the QSTFST ratio that are greater than 2 are bold faced. Values in parenthesis are the lower and upper quartiles associated with the medians and the SD of the FST values, obtained on 50 replicates.
  dcor Q ST F ST Q ST/FST
m rank 5
0.001 0.05 (0.03-0.09) 0.020 (0.019-0.022) 0.657 (0.020) 0.03 (0.029-0.035)
0.010.36 (0.17-0.78)0.056 (0.031-0.156)0.051 (0.001)1.13 (0.60-3.11)
0.05 0.53 (0.23-1.05) 0.078 (0.024-0.249) 0.010 (0.003) 7.8(2.48-24.7)
0.10.41 (0.26-0.90)0.057 (0.025-0.176)0.005 (0.002) 12.3(5.39-38.1)
  rank 3
0.0010.98 (0.62-1.52)0.10 (0.05-0.20)0.609 (0.022)0.16 (0.08-0.34)
0.01 2.03 (1.29-2.73) 0.48 (0.32-0.59) 0.048 (0.002) 10.1(6.79-12.3)
0.051.76 (0.83-2.41)0.36 (0.15-0.47)0.010 (0.001) 38.2(16.3-48.5)
0.1 1.55 (0.62-1.03) 0.26 (0.09-0.35) 0.005 (0.000) 58.8(19.9-77.3)


In this study, I show two important results. First, the interplay between gene flow and genetic constraints among phenotypic traits can generate substantial phenotypic divergence in traits under homogeneous selection among populations. Gene flow thus induces phenotypic divergence in those traits, contrary to its more general role of preventing such divergence. This divergence is however nonadaptive in the sense that it is not linked to habitat ecological divergence. Gene flow plays the driving role in the evolution of the correlated divergence as it is the mechanism that sets and insures persistence of the departure from optimality of the traits under heterogeneous selection. Persistent selection gradients on divergently selected traits force correlated traits under homogeneous selection to respond to this indirect selection and thus diverge as well. Second, the net effect of the genetic constraints rooted in mutational pleiotropy on the migration–selection balance is a decrease of the overall population phenotypic divergence and an increase of the migration load. I further show that maladaptation of the diverging populations increases with the dimensionality of the habitat divergence as ecological divergence in more dimensions is more likely to generate the conditions for large constraining effects on trait divergence. Pleiotropy imposes two types of evolutionary constraints. First, genetic correlations with possibly many traits under homogeneous selection limit adaptation of a trait under divergent selection as the strength of indirect selection increases with gene flow. Second, phenotypic divergence also decreases as the strength of selection on correlated traits increases (see Fig. 7) which limits the amount of genetic variation available for adaptation.

Gene flow, however, has a nonmonotonic effect on correlated divergence as past a certain value that depends on the strength of selection and genetic constraints, it counteracts selection enough to cancel population differentiation (see Figs. 2–4). Nonetheless, moderate-to-high migration rates, in the range 0.0001 ≤m≤ 0.1, in conjunction with strong genetic correlations (e.g., rμ≥ 0.5 for two traits), and mild-to-weak selection (ω2≥ 10) caused significant correlated divergence among populations, from below one phenotypic unit (or about one phenotypic standard deviation) for habitat divergence in two dimensions (Fig. 3) to two to four phenotypic units for habitat divergence in higher dimensions (Figs. 4 and 5). These migration rates are in the range that is most biologically relevant. They correspond to effective number of migrants from 0.1 to 100, which is in line with estimates of gene flow from molecular markers (see Morjan and Rieseberg 2004). Furthermore, empirical estimates of the strength of quadratic selection on phenotypic traits are also close to the values used in this study with a median estimate of the curvature of the selection surface of |γ| = 0.1 which roughly translates into ω2≈ 10 (Kingsolver et al. 2001) (Estes and Arnold 2007 more precisely estimate ω2= 11.3 ± 6.5 and a mean distance to the optimum of inline image from data in Kingsolver et al. (2001)). Although all these values are subject to large estimation errors, they concur with the fact that the ecological conditions required for the evolution of correlated phenotypic divergence among populations may commonly occur in nature.

Empirical data also suggest that the genetical conditions for the existence and expression of constraints on adaptation exist. For example, in a review of 1798 morphological, behavioral, and life-history trait combinations, Roff (1996) showed that the average absolute genetic correlation between traits was about 0.5 with a rather flat distribution in the range from 0 to 1 (although significantly not uniform). High genetic correlations are thus no exception, especially among morphological traits. Further evidence for the existence of genetic constraints come from studies of G-matrices of wild or domesticated species (reviewed by Schluter 2000; Kirkpatrick 2009). Most show that the distribution of genetic variance in multivariate phenotypic space is highly uneven; most of the variance is usually concentrated on a very small subset of phenotypic dimensions, usually below 1/3 of the quantitative traits measured, down to almost a single “effective” dimension (see Kirkpatrick 2009). The generality of these findings is not settled yet but they show that natural systems are likely quite constrained with one combination of traits (i.e., the first eigenvector) that concentrates most of the genetic variability available to evolution. Our results are in line with those findings as the effective dimensionality of systems with five traits varies between 2 and 3.5 and are consequently more likely to constrain adaptation.

Despite the fact that matrices of reduced rank represent strongly constrained genetic systems, there is a large variation in the correlated divergence of the populations with different M-matrices. This variation largely depends on the match between the distribution of the genetic co-variances among traits (i.e., the G-matrices) and the orientation of the selection gradients in multivariate trait space, as expressed by variation in the angle between gmax and the LoD and the projection of G onto the LoD. The variation is caused by differences among M-matrices in the sum of the genetic covariances of trait T2 with the traits under heterogeneous selection (see Fig. 5). Variation in the sum is given by different combinations of positive and negative correlations and is the result of the use of a Wishart distribution for random matrices where positive and negative covariances are expected to be equally represented with a mean of zero and probability density function that depends on the rank of the matrix; the average matrix correlation varies proportionally to inline image (see Martin and Lenormand 2006, their Appendix 2). The level of genetic constraints is thus higher in matrices of reduced rank because they concentrate the same total amount of genetic variation in a smaller portion of the phenotypic space and, in this study, that subspace was chosen to be more strongly associated with trait T2. Matrices of reduced rank are thus more likely to generate an evolutionary response in the direction of the trait under homogeneous selection than matrices of full rank.


This study explores the effect of indirect selection in the ideal situation where populations first evolved in allopatry before allowing migration to change the equilibrium state toward migration–selection balance. The departure from optimality caused by gene flow on the traits under divergent selection depends on the proportion of surviving migrants/hybrids left in the populations after selection and mixing. Greater departures from optimum values may arise under different ecological scenarios. Examples include populations diverging in parapatry after a recent change in habitat conditions or nonequilibrium dynamics such as extinction/recolonization. Divergence in parapatry has been frequently reported in the literature (e.g., Boulding and Hay 1993; Grant and Grant 2002; Nosil and Crespi 2004; Berner et al. 2008). It is expected that under this scenario, the correlated response in uniformly selected traits would be greater at the beginning of population divergence when the selection gradients on the diverging traits are stronger (e.g., Schluter 1996). Additional simulations for populations diverging in parapatry confirm this expectation (see Fig. S11). The adaptive trajectory of a population is thus expected to be more biased away from the selection gradient early in divergence than late (see Guillaume and Whitlock 2007, for examples of such trajectories). Recolonization of extinct populations with migrants from different habitats would lead to the same kind of scenario, bearing maladaptation of the seeding migrants. Populations at nonequilibrium are thus likely to show more pronounced patterns of correlated divergence than equilibrium populations.

Other effects linked to migration and gene flow can lead to increased diversification among populations. One is caused by nonrandom dispersal, or phenotype-dependent dispersal linked to matching habitat choice (e.g., Edelaar et al. 2008) where individuals either actively choose their habitat depending on their phenotype or not, when for instance high-quality individuals are more likely to settle in high quality habitats (as for the great tits of Wytham Woods, UK, see Garant et al. 2005). Another one may be caused by asymmetrical migration rates among populations. An example is given by two populations of great tits Parus major inhabiting a similar insular habitat in the Netherlands but receiving different amounts of immigrants from a mainland population. Postma and van Noordwijk (2005) showed that the divergence in clutch size between these two populations was caused by greater maladaptation in the population with higher immigration rate whereas local adaptation was favoring smaller clutch sizes in the other population. A key aspect of these two great tit studies (Garant et al. 2005; Postma and van Noordwijk 2005) is that neither habitats differed in selection or local optimum, most of the effects were driven by demography and dispersal. Demographic effects were not incorporated here but may have important consequences on microevolution and thus deserve better treatment in future studies. In particular, positive effects of gene flow on adaptation (e.g., heterosis, increased genetic variance) may appear in small and isolated populations that might counterbalance some of the negative effects of the sort modeled here.


The genetic basis of the traits was modeled with two strong assumptions. First, alleles were completely pleiotropic, that is each allele affected each trait modeled. There are empirical reasons to believe this is not always the case and empirical patterns suggest that genetic correlations among traits are caused by a limited number of genetic factors (e.g., Gardner and Latta 2007; Wagner et al. 2008; Wang et al. 2010). The effects of varying, or modular, pleiotropy on genetic correlations and constraints are not precisely known to date. If genetic pleiotropic effects are sequestrated within modules of quantitative traits of shared functionality, then it is conceivable that divergent selection may affect isolated modules of traits sharing ecological factors that commonly diverge among habitats. The effects of indirect selection might thus remain of small extent and limited to modules of traits with common functionality. This however strongly depends on how selection interacts with patterns of genetic covariation among traits, for which empirical data are still greatly missing.

Second, the mutation model was chosen as the continuum-of-allele model. Consequences of this are twofold. First, the breeding values do not follow a Gaussian distribution that causes discrepancies with predictions from the GAM, as explained in the results section (see also Yeaman and Guillaume 2009). Second, recombination has very limited effects on the results because migration tends to create condensed genetic architectures with a reduced number of loci involved in the population divergence as gene flow increases (results not shown, see also Yeaman and Guillaume 2009, S. Yeaman and M. Whitlock, unpubl. data). Outstanding questions are then whether tightly linked, nonpleiotropic loci would lead to similar patterns of correlated divergence and whether migration is able to maintain linkage disequilibria in the face of recombination in randomly mating populations. Data suggest that there is a continuum of effects from a pleiotropic locus, equivalent to loci with zero recombination, to freely recombining loci. Data from QTL studies for instance show that even if pairs of traits are commonly affected by only a few shared genetic factors (1-3), their correlation can be quite high (see Gardner and Latta 2007). To a certain limit, pleiotropy thus seems undistinguishable from tight linkage although the effects of the latter will decrease over time due to recombination. Linkage, however, is predicted to also generate nonuniformity of the eigenvalues of the G-matrix of the sort observed in natural systems, under neutral evolution (see Griswold et al. 2007). This source of genetic correlations might however not strongly constrain adaptation as selection may act more efficiently against them and align the G-matrix to the selection surface. Selection might thus dominate the dynamics of the system when correlations are due to linkage rather than pleiotropy, but this remains to be demonstrated.

Finally, genetic correlations among traits are not only the product of mutational correlations but may also evolve under correlational selection (or ridge selection). G-matrices may adopt a configuration of covariances that reflects the topology of the selection surface and have an orientation similar to the ridge in the selection surface (see e.g., Guillaume and Whitlock 2007). Additional simulations have shown that this type of genetic correlation (assuming mutational correlation is null) does not translate into strong constraints on adaptation but rather decreases the migration load and favors population divergence at equilibrium (see also Tufto 2010, for similar conclusions). A small correlated divergence may emerge under intermediate correlational selection (F. Guillaume, unpubl. data). The rational behind these results is that selection for higher phenotypic correlation among traits increases selection across the ridge in the selection surface and thus decreases gene flow by counter selecting hybrids falling outside the ridge, leading to a lower migration load and greater phenotypic divergence among populations.


Finding and demonstrating occurrences of migration-induced correlated divergence in the wild is challenging because of the difficulties of identifying the ecological factors that underlie phenotypic trait variation; not all factors can be known and measured, and traits under homogeneous selection may be erroneously regarded as traits under direct selection and ascribed to ecological factors causing divergence in correlated traits. Several approaches may however help disentangle correlated phenotypic divergence from trait divergence caused by divergent selection, in the presence of gene flow. First, the GAM analytical framework can be used to generate general predictions of the equilibrium trait divergence among habitats provided that the relevant parameters are accurately estimated (see e.g., Hendry et al. 2001; Moore et al. 2007). Comparing predictions while setting all trait covariances to zero in the G and P matrices with their observed values will show how genetic covariances might affect trait divergence (Hendry et al. 2001; Agrawal and Stinchcombe 2009). Second, measuring selection differentials among pairs of populations in natural or experimental environments may indicate traits for which phenotypic “convergence” is favored versus traits for which “divergence” is favored (e.g., for divergent selection: Bolnick and Nosil 2007; Nosil 2009). Evidence of convergent selection would be consistent with the presence of correlated phenotypic divergence between populations. Also, measuring multivariate selection surfaces in the wild (e.g., Lande and Arnold 1983) may indicate the concordance between ecological and trait divergence. Third, experimental manipulation of gene flow is probably the best way of demonstrating a causation of gene flow on adaptation (e.g., Riechert 1993; Nosil 2009) and on the occurrence of correlated trait divergence. Typical patterns of correlated divergence will be revealed by its increasing, although nonmonotonic relationship with migration rates. The absence of divergence in a phenotypic trait in the absence of gene flow will point at the source of the observed divergence in the presence of gene flow.

Finally, how likely is that we may be able to detect the effects of indirect selection on phenotypic divergence? The results suggest that correlated divergence may be picked up by QSTFST contrasts under conditions of large migration rates, intermediate to weak selection, and high genetic constraints. For the limited case of two correlated phenotypic traits, both QST and FST have very low values, on the order of 10−2 and below, too low to be accurately detected (see Whitlock and Guillaume 2009). Divergence in multiple traits, with strongly constrained genetic systems however leads to very high contrasts between QST and FST and to the sure rejection of neutral divergence under a large range of migration rates. This would generate false positive in tests of detection of selection in the sense that direct selection is not causing the observed divergence. This conclusion is true for any form of correlated divergence and although it has been previously acknowledged (e.g., Whitlock 2008), it remains unclear what proportion of published QSTFST values is influenced by indirect versus direct selection. However, the novelty here is that the divergence is maintained by migration–selection balance among populations. QSTFST contrasts will thus vary with migration rates and traits that diverge less among more isolated populations might point at the nature of the divergence among populations with higher gene flow.


In summary, I have shown that gene flow may induce large divergence among populations in traits experiencing homogeneous spatial selection. This divergence is thus not the product of direct ecological divergence among populations but stems from the genetic constraints that exist between genetically correlated traits and evolves as a correlated response to direct selection on traits under heterogeneous selection. Habitats diverging in phenotypic space of higher dimensionality are more likely to generate very large correlated divergence, large enough to be picked up by QSTFST analysis under a large range of migration rates. Correlated divergence, however, comes accompanied by an increased migration load and thus reduced local adaptation of the populations. As such, genetic constraints in multidimensional trait space are likely to negatively affect species in their geographical/ecological range and rate of formation (but see Nosil et al. 2009, for a contrasted view). Such consequences of multivariate selection have not attracted much attention until very recently (e.g., Gomulkiewicz and Houle 2009; Futuyma 2010) and our view of the process of population divergence and species range limit remains framed within a univariate theoretical framework (e.g., Kirkpatrick and Barton 1997). Nonetheless, the question is not so much theoretical than empirical. Apart from the fact that natural systems seem to meet the genetical conditions for the evolution of strong correlated phenotypic divergence and constrained local adaptation, the question of whether the ecological conditions for such constraints are met in natural systems is unsettled yet. The outstanding empirical questions we need to address are, first, whether the ecological and phenotypic dimensionality of habitat divergence are concordant, second, how do selection gradients align with G (or M) matrices in the wild, and finally, whether traits under divergent selection are commonly genetically associated with traits under homogeneous selection.

Associate Editor: L. Kruuk


I thank M. Whitlock, S. Yeaman, O. Ronce, P. Nosil, and K. Elmer for their helpful comments on the manuscript. G. Martin provided an R script to generate random M matrices. I am also grateful to three anonymous reviewers and L. Kruuk for their constructive comments and suggestions. This study was supported by a grant from the Swiss National Science Foundation (no. PZ00P3_121697).