EVOLVABILITY OF INDIVIDUAL TRAITS IN A MULTIVARIATE CONTEXT: PARTITIONING THE ADDITIVE GENETIC VARIANCE INTO COMMON AND SPECIFIC COMPONENTS

Authors


Abstract

Genetic covariation among multiple traits will bias the direction of evolution. Although a trait's phenotypic context is crucial for understanding evolutionary constraints, the evolutionary potential of one (focal) trait, rather than the whole phenotype, is often of interest. The extent to which a focal trait can evolve independently depends on how much of the genetic variance in that trait is unique. Here, we present a hypothesis-testing framework for estimating the genetic variance in a focal trait that is independent of variance in other traits. We illustrate our analytical approach using two Drosophila bunnanda trait sets: a contact pheromone system comprised of cuticular hydrocarbons (CHCs), and wing shape, characterized by relative warps of vein position coordinates. Only 9% of the additive genetic variation in CHCs was trait specific, suggesting individual traits are unlikely to evolve independently. In contrast, most (72%) of the additive genetic variance in wing shape was trait specific, suggesting relative warp representations of wing shape could evolve independently. The identification of genetic variance in focal traits that is independent of other traits provides a way of studying the evolvability of individual traits within the broader context of the multivariate phenotype.

Additive genetic variances and covariances, summarized by the G-matrix, play a fundamental role in phenotypic evolution (Lande 1979; Lande and Arnold 1983). For multitrait phenotypes under directional selection, G might bias the direction of evolution away from the direction of selection (Lande 1979; Cheverud 1984; Phillips and Arnold 1989; Bjorklund 1996). Furthermore, the direction of neutral phenotypic evolution will be determined by G (Lande 1976, 1979; Arnold et al. 2001; Phillips et al. 2001). Predicting the specific direction of phenotypic evolution therefore depends on understanding G (Cheverud 1984; Charlesworth 1990; Arnold 1992; Schluter 1996; Hansen and Houle 2008; Kirkpatrick 2009; Walsh and Blows 2009).

Certain phenotypes might be evolutionarily inaccessible to a population if genetic variation is lacking in that direction. The prevalence of such absolute genetic constraints, defined by the absence of genetic variance for a particular trait combination (Mezey and Houle 2005), is not known (Kirkpatrick 2009). Although estimates of univariate genetic variances are typically greater than zero (Mousseau and Roff 1987; Lynch and Walsh 1998), suggesting absolute genetic constraints will be rare, individual trait genetic variances are likely to provide a misleading picture of how multiple traits can respond to selection. As first highlighted by Dickerson, absolute genetic constraints can exist even when genetic variance is present for each individual trait (Dickerson 1955; Reeve 2000; Blows 2007; Agrawal and Stinchcombe 2009; Walsh and Blows 2009). Even in the absence of absolute genetic constraints, the uneven distribution of genetic variance among trait combinations in multivariate trait space (Kirkpatrick and Lofsvold 1992; Schluter 2000; Hansen et al. 2003; Mezey and Houle 2005; Hine and Blows 2006; McGuigan and Blows 2007) can generate relative genetic constraints, biasing the response for certain directions of selection (Schluter 1996; Hansen and Houle 2008; Agrawal and Stinchcombe 2009; Kirkpatrick 2009; Walsh and Blows 2009).

When additive genetic variance is present for individual traits, absolute and relative multivariate genetic constraints arise as a consequence of the genetic associations among traits. Historically, such genetic constraints have been inferred through interpretation of pairwise genetic correlations, particularly in a life-history context (Houle 1991; Roff and Fairbairn 2007). However, such an approach fails to recognize the influence of the broader context of a phenotype composed of many traits (Pease and Bull 1988; Charlesworth 1990; Fry 1993). Furthermore, although interpretation of correlations of ±1 and 0 is clear, it is not readily apparent how correlations of other magnitudes might affect evolutionary trajectories (Conner 2003; Brakefield 2006; Allen et al. 2008; Hansen and Houle 2008; Agrawal and Stinchcombe 2009). Thus, multivariate approaches are required to identify genetic constraints.

The alternative to testing pairwise genetic correlations is the statistical analysis of G, which involves testing hypotheses concerning matrix properties, particularly size, shape, and orientation. Several new approaches have been developed for the analysis of these properties of G. Current empirical evidence suggests the majority of genetic variance in multitrait phenotypes typically resides in far fewer dimensions than the number of phenotypes measured (Hine and Blows 2006; Meyer and Kirkpatrick 2008; Kirkpatrick 2009, but see Mezey and Houle 2005). Reduced rank of G suggests absolute genetic constraints, although limited statistical power to detect very small components of genetic variance suggests caution in this interpretation (Kirkpatrick 2009). The angle between the direction of selection and the direction of the predicted response to selection indicates the effect of G in biasing evolution, resulting in relative genetic constraint (Hansen and Houle 2008; Marroig et al. 2009; Walsh and Blows 2009). For individual traits, genetic covariation might result in responses in the opposite direction to selection (e.g., Van Homrigh et al. 2007; Smith and Rausher 2008). Similar metrics have also been developed to consider how relative constraints contained in G might bias the response to a given random direction of selection (Cheverud and Marroig 2007; Hansen and Houle 2008; Wagner et al. 2008; Agrawal and Stinchcombe 2009; Calsbeek and Goodnight 2009; Marroig et al. 2009). These latter approaches are important for understanding how populations might respond to changes in selection, whether due to colonization of a new habitat or changes in their current habitat through factors such as climate change or the introduction of new prey, predators, or competitors.

Although the multivariate context in which a trait exists is crucial for understanding evolutionary constraints, it might often be the case that the evolutionary potential of one (focal) trait, rather than the suite of traits as a whole, is of interest in a particular study. Focus on the evolutionary potential of an individual trait is the basis of Robertson's secondary theorem of natural selection (Robertson 1966), which considers the influence of the genetic covariance (σA) between a trait (z) and fitness (ω) on the response of the trait (Δz) to selection

image(1)

Robertson's theorem shows that only the part of the genetic variance correlated with fitness will determine the response of that trait to selection and, therefore, that genetic variation in the trait that is genetically independent of fitness plays no role in adaptive evolution. The idea that some part of the variation in a trait is shared with other traits (such as fitness) is also central to the concepts of integration and modularity (Olson and Miller 1958; Cheverud 1982, 1984; Wagner and Altenberg 1996; Magwene 2001; Mitteroecker and Bookstein 2007). Highly integrated traits form modules, share common variance, and thus each trait within a module might be associated with little independent variation. Statistical approaches for characterizing integration and defining modules therefore depend on defining independent versus shared components of variance (Mitteroecker and Bookstein 2007; Magwene 2008, 2009).

Hansen and colleagues suggested that estimating the genetic variance in a trait that is independent of other traits provides an informative approach for studying genetic constraints (Hansen 2003; Hansen et al. 2003; Hansen and Houle 2008). They showed how the genetic variance in a trait, y, that is independent of the genetic variance in other traits, x, can be estimated as (Hansen and Houle 2008)

image(2)

Hansen et al. (2003); Hansen and Houle (2008) referred to this trait-specific variance as the conditional genetic variance, Gy|x, which is determined by the total additive genetic variance in y, Gy the covariances between y and the traits in x (where Gyx and Gxy are row and column vectors of these covariances), and the additive genetic variance in the traits in x, Gx. In defining the conditional genetic variance, Hansen and colleagues identified the impact of this type of genetic constraint on evolvability of traits (Hansen 2003; Hansen et al. 2003; Hansen and Houle 2008). Hansen et al. (2003) demonstrated that estimated mean-standardized genetic variance in individual Dalechamia blossom traits was reduced when variation in common with other traits was removed through the application of (2), suggesting limits on independent evolvability of traits.

The portion of genetic variation that is shared among traits, the common genetic variance, is the portion of variance for which any selection will elicit correlated responses across the traits. The impact of this genetic covariation among the traits on their evolutionary trajectories will depend on the orientation of selection relative to the genetic variation (Cheverud and Marroig 2007; Hansen and Houle 2008; Smith and Rausher 2008; Wagner et al. 2008; Agrawal and Stinchcombe 2009; Calsbeek and Goodnight 2009; Marroig et al. 2009; Walsh and Blows 2009). The genetic variation that is independent of other traits, the specific (conditional) genetic variance, is available for traits to respond independently to selection, and will not drive correlated evolution. Trait-specific genetic variation might be particularly important when the focal trait is under directional selection whereas other, correlated traits are under stabilizing selection, in which case evolution will be dependent on the trait-specific variation (Hansen and Houle 2008).

The approach developed by Hansen et al. (Hansen et al. 2003) involved the application of (2) to estimates of G that were obtained from separate linear models, and thus did not provide a framework for testing hypotheses about specific versus common genetic variation. In this article, we develop an approach that estimates the independence of genetic variance in individual traits, as advocated by Hansen (2003; Hansen et al. 2003), but within a hypothesis-testing framework. Factor analysis, a common approach for studying phenotypic integration (Mitteroecker and Bookstein 2007), partitions variation to trait-specific and common factors. We implement factor analysis within a mixed-model framework to estimate the specific versus common additive genetic variance in two sets of traits in the Australian rainforest fly, Drosophila bunnanda. For both cuticular hydrocarbons (CHCs) (Van Homrigh et al. 2007) and wing shape (McGuigan and Blows 2007), we previously inferred the presence of absolute genetic constraints on evolution, based on the observation that the statistically supported rank of G was less than the number of traits measured. By considering the distribution of trait-specific versus common genetic variance, we rephrased the question of constraint to consider whether there was independent genetic variation associated with individual traits that might allow them to respond independently to selection.

Methods

FACTOR-ANALYTIC MODELING OF SPECIFIC AND COMMON GENETIC VARIANCE

Factor-analytic modeling is perhaps the best developed statistical approach to determining how many dimensions of G contain significant genetic variance, taking into account the amount of genetic variance, the strength of genetic covariance among traits, and sampling variance (Kirkpatrick and Meyer 2004; Meyer and Kirkpatrick 2005; Hine and Blows 2006; Meyer and Kirkpatrick 2008). The factor-analytic approach involves modeling a reduced-rank covariance matrix (inline image) for the random effect representing the additive genetic variance (e.g., the sire level term in a paternal half-sibling breeding design). The reduced rank covariance matrix, inline image, is given as

image(3)

where inline imageis a p×m lower triangular matrix of constants representing factor loadings of the m latent factors. This model, which is analogous to a principal components analysis, explicitly assumes that all genetic variance is shared among traits, and that trait-specific genetic variances are zero. As in any principal components analysis, the reduced rank covariance matrix, inline image, can be represented by its eigenvalues and eigenvectors.

In this article, we were interested in estimating and interpreting trait-specific genetic variances. We therefore modeled a factor-analytic covariance structure that included trait-specific variances as well as the common variance–covariance matrix. Under this covariance structure, the reduced-rank additive genetic (sire) covariance matrix is given by

image(4)

where inline image is a p×p diagonal matrix of the specific variances for each trait. This model is analogous to a factor analysis. To be consistent with the factor analysis terminology, we refer to the independent variances (inline image) as the specific genetic variance, and the remaining genetic variance, captured by the factors in inline image, as the common genetic variance (Smith et al. 2001; Meyer 2009). The specific genetic variances are equivalent to Hansen's conditional genetic variances (Hansen 2003; Hansen et al. 2003). The unconditional genetic variance of Hansen (2003); Hansen et al. (2003); Hansen and Houle (2008), which equates to the total genetic variance, is not directly estimated in factor analytic models with reduced rank. This unconditional genetic variance corresponds to the genetic variance estimated when the covariance structure of G is modeled in an unconstrained manner (Supporting information). We interpret the trait autonomous genetic variance, the proportion of the total (unconstrained) additive genetic variance that was trait-specific (Hansen and Houle 2008).

The data analyzed in this article come from a standard quantitative genetic breeding design, paternal half-siblings. The mixed model employed to analyze this data took the form

image(5)

where X is the design matrix for the fixed effects (the fixed effects, B, sampling day and mating success, are described below), and Zd and Zs are the design matrices for the random effects of dam and sire, respectively. The variance at the dam level, inline imaged, was modeled using an unstructured covariance matrix, whereas the variance at the sire level, inline images, was modeled using an unstructured covariance matrix, and the factor-analytic covariance structures given in (3) and (4). Because the factor-analytic covariance structures were fit within a mixed model, individual elements (such as the trait-specific additive genetic variances) could be tested for significance using a series of nested log-likelihood ratio tests. All analyses were implemented under the MIXED procedure in SAS (version 9.1, SAS Institute Inc., Cary, NC) using restricted maximum likelihood.

We took three steps to analyzing the data. First, we determined whether there was statistically detectable additive genetic variance for each of the traits under consideration. If we lacked the power to detect additive genetic variance it would confound interpretation of statistical analyses to partition additive genetic variance into trait-specific versus common components. For each trait, (5) was fit, and a log-likelihood ratio test used to determine whether the model in which the sire-level (genetic) variance was estimated was a better fit than a model in which this variance component was held to zero (using the PARMS statement in Proc MIXED); tests were one-tailed to account for the fact that variances cannot be less than zero (Littell et al. 1996). Second, for the suite of traits under consideration, we determined the rank of G (the sire level covariance matrix) under the hypotheses: (1) no specific variance, as in (3) and; (2) specific variance, as in (4). To fit the covariance structures corresponding to models (3) and (4) we, respectively used the TYPE = FA0(m) and TYPE = FA(m) statements at the sire level of (5) (see Supporting information). Log-likelihood ratio tests were applied to determine which value of (m) best explained the data, that is, what the statistically supported number of dimensions of G were (Hine and Blows 2006).

Finally, we used the Akaike information criterion (AIC) to identify the best overall model fit from the FA(m) and FA0(m) models. This comparison is a test of whether modeling-specific variances improved model fit over a model in which specific variances were assumed to be zero, and thus whether specific additive genetic variance accounted for significant variation in the suite of traits. This test of specific variances was complimented with log likelihood ratio tests of significance of individual trait-specific variances, conducted using the PARMS statement in Proc MIXED for the best FA(m) model to hold the specific variance to zero, testing one trait at time. Only traits with nonzero estimates of specific variance (Table 1) were tested for significance. The AIC comparison of model fit between FA(m) and FA0(m) models provides a more sensitive test of specific variances in the trait set because it tests the hypothesis that there is specific variance across all traits simultaneously. We were unable to estimate confidence intervals around the variance components within Proc MIXED.

Table 1.  Additive genetic variances for each trait.
TraitTotal VA1Common VA2Trait-specific VA2CD Common VA3CD Trait-specific VA3
  1. 1Estimated in an unconstrained univariate model.

  2. 2Estimated from the best-fit model (Tables 2 and 4) for each trait set.

  3. 3Estimated from the best-fit model (Table 6) for the 12 condition-dependent traits.

  4. 4P<0.05; 5P=0.0716; 6P=0.0590.

2-Me-C240.47440.4190.0000.4490.000
C25:1 (A)0.27940.0420.0000.0240.000
C25:1 (B)0.38240.1520.03540.1360.031
C25H48(B)0.27940.1080.0490.2040.021
7,11-C27:20.39840.2070.0000.1110.005
C27:10.1340.0470.0000.0220.000
C27H50 (A)0.25140.0630.08740.0800.095
2-Me-C280.0930.0760.0000.1500.000
2-Me-C300.1120.0200.0320.1320.024
size0.1740.1130.0650.1230.000
RW10.65040.0830.55340.0730.622
RW20.46440.0250.4174  
RW30.47040.0660.44340.4810.000
RW40.51840.2480.281  
RW50.1200.0110.110
RW60.23950.0020.2036  
RW70.39140.2450.106
RW80.1390.0110.112  

SAMPLE COLLECTION AND MEASUREMENT

The experimental design and data collection has been described elsewhere (McGuigan and Blows 2007; Van Homrigh et al. 2007). Briefly, genetic parameters were estimated from paternal half-siblings generated by mating 125 sires each to four dams, and sampling two male offspring per family. Males were sampled following participation in binomial mate choice trials. The competing male in these trials came from the same population, was raised under the same conditions, and was the same age as the focal male from the half-sibling breeding design (Van Homrigh et al. 2007). The choosing females came from the half-sibling breeding design, but was always unrelated to the focal (half-sibling) male in the mate choice trial (McGuigan et al. 2008). As males that were successful or unsuccessful in these trials differed in trait means (Van Homrigh et al. 2007) male mating success was included as a fixed effect in the analyses. Males were sampled over three successive days (flies were 6-, 7-, or 8-day old), and sampling day was also included as a fixed effect in the analyses.

Following the mate choice trials, males from the genetic design were individually washed in hexane to collect CHCs, which were analyzed on a gas chromatograph. Peak area was calculated for 10 peaks and the area under each peak divided by the total CHC content, and log-contrast scores calculated using C25 as the divisor (Van Homrigh et al. 2007). The remaining nine traits, which are under sexual selection through female mate choice (Van Homrigh et al. 2007), were analyzed here. Following CHC extraction, the right wing was collected from each male, and photographed using a Leica MZ6 microscope with a Panasonic digital camera and the software Video Pro (Leading Edge Pty Ltd, Marion, SA, Australia). Nine landmarks (McGuigan and Blows 2007) were recorded using tpsDig version 2.04 (Rohlf 2005). Landmark coordinates were aligned, centroid size estimated, and shape variation summarized as relative warps (RWs) using the default settings in tpsRelw version 1.42 (Rohlf 2005). Centroid size is the square root of the sum of squared distances of each of the nine landmarks to their centroid, and RWs are eigenvectors of the aligned Procrustes residuals, summarizing variation about the mean wing shape (Rohlf 1999). Here, we analyze the first eight RWs (accounting for 90% of the phenotypic variation in wing shape: Table S1) and centroid size. A total of 122 sires and 642 sons were analyzed. All traits (CHC log-contrast scores, centroid size, and RWs) were standardized (mean = 0, standard deviation = 1) prior to analyses to ensure that the different trait types (CHCs and wings) were on the same scale, and to facilitate convergence in the mixed model analyses (note that under this standardization the mixed model partitions the phenotypic correlation matrix).

We applied the three-step analytical approach described above to three combinations of these traits. First, we analyzed the nine CHCs. Traits that interact to perform a single function might reflect this functional integration through a shared genetic basis (Cheverud 1982, 1984; Klingenberg 2008). In D. bunnanda, females exhibit a preference for a specific combination of male CHCs (Van Homrigh et al. 2007; McGuigan et al. 2008). Male CHCs therefore perform a common function (sexual signaling), and tight coordination of relative expression of the different compounds is required to achieve mating success (McGuigan et al. 2008). Production of different CHC compounds depend on common biochemical pathways (Stanley-Samuelson and Nelson 1993; Howard and Blomquist 2005), and such developmental integration is also expected to be reflected in the genetic covariance structure (Cheverud 1982, 1984; Klingenberg 2008). We therefore expected that much of the genetic variation male CHCs will be common, shared by the nine compounds, with limited genetic autonomy.

Second, we analyzed wing size and the eight RWs. The Drosophilid wing has emerged as an important model in evolutionary biology (Klingenberg and Zaklan 2000; Phillips et al. 2001; Houle et al. 2003; Mezey et al. 2005; Weber et al. 2008). The function of wings in flight strongly suggests functional integration, although different aspects of wing shape might have other functions and experience different selection pressures (McGuigan 2009). Developmental studies have identified loci with general effects on wings (Stark et al. 1999; de Celis 2003; Crozatier et al. 2004), indicating developmental integration and suggesting common genetic variance is likely to dominate the wing G. However, artificial selection experiments have revealed that different aspects of Drosophila wings are genetically independent (Cavicchi et al. 1981; Thompson and Woodruff 1982; Weber 1992). Furthermore, QTL with trait-specific effects have been inferred (Zimmerman et al. 2000; Mezey et al. 2005). We therefore expected to observe both common and specific additive genetic variation in wings of D. bunnanda.

Finally, we analyzed a combination of CHC and wing traits. Sexual selection hypotheses predict that expression of sexually selected traits depends on loci with effects on overall viability or condition (Rowe and Houle 1996; Hunt et al. 2004). This genic capture of loci affecting condition ensures the maintenance of genetic variance in sexually selected traits, but also the genetic covariation of different suites of sexually selected traits (Rowe and Houle 1996). However identifying genetic variation in condition (and the genetic variance in a sexual trait that is due to condition rather than loci contributing only to the sexual signal) is difficult, partly because condition itself is not amenable to measurement (Tomkins et al. 2004). The genetic variance that is common to a suite of sexually selected traits (such as CHCs) might reflect the contribution of condition, but equally might reflect the fact that CHCs share part of their biochemical production pathways. However, if developmentally disparate sets of traits, such as CHCs and wings, share a common genetic factor it is likely to be condition. In D. bunnanda, the nine CHC traits (Van Homrigh et al. 2007) and two of the wing-shape traits (RW1 and RW3: McGuigan 2009) are condition-dependent sexually selected traits. We considered these 11 traits along with wing size, a condition index, to determine whether there was evidence that these disparate trait sets share common genetic variation, consistent with the expectations of the condition dependence hypothesis.

Results

GENETIC VARIATION IN CHCs

Additive genetic variation was detected for six of the nine CHCs (Table 1; Table S2), suggesting sufficient statistical power to test hypotheses about the partitioning of additive genetic variation into specific versus common genetic factors. We then determined the FA(2) and FA0(3) models best described the additive genetic variation in CHCs (Table 2). The AIC supported FA(2) as the model that best described the data (Table 2), indicating trait-specific additive genetic variation in CHCs. Four of the nine CHCs were associated with nonzero estimates of specific additive genetic variance, although only two of these, C25:1(B) and C27H50(A), had specific variances that were statistically different from zero (Table 1). The autonomous genetic variation in individual CHCs ranged from 0% to 34.66% (proportion of the total genetic variance that was trait-specific [Hansen and Houle 2008]; Table 1). Overall, the specific variances accounted for only 9.11% of the total additive genetic variation in CHCs, indicating that most of the genetic variance was due to alleles with pleiotropic effects across multiple compounds.

Table 2.  Genetic model fit for cuticular hydrocarbons. Model-fit information (number of parameters estimated, the −2 log likelihood score, and the Akaike information criterion (AIC) score) from REML mixed models of genetic variation in the nine CHCs when specific variances for each trait were explicitly estimated [FA(m)] or zero trait-specific variances were assumed [FA0(m)]. The difference in −2 log likelihood scores gives a statistic with a chi-square distribution; degrees of freedom are equal to the difference in the number of parameters estimated by each model. The genetic variance explained by each model is given as the percentage of the total additive genetic variance (VA), which was determined from a model with unconstrained sire-level variances. Best AIC model fit shown in bold.
ModelParameters−2 Log likelihoodAICChi-square statistic% VA
  1. 1P<0.05 for log likelihood ratio test.

  2. 2This model does not contain a sire variance–covariance matrix, and is the lowest level in the hierarchical comparison of log likelihood statistics for both the FA(1) and the FA0(1) models.

No sire variance2 907364.687544.7
FA(1)1087273.027483.091.663139.97%
FA(2)1167246.837468.826.190159.26%
FA(3)1237239.067475.1 7.77488.05%
FA0(1) 997316.447514.448.239118.10%
FA0(2)1077281.967496.034.484132.43%
FA0(3)1147251.987480.029.989155.83%
FA0(4)1207243.827479.8 8.14858.56%
FA0(5)1257236.147486.1 7.67781.22%

Quantitative genetic experiments typically suffer from a lack of power to detect small effects, and share with all analyses the impossibility of demonstrating that something does not exist. Our analyses provided statistical evidence for small components of variance; we were able to demonstrate that the specific variance in C25:1 (B) was greater than zero, despite this variance component comprising less than 10% of the additive genetic variance in C25:1 (B), and 0.1% of the total phenotypic variance in the CHCs. Although it is not possible to demonstrate that traits for which the trait-specific variance was estimated to be zero would remain so in an experiment of larger size, these variances are likely to be very small, and their contribution to phenotypic evolution is therefore likely to be limited.

The two eigenvectors of the FA(2) model were very similar to the first two eigenvectors of the FA0(3) model (Table 3: vector correlations between gS1 and gN1, and between gS2 and gN2 were 0.99). The eigenvalues of the corresponding vectors were also similar, with gS1 and gN1 accounting for 33% and 35% of the variance, respectively, and gS2 and gN2 accounting for 17% and 16%, respectively (Tables 2 and 3). C27H50(A), the trait with the largest specific variance, contributed strongly to the third eigenvector of the FA0(3) model (Table 3). gN3 accounted 4% of the total additive genetic variance, as did the specific variance of C27H50(A) (Table 1).

Table 3.  Eigen analyses of genetic variation in cuticular hydrocarbons. Trait loadings on eigenvectors of each G, the additive genetic variance (eigenvalue) associated with each eigenvector, and the percentage of the total additive genetic variance (from the unconstrained model) explained by each eigenvector.
 FA(2)FA0(3)
gS1gS2gN1gN2gN3
VA 0.7544 0.3787 0.7876 0.3706 0.1008
%VA total 33.45% 16.79% 34.93% 16.44% 4.47%
2-Me-C24 0.707−0.331 0.644−0.395−0.149
C25:1(A) 0.224 0.102 0.248 0.081 0.060
C25:1(B) 0.448 0.023 0.450−0.092−0.349
C25H48(B) 0.055−0.529−0.010−0.511 0.187
7,11-C27:2 0.261 0.641 0.315 0.608−0.112
C27:1−0.023 0.351 0.035 0.366 0.186
C27H50(A) 0.287 0.037 0.371 0.044 0.859
2-Me-C28 0.298 0.152 0.281 0.129−0.159
2-Me-C30 0.075 0.205 0.062 0.211−0.083

GENETIC VARIATION IN WINGS

There was significant additive genetic variation for six of the nine wing traits (Table 1; Table S2), again allowing us to interpret partitioning of genetic variances into trait-specific versus common factors. We determined an FA(1) or FA0(3) model best explained the additive genetic variation in wings under specific variance or no specific variance assumptions, respectively (Table 4). The AIC indicated the FA(1) model better described the data than the FA0(3) model (Table 4), providing evidence for trait-specific additive genetic variance in this suite of wing traits. All nine traits were associated with nonzero estimates of specific variance, with statistical support for four of these variance components (Table 1). The autonomy of individual wing traits ranged from a low of 27.11% (RW7) up to 94.26% (RW3) (Table 1). Overall, trait-specific variance accounted for 72.33% of the total genetic variance in wing shape. These results indicate that most of the segregating alleles affecting wings in this population of D. bunnanda have unique effects on particular aspects of wing shape.

Table 4.  Genetic model fit for wings. The number of parameters estimated, the −2 log likelihood score, and the Akaike Information Criterion (AIC) score for REML mixed models of the genetic variation in the nine wing traits when specific variances for each trait [FA(m)] were explicitly estimated or zero trait-specific variances were assumed [FA0(m)]. The difference in −2 log likelihood scores gives a statistic with a chi-square distribution; degrees of freedom are equal to the difference in the number of parameters estimated by each model. The genetic variance explained by each model is given as the percentage of the total additive genetic variance (VA), which was determined from a model with unconstrained sire-level variances. Best AIC model fit is shown in bold.
ModelParameters−2 Log likelihoodAICChi-square statistic% VA
  1. 1P<0.05.

  2. 2This model does not contain a sire variance–covariance matrix, and is the lowest level in the hierarchical comparison of log likelihood statistics for both the FA(1) and the FA0(1) models.

No sire variance2 90 1,6181.661,6361.7  
FA(1)108 1,6096.901,6312.984.753197.69%
FA(2)116 1,6093.291,6323.3 3.61998.77%
FA0(1) 99 1,6136.951,6334.944.710133.78%
FA0(2)107 1,6113. 061,6327.623.350153.50%
FA0(3)114 1,6099.501,6327.514.098169.76%
FA0(4)120 1,6094.141,6334.1 5.36180.20%

The first eigenvector of G from the FA(1) and the FA0(3) models were very similar (vector correlation between gS1 and gN1 was 0.98; Table 5). This leading eigenvector accounted for 26%[FA(1)] or 34%[FA0(3)] of the additive genetic variance, and was determined by opposing contributions from traits with the lowest autonomy, RW4, RW7, and size (Tables 1 and 5). As was observed in the analysis of CHCs, the subsequent eigenvectors of the FA0(3) model were associated with traits with large specific variance; gN2 was dominated by RW1 and gN3 determined by RW2 and RW3 (Table 5).

Table 5.  Eigen analyses of the genetic variation in wings. Trait loadings on eigenvectors of each G, the additive genetic variance (eigenvalue) associated with each eigenvector, and the percentage of the total additive genetic variance (from the unconstrained model) explained by each eigenvector.
 FA(1)FA0(3)
gS1gN1gN2gN3
VA 0.80301.0762 0.6175 0.5146
%VA total 25.37% 34.00% 19.51% 16.26%
Centroid Size 0.374 0.280 0.014 0.192
RW1 0.322 0.450−0.743−0.133
RW2 0.178 0.203 0.277 0.639
RW3−0.286−0.346−0.442 0.621
RW4−0.556−0.571−0.291−0.224
RW5 0.115 0.079−0.006 0.126
RW6 0.047 0.054 0.142−0.203
RW7 0.553 0.459−0.050−0.119
RW8 0.115 0.111−0.262 0.171

GENETIC VARIATION IN CONDITION-DEPENDENT CHCs AND WINGS

Sexually selected traits have been predicted to maintain additive genetic variation through the genic capture of condition (Rowe and Houle 1996). Here, we determined whether there was common genetic variance shared between two developmentally disparate sets of sexually selected traits, the nine CHCs and three wing traits. FA0(5) was the best FA0(m) model of genetic variances in the condition-dependent traits (Table 6). Log-likelihood ratio test supported FA(2) as the best FA(m) model, but the FA(3) model was only marginally rejected, and this three-rank model best met the AIC (Table 6). The log-likelihood ratio test of rank is conservative, and here we interpret FA(3) as the best FA(m) model. Both the FA(3) and FA0(5) models accounted for 80% of the additive genetic variance. In the FA(3) model, 23% of the total additive genetic variance was due to trait-specific factors. As expected from the separate analyses of CHCs and wings, the best trait-specific model [FA(3)] best fit the data (Table 6), indicating specific variance.

Table 6.  Genetic model fit for condition-dependent cuticular hydrocarbons and wing traits. The number of parameters estimated, the −2 log likelihood score, and the Akaike information criterion (AIC) score for REML mixed models of the genetic variation in the nine CHCs, wing size, RW1, and RW3 when specific variances for each trait [FA(m)] were explicitly estimated or zero trait-specific variances were assumed [FA0(m)]. The difference in −2 log likelihood scores gives a statistic with a chi-square distribution; degrees of freedom are equal to the difference in the number of parameters estimated by each model. The genetic variance explained by each model is given as the percentage of the total additive genetic variance (VA), which was determined from a model with unconstrained sire-level variances. Best AIC model fit is shown in bold.
ModelParameters−2 Log likelihoodAICChi-square statistic% VA
  1. 1P<0.05; 2P=0.1036.

  2. 3This model does not contain a sire variance–covariance matrix, and is the lowest level in the hierarchical comparison of log likelihood statistics for both FA(1) and FA0(1) models.

No sire variance3156 1,2657.351,2969.4  
FA(1)180 1,2526.681,2880.7130.677120.60%
FA(2)191 1,2496.201,2870.2 30.478139.52%
FA(3)201 1,2480.341,2868.3 15.863279.90%
FA(4)210 1,2470.611,2876.6  9.72398.45%
FA0(3)189 1,2519.981,2898 34.302152.92%
FA0(4)198 1,2492.301,2884.3 27.680162.62%
FA0(5)206 1,2471.171,2881.2 21.128179.63%
FA0(6)213 1,2461.891,2887.9  9.28793.81%

The estimates of specific variance for each trait were very similar in this 12-trait model to the nine-trait CHC only or wing only analyses (Table 1; Table S2). There were several exceptions to this. Most notably, RW3 was associated with statistically nonzero-specific variance in the wing analysis, but not in this 12-trait condition dependence model (Table 1). We held the specific sire variance in RW3 to the value estimated in the wing-only model, and used a log likelihood ratio test to determine that this was a worse fit of the data than the estimated (zero) specific variance (χ2= 5.432, df = 1, P= 0.0099). Thus, the previously observed specific variance in RW3 appears to be attributable to segregating alleles that also affect CHCs, providing evidence for an underlying common genetic factor, interpretable as condition. Although involving traits for which we lacked the statistical power to demonstrate nonzero trait-specific variance, changes in the estimated trait-specific variance in two traits further supported this interpretation. About one-third of the additive genetic variance in wing size was trait specific in the wing-only model, but none was in the condition dependence model (Table 1). Similarly, the trait-specific component of variance in C25H48(B) under the 12-trait model was one-half of that estimated in the nine-CHC model (Table 1). Again, traits with high specific variances dominated lower eigenvectors of the FA0(5) model (Table 7).

Table 7.  Eigen analyses of genetic variation in condition-dependent traits. Trait loadings on eigenvectors of each G, the additive genetic variance (eigenvalue) associated with each eigenvector and the percentage of the total additive genetic variance (from the unconstrained model) explained by each eigenvector.
 FA(3) modelFA0(5) model
gS1gS2gS3gN1gN2gN3gN4gN5
VA 1.0536 0.5186 0.4129 1.1801 0.6821 0.4192 0.3583 0.1333
% total VA 30.26% 14.89% 11.86% 33.89% 19.59% 12.04% 10.29% 3.83%
2-Me-C24 0.524−0.509 0.249 0.491−0.223−0.464−0.285 0.059
C25:1(A) 0.121−0.118 0.057 0.136 0.138−0.089−0.150−0.138
C25:1(B) 0.304−0.262 0.083 0.269 0.014−0.267−0.314−0.405
C25H48(B) 0.228 0.264 0.523−0.015−0.336−0.258 0.118 0.113
7,11-C27:2 0.082−0.201−0.447 0.196 0.318 0.296−0.362−0.364
C27:1−0.002 0.107−0.200 0.073 0.195 0.242 0.084−0.092
C27H50(A) 0.270−0.078−0.036 0.252−0.032 0.014−0.256 0.666
2-Me-C28 0.358−0.013−0.191 0.321−0.148 0.157−0.061−0.130
2-Me-C30 0.267 0.239−0.258 0.177−0.203 0.357 0.015−0.086
Size 0.075−0.019−0.533 0.140 0.024 0.487−0.323 0.369
RW1 0.123−0.305−0.141 0.623 0.312 0.002 0.668 0.076
RW3 0.517 0.616−0.079 0.140−0.719 0.322 0.160−0.225

Discussion

The genetic basis of traits, and in particular, genetic covariation among traits, is expected to constrain the direction and rate of phenotypic evolution (Lande 1979; Cheverud 1984; Phillips and Arnold 1989; Bjorklund 1996). Therefore, information about the genetic independence of individual traits is necessary for identification of genetic limits to evolution (Hansen 2003; Hansen et al. 2003; Hansen and Houle 2008). In this article, we developed an analytical approach to partitioning genetic variance to independent (trait-specific) versus nonindependent (common) variance. When implemented in a hypothesis-testing restricted maximum likelihood framework, factor analytic modeling provides a powerful and readily accessible approach for partitioning additive genetic variation to address questions about the effect of a shared genetic basis on phenotypic evolution.

Factor analysis is not a novel tool in evolutionary studies, although typically only the common factors have been interpreted in biological applications of factor analysis, with particular focus on the effect of different common factors on subsets of traits (e.g., Zelditch 1987). Mitteroecker and Bookstein (2007) recently detailed the application of factor analysis for studying modularity and integration. Identifying shared versus specific additive genetic factors is also a common approach for inferring the contribution of genes underlying a suite of traits, particularly in human studies (Martin and Eaves 1977; Bauman et al. 2005; Hansen et al. 2006; Martin et al. 2009). Meyer (2009) recently described the application of factor analysis to characterizing genotype by environment variation, partitioning effects of genotype to those common across environments versus specific to certain environments. We suggest that factor analysis is an under-utilized tool in evolutionary quantitative genetics, and that considering the extent to which individual traits are associated with trait-specific genetic factors will generate novel insights into both their development and their evolution.

In this article, we addressed the question of possible evolutionary constraints in two different sets of traits measured on the same male D. bunnanda. Only 9% of the total additive genetic variation in CHCs was observed to be trait specific, and only four compounds were associated with any unique additive genetic variation. This result suggests that individual CHCs will be constrained to evolve in concert with other CHCs. Manipulative experiments are required to test such predictions concerning the evolvability of traits. In single traits for example, artificial selection on one CHC should result in strong correlated responses in other CHCs. Similarly in a multivariate context, artificial selection for combinations of CHCs should result in some individual traits evolving in the direction opposite to the selection applied on them (Van Homrigh et al. 2007; Smith and Rausher 2008), although this has rarely been observed.

Theoretical prediction of phenotypic evolution relies on certain simplifying assumptions about the underlying genetic details (Lande 1979, 1980; Turelli 1988; Reeve 2000). The importance of understanding the basis of genetic covariation has long been recognized, but determining these details presents a major empirical challenge (Barton and Turelli 1987; Reeve 2000; Johnson and Barton 2005), and relatively little progress has been made in characterizing the allelic variation that causes genetic covariation (Wolf et al. 2006; Gardner and Latta 2007; Kelly 2009; Kenney-Hunt and Cheverud 2009). The observed predominance of common variance suggests that most of the CHC additive genetic variation in this population of D. bunnanda is due to either segregating alleles with different pleiotropic effects on more than one CHC, or to linkage disequilibrium among loci uniquely affecting individual CHCs. Development of molecular tools in the D. serrata species group (Frentiu et al. 2009) opens the way for a coupling of quantitative and molecular genetics to address this question of the genetic details underlying G.

The partitioning of variance in wing shape contrasted sharply with that in CHCs. Most (72%) of the additive genetic variation in wing shape was due to trait-specific factors. This observation suggests a ready response to selection to generate new combinations of RW wing-shape traits. Because RWs of wing shape in these male D. bunnanda describe variation in the position of x and y coordinates of multiple landmarks (Table S1), this result specifically indicates independent genetic variance for coordinated shifts of the nine landmarks in two dimensions. In studies of morphology, principal components (RWs) of aligned landmarks are a common geometric morphometric approach for characterizing shape, and understanding how such traits behave is crucial for their use in evolutionary studies. The low level of genetic covariation among RWs (inferred from the low level of common variance) implicates strong similarity of P and G: RWs that summarize trait covariation at the phenotypic level also appear to capture major patterns of trait covariation at the genetic level (Cheverud 1988), leading to the estimation of independent genetic variation for each phenotypic factor. Congruence of P and G was previously demonstrated for these same D. bunnanda males when wing shape was characterized as interlandmark distances (McGuigan and Blows 2007). Nonetheless, the analysis also supported one common factor, suggesting that some genetic factor affected more than one wing RW, and that G and P do differ to some extent.

Drosophila wings have emerged as a powerful model in evolution (Zimmerman et al. 2000; Houle et al. 2003; Mezey and Houle 2005; Hansen and Houle 2008; Weber et al. 2008). The inference in this study that male wing shape was primarily determined by genetic factors whose effects are restricted to specific aspects of shape is consistent with other types of evidence. Different wing shape traits respond readily to divergent artificial selection (Cavicchi et al. 1981; Thompson and Woodruff 1982; Weber 1992; see also Houle et al. 2003). Wing shape appears to be highly polygenic (Weber et al. 1999, 2001, 2005, 2008; Zimmerman et al. 2000), with loci tending to have specific effects only on certain aspects of shape (Zimmerman et al. 2000; Mezey et al. 2005). In particular, in a study in which wing shape was characterized as principal components of 12 landmarks (including the same nine as in this study), Mezey et al. (2005) identified 13 wing-shape QTL with trait-specific effects, and eight QTL that affected two or three PCs. Together these data suggest that RWs of wing shape are likely to be affected by loci with effects restricted to particular aspects of shape, making it plausible that considering individual RWs alone could lead to reasonable predictions of the evolutionary dynamics of wing shape.

There was some evidence from our analysis that the two different sexually selected trait sets, CHCs and wings, shared some genetic variation. Specifically, when considered with the CHCs, RW3 was not associated with any trait-specific variance. It is expected that the common genetic factor leading to this trait covariation is condition, that is, the acquisition and allocation of resources. Wing shape is strongly dependent on larval nutrition, particularly the availability of lipids (Panakova et al. 2005), which are the constituents of the CHCs. CHCs are known to be involved in resource allocation trade-offs with a number of functions in insects (Stanley-Samuelson and Nelson 1993). With further information on which to base specific hypothesis about the relationships among traits (or sets of traits), it is possible to use factor analysis to identify the causal common factors underlying such genetic covariation (Mitteroecker and Bookstein 2007; Magwene 2009).

The unsuitability of pairwise genetic correlations has long been recognized (Dickerson 1955; Pease and Bull 1988; Charlesworth 1990), and increasingly attention has focused on multivariate approaches (Kirkpatrick and Meyer 2004; Meyer and Kirkpatrick 2005; Mezey and Houle 2005; Hine and Blows 2006; Blows 2007; Cheverud and Marroig 2007; Wagner et al. 2008; Calsbeek and Goodnight 2009). The factor analytic modeling approach developed in this article is a method for estimating G, which can then be examined using recently developed metrics for characterizing relative evolutionary constraints (e.g., Cheverud and Marroig 2007; Hansen and Houle 2008; Calsbeek and Goodnight 2009; Kirkpatrick 2009). By portioning the variation in a suite of traits to specific versus shared variance, it is possible to consider the potential effect of trait covariation on the evolvability of the suite of traits as a whole, but also to focus on individual traits, having taken into account their phenotypic context. A focus on individual traits retains considerable intuitive appeal to many biologists (e.g., Conner 2007; Preziosi and Harris 2007). Two recent studies contrasted predicted evolutionary responses with and without additive genetic covariation (Smith and Rausher 2008; Agrawal and Stinchcombe 2009). However, genetic variances and covariances are not independent, an issue that holding covariation to zero does not address. In particular, pleiotropic alleles contribute both to the covariance among traits, and to trait variances (Lande 1980). We suggest that partitioning genetic variance into specific and common components provides a more informative approach for determining the effect of a shared genetic basis in biasing the evolution of individual traits.

CONCLUSIONS

The development of statistical tools that allow a univariate focus in a multivariate context might have considerable value for understanding how genetics bias the evolution of phenotypes (Hansen et al. 2003; Walsh and Blows 2009). Given the prevalence of pleiotropy and the apparent modularity of phenotypes (Wagner et al. 2007; Klingenberg 2008) it seems likely that some part of the genetic variance in most traits will be shared with some other trait. Further, key predictions of sexual selection (Rowe and Houle 1996; Whitlock and Agrawal 2009) and life-history (Riska 1986; Houle 1991; de Jong and van Noordwijk 1992; Roff and Fairbairn 2007) theories are that functionally and developmentally unrelated traits will share a genetic basis through their co-dependence on common factors of resource acquisition and allocation. Testing these predictions depends on partitioning genetic variance to shared and unique factors, which can be accomplished in a robust hypothesis-testing framework using factor analytic modeling.


Associate Editor: G. Marroig

ACKNOWLEDGMENTS

We gratefully acknowledge the work of A. Van Homrigh and M. Higgie in applying the half-sibling breeding design reported here, and collecting the CHC data. This manuscript benefited from the constructive criticism of two anonymous reviewers and G. Marroig.