O R I G I NA L A RT I C L E doi:10.1111/j.1558-5646.2010.00968.x
EVOLVABILITY OF INDIVIDUAL TRAITS IN A MULTIVARIATE CONTEXT: PARTITIONING THE ADDITIVE GENETIC VARIANCE INTO COMMON AND SPECIFIC COMPONENTS Katrina McGuigan1,2 and Mark W. Blows1 1
School of Biological Sciences, The University of Queensland, Brisbane, 4072, Australia 2
E-mail:
[email protected]
Received September 10, 2009 Accepted January 8, 2010 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.
KEYWORDS:
Factor analysis, G-matrix, modularity, sexual selection.
Additive genetic variances and covariances, summarized by the Gmatrix, 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 C
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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
C 2010 The Society for the Study of Evolution. 2010 The Author(s). Journal compilation Evolution 64-7: 1899–1911
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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 lifehistory 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.
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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 (z) to selection z = σ A (z, ω).
(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) G y|x = G y − G yx G−1 x Gxy .
(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.
C O M M O N V E R S U S S P E C I F I C A D D I T I V E G E N E T I C VA R I A N C E
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 ( ˆ ) 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, ˆ , is given as ˆ (3) = T where is 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 com ponents analysis, the reduced rank covariance matrix, ˆ , 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 factoranalytic 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 ˆ (4) = T + where 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 () as the specific genetic variance, and the remaining genetic variance, captured by the factors in T , 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 y = α + XB + Zd δd + Zs δs + ε
(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
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dam and sire, respectively. The variance at the dam level, δd , was modeled using an unstructured covariance matrix, whereas the variance at the sire level, δs , 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 hypothe-
Table 1.
ses: (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 traitspecific 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.
Additive genetic variances for each trait.
Trait
Total V A 1
Common V A 2
Trait-specific V A 2
CD Common V A 3
CD Trait-specific V A 3
2-Me-C24 C25:1 (A) C25:1 (B) C25 H48 (B) 7,11-C27:2 C27:1 C27 H50 (A) 2-Me-C28 2-Me-C30
0.4744 0.2794 0.3824 0.2794 0.3984 0.134 0.2514 0.093 0.112
0.419 0.042 0.152 0.108 0.207 0.047 0.063 0.076 0.020
0.000 0.000 0.0354 0.049 0.000 0.000 0.0874 0.000 0.032
0.449 0.024 0.136 0.204 0.111 0.022 0.080 0.150 0.132
0.000 0.000 0.031 0.021 0.005 0.000 0.095 0.000 0.024
size RW1 RW2 RW3 RW4 RW5 RW6 RW7 RW8
0.174 0.6504 0.4644 0.4704 0.5184 0.120 0.2395 0.3914 0.139
0.113 0.083 0.025 0.066 0.248 0.011 0.002 0.245 0.011
0.065 0.5534 0.4174 0.4434 0.281 0.110 0.2036 0.106 0.112
0.123 0.073
0.000 0.622
0.481
0.000
1
Estimated in an unconstrained univariate model.
2
Estimated from the best-fit model (Tables 2 and 4) for each trait set.
3
Estimated from the best-fit model (Table 6) for the 12 condition-dependent traits.
4
P
