#### Univariate changes in sexual dimorphism

The univariate breeder's equation, , describes the phenotypic response to selection in a single trait. Heritability, , is the additive genetic variance of the trait, (), divided by the total phenotypic variance, (). The additive genetic variance is the variance due to the average effects of alleles, and phenotypic variance is the square of the phenotypic standard deviation. The selection differential, *s*, is defined as the difference in the phenotypes between the mean of the entire population and the mean after selection (Falconer & Mackay, 1996). Thus, the response to selection, *R*, is the difference in the mean of the entire population before selection and the mean of the offspring of the selected parents (Falconer & Mackay, 1996; Lynch & Walsh, 1998).

At first, it may seem mysterious that males and females could have different additive genetic variances () because they share almost the entire genome. However, as genetic variances can be imagined as the sum of locus-specific allelic variances, sexual dimorphism in allele frequencies, additive effects and dominance effects can all contribute. Empirically, alleles can have different additive effects or different dominance coefficients in males versus females (Fry, 2010). New mutations may have sex-specific effects that lead to allele frequency differences between the sexes. Such sex-specific effects may arise due to sex-linkage; although the breeder's equation (3) assumes autosomal linkage of genes, sex chromosomes can often affect the expression of autosomal genes. Several studies have found statistically significant sex-specific differences in heritabilities or additive genetic variances (e.g. Mousseau & Roff, 1989; Wilcockson *et al*., 1995; Ashman, 1999, 2003; Mignon-Grasteau, 1999; Jensen *et al*., 2003; Rolff *et al*., 2005; Fedorka *et al*., 2007; Zillikens *et al*., 2008; Gershman *et al*., 2010; Stillwell & Davidowitz, 2010).

The mean change in a male character () or female character () is governed by the sex-specific genetic variances and the degree of correlation between the sexes for the shared trait. The change in the sexual dimorphism (Δ*SD*) may be described as the difference in the response to selection for males and females, measured as changes in trait means:

- (3)

As equation 3 is complicated, it is easier to see the effect of by making a few simplifications. To illustrate, we follow the Cheverud *et al*. (1985) decomposition of the univariate breeder's equation for sexual dimorphism, which assumed that heritabilities and phenotypic variances are the same in each sex ( and ):

- (4)

Because of the negative sign in front of , large positive correlations always have a constraining effect on positive, male-biased changes in sexual dimorphism, that is, such that Δ*SD* > 0 (NB: for simplicity in this paper we only consider positive changes in sexual dimorphism due to stronger selection in males; however, female-biased changes may be easily substituted). The absolute constraining effect of rests upon the assumption of identical genetic variances (Leutenegger & Cheverud, 1982; Slatkin, 1984; Cheverud *et al*., 1985; Leutenegger & Cheverud, 1985; Reeve & Fairbairn, 1996; Lynch & Walsh, 1998; Reeve & Fairbairn, 2001; Bonduriansky & Chenoweth, 2009; Poissant *et al*., 2009).

The negative correlation between and the extent of sexual dimorphism predicted by equation 4 has received empirical support. A study of antler flies revealed an overall negative relationship between and morphological sexual dimorphism (Bonduriansky & Rowe, 2005). When the intersexual correlation for a shared trait was negative or low, the trait was more dimorphic. By contrast, when the intersexual correlation was positive and substantial, the trait was less dimorphic. A recent survey confirms that this negative relationship extends to a variety of taxa and trait types (Poissant *et al*., 2009). Yet, there is a great deal of scatter in this relationship, particularly at low sexual dimorphism values (e.g. Bonduriansky & Rowe, 2005; Poissant *et al*., 2009). Low phenotypic dimorphism could be associated with high or low . The intersexual genetic correlation may be small when the intersexual covariance is small, or when the sex-specific genetic variances are large, or both - suggesting why the correlation between and phenotypic sexual dimorphism can sometimes be weak (Poissant *et al*., 2009). Simulations confirm that high positive can be associated with both low and high sexual dimorphism, as measured by means and phenotypic standard deviations (Reeve & Fairbairn, 2001). Furthermore, although temporary decreases in are necessary for the evolution of sexual dimorphism, the equilibrium magnitude of ultimately depends upon differences in selection, mutation and the number of concordantly and discordantly selected alleles between the sexes (Bonduriansky & Chenoweth, 2009). For instance, sex-specific selection may make some loci sex-limited in expression to accommodate greater phenotypic sexual dimorphism. Sex-limited expression can potentially increase because the intersexual covariance in the numerator will depend only upon the remaining concordantly selected alleles. However, if the new sex-limited loci are still polymorphic, male variance increases in the denominator may instead decrease the overall value of . The interplay between concordantly and discordantly selected alleles may further explain why and sexual dimorphism need not be negatively correlated.

Few studies have explicitly examined sex-specific differences in additive genetic variances and shown significant differences on a trait-by-trait basis. A recent analysis (Wyman and Rowe, *unpublished data*) showed that the overall mean difference in male- and female- specific heritabilities was not statistically different from zero (although extreme differences did occur for certain traits). If the same traits with monomorphic heritabilities also have high , genetic constraints may indeed be widespread in single traits that covary little with other traits.

#### Multivariate changes in sexual dimorphism

Here, we move from a single trait view to the multivariate framework proposed by Lande and Arnold (Lande, 1979; Lande & Arnold, 1983). Because of correlations among traits, selection on one trait can cause an indirect response to selection in other traits. The multivariable response to selection is modelled as:

- (6)

where **G** is the additive genetic variance-covariance matrix for the traits under consideration whereas *β* is the vector of the selection gradients for each trait. *β* results from multiplying the inverse of the phenotypic variance-covariance matrix, , and the vector of selection differentials, **s**.

The change in sexual dimorphism is defined as the difference in the mean changes in males and females, . **ΔSD** expanded yields:

- (8)

Because of the complexity of this equation, Lande (1980) made a few simplifying assumptions to elucidate its meaning. First, he assumed monomorphic **G** matrices. Second, he assumed that , resulting in:

- (9)

In other words, the extent of change in sexual dimorphism is governed by the similarity of the **G** and **B** matrices, in addition to differences in selection gradients. As a result, if male and female **G** matrices are the same and if the intersexual covariance is complete, the **B** matrix is simply the **G** matrix; under these circumstances, no divergence is possible. However, what kind of support exists for monomorphic **G** matrices and a symmetrical **B** matrix?

#### Sexual dimorphism in **G matrices**

In the univariate formulation, a single variance value only has a magnitude, whereas in the multivariate formulation, the **G** matrix has both a magnitude (i.e. eigenvalues) and an orientation (i.e. eigenvectors). Eigenvectors describe a direction in multi-trait space, whereas eigenvalues describe the variation in that direction. Eigenvectors reveal the direction where the genetic variance is oriented, whereas eigenvalues reveal how fast the predicted response to selection along that direction will be (Walsh & Blows, 2009).

Because of this distinction, comparing **G** matrices between the sexes is more involved than statistically comparing a pair of heritabilities or variances. It is possible to do an element-by-element comparison of the heritabilities and genetic correlations in the male and female **G** matrices (e.g. Steven *et al*., 2007; Leinonen *et al*., 2010), but such one-to-one comparisons between matrices can be misleading (Walsh, 2007). So, better yet is to take advantage of the multivariate framework and compare eigenvectors and eigenvalues between the sexes. For instance, the Flury hierarchy analysis compares a set of eigenvectors and eigenvalues for their shapes and relative sizes (Phillips & Arnold, 1999). The Flury analysis proceeds by testing a nested set of hypotheses in an ascending manner, or by comparing the various hypotheses with the hypothesis of no relationship. **G** matrices may be identical, sharing the same eigenvectors and same eigenvalues. Or, they may be proportional, sharing the same eigenvectors but having eigenvalues that differ by a constant factor. Alternatively, **G** matrices may share the same eigenvectors but have nonproportional eigenvalues (i.e. full common principal components (CPC)) or share a subset of eigenvectors (i.e. partial CPC). Finally, **G** matrices may not share any eigenvectors and be unrelated (Phillips & Arnold, 1999). However, although intuitive, the Flury hierarchy tends to overestimate matrix differences and may miss underlying similarities (e.g. Houle *et al*., 2002; Mezey & Houle, 2003). For instance, two matrices might share a common eigenvector but for different associated eigenvalues (e.g. the largest eigenvalue in one matrix and the second largest eigenvalue in the other matrix).

The difference between genetic variance in univariate and multivariate space can be seen by comparing two vectors in one- and two-dimensional space (Fig. 1). In one-dimensional space, male and female heritabilties lie along a number line. Their difference is one factor that contributes to the change in the extent of sexual dimorphism. When , the heritabilities cancel out (Fig. 1a); the evolution of sexual dimorphism depends only upon differences in the strength of sex-specific selection and the intersexual genetic covariance of the shared single trait (equation 4). In two-dimensional space (i.e. two different traits under consideration), sexual dimorphism in genetic variances may be measured by the direction of the greatest genetic variance (, or principal component (PC) 1 of the **G** matrix). for the male and female **G** matrices may have the same magnitudes but point in different directions (Fig. 1b). By considering a second trait, the multivariate view illustrates the potential for sexual dimorphism to evolve through orientation differences even though the total amount of genetic variation is the same. Similarly, even if the male and female point in the same direction, differences in their magnitude will provide genetic variance for the evolution of sexual dimorphism (Fig. 1c). Thus, sexual dimorphism in magnitude and/or direction of the genetic variances – along with the covariance structure and sex-specific selection – contributes to phenotypic divergence. The potential for a multivariate view to accommodate more avenues for evolving sexual dimorphism contrasts with the typical view of multivariate quantitative genetics, so that including additional traits typically leads to more constraints (Walsh & Blows, 2009).

Sexual dimorphism in **G** occurs through sex-specific differences in genetic variances and covariances. As with sex-specific heritabilities, sexual dimorphism in dominance, allele frequencies and additive effects may contribute to sex-limited patterns of gene expression and pleiotropy to alter eigenstructures between the sexes.

Several studies have shown that sexual dimorphism is common in sex-specific **G** matrices (e.g. Holloway *et al*., 1993; Guntrip *et al*., 1997; Ashman, 2003; Jensen *et al*., 2003; Rolff *et al*., 2005; McGuigan & Blows, 2007; Sakai *et al*., 2007; Steven *et al*., 2007; Campbell *et al*., 2010; Dmitriew *et al*., 2010; Lewis *et al*., 2011) (see also Steven *et al*., 2007; Barker *et al*., 2010 for further discussion). When the Flury hierarchical testing procedure is applied, by and large, **G** matrices across populations, experimental treatments, species and sexes share some subset of eigenvectors (Arnold *et al*., 2008) suggesting conservation. In particular, the sexes appear to share all or some principle components in 78% of comparisons (Arnold *et al*., 2008). For instance, **G** matrices between females and hermaphrodites of the gynodioecious plant species, *Fragaria virginiana* (Ashman, 2003) and *Schiedea salicaria* (Campbell *et al*., 2010), shared all eigenvectors but not eigenvalues so that they differed in shape but not orientation.

Conservation of eigenvectors means that the direction along which males and females may respond to selection is the same. In either the full or partial CPC case, if eigenvalues differ for shared eigenvectors, the total response to identical selection will differ in males versus females along that shared direction – as in the univariate case. If the eigenvectors are completely shared, then comparing univariate variances would be sufficient. However, even in the full CPC case, a multivariate approach will provide new insights through the **B** matrix (see more below). If the eigenvectors are partially shared, then the multivariate approach has revealed potentially interesting multidimensional trait axes, so that responses to selection will be different between the sexes because of the distinct eigenvectors.

Despite the widespread conservation of eigenvectors, comparisons of **G** between the sexes also revealed that they are never equal (same eigenvectors and same eigenvalues) or proportional (same eigenvectors but with proportional eigenvalues) (Arnold *et al*., 2008) – in contrast to comparisons of **G** between experimental treatments, populations, or species (Arnold *et al*., 2008). Moreover, distinct, unrelated eigenstructures are by no means uncommon, representing 22% of between-sex comparisons (Arnold *et al*., 2008). For instance, male and female **G** matrices were completely unrelated in the plant *Silene latifolia* (Steven *et al*., 2007) and the house sparrow *Passer domesticus* (Jensen *et al*., 2003). Sex-specific **G** matrices that do not share any eigenvectors mean that the response to selection may proceed along completely different axes in each sex so that changes in sexual dimorphism will occur, but due to the evolution of different trait sets in each sex (depending on the strength and direction of *β*). And although the Flury hierarchical approach tends to overestimate matrix differences (e.g. Houle *et al*., 2002; Mezey & Houle, 2003), there is no reason why this issue should afflict sex-based comparisons of the **G** matrix more than comparisons across environments or experimental treatments.

Because full rank **G** matrices have as many eigenvalues as there are number of traits measured, it is important to characterize the eigenvalues – for example, how many of the eigenvalues describe a large or statistically significant proportion of the total variance? The number of statistically supported eigenvalues will indicate the number of directions that evolution may proceed along in multivariate space. Recent studies suggest that many **G** matrices seem to be ill-conditioned, such that most of the genetic variance is explained by the first 1 or 2 eigenvalues (Kirkpatrick, 2009; Walsh & Blows, 2009; Simonsen & Stinchcombe, 2010) – although determining the contribution of sampling biases to these results (e.g. Hill & Thompson, 1978; Hayes & Hill, 1981) is an ongoing challenge. Thus, although many traits may be measured, the response to selection may only proceed along 1 or 2 dimensions. However, in terms of the evolution of sexual dimorphism, it may not be necessary for sex-specific differences to proceed along all potential directions. Sexual dimorphism can evolve because males and females can differ with regard to the effective number of dimensions, the amount of genetic variance explained, the orientations and/or the magnitudes. For example, McGuigan & Blows (2007) compared male and female differences in genetic dimensions and found that females possessed a greater number of effective dimensions than males in *Drosophila bunnanda*. In *Drosophila serrata*, males and females diverged in different sets of multivariate axes with respect to population divergence (Chenoweth & Blows, 2008). Although empirical studies may underestimate the true dimensionality of **G** matrices, there is no *a priori* reason why dimensionality should be consistently over- or under-estimated in one sex relative to the other.

*B* matrix and

In addition to the sex-specific **G** matrices, the intersexual covariance **B** matrix is important to understanding the evolution of sexual dimorphism. Because **B** keeps track of additional genetic covariances across different traits in each of the sexes, it will typically modify the predicted response to selection provided by the univariate quantity alone.

Because equation 10 describes two traits, it has additional intersexual covariance terms (10d) that are not present in the univariate breeder's equation. Interestingly, these additional intersexual covariances can have a negative or positive sign, which will, respectively, hinder or facilitate positive, male-biased changes in sexual dimorphism for trait K (i.e. because ). By contrast, positive covariances may only hinder positive changes in sexual dimorphism in the breeder's equation. Finally, the difference between the last two intersexual covariance terms (10d) is a measure of how asymmetric the **B** matrix is. If these two terms are very different, **B** can affect one sex more strongly than the other.

Several studies have reported the complete **B** intersexual covariance matrix (Meagher, 1999; Steven *et al*., 2007; Campbell *et al*., 2010; Lewis *et al*., 2011; Gosden *et al*., 2012). In general, it appears that according to point estimates from the matrix. For example, in Lewis *et al*. (2011), male longevity has a positive covariance with female size, but female longevity has a negative covariance with male size in *Plodia interpunctella*. The asymmetry of **B** was apparent in a study of seven cuticular hydrocarbons in *Drosophila serrata* by demonstrating that the above- and below- diagonal elements of **B** had a smaller correlation than a similar comparison between the and matrices (Gosden *et al*., 2012). The remaining studies of **B** in plants – *Silene latifolia* (Meagher, 1999; Steven *et al*., 2007) and *Schiedea adamantis* (Campbell *et al*., 2010) – also support differences in the male-to-female versus female-to-male patterns of covariation among traits. In *Silene latifolia*, male leaf length and female calyx width had a positive covariance, but female leaf length and male calyx width had a negative covariance (Steven *et al*., 2007). In *Schiedea adamantis*, the off-diagonal covariances are all positive, but with the size of the covariances differing greatly below and above **B**'s diagonal; this difference caused the genetic correlation between female terminal capsule weight and hermaphrodite terminal carpel weight to be twice the genetic correlation in the converse direction. Formal hypothesis tests for determining asymmetry between individual elements of **B** or for the entire matrix await further development.

Factors such as genomic imprinting and sex-limited expression may play a role in creating asymmetries in **B**. It may also be that asymmetries actually point to the prior efficacy of sex-specific selection in producing sexual dimorphism. Correlational selection can cause the **G** matrix to point in the same direction of the selection as it accumulates mutations oriented in this same direction (Roff & Fairbairn, 2012), making the direction of nonrandom with respect to the direction of selection (Schluter, 1996). In a similar manner, the pattern of intersexual covariances may have been altered to accommodate sex-specific selection (Barker *et al*., 2010; Delph *et al*., 2011).

#### Putting **G, B, and ***β* together

For example, although marked differences between and can predict a large degree of phenotypic sexual dimorphism, it is only one part of the whole picture. Two closely related plants, *Schiedea adamantis* and *Schiedea salicaria*, have been studied for and differences. Both species are gynodioecious (possessing hermaphrodite and female individuals), but *S. adamantis* has a higher proportion of females than *S. salicaria*. Interestingly, sexes of the less sexually dimorphic species, *S. salicaria*, shared no principal components (Campbell *et al*., 2010), whereas sexes of the more dimorphic species, *S. adamantis*, shared all principal components (Sakai *et al*., 2007). Furthermore, Campbell *et al*. (2010) found no evidence that the intersexual genetic correlations for homologous traits were lower in the more sexually dimorphic species compared with the less dimorphic species (Campbell *et al*., 2010), suggesting that the between-sex imposed constraints were not fundamentally different between the two species. As a result, the greater sexual dimorphism in *S. salicaria* may have occurred through eigenvalue differences between and , or through the greater asymmetry of **B**. Alternatively, sex-specific selection may have been altogether absent or not aligned with the sex-specific **G** and/or **B** matrices in *S. salicaria* resulting in less sexual dimorphism. The work in *Schiedea* illustrates how understanding the components of equation 7 can elucidate the factors that permitted sexual dimorphism to evolve.