SEX LINKAGE, SEX-SPECIFIC SELECTION, AND THE ROLE OF RECOMBINATION IN THE EVOLUTION OF SEXUALLY DIMORPHIC GENE EXPRESSION

Abstract

Sex-biased genes -- genes that are differentially expressed within males and females -- are nonrandomly distributed across animal genomes, with sex chromosomes and autosomes often carrying markedly different concentrations of male- and female-biased genes. These linkage patterns are often gene- and lineage-dependent, differing between functional genetic categories and between species. While sex-specific selection is often hypothesized to shape the evolution of sex-linked and autosomal gene content, population genetics theory has yet to account for many of the gene- and lineage-specific idiosyncrasies emerging from the empirical literature. With the goal of improving the connection between evolutionary theory and a rapidly growing body of genome-wide empirical studies, we extend previous population genetics theory of sex-specific selection by developing and analyzing a biologically informed model that incorporates sex linkage, pleiotropy, recombination, and epistasis, factors that are likely to vary between genes and between species. Our results demonstrate that sex-specific selection and sex-specific recombination rates can generate, and are compatible with, the gene- and species-specific linkage patterns reported in the genomics literature. The theory suggests that sexual selection may strongly influence the architectures of animal genomes, as well as the chromosomal distribution of fixed substitutions underlying sexually dimorphic traits.

Sexual dimorphism is common among animal species, and is thought to reflect sexually discordant natural or sexual selection (Darwin 1871; Andersson 1994). The idea that sexual dimorphism reflects differential adaptation – that sex-specific selection drives evolutionary divergence between the sexes – is clearly articulated by Trivers (1972), who states:

One can, in effect, treat the sexes as if they were different species, … female “species” usually differ from male species in that females compete among themselves for such resources as food but not for members of the opposite sex, whereas males ultimately compete only for members of the opposite sex, all other forms of competition being important only insofar as they affect this ultimate competition. (Trivers 1972, p. 153)

From a population genetics perspective, the “sexes as species” analogy is less apt. Excluding the few genes that are limited to a single sex (e.g., Y-linked genes in males; W-linked genes in females), males and females carry the same set of genes and make equal reproductive contributions to each generation. Males and females may be exposed to different patterns of natural and sexual selection, yet “gene flow” between the sexes is unrestricted and should constrain intersexual divergence. While it is clear that sexual dimorphism does evolve, the genetic basis and sequence of evolutionary events that permit males and females to diverge is not well known.

The evolution of sexual dimorphism has been conceptualized with two models (Darwin 1871; Fisher 1958; Rhen 2000; Coyne et al. 2008). Under a “sex-limited” model, sexual dimorphism evolves by selection of mutations with sex-limited phenotypic effects. If sex-limited mutations are available, the evolution of sexual dimorphism is not problematic. For most mutations, however, at least some degree of expression is expected in both sexes (see Lande 1980; Rice 1984; Rhen 2000; Morrow et al. 2008; Poissant et al. 2010; though the magnitude may be sex-specific: e.g., Mackay 2001). Consequently, the evolution of sexual dimorphism might require genetic substitutions at multiple loci, with genetic interactions underlying inter-sexual divergence. Such a multi-step process could potentially involve sexually antagonistic divergence – correlated evolution between the sexes that increases adaptation of one sex and reduces it for the other – followed by the evolution of sex-limited expression of sexually antagonistic traits (Lande 1980; Rice 1984; Bonduriansky and Chenoweth 2009; van Doorn 2009; Stewart et al. 2010).

Large proportions of animal genomes are differentially expressed between the sexes (Parisi et al. 2003; Ellegren and Parsch 2007), with “sex-biased” gene expression potentially decoupling male and female development and permitting adaptive sexual differentiation (Williams and Carroll 2009). The genomic distribution (i.e., chromosomal linkage patterns) of sex-biased genes may also provide clues about the evolutionary processes and genetic basis underlying sex-specific divergence (Mank 2009). Despite substantial variability between species, there is an emerging consensus that sex-biased genes are non-randomly distributed between sex chromosomes and autosomes (e.g., Reinke et al. 2000; Wang et al. 2001; Parisi et al. 2003; Ranz et al. 2003; Khil et al. 2004; Kaiser and Ellegren 2006; Storchova and Divina 2006; Sturgill et al. 2007; Mueller et al. 2008; Mank and Ellegren 2009; Meisel et al. 2009; Mořkovský et al. 2010). Such patterns may suggest an important role for sexual antagonism during the evolution of sexual dimorphism (e.g., Parisi 2003; Rogers et al. 2003; Oliver and Parisi 2004; Connallon and Knowles 2005; Ellegren and Parsch 2007; Gurbich and Bachtrog 2008; Mank 2009; Innocenti and Morrow 2010) – an interpretation inspired by population genetics theory that predicts unique evolutionary fates for sex-linked versus autosomal sexually antagonistic mutations (mutations beneficial to one sex and deleterious to the other; Pamilo 1979; Rice 1984; Patten and Haig 2009; Fry 2010).

Nevertheless, the connection between sexually dimorphic gene expression patterns and population genetics theory is incomplete for several reasons. First, although sexual antagonism is often invoked, it does not by itself generate sexual dimorphism, and previous theory has mostly neglected epistasis between sexually antagonistic alleles and mutations that modify intersexual genetic correlations (but see Rice 1984). The role of epistasis is likely to be profound: recent research reveals that interactions between different cis-regulatory motifs often underlie sexually dimorphic expression (Williams and Carroll 2009). Second, genetic correlations between different traits will also constrain adaptive evolution, yet it is unclear how strongly pleiotropy might hinder the evolution of sexual dimorphism or whether such constraints might differentially impact sex chromosomes and autosomes. Third, the current theory is almost entirely deterministic and focuses on conditions where selection favors the invasion of a rare allele, rather than the probability and rate of sex-specific divergence, which is governed by selection, mutation and genetic drift (Crow and Kimura 1970; Ohta 1992). Contrasts between chromosomes should be based on the relative invasion probabilities of individual mutations, and waiting times for the evolution of sexual dimorphism. Finally, the relationship between phenotype and fitness has not been consistently articulated by theory, in some cases leading to opposing predictions about the relationship between sex linkage and sexual antagonism (e.g., Rice 1984; Patten and Haig 2009; Fry 2010). Fortunately, there is a considerable body of research linking gene expression variation to fitness variation (see below), which can be used to refine sex-specific selection models and incorporate biologically plausible parameterization.

With the goal of integrating empirical patterns of sex-biased gene expression and population genetics theory, we developed and analyzed a series of two-locus models for the evolution of sex-biased expression in response to selection for gene expression divergence. We separately consider three general evolutionary routes toward sexually dimorphic gene expression: (1) the fixation of sex-limited and tissue-specific mutations that are unconstrained by pleiotropy or sexual antagonism and directly generate sexual dimorphism; (2) the sequential fixation of sexually antagonistic or pleiotropically constrained alleles and alleles at modifier loci, which interact epistatically to generate sexual dimorphism or tissue-specific divergence; and (3) The co-invasion and simultaneous fixation of linked alleles that are not individually favored by selection, but are beneficial in combination. For each scenario, we analytically characterize the necessary conditions and timescale of sex-biased gene evolution under X-, Z- and autosomal linkage. The theory provides a general framework for the evolution of sex-biased gene expression and generates a set of specific and testable predictions.

Model

To accommodate the potential impact of both linkage and epistasis, we first developed two-locus, bi-allelic population genetic models with arbitrary fitnesses assigned to each genotype and sex (Table 1). Since we are interested in contrasting patterns of evolution on sex chromosomes and autosomes, the loci are assumed to be both X-linked, both Z-linked, or both autosomal. We discuss the potential role of inter-chromosomal interactions within the discussion. The distance between interacting loci is presented as a function of the recombination rate between them. Since recombination is potentially sexually dimorphic, sex-specific recombination rates between the loci are given by parameters r_f and r_m for females and males, respectively. Recursions for the four possible haplotypes (A₁B₁, A₂B₁, A₁B₂, A₂B₂) are developed in Appendix 1, and follow the sequence of (i) birth, (ii) selection, (iii) recombination, and (iv) syngamy.

Table 1.

Genotypes and corresponding sex-specific fitness parameters

	Diploid Loci: Autosomes, Z-linked Loci in Males, X-linked Loci in Females
Genotype:	A1A1B1B1	A1A2B1B1	A1A1B1B2	A1A2B1B2	A2A2B1B1	A1A1B2B2	A2A2B1B2	A1A2B2B2	A2A2B2B2
Female fitness:	f11	f21	f12	f22C, f22R	f31	f13	f32	f23	f33
Male fitness:	m11	m21	m12	m22C, m22R	m31	m13	m32	m23	m33
	Z-linked in Females					X-linked in Males
Genotype:	A1B1	A2B1	A1B2	A2B2		A1B1	A2B1	A1B2	A2B2
Female fitness:	f1	f2	f3	f4		m1	m2	m3	m4

The haplotype configuration A₁B₁/A₂B₂ corresponds to fitness f_22C and m_22C; haplotype configuration A₁B₂/A₂B₁ corresponds to f_22R and m_22R.

Because the population mutation rate in animals – e.g., 4N_eu per autosomal locus, per generation – is expected to be less than one, we assume that each locus will initially be fixed or nearly fixed for a common allele (Crow and Kimura 1970). Consequently, evolutionary outcomes of selection at single loci or pairs of loci will be governed by patterns of selection acting on rare genetic variants. Under this assumption, the strength of selection, and probability of invasion for rare alleles or haplotypes can be addressed analytically with local linearized stability criteria. A common genotype is susceptible to the evolutionary invasion of a rare allele or haplotype when the leading eigenvalue for the system, that is the largest eigenvalue of the Jacobian matrix for the set of recursions, is greater than one. Furthermore, the leading eigenvalue provides information about the probability of invasion versus loss of the rare allele, as described further below. For each system of inheritance – autosomal, X- and Z-linkage – there are three potential leading eigenvalues, which are each presented in Table 2. Details of the recursions, as well as the stability analysis, can be found in Appendices 1 and 2.

Table 2.

Candidate leading eigenvalues for the two-locus models of sex-specific selection, with alleles A₁ and B₁ fixed.

	Locus A	Locus B	Both loci (A and B interacting)
Autosome Linkage:	${\lambda }_A=\frac{f_{21}}{2f_{11}}+\frac{m_{21}}{2m_{11}}$	${\lambda }_B=\frac{f_{12}}{2f_{11}}+\frac{m_{12}}{2m_{11}}$	${\lambda }_{\mathit{AB}}=\frac{f_{22C}(1-r_f)}{2f_{11}}+\frac{m_{22C}(1-r_m)}{2m_{11}}$
X-Linkage:	${\lambda }_A=\frac{f_{21}}{4f_{11}}\left(1+\sqrt{1+8\frac{f_{11}{m}_2}{f_{21}{m}_1}}\right)$	${\lambda }_B=\frac{f_{12}}{4f_{11}}\left(1+\sqrt{1+8\frac{f_{11}{m}_3}{f_{12}{m}_1}}\right)$	${\lambda }_{\mathit{AB}}=\frac{f_{22C}(1-r_f)}{4f_{11}}\left(1+\sqrt{1+8\frac{f_{11}{m}_4}{f_{22C}{m}_1(1-r_f)}}\right)$
Z-Linkage:	${\lambda }_A=\frac{m_{21}}{4m_{11}}\left(1+\sqrt{1+8\frac{m_{11}{f}_2}{m_{21}{f}_1}}\right)$	${\lambda }_B=\frac{m_{12}}{4m_{11}}\left(1+\sqrt{1+8\frac{m_{11}{f}_3}{m_{12}{f}_1}}\right)$	${\lambda }_{\mathit{AB}}=\frac{m_{22C}(1-r_m)}{4m_{11}}\left(1+\sqrt{1+8\frac{m_{11}{f}_4}{m_{22C}{f}_1(1-r_m)}}\right)$

FITNESS AS A FUNCTION OF GENE EXPRESSION

Throughout the analysis, we focus on evolutionary divergence in gene expression due to cis-regulatory substitutions. This particular focus is empirically justified by the disproportionate contribution of cis- relative to trans-regulatory changes during species divergence for gene expression and morphology (e.g., Wittkopp et al. 2004; Wray 2007; Carroll 2008; Wittkopp et al. 2008; Graze et al. 2009), as well as the important role of cis-regulatory interactions during sex-biased gene regulation and morphological divergence between the sexes (Williams and Carroll 2009; also see Loehlin et al. 2010a, 2010b). An additional justification is conceptual: because trans-acting mutations are likely to influence many more downstream targets than cis- mutations, disruptive effects of trans-acting mutations are generally expected to be deleterious and will play a relatively small role during evolutionary divergence (Stern 2000; Prud'homme et al. 2007; Wray 2007). Nevertheless, as shown below, the evolutionary trajectories of pleiotropic cis-regulatory mutations can differ between sex chromosomes and autosomes, and these results provide insight into the evolutionary dynamics of hypothetical trans-acting mutations with sex-specific fitness consequences. We return to this subject in the discussion.

To parameterize the fitness effects of gene expression mutations, we consider the following model for the expression-fitness landscape. We assume that there is an ancestral state where selection in males and females favors the same pattern of expression and that the population has evolved to this gene expression optimum, with gene expression maintained by stabilizing selection near the fitness peak. Following a change in the environment (ecological; social; or genetic), the fitness landscape may diverge between the sexes, thereby displacing one sex from its optimum and potentially generating directional selection within that sex toward the new fitness peak (Figure 1; stabilizing selection persists for the other sex).

Figure 1. Divergent expression-fitness landscapes generate conflicting selection over gene expression.

The vertical gray lines represent the wild-type gene expression at the time point when the fitness landscapes diverge between the sexes. Arrows indicate the net direction of selection: stabilizing selection in the top panels, and selection for increased expression in the bottom panels. (A) Sex-specific selection without pleiotropy: selection favors a different level of gene expression in males and females. (B) Sex-specific selection with pleiotropy. A gene is expressed in multiple tissues or during multiple time periods during development. Expression-fitness functions overlap between the sexes within some contexts (context 1), and are divergent in others (context 2). Note that although we display symmetrical fitness landscapes (i.e., deviations above and below expression optima are equally costly), this need not necessarily be the case. The theory presented here permits asymmetries within or between male and female fitness landscapes.

Prior theory and data suggest that gene expression-fitness landscapes are likely to be concave in the vicinity of their expression optimum (e.g., Wright 1934; Charlesworth 1979; Kacser and Burns 1981; Phadnis and Fry 2005; Dekel and Alon 2005; Lunzer et al. 2005; Kalisky et al. 2007; Bedford and Hartl 2009; Zhang et al. 2009). There are two lines of evidence to suggest that fitness surfaces are generally concave:

• (1). The best supported theory for allelic dominance (Wright's physiological theory: Wright 1934; metabolic control theory: Kacser and Burns 1981; empirical support based on mutant fitness effects – e.g., Charlesworth 1979; Orr 1991; Phadnis and Fry 2005 – and direct gene expression-based assays – e.g., Kacser and Burns 1981; Middleton and Kacser 1983; Hartl et al. 1985) is based on the premise that fitness benefits of gene expression follow a saturating function. Because expression of a gene is expected to also carry one or more costs – i.e., energetic and metabolic tradeoffs (Kacser and Beeby 1984; Hurst and Randerson 2000; Wagner 2005, 2007), competition for translation (Gout et al. 2010), interference between molecular pathways (Lion et al. 2004), or toxicity (Clark 1991) – it is a mathematical necessity that any composite fitness function that incorporates both benefits and costs of gene expression will be concave within the vicinity of the gene expression optimum (see Supplementary Materials). A balance between benefits and costs is particularly relevant for the expression and evolution of sex-biased genes, which appear to be condition-dependent, implying a significant cost of gene expression (Wyman et al. 2010).

• (2). Direct experimental measurement of benefits, costs and total fitness as a function of the gene expression phenotype, supports a concave fitness surface (Dean et al. 1986; Dykhuizen et al. 1987; Dean 1989; Papp et al. 2003; Dekel and Alon 2005). While the concavity assumption is ultimately an issue that can only be resolved by additional experimental work, the current evidence strongly suggests that fitness is a concave function of a gene's expression level. We therefore adopt this concave relationship throughout our analysis, with the caveat that our predictions are subject to revision following future empirical findings.

If we assume that the fitness topologies of males and females diverge by small increments relative to time, and that the majority of mutations have small effects on gene expression relative to the landscape, then we can model the fitness effects of mutations along a concave fitness surface. Figure 2 presents a conceptual relationship between concave fitness surfaces and their relationship to selection and dominance parameters. Fitness (w) as a function of gene expression (x) can be formally developed with the relationship:

$$w\left(x\right)=\exp \{-c{|\widehat{x}-x|}^k\}$$

where x̂ is the gene expression optimum that maximizes fitness, and c is a positive constant (c << 1). Note that this function has an inflection point as expression deviates strongly from the optimum. Within the vicinity of the optimum, the relevant parameter space that we consider here, the fitness landscape will be concave and will not differ substantially from a parabolic function. The fitness landscape is concave when k > 1. To simplify the presentation, we present derivations using a second order function (k = 2, which is a Gaussian fitness function) and present general results for arbitrary values of k > 1.

Figure 2. Conceptual relationships between gene expression variation and dominance along a concave fitness surface.

The solid line represents a hypothetical expression-fitness function with the expression optimum occurring at the filled circle. Mutations alter expression by Δx when heterozygous and 2Δx when homozygous. The fitness effects of heterozygous and homozygous mutations depend upon the position along the fitness surface. For a population that is fixed for alleles expressing at level b (at the optimum), mutations decrease fitness and the dominance coefficient for a deleterious mutation is h_d = α₁/(α₁ + α₂) < 1/2. For a population at position a or c on the fitness surface, beneficial mutations approaching position b will have dominance coefficients h_b = α₂/(α₁ + α₂) > 1/2.

Beneficial mutations

Consider a wild-type allele A₁, which is fixed within a population and which causes gene expression at a level x₁. Following a shift in the environment, x₁ no longer resides at the fitness optimum such that x₁ ≠ x̂. A₂ is a rare mutation that changes gene expression from x₁ to x₁ + Δx when heterozygous, and to x₁ + 2Δx when homozygous. For simplicity, assume that wild-type expression is below the optimum, causing selection to favor alleles that increase expression (since the model is symmetrical, the opposite case yields the same results). Assuming that the homozygous genotype does not overshoot the expression optimum, such that x̂ ≥ x₁ + 2Δx and Δx > 0, the fitness of the three genotypes will be: w(A₁A₁) = exp{−c(x̂−x₁)²}, w(A₁A₂) = exp{−c(x̂−x₁−Δx)²}, and w(A₂A₂) = exp{−c(x̂−x₁−2Δx)²}. The dominance coefficient of A₂ over A₁ is therefore:

$$h_b=\frac{1-\exp \left\{c\mathrm{\Delta }x(2\widehat{x}-2x_1-\mathrm{\Delta }x)\right\}}{1-\exp \left\{4c\mathrm{\Delta }x(\widehat{x}-x_1-\mathrm{\Delta }x)\right\}}\approx \frac{2(\widehat{x}-x_1-\mathrm{\Delta }x)+\mathrm{\Delta }x}{4(\widehat{x}-x_1-\mathrm{\Delta }x)}=\frac{1}{2}+\frac{\mathrm{\Delta }x}{4(\widehat{x}-x_1-\mathrm{\Delta }x)}$$

which can hypothetically range between 0.5 < h_b < 0.75 (in this parameterization, the additive case has h_b = ½). Extending the theory for an arbitrary k^th-order fitness function, dominance of beneficial mutations will range between 0.5 < h_b < (2^k – 1)/2^k.

Deleterious mutations

When the population resides at the fitness peak (i.e., the wild type genotype produces optimal gene expression), all mutations are deleterious because they cause gene expression to deviate from the optimum. Fitness of a deleterious mutation in heterozygous and homozygous state (respectively) follows the functions: w(A₁A₂) = exp{−c(Δx)²}, and w(A₂A₂) = exp{−c(2Δx)²}. For this second-order function, dominance of each deleterious mutation is h_d ≈ ¼. For a generalized k^th-order function, dominance of deleterious mutations is h_d ≈ 1/2^k.

Pleiotropy

Previous models generally assume that sexually antagonistic selection arises from opposing directional selection between the sexes (e.g. Fig. 1A). An alternative possibility arises for genes that are expressed in multiple tissues or developmental stages. Mutations for such genes are likely to have pleiotropic fitness effects (e.g. Mank et al. 2008), with gene expression altered in more than one context of expression. When selection favors gene expression divergence within a single context, but mutations alter expression in multiple contexts, gene expression evolves upon a complex fitness landscape with multiple, context-specific peaks (Fig. 1B).

Pleiotropy may often be associated with gene expression-altering mutations because most genes are expressed in multiple tissues or contexts (e.g., Chintapalli et al. 2007), yet the population genetic consequences of pleiotropy have yet to be explicitly analyzed within the theoretical context of sex-specific selection (Fitzpatrick 2004; Mank 2009). To incorporate pleiotropy into the theory, we follow Curtsinger et al. (1994) and model total fitness per genotype as a multiplicative function of fitness within each context of expression (e.g., tissue or developmental stage). Thus, for a genotype ‘g’ expressed in n contexts or tissues, total fitness is $w_g={\displaystyle \prod _{i=1}^n}{w}_i$. The model is roughly equivalent to a linear fitness model where the fitness benefits and costs of a mutation are small for each context (1 – w_i << 1). For example, a bi-allelic locus with one allele (A₁) favored in j contexts and the other favored in one context (A₂), will follow the fitness scheme: w(A₁A₁)=1−s; $w\left(A_1{A}_2\right)=(1-\mathit{sh}){\displaystyle \prod _{i=1}^j}(1-t_i{h}_i)\approx 1-\mathit{sh}-{\displaystyle \underset{i=1}{\overset{j}{\Sigma }}}{t}_i{h}_i$; and $w\left(A_2{A}_2\right)={\displaystyle \prod _{i=1}^j}(1-t_i)\approx 1-{\displaystyle \underset{i=1}{\overset{j}{\Sigma }}}{t}_i$, where s and h represent selection and dominance coefficients with respect to the tissue that is under directional selection, and t₁−t_j and h₁−h_j are the selection and dominance coefficients with respect to the j tissues that are under stabilizing selection. Note that this is probably the simplest model for pleiotropy, and is limited to the case where the genotype is the unit of selection.

PROBABILITY OF INVASION

For a population with common alleles A₁ and B₁ fixed or nearly fixed (thus, stability is assessed at the equilibrium [A₁] = [B₁] = 1), the leading eigenvalue for the system of recursions provides information about the strength and direction of selection acting on rare alleles or haplotypes. The leading eigenvalue minus one (λ_L – 1) provides the equivalent of a selection coefficient at the particular point at which the system is analyzed (e.g., for a rare mutation; Otto and Bourguet 1999; Otto and Yong 2002). For our purposes, leading eigenvalues can be used to approximate the probability of invasion for new mutations. Such probabilities can potentially refer to individual alleles (A₂ or B₂), or of haplotypes (A₂B₂). For 1 >> λ_L – 1 >> 1/N_e, where N_e is the effective population size, the probability of establishment versus loss of a new mutation is:

$$P=\frac{2({\lambda }_L-1)}{{{\lambda }_L}^2}\approx 2({\lambda }_L-1)$$ (1)

(i.e., by branching process; Haldane 1927; Otto and Day 2007). The approximation is used in subsequent results, and its accuracy is verified by simulation (see Figs. S1-S2).

TEMPORAL DYNAMICS

To describe the relative rate at which sexual dimorphism evolves under different modes of inheritance, we develop expressions to characterize the mean waiting time until sex-specific expression divergence. Under the assumption of strong selection and weak mutation (SSWM; Gillespie 1984, 1991), where the strength of selection acting on a rare allele or haplotype is much greater than the reciprocal of the population size (1 >> λ_L − 1 >> 1/N_e, as described above), and the rate of mutation is much smaller than 1/N_e (i.e., N_eu << 1), the temporal dynamics of individual substitutions are dominated by the waiting time until invasion of an allele or haplotype (the time required for an allele to arise within the population and then escape stochastic loss by genetic drift), and the transit time for each selective sweep is negligible.

For models involving single-step evolutionary transitions (e.g., the invasion of alleles or haplotypes with sex-limited and tissue-specific effects), we calculate the mean waiting time until the invasion of a derived allele or haplotype. This basic approach can also be applied to sequential invasion models with epistasis (Weinreich and Chao 2005) – in our case, where initial invasion of a sexually antagonistic or antagonistic pleiotropic allele generates selection for a modifier allele causing sex- or tissue-specific divergence. As with the single-step scenario, SSWM assumptions permit us to approximate the two-step fixation process by analyzing the waiting time for each invasion event. The expected waiting time until both substitutions is equal to the sum of individual waiting times until invasion. These approaches are used widely in the theoretical literature, and are accurate under SSWM conditions (e.g., Stephan 1996; Weinreich and Chao 2005; Kim 2007).

Results

EVOLUTIONARY ROUTES TOWARD SEXUALLY DIMORPHIC GENE EXPRESSION

Sexually dimorphic gene expression can evolve by three basic evolutionary pathways. For mutations with sex-limited and tissue-specific effects on gene expression, sexual dimorphism can evolve by single, unconstrained genetic substitutions, provided that the mutation alters expression within the specific context where divergence is favored. For mutations that are constrained by sexual antagonism or pleiotropy, sex-specific adaptation may arise as a consequence of multiple genetic substitutions involving mutations whose epistatic effects generate a sex- and/or tissue-specific phenotypic response.

The fixation of combinations of mutations, although more restrictive than the fixation of sex-limited alleles, can occur either sequentially or simultaneously. If one of the mutations at the two loci is individually favored in the wild-type genetic background, it can increase from low initial frequency. Formally, this occurs when λ_A > 1 or λ_B > 1 (see Table 2). As one allele increases in frequency, selection will favor alleles at the other locus that interact positively with the invading allele such that the combination of alleles improves net fitness across the sexes. Such a two-step process leads to the eventual fixation of positively interacting pairs of alleles that optimize expression divergence between the sexes.

If mutations at the two loci are individually disfavored by selection (i.e., λ_A < 1 and λ_B < 1), but are favored in combination, sexual dimorphism can potentially evolve if the positive epistatic interactions between mutations are strong relative to the recombinational distance between the loci (Crow and Kimura 1965; Weinreich and Chao 2005). Formally, this will occur when λ_AB > 1 (Table 2). In terms of gene expression evolution, an allele combination might be favored because it decouples gene expression variation between the sexes or between different tissues, and thereby reduces evolutionary constraints imposed by sexually antagonistic selection or antagonistic pleiotropy. Both types of epistasis could arise from fitness interactions between mutations at sex- or tissue-specific and non-sex-specific cis-regulatory elements (see Williams and Carroll 2009).

The opportunity for each of these evolutionary transitions can systematically differ between sex chromosomes and autosomes. Differential accumulation of sex-linked and autosomal substitutions is a function of several factors: (1) whether males or females are selected to diverge, (2) whether alleles are sex-limited in expression or expressed in both sexes, (3) whether mutations have pleiotropic effects or instead alter expression within single tissues, and (4) whether the relative rate of recombination differs between sex chromosomes and autosomes. Below, we present these results by contrasting X-linked and autosomal inheritance. These results also apply to Z versus autosome contrasts, which represent a mirror image, with results reversed across sexes.

THE FIXATION OF SEX-LIMITED, TISSUE-SPECIFIC MUTATIONS

If a mutation only alters gene expression within the tissue and sex where divergence is beneficial, there will be no constraint to its invasion and eventual fixation. With recurrent mutation to the favored allele, the substitution will eventually become fixed, with the waiting time determined by the mutation rate and strength of selection in favor of the mutation. For an autosomal allele, the mean waiting time until fixation will be:

$$T_{\mathit{fix}}=\frac{1}{N_A{u}_A\mathrm{Pr}\left(\mathit{fix}\right)}+\tau$$ (2)

where N_A is the autosomal effective size (N_A ≈ 2N_e), u_A is the autosomal rate of mutation to the sex-limited allele, Pr(fix) is the probability that the allele eventually becomes fixed, and τ represents the transit time of the allele – i.e., the time it takes for the allele to change from frequency 1/N_A to frequency one. Given the simplifying assumption of strong selection/weak mutation (SSWM: 1 >> sh > 1/N_e > u; Gillespie 1984, 1991), the second term has little impact and can be ignored (e.g., Stephan 1996; Gillespie 2000; Weinreich and Chao 2005; Kim 2007). Furthermore, the probability of fixation for a sex-limited beneficial mutation is approximately equal to the heterozygous selection coefficient in favor of that mutation (sh_b, the result for sex-limited selection on the allele). Eq. (2) can therefore be modified to:

$$T_{\mathit{Afix}}=\frac{1}{N_A{u}_A{\mathit{sh}}_b}$$ (3a)

Under the same conditions, except with X-linked inheritance, the mean waiting times for male-limited and for female-limited beneficial substitutions (respectively) are:

$$T_{\mathit{Xfix}}\left(m\right)=\frac{3}{2N_X{u}_{X}s}$$ (3b)

and

$$T_{\mathit{Xfix}}\left(f\right)=\frac{3}{4N_X{u}_X{\mathit{sh}}_b}$$ (3c)

where N_X represents the X-linked effective size, and u_X the X-linked mutation rate per generation with respect to the allele. These results represent reciprocals of the instantaneous rate of adaptive substitution, which are often used in theories of molecular evolution, including contrasts between sex chromosomes and autosomes (Charlesworth et al. 1987; Kirkpatrick and Hall 2004; Vicoso and Charlesworth 2009a).

The specific waiting times until substitution on the X and autosomes is sensitive to the effective mutation rate on each chromosome (the ratio N_Xu_X/N_Au_A is expected to be ¾, but may vary as a result of demography, life-history, or breeding system; Charlesworth 2001, 2009; Ellegren 2007, 2009; Hedrick 2007). Nevertheless, the relative rate of fixation for male-beneficial mutations will always be more sensitive to X-linkage and dominance. Under the null expectation, 4N_Xu_X = 3N_Au_A, the relative waiting time until female-beneficial substitution is the same on the X and autosomes: T_Xfix(f)/T_Afix = 1. The relative time until male-beneficial substitution is T_Xfix(m)/T_Afix = 2h_b, where the substitution rate is always more rapid on the autosomes under the concave fitness landscape model (i.e., h_b > 1/2 when k > 1). X-linkage facilitates female-beneficial substitution as long as 4N_Xu_X > 3N_Au_A, and male-beneficial substitution when 2N_Xu_X > 3N_Au_Ah_b, with the latter requirement always more restrictive for h_b > 1/2. These results are in agreement with several previous studies concerned with “faster-X” molecular evolution (i.e., Charlesworth et al. 1987; Kirkpatrick and Hall 2004; Vicoso and Charlesworth 2009a).

INVASION OF PLEIOTROPIC MUTATIONS, FOLLOWED BY MODIFIERS OF PLEIOTROPY

Adaptive divergence may be constrained if expression-altering mutations also affect tissues evolving under stabilizing selection. Consider the evolution of a pleiotropic locus A, at which a derived mutation (A₂; A₁ represents the wild type) is beneficial within one tissue and deleterious within another. For simplicity, we assume that mutations have sex-limited effects on gene expression; the evolutionary genomic consequences of sexual antagonism are considered separately. Table 3 presents the fitness parameterization under a model of antagonistic pleiotropy, where coefficients s and t are positive and small (0 < s, t << 1), and the dominance coefficient, h_d = 1/2^k (0 < h_d < 0.5 for k > 0).

Table 3.

Fitness parameterization under the antagonistic pleiotropy model.

		Genotype
		A1, A1A1	A1A2	A2, A2A2
Female-specific fitness effects
	Female fitness	f11 = 1 − s	f21 = (1 − hds)(1 − hdt)	f31 = 1 − t
	Male fitness (A)	m11 = 1	m21 = 1	m31 = 1
	Male fitness (X)	m1 = 1	--	m2 = 1
Male-specific fitness effects
	Female fitness	f11 = 1	f21 = 1	f31 = 1
	Male fitness (A)	m11 = 1 − s	m21 = (1 − hds)(1 − hdt)	m31 = 1 − t
	Male fitness (X)	m1 = 1 − s	--	m2 = 1 − t

Under autosomal linkage and sex-specific selection in either sex, a derived allele A₂ will invade when:

$$s>\frac{{\mathit{th}}_d}{1-h_d}$$ (4a)

will become fixed when:

$$s>\frac{t(1-h_d)}{h_d}$$ (4b)

and will remain polymorphic when:

$$\frac{{\mathit{th}}_d}{1-h_d}<s<\frac{t(1-h_d)}{h_d}$$ (4c)

(these results are in agreement with those of Curtsinger et al. 1994). If A₂ remains polymorphic, it will converge to the equilibrium frequency:

$$\widehat{p}\approx \frac{s(1-h_d)-{\mathit{th}}_d}{(s+t)(1-2h_d)}$$ (4d)

These conditions of invasion, fixation, and polymorphism are the same under female-specific selection at an X-linked locus. Under male-limited selection and X-linkage, the conditions for invasion and eventual fixation are the same (s > t), and there in no parameter combination permitting a stable polymorphism.

When A₂ is favored, a single copy will invade the population with probability ~2(λ_A − 1), where λ_A is presented in the first row of Table 2. Under SSWM conditions, the mean waiting time until A₂ invades on an autosome will be:

$$T_{\mathit{aut}}\left(A_2\right)=\frac{1}{2N_A{u}_A({\lambda }_A-1)}$$ (5a)

The expected waiting time under X-linkage will be:

$$T_X\left(A_2\right)=\frac{1}{2N_X{u}_X({\lambda }_X-1)}$$ (5b)

The invasion of mutations with deleterious pleiotropic effects can subsequently select for a modifier allele (B₂) that limits expression divergence to the appropriate tissue, thereby maximizing fitness with respect to gene expression across different tissues. Assuming allelic interactions in cis-, an A₂B₂ (coupling) haplotype will cause tissue-specific expression divergence, and A₂B₁ haplotypes will be associated with both beneficial and costly expression divergence (i.e., A₂B₂ haplotypes eliminate the pleiotropic cost, whereas A₂B₁ haplotypes do not; for a complete parameterization of the 2-locus system, see Table S1). As before, invasion criteria for the B₂ allele can be addressed with local linearized stability criteria. Autosomal and X-linked Jacobian matrices were recalculated for an arbitrary equilibrium frequency of A₂, and stability was determined with the modifier allele absent ([B₂] = 0; Appendix 3). Under autosomal inheritance and tight linkage between A and B (r_m, r_f << 1/2, as expected for two cis-regulatory loci for the same gene), the leading eigenvalue with respect to the B locus is:

$${\lambda }_B\approx \frac{(1-\widehat{p})f_{22C}(1-r_f)+\widehat{p}{f}_{32}}{2{\stackrel{-}{w}}_f}+\frac{(1-\widehat{p})m_{22C}(1-r_m)+\widehat{p}{m}_{32}}{2{\stackrel{-}{w}}_m}$$ (6a)

where w̄_f and w̄_m represent mean female and male fitness as a function of the frequency of A₂, with B₁ fixed. For X-linked loci, the leading eigenvalue is:

$${\lambda }_B\approx \frac{(1-\widehat{p})f_{22C}(1-r_f)+\widehat{p}{f}_{32}}{4{\stackrel{-}{w}}_f}\left(1+\sqrt{1+8\frac{{\stackrel{-}{w}}_f{m}_4}{{\stackrel{-}{w}}_m[(1-\widehat{p})f_{22C}(1-r_f)+\widehat{p}{f}_{32}]}}\right)$$ (6b)

An analysis of these eigenvalues indicates that, given invasion of A₂ prior to the introduction of B₂, and if an A₁B₂ haplotype is not strongly deleterious, selection will generally favor the invasion of the modifier allele. As the net rate of recombination between the loci increases relative to the strength of selection, invasion of a modifier mutation is not guaranteed unless it is neutral in an A₁ genetic background, or A₂ is common within the population (see Appendix 4 for details). Given our focus on epistatic interactions in cis-, we focus our analysis of the sequential invasion model under a scenario of relatively tight linkage. We revisit this issue within the discussion.

The waiting time until B₂ invades (following invasion and equilibrium of the A₂ allele) is sensitive to the equilibrium frequency of A₂, the rate at which A₂B₂ haplotypes are formed by mutation and recombination, and the strength of selection in favor of the A₂B₂ haplotype, once created. Accounting for each of these factors (see Appendix 5), the mean waiting time until B₂ invades will fall within the boundary:

$$\frac{N_A\widehat{p}(1-\widehat{p})(r_m+r_f)+2}{2N_A\widehat{p}{v}_A({\lambda }_B-1)[N_A(1-\widehat{p})(r_m+r_f)+2]}<T_{\mathit{aut}}\left(B_2\right)<\frac{1}{2N_A\widehat{p}{v}_A({\lambda }_B-1)}$$ (7a)

under autosomal inheritance, and:

$$\frac{2N_X\widehat{p}(1-\widehat{p})r_f+3}{2N_X\widehat{p}{v}_X({\lambda }_B-1)[2N_X(1-\widehat{p})r_f+3]}<T_X\left(B_2\right)<\frac{1}{2N_X\widehat{p}{v}_X({\lambda }_B-1)}$$ (7b)

under X-linked inheritance (where N_A and N_X refer to the autosomal and X-linked effective size, v_A and v_X represent chromosome-specific mutation rates at the B locus, and λ_B is characterized by eqs. (6a) or (6b)). When A₂ is rare and A₁B₂ is neutral, the waiting time approaches the term on the left. As the equilibrium frequency of A₂ and the deleterious effect of A₁B₂ increase, the waiting time approaches the term on the right (Appendix 5).

Permissive conditions for gene expression divergence, along with the expected waiting times until A₂ and B₂ invasion, are presented in Figure 3. These representative results show that X-linkage will constrain male-specific evolutionary divergence, but will not constrain female-specific divergence. As with the sex- and tissue-limited model presented previously, X-linkage will actually enhance the rate of female-specific adaptation as long as 4N_Xu_X > 3N_Au_A, and will be equal when 4N_Xu_X = 3N_Au_A.

Figure 3. Sequential invasion conditions and waiting times until tissue-specific gene expression divergence under antagonistic pleiotropy.

Pleiotropically-expressed alleles (A₁ and A₂) follow the fitness parameterization from Table 3. Opportunities for invasion are described by the shaded bar at the base of each panel. The curves represent the mean time until derived alleles invade at the pleiotropic and modifier loci, with the uniform curve representing the X-linked scenario and the circle-bearing curve representing autosomal linkage. Results were obtained using the SSWM approximations presented in the text. Results are shown for k = 2 (h_d = ¼), t = 0.01, and utilize the permissive left hand term of eqs. (7a-7b); mutation rates per locus and per sex are u = v = 10⁻⁸; X and autosome effective sizes are N_A = 10⁶ = 4N_X/3. Results use dimorphic recombination parameters r_f = 0.001 and r_m = 0, yet sexually monomorphic recombination yields a nearly identical pattern.

These patterns are a natural consequence of evolution along a concave fitness surface. Concave fitness landscapes generate dominance reversals: deleterious fitness effects act recessively, whereas beneficial effects are dominant (see above; Fig. 2). In females, which are diploid throughout the genome, the expression of beneficial and deleterious fitness effects of individual mutations does not differ between the X and autosomes. However, female-limited selection can be more effective on the X because each X-linked locus evolves within a female genome with two-thirds probability, in contrast to one-half probability for autosomes. In males, benefits of expression divergence are marginally enhanced by X-linked inheritance because they are dominant, whereas costs of expression divergence are strongly enhanced by X-linkage because they act recessively. For male-specific adaptive divergence, pleiotropy should generate a strong bias off of the X because pleiotropic constraints in males are exacerbated by hemizygous expression. The severity of the bias will increase with the curvature of the fitness landscape (i.e., as k increases).

INVASION OF SEXUALLY ANTAGONISTIC MUTATIONS, FOLLOWED BY MODIFIERS OF SEX-LIMITATION

Mutations that are expressed in both sexes can give rise to sexual antagonism if they are favored in one sex but disfavored in the other. The invasion of each sexually antagonistic allele can subsequently generate selection for alleles at secondary loci that limit expression divergence to the sex exposed to directional selection. Because invasion at these secondary loci are limited by prior invasion at the sexually antagonistic locus, the opportunity for sequential invasion depends on whether or not each sexually antagonistic allele can invade and trigger the two-step process of divergence. The specific conditions for invasion have been described in several previous studies (e.g., Kidwell 1977; Pamilo 1979; Rice 1984; Albert and Otto 2005; Patten and Haig 2009; Fry 2010). We summarize these results below, with relevant eigenvalues (Table 2) evaluated using the fitness landscape model described above and in Table 4.

Table 4.

Fitness parameterization under the sexual antagonism model.

		Genotype
		A1, A1A1	A1A2	A2, A2A2
Selection for expression divergence in females
	Female fitness	f11 = 1 − sf	f21 = 1 − hdsf	f31 = 1
	Male fitness (A)	m11 = 1	m21 = 1 − hdsm	m31 = 1 − sm
	Male fitness (X)	m1 = 1	--	m2 = 1 − sm
Selection for expression divergence in males
	Female fitness	f11 = 1	f21 = 1 − hdsf	f31 = 1 − sf
	Male fitness (A)	m11 = 1 − sm	m21 = 1 − hdsm	m31 = 1
	Male fitness (X)	m1 = 1 − sm	--	m2 = 1

When divergence is favored in males, an autosomal male-beneficial allele can invade when:

$$s_m>\frac{s_f{h}_d}{1-h_d(1-s_f)}$$ (8a)

Such an allele will go to fixation when:

$$s_m>\frac{s_f(1-h_d)}{h_d(1-s_f)}$$ (8b)

The system will remain polymorphic when:

$$\frac{s_f{h}_d}{1-h_d(1-s_f)}<s_m<\frac{s_f(1-h_d)}{h_d(1-s_f)}$$ (8c)

Given our constraint that h_d < 0.5, the male-beneficial polymorphism will converge to the equilibrium frequency:

$$\widehat{p}\approx \frac{s_m(1-h_d)-s_f{h}_d}{(s_m+s_f)(1-2h_d)}$$ (8d)

When females are selected to diverge, the same results apply, with s_m and s_f substituted.

The invasion of an X-linked, male-beneficial mutation can occur when:

$$s_m>\frac{2s_f{h}_d}{1+s_f{h}_d}$$ (9a)

These alleles will go to fixation when:

$$s_m>\frac{2s_f(1-h_d)}{1-s_f{h}_d}$$ (9b)

An X-linked, female-beneficial allele can invade when:

$$s_f>\frac{s_m}{2-h_d(2-s_m)}$$ (9c)

and will become fixed when:

$$s_f>\frac{s_m}{h_d(2-s_m)}$$ (9d)

X-linked polymorphism will be maintained under the condition:

$$\frac{2s_f{h}_d}{1+s_f{h}_d}<s_m<\frac{2s_f(1-h_d)}{1-s_f{h}_d}$$ (9e)

with the male- and female-beneficial alleles converging to the respective equilibrium frequencies:

$${\widehat{p}}_m\approx \frac{s_m-2s_f{h}_d}{2s_f(1-2h_d)}$$ (9f)

and

$${\widehat{p}}_f\approx \frac{2s_f(1-h_d)-s_m}{2s_f(1-2h_d)}$$ (9g)

The equilibria each require that h_d < 0.5, as expected for the concave fitness landscape model.

The expected time until invasion of a derived, sexually antagonistic allele (A₂) is given by eqs. (5a) and (5b), with the relevant eigenvalues (λ_A) parameterized using fitness values from Table 4. Coupling haplotypes with derived sexually antagonistic and modifier alleles (A₂B₂) will cause sex-specific expression divergence, while repulsion haplotypes (A₂B₁) generate expression divergence in both sexes (for a complete parameterization of the 2-locus system, see Table S2). As with antagonistic pleiotropy (again, assuming that A₁B₂ haplotypes are not strongly deleterious), most conditions favoring invasion of a sexually antagonistic allele will also subsequently favor invasion of a linked, cis-regulatory modifier of sex-limitation (see Appendix 4). The general stability conditions and waiting times until the B₂ allele invades (conditional on A₂ reaching an equilibrium frequency greater than zero) are given by eqs. (6a - 7b); see Appendix 3 and 4 for the additional case of relatively loose linkage.

Results for the sexually antagonistic modifier model show that gene expression divergence, involving the sequential invasion of sexually antagonistic and modifier alleles, will generally be constrained by X-linked inheritance (Figure 4). The conditions permitting such a process are broader on the autosomes. Furthermore, the mean waiting time until the resolution of sexual antagonism (i.e., the invasion and fixation of A₂ and B₂ alleles) will be abbreviated on the autosomes relative to the X. These results apply to both male- and female-specific expression divergence. The underlying mechanism for the bias off the X is similar to the case of male-specific divergence under antagonistic pleiotropy: sexually antagonistic fitness costs (to the sex exposed to stabilizing selection) are exacerbated by X-linkage and minimized by autosomal linkage, which limits opportunities for the invasion of sexually antagonistic alleles on the X. These results are in agreement with a recent analysis by Fry (2010), which focused on opportunities for stable polymorphism on the X and autosomes, and runs counter to the intuition that sexual antagonism will generally lead to an enrichment of sex-linked, sex-biased genes.

Figure 4. Sequential invasion conditions and waiting times until sex-specific gene expression divergence under sexual antagonism.

Opportunities for invasion are described by the shaded bar at the base of each panel. The curves represent the mean time until derived alleles invade at the sexually antagonistic and modifier loci, with the uniform curve representing the X-linked scenario and the circle-bearing curve representing autosomal linkage. Results were obtained using the SSWM approximations presented in the text. Sexually antagonistic alleles (A₁ and A₂) follow the fitness parameterization from Table 4. Results are shown for k = 2 (h_d = ¼), s_f = 0.01 for the case of directional selection in males, and s_m = 0.01 for directional selection in females; mutation rates per locus and per sex are u = v = 10⁻⁸; X and autosome effective sizes are N_A = 10⁶ = 4N_X/3. Results use dimorphic recombination parameters r_f = 0.001 and r_m = 0, yet sexually monomorphic recombination yields a nearly identical pattern.

SIMULTANEOUS INVASION OF EPISTATICALLY BENEFICIAL MUTATIONS

Strong selective constraints can limit opportunities for gene expression divergence. If individual mutations evolve under a net purifying selection (averaged across tissues or across the sexes; formally, when λ_A < 1 and λ_B < 1; Table 2), and purifying selection is strong enough to prevent the fixation of individual mutations by genetic drift (1 − λ_A, 1 − λ_A >> 1/N_e), then sex-specific evolutionary divergence requires the simultaneous substitution of positively interacting pairs of mutations. These pairs of mutations can simultaneously invade when selection in favor of a double-mutant haplotype is strong relative to the probability of recombination between the mutations (Crow and Kimura 1965; Weinreich and Chao 2005; Kim 2007).

The specific parameter conditions that are conducive to simultaneous invasion can systematically differ between sex chromosomes and autosomes, with invasion conditions for the derived haplotype depending on: (i) whether the haplotype is favored in males or in females; (ii) whether it is X-linked or autosomal; and (iii) the degree of linkage (1 − r_f, 1 − r_m) between the two loci. The eigenvalue λ_AB (Table 2) characterizes whether an A₂B₂ haplotype can invade a population fixed for A₁B₁. Assuming that each A₂B₂ haplotype causes expression divergence within the appropriate sex and tissue (i.e., sexually antagonistic or pleiotropic fitness costs are absent in the beneficial haplotype A₂B₂; under male-specific selection, fitness is defined as m₁₁ = m₁ = 1 − s_m, m_22C = 1 − h_ds_m, and m₄ = 1; under female-specific selection, fitness is defined as f₁₁ = 1 − s_f, and f_22C = 1 − h_ds_f; these are used to evaluate λ_AB), simultaneous invasion of a male-beneficial haplotype will be favored on the autosomes when:

$$s_m>\frac{r_m+r_f}{1+r_f-h_d(1-r_m)}$$ (10a)

Simultaneous invasion of an autosomal, female-beneficial haplotype can occur when:

$$s_f>\frac{r_m+r_f}{1+r_m-h_d(1-r_f)}$$ (10b)

Under X-linkage, simultaneous invasion of a male-beneficial haplotype can occur when:

$$s_m>\frac{2r_f}{1+r_f}$$ (10c)

Invasion of a female-beneficial haplotype is favored when:

$$s_f>\frac{r_f}{1-h_d(1-r_f)}$$ (10d)

The waiting time until simultaneous invasion of a pair of linked alleles depends on both the probability of invasion of an A₂B₂ haplotype, as well as the rate at which A₂B₂ haplotypes are created by mutation or recombination. Assuming 1 >> λ_AB − 1 >> 1/N_e, the probability of invasion and eventual fixation of a male-beneficial haplotype will be:

$$\mathrm{Pr}(A_2{B}_2\mid m,\mathrm{aut}.)\approx 2({\lambda }_{\mathit{AB}}-1)=\frac{(1-h_d{s}_m)(1-r_m)-(1-s_m)(1+r_f)}{(1-s_m)}$$ (11a)

on the autosomes, and:

$$\mathrm{Pr}(A_2{B}_2\mid m,\mathrm{X})\approx \frac{1-r_f}{2}\left(1+\sqrt{1+8\frac{1}{(1-s_m)(1-r_f)}}\right)-2$$ (11b)

on the X. The invasion probability of a female-beneficial haplotype will be:

$$\mathrm{Pr}(A_2{B}_2\mid f,\mathrm{aut}.)\approx \frac{(1-h_d{s}_f)(1-r_f)-(1-s_f)(1+r_m)}{(1-s_f)}$$ (11c)

on the autosomes, and:

$$\mathrm{Pr}(A_2{B}_2\mid f,\mathrm{X})\approx \frac{(1-h_d{s}_f)(1-r_f)}{2(1-s_f)}\left(1+\sqrt{1+8\frac{(1-s_f)}{(1-h_d{s}_f)(1-r_f)}}\right)-2$$ (11d)

on the X.

To calculate the rate at which A₂B₂ haplotypes are generated, we assume (following previous theory; see Weinreich and Chao 2005; Kim 2007) that the rate of recombination between repulsion haplotypes, A₂B₁ and A₁B₂, is negligible (as expected when r_m, r_f << 1, a requirement for invasion, and when A₂B₁ and A₁B₂ are rare, as expected under purifying selection), and that parameters of mutation and purifying selection are the same at locus A and B. Given these simplifications, the rate at which A₂B₂ haplotypes are created is 2u²/ω, where u is the mutation rate at each locus, per generation, and ω is the net strength of purifying selection acting on each deleterious haplotype. Incorporating the rate of creation and probability of fixation for A₂B₂, the mean waiting time until invasion will be:

$$T(A_2{B}_2\mid i,j)=\frac{{\omega }_j}{2N_j{u}^2\mathrm{Pr}(A_2{B}_2\mid i,j)}$$ (12)

where j can refer to X or autosomal linkage, and i refers to male or female selection (subscripts m and f). Characterizing the net strength of purifying selection on deleterious haplotypes requires a detailed specification of the nature of sex-specific selection acting on A₂B₁ and A₁B₂ haplotypes. Although space limitation precludes a detailed analysis of all possibilities for the X and autosomes, we present waiting time results under different ratios: R_ω = ω_X/ω_A. This is the most salient factor affecting the relative rate of A₂B₂ creation on each chromosome.

Invasion conditions and waiting times, under the simultaneous substitution model, are presented in Figs. 5 and 6. In species where recombination only occurs in females (e.g., Drosophila), the invasion conditions of female-beneficial haplotypes are roughly the same between the X and autosomes, whereas male-beneficial haplotypes are constrained by X-linkage (Fig. 5). In species with recombination in both sexes (e.g., mammals), X-linkage expands the invasion conditions of both female- and male-beneficial haplotypes (Fig. 6). These species-specific results reflect an underlying constraint that will often prevent pairs of mutations from invading simultaneously: rare, beneficial allelic combinations can spread if they tend to be co-inherited. Recombination between loci breaks apart beneficial genetic combinations, thereby preventing the invasion of adaptive genetic complexes. A closer look at conditions (10a – 10d) provides an explanation for the effect of recombination on the relative haplotype invasion conditions on the X and autosomes. When there is no recombination in males (r_m = 0), the minimum selection coefficient required for a female-beneficial haplotype to invade is identical between the X and autosomes; with male-specific selection, autosomal linkage is more conducive to invasion when h_d < 0.5, approximately (assuming that r_f << 1). As r_m increases, recombination becomes effectively higher on the autosomes, the minimum conditions necessary for invasion will increase for autosomal haplotypes, and X-linkage will facilitate the invasion of both male- and female-beneficial allelic combinations.

Figure 5. Simultaneous invasion conditions and waiting times for epistatically beneficial haplotypes when males do not recombine.

Opportunities for invasion are described by the shaded bar at the base of each panel. The curves represent the mean time until double-mutant haplotypes invade, with the uniform curve representing the X-linked scenario and the circle-bearing curve representing autosomal linkage. The net autosomal purifying selection against A₂B₁ or A₁B₂ haplotypes is ω_A = 0.001, with results for two ω_X/ω_A ratios shown. Results are shown for k = 2 (h_d = ¼), s_f = 0.01 for the case of directional selection in males, and s_m = 0.01 for directional selection in females; mutation rates per locus and per sex are u = v = 10⁻⁷; X and autosome effective sizes are N_A = 10⁶ = 4N_X/3. Results were obtained using the SSWM approximations presented in the text.

Figure 6. Simultaneous invasion conditions and waiting times for epistatically beneficial haplotypes when both sexes recombine.

Opportunities for invasion are described by the shaded bar at the base of each panel. The curves represent the mean time until double-mutant haplotypes invade, with the uniform curve representing the X-linked scenario and the circle-bearing curve representing autosomal linkage. The results follow the same parameterization as presented in Figure 5, with r_m = r_f, and were obtained using the SSWM approximations presented in the text.

The waiting time until beneficial haplotypes become fixed is sensitive to the strength of positive selection, the effective rate of recombination, and the pattern of net purifying selection against individual derived alleles (Figs. 5, 6). When the strength of purifying selection against disfavored alleles is greater on the X (i.e., ω_X/ω_A > 1), the underlying conditions permitting invasion will be unaffected. However, the waiting time until invasion will increase on the X, due to a decrease in the rate at which A₂B₂ haplotypes arise within the population.

Because our model invokes an interaction between two pre-specified loci, and the ancestral population (A₁B₁ fixed) is initially two mutation steps away from a beneficial haplotype, the waiting time for simultaneous invasion can be relatively long. Nevertheless, the actual rate of epistatic coevolution might still be substantial. Simultaneous invasion is limited by the rate at which beneficial haplotypes arise within a population. This, in turn, depends on two factors: (1) the number of genes, at any given time, that are exposed to sex-specific selection for expression divergence; and (2) the number of epistatically interacting mutational combinations that influence sex-specific expression variation at each gene. If there are a large number of genes under sex-specific disruptive selection, or many epistatically beneficial allele combinations for each gene, then epistatic transitions toward sexually dimorphic expression may be common.

Discussion

The ubiquity of sexual reproduction (Bell 1982; Rice 2002), along with widespread observations of selection differences between males and females (Andersson 1994; Arqvist and Rowe 2005), has inspired a large and growing body of evolutionary theory. Sex-specific selection has implications for several important topics in evolutionary biology, including the maintenance of genetic variation for fitness, the genomic architecture of species differences and sex-specific traits, the opportunity for and rate of adaptation in sexually reproducing populations, and the evolution of female mating biases (e.g., Manning 1984; Kondrashov 1988; Koeslag and Koeslag 1994; Whitlock 2000; Agrawal 2001; Siller 2001; Rice and Chippindale 2001; Lorch et al. 2003; Fedorka and Mousseau 2004; Albert and Otto 2005; Scotti and Delph 2006; Pischedda and Chippindale 2006; Hadany and Beker 2007; Candolin and Heuschele 2008; Calsbeek and Bonneaud 2008; Bonduriansky and Chenoweth 2009; Van Doorn 2009; Whitlock and Agrawal 2009; Cox and Calsbeek 2009; Cox and Calsbeek 2010; Connallon 2010; Connallon et al. 2010; Blackburn et al. 2010).

The theory presented here builds upon several independent contributions, including the population genetics of sexual antagonism (e.g., Owen 1953; Haldane 1962; Kidwell et al. 1977; Pamilo 1979; Patten and Haig 2009; Fry 2010; and especially Rice 1984) and antagonistic pleiotropy (e.g., Curtsinger 1994; Prout 1999), X versus autosome molecular evolution (Charlesworth 1987; Kirkpatrick and Hall 2004; Vicoso and Charlesworth 2009a), the evolution of peak shifts (e.g., Crow and Kimura 1965; Weinreich and Chao 2005; Kim 2007), and the evolutionary and physiological basis of allelic dominance (e.g., Wright 1934; Kacser and Burns 1981; Dekel and Alon 2005; Gout et al. 2010). Below, we discuss the potential role of sex-specific selection during the evolution of sex-biased gene expression, by: (i) outlining the predictions of our models within the context of previous theory; and (ii) contrasting observed genomic patterns of sex-biased gene expression with theoretical predictions.

SEX-SPECIFIC SELECTION AND THE EVOLUTION OF SEX-BIASED GENE EXPRESSION

Sex-biased gene expression might evolve through adaptive shifts by males or by females. For example, a male-biased gene might evolve in response to stabilizing selection in males and selection for decreased expression in females, or through stabilizing selection in females and selection for increased expression in males. In other words, the observation of sex-biased expression is consistent with selection for either male or for female gene expression divergence, and additional information is required in order to equate sex-biased expression with male- or female-specific adaptive divergence (Connallon and Knowles 2005). To interpret genomic patterns of sex-biased expression within the theoretical framework developed here, the frequency of each evolutionary route toward male- and female-biased gene expression must be known. Within Drosophila, the answer appears to be clear: the evolution of male-biased gene expression typically involves expression increases in males (male divergence); female-biased gene expression involves expression increases in females (female divergence; Vicoso and Charlesworth 2009b). Although we acknowledge that additional research will be required to assess the generality of this pattern, the following discussion is based on this clear pattern from Drosophila. That is, our interpretation of the data assumes that male-biased genes evolve by directional selection for male expression divergence, and female-biased genes evolve by directional selection for female expression divergence. Our theoretical results are valid whether or not this assumption holds true, yet interpretation of the empirical data, in light of the theory, is subject to revision pending future research.

Given the empirical relationship between sex-biased gene expression and sex-specific divergence described above, sex-specific selection theory makes five predictions about the genomic distribution of male- and female-biased genes:

• (1) When mutations underlying sex-specific expression divergence have sex-limited and tissue-specific effects, genes that are highly expressed in the heterogametic sex (males in species with an X, females with a Z) will more readily evolve on autosomes. Genes that are highly expressed in the homogametic sex (XX females or ZZ males) may become preferentially sex linked when the effective mutation rate is relatively large per sex-linked locus (4N_Xu_X > 3N_Au_A or 4N_Zu_Z > 3N_Au_A) – a condition that is likely to vary between different animal lineages (for recent reviews of the theory and data, see Charlesworth 2009; Ellegren 2009). This prediction has obvious parallels with those of sex-linked versus autosome molecular evolution theory (see Kirkpatrick and Hall 2004; Vicoso and Charlesworth 2009a).

• (2) Pleiotropy greatly accentuates the patterns predicted under a model of sex- and tissue-limited mutation. Genes that are highly expressed within the homogametic sex are expected to accumulate at similar rates on the sex chromosomes and autosomes, and may disproportionately accumulate on the X or Z when 4N_Xu_X > 3N_Au_A or 4N_Zu_Z > 3N_Au_A. Sex linkage severely constrains adaptive gene expression divergence in the heterogametic sex, by reducing the range of parameters that are conducive to evolutionary divergence and by extending the mean waiting time until favored alleles invade in the population. To our knowledge, this result has not previously been reported in the population genetics literature.

• (3) Sexual antagonism over a gene's expression level can represent a greater evolutionary constraint for sex-linked loci. Under a model of sequential coevolution between sexually antagonistic alleles and modifiers of sex-limited expression, sex chromosomes are less hospitable to both male- and female-biased genes. This result contradicts the widespread intuition that sex linkage should generally promote the evolution of sex-biased expression from an initially sexually antagonistic state – an expectation based on the assumption of constant allelic dominance for beneficial and deleterious mutations (see Rice 1984; Patten and Haig; Fry 2010). Concave fitness landscapes represent an interesting and possibly widespread example of context-dependent dominance, where deleterious alleles will be partially recessive and beneficial alleles will be partially dominant. Fry (2010) recently demonstrated that such dominance reversals will accommodate an excess of balanced sexually antagonistic polymorphism on the autosomes relative to the X. Our study extends this result to show that, if sex-biased expression divergence involves the sequential invasion of sexually antagonistic alleles, followed by modifiers of sex-limited expression divergence, sex-biased genes are likely to accumulate more readily on autosomes (i.e., the invasion conditions are more permissive, and the waiting time until resolution is shorter).

• (4) The relative rate of recombination in males vs. females can have a major influence on the evolution and genomic distribution of sex biased genes. In species where recombination occurs in one sex only (i.e., the homogametic sex; Haldane 1922; Huxley 1928; Lenormand and Dutheil 2005), the probability of invasion for sex-linked relative to autosomal genetic combinations that cause sex-biased expression will decrease when they benefit the nonrecombining sex, and increase when they benefit the recombining sex. When both sexes recombine, sex linkage will facilitate the evolution of both male- and female-biased gene expression. This finding also provides a novel prediction for evolutionary theories of adaptive peak shifts, which had not previously considered sex-linked inheritance (e.g., Crow and Kimura 1965; Weinreich and Chao 2005; Kim 2007).

• (5) The relative contribution of linkage and epistasis to genomic patterns of sexually dimorphic expression will ultimately depend on the strength of evolutionary constraints influencing the evolution of interacting loci. As such, our results can be viewed along a continuum of evolutionary constraint. Under weak constraint, individual derived alleles can reach a high population frequency or become fixed in the population. This initial invasion event can also generate selection for epistatic modifiers, and physical linkage between epistatically interacting loci becomes less relevant to the evolutionary dynamics. Under strong constraints, antagonistically selected alleles persist as rare balanced polymorphisms or as ephemeral deleterious mutations. Coadaptation between these alleles and expression modifiers is facilitated – and in some cases may require – tight physical linkage between the interacting loci. This suggests that regional genomic patterns of linkage and recombination will most strongly influence the evolution of sex-biased genes from strongly constrained precursors. Weakly constrained genes may show less sensitivity to physical linkage.

SEX-SPECIFIC SELECTION AND THE OBSERVED CHROMOSOMAL DISTRIBUTION OF SEX-BIASED GENES

Observed chromosomal distributions of sex-biased genes are compatible with the theory outlined here. One of the most striking empirical patterns is the positive association between sex linkage and preferential expression within the homogametic sex. Such linkage patterns have been reported in species of Caenorhabditis elegans, Drosophila, mouse, silkworm, and chicken (e.g., Reinke et al. 2004; Ranz et al. 2003; Khil et al. 2004; Arunkumar et al. 2009; Kaiser and Ellegren 2006). This particular pattern is exclusively predicted by models of sex-specific selection, and can potentially arise via multiple evolutionary pathways (e.g., through invasion of sex-limited mutations and coevolution of epistatically interacting, linked mutations).

Genes that are preferentially expressed within the heterogametic sex exhibit inconsistent chromosomal distributions between species, yet these lineage-specific patterns are both informative and compatible with models of sex-specific selection. In Drosophila, where males do not recombine, there is a deficit of X-linkage for genes with male-biased expression and male-specific function (e.g., accessory gland proteins; Swanson et al. 2001; Parisi et al. 2003; Mueller et al. 2005; Sturgill et al. 2007). In birds and mammals, where recombination rates are relatively similar between the sexes (Groenen et al. 2000; Stauss et al. 2003; Lenormand and Dutheil 2005; Hansson et al. 2005; Akesson et al. 2007; Hale et al. 2008; Backström et al. 2008; Stapley et al. 2008; Jaari et al. 2009), genes that are preferentially expressed in the heterogametic sex are enriched on the X or Z (Wang et al. 2001; Lercher et al. 2003; Khil et al. 2004; Mořkovský et al. 2010). These lineage-specific patterns suggest an important role for epistatic coevolution between linked cis-regulatory loci during the evolution of sex-biased expression, and also agree nicely with widespread observations of cis- interactions mediating sex-specific regulation during development (Williams and Carroll 2009).

Whether or not interacting mutations are fixed sequentially or simultaneously, a negative relationship is expected between sex linkage and the relative pleiotropy of male-biased genes. To date, two studies have been published that are relevant to this prediction. Fitzpatrick (2004) analyzed the chromosomal distribution of 63 Drosophila “sexually selected” genes associated with courtship behavior, mating receptivity and seminal fluid, and found that the number of X-linked genes was proportional to the size of the X (as a function of euchromatin), suggesting a random chromosomal distribution of sexually selected genes. However, many genes in the study were detected through their mutant phenotypes, and X-linked mutations are often easier to detect (e.g., Haldane 1935). Using a random set of visible mutations to establish a baseline expectation of X-linkage (Table A1 of Fitzpatrick 2004), the proportion of sex-linked, sexually selected genes appears to be lower than expected by chance (Table A1 set: 39% X-linked; sexually selected set: 27% X-linked). Since mutations at these sexually selected genes are generally pleiotropic, the pattern agrees with the theoretical prediction regarding X-linkage and pleiotropy. In a human study using gene expression data from 14 different tissues, Lercher et al. (2003) found that expression breadth (as a metric of pleiotropy) was significantly lower for X-linked relative to autosomal genes. An analysis of male-specific tissues (genes exclusively expressed in the prostate) enhances this bias towards low pleiotropy on the X (again, as predicted by the theory).

Two additional properties of X and Z chromosomes are likely to influence linkage patterns of sex-biased genes. Sex chromosomes are inactivated during male meiosis in mammals and Drosophila (Lifschytz and Lindsley 1972; Solari 1974; Hense et al. 2007; Turner 2007), and during female meiosis in the chicken (Schoenmakers et al. 2009). There is now considerable evidence from Drosophila that this process of meiotic sex chromosome inactivation (MSCI) constrains the evolution and retention of sex-linked male-biased genes, particularly those expressed in testis (Bétran et al. 2002; Arbeitman et al. 2002; Parisi et al. 2003; Wu and Xu 2003; Meisel et al. 2009; Vibranovski et al. 2009a, 2009b). Constraints against sex-biased gene expression may also arise because the heterogametic sex carries a single copy of each X- or Z-linked gene. To the extent that gene expression levels are constrained by dosage compensation, highly expressed genes within the heterogametic sex are expected to be disproportionately autosomal (Rogers et al. 2003; Vicoso and Charlesworth 2006), a prediction with some support from Drosophila (Vicoso and Charlesworth 2009b; Bachtrog et al. 2010). Because these processes tend to skew male-biased genes toward autosomes, evolutionary constraint imposed by MSCI and dosage compensation will tend to reinforce linkage patterns driven by sex-specific selection.

A distinct lack of non-randomness between X and autosome gene content has been reported for the mosquito Anopheles gambiae (Hahn and Lanzaro 2005). This result may be shaped by a suite of lineage-specific factors, including MSCI (reported to occur in Anopheles; McKee and Handel 1993), dosage compensation (unconfirmed), as well as linkage and recombination. Recombination occurs in male and female Anophelines – possibly at similar rates (Benedict et al. 2003) – which could effectively depress the recombination rate per generation on the X and generate a more favorable environment for male-biased genes compared to Drosophila. On the other hand, sample size may affect these conclusions, as relatively few genes included in the analysis are X-linked. Considering the entire dataset and using a two-fold gene expression cut-off to define male- and female-biased expression (M/F > 2 and M/F < 0.5, respectively; data obtained from the SEBIDA database: Gnad and Parsch 2006), A. gambiae shows a slight enrichment in female-biased genes (X = 8.3 %; n = 782 female-biased genes) and a deficit of male-biased genes (X = 5.0 %; n = 481 male-biased genes) relative to the genome-wide expectation (X = 7.1 % for the entire dataset; n = 4281 total genes). However, given the low proportion of X-linkage, neither difference is statistically significant.

THE SHAPE OF THE EXPRESSION-FITNESS LANDSCAPE

Predictions of the theory depend on the shape of the relationship between gene expression and fitness, which we have assumed follows a concave function, at least within the region of trait space where the evolution of sexually dimorphic expression occurs. If most gene expression evolution occurs within the vicinity of a local fitness optimum, then the assumption of fitness concavity does not appear to be controversial, and indeed, several lines of empirical and theoretical evidence support it (e.g., Wright 1934; Charlesworth 1979; Kacser and Burns 1981; Phadnis and Fry 2005; Dekel and Alon 2005; Lunzer et al. 2005; Kalisky et al. 2007; Bedford and Hartl 2009; Zhang et al. 2009; see above). This assumption is further justified by analogy to mathematical theories of DNA adaptation, which suggest that beneficial substitutions will follow a diminishing returns function (Gillespie 1984; see Orr 2005 for a clear discussion of this body of work). The specific degree of fitness concavity, which we incorporate into our model with parameter k (where the fitness function is concave whenever k > 1, and the degree of concavity increases with k), cannot currently be inferred empirically. We simply note that the actual value of k has little impact on qualitative predictions of the models as long as k > 1 (see supplementary Figures S3-S6 for analyses with different values of k). Thus, to the extent that selection for sex-specific divergence occurs on a concave expression-fitness function, our qualitative predictions will be robust to the actual degree of concavity.

The concavity assumption may be violated for genes that, prior to adaptation, are severely distant from a local fitness optimum. Although concavity near the optimum is expected, linear (k = 1) or convex (0 < k < 1) fitness landscapes could potentially arise within other regions of trait space. Given our analysis above, convex fitness surfaces will cause a shift toward recessive fitness effects of beneficial mutations (in terms of our model, h_b < 0.5 when 0 < k < 1). Under this condition, the probability of invasion for a male-beneficial mutation will be enhanced by X-linkage, a scenario that is more in line with Rice's (1984) classic analysis of sexually antagonistic alleles.

Although we focus on cis-regulatory substitutions, our results can easily be generalized to incorporate a possible role of trans-regulatory mutations during gene expression evolution. As with cis-regulatory modifiers of sex- or tissue-limitation, the invasion of pleiotropic or sexually antagonistic alleles can potentially generate selection in favor of trans-acting modifier alleles (this is similar to the scenario studied by Rice 1984). To the extent that trans-acting modifiers are unconstrained by pleiotropy, sequential coevolution between cis- and trans- alleles may be likely. This also opens up the possibility of epistatic coevolution between loci on different chromosomes (including coevolution between the X, Y, and autosomes), which may be important during the evolution of reproductive isolation between species (see Ortiz-Barrientos et al. 2007). However, simultaneous invasion of positively interacting cis- and trans- mutations will generally be impossible if such mutations are not closely linked. Thus, cis-/cis- epistasis should permit a broader range of evolutionary transitions relative to cis-/trans- or trans-/trans- interactions. Because trans-regulatory mutations are also expected to be more pleiotropic, they should provide little opportunity for adaptation compared to cis-regulatory mutations (Stern 2000; Prud'homme et al. 2007; Wray 2007). Nevertheless, current genomic analyses of sex-biased gene expression cannot rule out a role of trans-substitutions during the evolutionary origin of sex-biased expression, and future empirical research will be required to further address this issue.

Conclusion

Sex-specific selection, particularly selection for gene expression divergence between males and females, can play a prominent role in shaping the genomic distributions of sex-biased genes and genes underlying sexual dimorphism. We show that, while the underlying patterns of selection can be complex, the genomic predictions of sex-specific selection hypotheses are generally consistent with empirical distributions of sex-biased genes, though the current data applies to a relatively small (but growing) set of well-characterized species. Future genome-wide analyses of sex-specific genetic architecture, including the distributions of genes with sex-specific expression patterns in non-model organisms, will permit a better evaluation of these theoretical predictions.

One notable pattern emerging from this theory is that chromosomal distributions of sex-biased genes are likely to have both gene- and species-specific attributes. These predictions may help to inform future genomic analyses, and may point toward the specific evolutionary processes underlying a diversity of linkage patterns between gene categories and species.

Finally, we focus on selection acting on gene expression variation in an attempt to explain the evolution of a gene expression phenotype (i.e., sex-biased gene expression) that is ubiquitous at the molecular genomic scale. Intersexual divergence in gene expression may play a general role during the evolution of sexually dimorphic phenotypes, including sex-specific morphology, behavior and life history. On the other hand, the genetic basis of sexual dimorphism may critically depend on the types of genetic substitutions that are differentially favored between males and females. In principle, disruptive selection might favor protein sequence divergence between the sexes, yet it is difficult to see how conflicts over coding sequence would necessarily become resolved by the evolution of dimorphic gene expression alone. Additional theory, particularly theory involving the evolution of alternative splicing and gene duplication, will be required to address how sexual dimorphism might evolve from an initial condition of coding sequence conflict between the sexes. Experimental research will also be required to better understand the degree to which coding and non-coding mutations contribute to sex-specific fitness variation (for a promising step toward this difficult goal, see Innocenti and Morrow 2010).