The Resolution of Sexual Antagonism by Gene Duplication

Abstract

Disruptive selection between males and females can generate sexual antagonism, where alleles improving fitness in one sex reduce fitness in the other. This type of genetic conflict arises because males and females carry nearly identical sets of genes: opposing selection, followed by genetic mixing during reproduction, generates a population genetic “tug-of-war” that constrains adaptation in either sex. Recent verbal models suggest that gene duplication and sex-specific cooption of paralogs might resolve sexual antagonism and facilitate evolutionary divergence between the sexes. However, this intuitive proximal solution for sexual dimorphism potentially belies a complex interaction between mutation, genetic drift, and positive selection during duplicate fixation and sex-specific paralog differentiation. The interaction of these processes—within the explicit context of duplication and sexual antagonism—has yet to be formally described by population genetics theory. Here, we develop and analyze models of gene duplication and sex-specific differentiation between paralogs. We show that sexual antagonism can favor the fixation and maintenance of gene duplicates, eventually leading to the evolution of sexually dimorphic genetic architectures for male and female traits. The timescale for these evolutionary transitions is sensitive to a suite of genetic and demographic variables, including allelic dominance, recombination, sex linkage, and population size. Interestingly, we find that female-beneficial duplicates preferentially accumulate on the X chromosome, whereas male-beneficial duplicates are biased toward autosomes, independent of the dominance parameters of sexually antagonistic alleles. Although this result differs from previous models of sexual antagonism, it is consistent with several findings from the empirical genomics literature.

SEX-SPECIFIC selection can favor evolutionary divergence between males and females (Darwin 1871; Andersson 1994), yet the evolution of sexual dimorphism is likely to be constrained by strong genetic correlations between the sexes (Poissant et al. 2010). Such “sexually antagonistic” disruptive selection can generate a genomic tug-of-war that may persist over long evolutionary timescales (Lande 1980; Stewart et al. 2010). Sexually antagonistic selection appears to be relatively common (Chippindale et al. 2001; Rice and Chippindale 2001; Bonduriansky and Chenoweth 2009; Cox and Calsbeek 2009; Van Doorn 2009) and can have important consequences for the maintenance of population genetic variation (Owen 1953; Haldane 1962; Kidwell et al. 1977; Rice 1984; Patten and Haig 2009; Fry 2010; Patten et al. 2010; Úbeda et al. 2010), the fitness costs of sexual reproduction (Whitlock and Agrawal 2009; Connallon et al. 2010), the evolution of sex-specific genetic architecture (Day and Bonduriansky 2004; Bedhomme and Chippindale 2007; Ellegren and Parsch 2007), and the indirect genetic benefits of mate choice (Chippindale et al. 2001; Albert and Otto 2005; Pischedda and Chippindale 2006; Blackburn et al. 2010; Connallon 2010; Cox and Calsbeek 2010).

While selection is expected to favor alleles that decouple intersexual genetic correlations and facilitate the evolution of sexual dimorphism (Lande 1980; Rice 1984; Rice and Chippindale 2001), little is currently known about the genetic basis and evolutionary dynamics underlying sexually dimorphic traits. The types of evolutionary transitions yielding sexual dimorphism may be context dependent and critically depend upon the nature of the underlying genetic conflict. For sexually antagonistic traits that are primarily influenced by gene expression variation, transitions toward sexual dimorphism may require the evolution of “sex-biased” gene expression, without any change to associated protein sequences (Connallon and Clark 2010b). This mechanism for resolving sexual antagonism might be common, given the ubiquity of sex-biased genes in animal genomes (Ellegren and Parsch 2007; Mank 2009a), as well as the association between sex-biased genes and sex-specific regulatory elements (Williams and Carroll 2009).

Sexual antagonism over protein-coding sequence (i.e., selection favoring different protein isoforms in males and females) is likely to require a more complex evolutionary resolution. Several authors have suggested that protein divergence between the sexes might require a multistep process of gene (or exon) duplication, paralog divergence, and sex-specific expression of the ancestral and derived loci (Ellegren and Parsch 2007; Bonduriansky and Chenoweth 2009; Van Doorn 2009; Stewart et al. 2010). This route toward sexual dimorphism seems reasonable from an engineering perspective: gene duplication may facilitate divergence between the sexes by generating male- and female-specific genes that are subsequently free to evolve independently. Nevertheless, such evolutionary transitions may be unlikely because they require multiple genetic substitutions to become favored by selection (Van Doorn 2009; Stewart et al. 2010).

The resolution of sexual antagonism by gene duplication may be further complicated by the initial conditions of selection and polymorphism at ancestral loci. Previous gene duplication models suggest that natural selection can immediately favor the evolutionary invasion of duplicates from a locus that is evolving by balancing selection (Spofford 1969; Otto and Yong 2002; Walsh 2003; Proulx and Phillips 2006; Innan and Kondrashov 2010). However, functional diversification by gene duplication is constrained for loci evolving under purifying selection (Walsh 1995, 2003). Such loci initially generate redundant (nondifferentiated) duplicates, which are favored by selection only after the occurrence of postduplication (“neofunctional”) mutations that cause functional divergence between paralogs. It is well known that sexually antagonistic selection can yield both initial conditions—balanced polymorphisms and net purifying selection across the sexes—and that the relative frequency of each outcome varies between different genomic regions, such as the sex chromosomes and autosomes (Kidwell et al. 1977; Pamilo 1979; Rice 1984; Patten and Haig 2009; Fry 2010). This implies that the probability of gene duplication should depend on specific parameter conditions or linkage relationships that characterize sexually antagonistic loci.

To address the potentially complex interactions between sex-specific selection, ancestral polymorphism, and linkage relationships between ancestral and derived duplicate pairs, we present a population genetic analysis of gene duplication as a mechanism for resolving sexual antagonism and promoting the evolution of sexual dimorphism. Our specific goals are (i) to describe conditions favorable for the invasion of gene duplicates from ancestral, sexually antagonistic loci, (ii) to characterize the waiting time until gene duplicates become established and maintained within a population, and (iii) to identify whether a nonrandom chromosomal distribution of ancestral and derived duplicate genes is likely to emerge as a consequence of sexual antagonism and its resolution. Our results are directly relevant to the temporal duration of sexual antagonism and the evolution of sexual dimorphism and have implications for the rate of gene duplication, the evolution of genome complexity, and the genetic basis of hybrid incompatibilities between species.

MODEL

Consider a scenario where sexual antagonism within a coding sequence is resolved through a three-step process. First, an ancestrally sexually antagonistic locus becomes duplicated. Next, the duplicate invades from low initial frequency (by selection or by genetic drift), with paralog differentiation into male- and female-beneficial variants occurring during or after the invasion phase. This second step is critical, as it includes the “fate determination” phase (Innan and Kondrashov 2010), after which selection will maintain both paralogs within the population. Finally, after invasion and functional differentiation, selection can favor the evolution of sex-limited expression and sex-specific specialization of each paralog. Assuming that whole-gene duplications are relatively infrequent [which seems reasonable on the basis of patterns of whole-gene copy number polymorphism (e.g., Emerson et al. 2008)], then the process of duplicate invasion and protein differentiation between paralogs (i.e., the transition to the fate determination phase) represents the limiting step during the resolution of sexual antagonism. The evolution of sex-limited expression of male- and female-beneficial gene copies will necessarily follow, given time and recurrent mutation. We therefore characterize the permissive conditions for reaching the fate determination phase and calculate the expected waiting time for this critical event to occur. We first present a detailed analysis of duplication, divergence, and invasion processes under diploid (autosomal) inheritance and then consider whether sex-linked inheritance of ancestral and/or derived copies of a duplicate pair will alter opportunities for gene duplicate invasion.

Initial conditions at ancestral and derived loci:

Consider an ancestral, biallelic, diploid locus (locus A) that evolves by sexually antagonistic selection, with the A_f allele favored in females and A_m favored in males. Given the selection and dominance parameterization of Table 1, the population will evolve to one of three possible evolutionary equilibria (e.g., Owen 1953; Kidwell et al. 1977). The female-beneficial allele (A_f) will become fixed (with frequency ) when

(1a)

The male-beneficial allele (A_m) will be fixed () when

(1b)

A_f and A_m will both segregate () when

(1c)

TABLE 1.

Fitness parameterization for a diploid locus under sexually antagonistic selection

	Genotypes at the ancestral (A) locus
	AfAf	AfAm	AmAm
Female fitness	1	1 − hfsf	1 − sf
Male fitness	1 − sm	1 − hmsm	1

Parameters s_m and s_f represent male- and female-specific selection coefficients (s_m, s_f > 0); sex-specific dominance coefficients are given by h_m and h_f for males and females, respectively (1 > h_m, h_f ≥ 0).

When selection is relatively weak (s_m, s_f ≪ 1), the conditions yielding a balanced polymorphism at a diploid locus are extremely restrictive except under “dominance reversal” conditions, where the dominant allele in males is A_m and the dominant allele in females is A_f (i.e., h_f, h_m < 0.5; Kidwell et al. 1977; Prout 1999; Fry 2010). The initial conditions at a diploid locus can therefore be simplified into two domains. Under constant dominance, the dominance of A_f relative to A_m is the same in males and females (i.e., h_f = 1 − h_m, given the parameterization of Table 1; see also Fry 2010), and parameter combinations yielding condition (1c) become so restrictive that the population will generally be fixed (or nearly fixed) for either A_f or A_m. Under dominance reversal conditions (h_f = h_m < 0.5), the region of parameter space permitting a balanced polymorphism increases. When condition (1c) is true and selection is weak, A_f approaches the equilibrium frequency:

(2)

Next, consider a derived locus (locus B, which is also diploid) with three possible allelic states: the ancestral allele, B₀, is nonfunctional; B_f represents the female-beneficial protein isoform of A (i.e., an A_f allele duplicated to the B locus); and B_m is the male-beneficial isoform (A_m duplicated to the B locus). The ancestral state has B₀ fixed within the population. Duplication events cause transitions from B₀ to B_f or B_m (at rate u_dup), and mutation events permit transitions between B_f and B_m (at rate u_b); knockout mutations permit transitions from B_f or B_m to B₀ (at rate u_n, with no back mutation).

Initial conditions at an X-linked locus:

Initial polymorphic conditions can differ between sexually antagonistic loci on the X chromosome and autosomes (Pamilo 1979; Rice 1984; Patten and Haig 2009; Fry 2010). Under X-linked inheritance, an A_f allele will be fixed () when

(2a)

A_m will be fixed () when

(2b)

A_f and A_m alleles will both segregate when

(2c)

X-linkage is particularly conducive to balanced polymorphisms when h_f < 0.5. Under condition (2c), h_f < 0.5, and weak selection, A_f approaches the equilibrium:

(3)

(Albert and Otto 2005). Conditions leading to (2c) evaporate as h_f → 0.5, and the equilibrium frequency of A_f is generally for ∼s_f > s_m and for ∼s_f < s_m.

Fitness parameterization across ancestral and derived loci:

Following Proulx and Phillips (2006), we assume that there is no selection on gene copy number. We note that, if there is stabilizing selection on dosage (e.g., Rifkin et al. 2005; Bedford and Hartl 2009), then the opportunity for gene duplication will become more restrictive relative to our analytical results. Stabilizing selection will increase the waiting time until duplicate fixation or (for strong stabilizing selection) eliminate the probability that sexual antagonism becomes resolved by gene duplication. This topic is revisited further below.

The fitness of each genotype is defined by the proportion of male- (A_m, B_m) vs. female-beneficial alleles (A_f, B_f) that are expressed within an individual (also see Proulx and Phillips 2006). Female fitness is defined by the function f(x₂) = 1 − (x₂)^ks_f, where x₂ is the frequency of male-beneficial relative to female-beneficial alleles expressed by the female (for example, assuming that A and B are equally expressed in females, the genotype A_fA_mB_fB₀ gives x₂ = ), and k = −ln(h_f)/ln(2). The parameter k is scaled to reflect the dominance relationships between the alleles, such that k = 1 represents the additive case (h_f = 0.5), k > 1 when A_m is recessive in females (h_f < 0.5), and k < 1 when A_m is dominant in females (h_f > 0.5). Under the constant dominance scheme, h_f = 1 − h_m, and consequently the male fitness function is m(x₂) = 1 − [1 − (x₂)^k]s_m. Under dominance reversal conditions m(x₂) = 1 − (1 − x₂)^ks_m. Examples of male and female fitness as a function of x₂ are presented in Figure 1. Note that these relationships between fitness and locus-specific expression were developed to simplify the presentation; appendix a provides general results for arbitrary fitness parameterizations.

Figure 1.—

Fitness functions under constant dominance (h_m = 1 − h_f) and dominance reversal conditions (h_m = h_f). The parameter k defines the curvature of the fitness function (see text for details). Examples show the case of s_m = s_f.

In principle, there are an infinite number of possible expression levels of ancestral relative to derived loci, and the model can be modified to account for any of these possibilities. For simplicity, we focus on two possible scenarios: (a) ancestral and derived loci are expressed in both sexes and (b) ancestral loci are expressed in both sexes and duplicate (derived) loci are expressed in a single sex. In most cases (unless otherwise noted), we assume that, if both loci are expressed, they are expressed at the same rate (i.e., if locus A is expressed at level x, and B is also expressed, then B is also expressed at level x). Violation of the latter assumption does not alter our results concerning the invasion conditions for duplicate alleles. The validity of the assumption will influence the absolute strength of selection on a duplicate (thus, if the duplicate is more highly expressed than the ancestral locus, positive selection on the duplicate can increase; if the ancestral locus has higher expression, positive selection will decrease).

Systematic biases between the expression of ancestral and derived paralogs can potentially arise when X chromosome dosage compensation mechanisms systematically bias expression differences between loci or between the sexes. To account for these potential biases, we define and present results for two specific scenarios. Without dosage compensation, each expressed functional gene copy is expected to have the same level of expression. With dosage compensation, the relative expression of X-linked and autosomal loci is balanced within each sex, with each gene copy having the same average level of expression in females; in males, each X-linked copy is expressed at twice the level of a single autosomal copy (see Mank 2009b; Vicoso and Bachtrog 2009). Additional details are presented in the relevant sections below.

Population genetic recursions and evolutionary stability:

We developed two sets of sex-specific population genetic recursions. For sufficiently weak selection and loose linkage between loci (e.g., loci on different chromosomes), we assume linkage equilibrium and characterize the evolutionary dynamics at the two loci as a function of the population frequencies of each allele (e.g., Takahasi and Tajima 2005). For physically linked loci, we developed exact recursions for each of the six haplotypes (A_fB₀, A_mB₀, A_fB_f, A_fB_m, A_mB_f, and A_mB_m). Models assume the order of events: (i) birth, (ii) selection, (iii) mutation and recombination, and (iv) random mating and syngamy.

Opportunities for duplicate gene invasion depend on the rate at which favorable variants arise within the population and the likelihood that each variant invades from low initial frequency. Because individual alleles or haplotypes have different probabilities of arising within the population (e.g., in a population fixed for A_fB₀, we can safely ignore the invasion of A_mB_m haplotypes, which are three mutation steps away), and each has a different invasion probability, it is useful to determine the evolutionary stability and strength of selection acting on individual alleles and haplotypes rather than the stability of the entire system (the latter is represented by the leading eigenvalue for the set of recursions). To separately consider the invasion of individual haplotypes, we focus on two linkage scenarios. The haplotype recursion equations are used to characterize evolution of tightly linked paralogs (r_m², r_f² ≈ 0, where r_m and r_f refer to the recombination rate between A and B, per male and female meiosis). Loose linkage approximations are used to characterize loci with a high recombinational distance relative to the strength of selection (e.g., r_avg ≫ s_m, s_f; for a similar analysis, see Otto and Bourguet 1999; Otto and Yong 2002; Takahasi and Tajima 2005).

At the ancestral equilibrium with B₀ fixed and the A locus at equilibrium [i.e., corresponding to conditions (1)–(3), above], an allele or haplotype can invade when its associated eigenvalue is greater than one (λ > 1). Conditional on this eigenvalue exceeding one, the strength of positive selection acting on the rare variant is given by s = λ − 1 (Otto and Bourguet 1999; Otto and Yong 2002; Connallon and Clark 2010a,b). Complete details of the recursions and stability analysis are presented in appendix a.

RESULTS

Invasion of gene duplicates under autosomal inheritance and constant dominance:

Under a diploid model with weak selection and constant dominance (h_m = 1 − h_f), essentially all parameter combinations yield an initial condition with A_fB₀ fixed ( when s_f > s_m) or A_mB₀ fixed ( when s_f < s_m). Given the symmetry of our model, we focus on the scenario where A_fB₀ is initially fixed and characterize the invasion conditions of derived haplotypes from this boundary (the case where A_mB₀ is initially fixed gives an analogous result, with the sexes reversed).

When the B locus is expressed in both sexes, the invasion criteria for male-beneficial alleles or haplotypes are restrictive. Duplicate invasion is favored when

(4)

which is identical to the single-locus invasion criteria for A_m over A_f (see above; Kidwell et al. 1977). In other words, if A_m will not invade in the ancestral, single-locus case [condition (1a)], then a duplicate B_m will also fail to invade if it is expressed in both sexes, in which case only male-limited duplications can invade.

When the B locus is male limited in expression, a B_m allele (or A_fB_m haplotype) is always favored by selection. Sex-limited expression can therefore facilitate duplicate invasion, yet there remains an additional constraint. Each duplication event initially produces an A_fB_f haplotype, which does not alter the proportion of male- and female-beneficial alleles per individual (x₂ = 0 for both A_fA_fB_fB₀ and A_fA_fB₀B₀). Mutations causing protein divergence between loci A and B are therefore required before selection can favor invasion and/or maintain the duplicate within the population.

Assuming that B_f alleles evolve neutrally within a population fixed for A_f (i.e., there is no stabilizing selection on gene dosage), there will be two pathways toward duplication and differentiation. A B_f allele may become fixed by genetic drift—with probability 1/(2N_e)—with a B_m substitution occurring after duplicate fixation. Conditional on B_f becoming fixed by drift, the eventual fate of the duplicate depends upon the probability of “neofunctionalization” (substitution of a B_m allele) relative to pseudogenization (substitution of a null allele), with the former event generating selection to maintain the duplicate. Because B_f is redundant when A_f is fixed, mutations eliminating B_f function will be neutral; male-beneficial mutations will improve male fitness by the amount ()^ks_m/(1 − s_m) when heterozygous with B_f (for a B locus with male-limited expression, this also represents the probability of fixation for a B_m allele). Under conditions of strong selection relative to mutation [1 ≫ ()^ks_m/(1 − s_m) ≫ 1/2N_e ≫ u_b + u_n (e.g., Gillespie 1991, 2000)], the fixation probability of B_m rather than a null allele is

(5)

where ρ = u_b/u_n, u_n is the null mutation rate, and u_b is the mutation rate between B_f and B_m (for a similar analysis, see Walsh 1995, 2003). Taking both steps into account (fixation of B_f by drift and B_m by selection), the expected waiting time until B_m becomes fixed is approximately

(6)

where u_dup is the gene duplication rate per meiosis, and f_m is the probability that the B locus is male limited in expression. The first term describes the mean time until neutral fixation of a B_f allele that subsequently transitions to B_m, and the second represents the lag time between fixation of B_f and substitution of B_m. The approximation ignores the transit time of the fixation process [i.e., the number of generations it takes for an invading mutation to spread from frequency 1/(2N_e) to 1], as this marginally contributes to the total waiting time.

For relatively large populations [i.e., 2N_e > 1/√(u_bs_m/3^k) (Weissman et al. 2009)], a B_f allele that is otherwise destined to become lost from the population may alternatively become “rescued” by mutation to B_m followed by positive selection. This represents a special case of “stochastic tunneling” (Komarova et al. 2003). Modifying the results of several previous studies (Iwasa et al. 2004; Weissman et al. 2009; Lynch 2010; Lynch and Abegg 2010), the mean waiting time until a male-beneficial duplicate becomes fixed by the tunneling process is

(7)

(for details, see appendix b). The mean waiting time until a male-beneficial duplicate becomes established by either process is T = 1/(1/T_drift + 1/T_res) (see Weinreich and Chao 2005; Lynch 2010; Lynch and Abegg 2010), which is negatively correlated with the population size (N_e), the proportion of male-beneficial relative to null mutations (u_b/u_n), and the strength of selection in males (s_m; Figure 1a; supporting information, Figure S1). Each parameterization set yields a long waiting time until duplicate invasion. This result is conservative, as stabilizing selection on gene dosage will further elevate the waiting time.

To incorporate dosage selection into the model, consider the case where a redundant duplicate reduces male fitness by the fraction s_d. If N_es_d ≫ 1, a B_f allele cannot become fixed by drift, but can invade by the tunneling process as long as selection on the coding sequence is stronger than selection on dosage [i.e., when λ(1 − s_d) > 1, where λ represents the eigenvalue for a duplicate when there is no dosage effect]. Under such a scenario, the mean waiting time until a B_m allele becomes established is on the order of , where the probability that a B_m allele becomes fixed is given by (see Weinreich and Chao 2005; Lynch and Abegg 2010). This waiting time can be several orders of magnitude greater than the scenarios presented above. If selection on dosage is stronger than selection on the coding sequence, duplicate invasion becomes impossible.

Enhanced invasion of gene duplicates under dominance reversal conditions:

Under diploid inheritance and dominance reversal conditions (h_m = h_f < 0.5; k > 1), two factors will reduce the waiting time until duplicate invasion. First, the invasion conditions for gene duplicates are much broader and do not require that duplicate loci are initially sex limited in expression. Second, the rate at which beneficial duplicates are generated by mutation and recombination is much higher when ancestral loci are polymorphic for sexually antagonistic alleles (A_f and A_m), and balanced polymorphism is much more likely under dominance reversal conditions (e.g., Kidwell et al. 1977).

Because the dominance reversal scenario represents a continuum between additive fitness effects [] and completely recessive deleterious effects [], it is informative to evaluate the invasion conditions at the extremes, with the undertanding that intermediate conditions for k yield intermediate invasion outcomes. As the system approaches additivity (h_f = h_m → ), most of the parameter space will not maintain a balanced polymorphism, and the previous waiting time approximations for boundary conditions apply (above). When the conditions for a balanced polymorphism are met (s_m ≈ s_f, where the female beneficial allele approaches a frequency of ∼), invasion still requires sex-limited expression of the B locus.

At the opposite extreme, where the beneficial variant for each sex is completely dominant (h_m = h_f = 0; A_f dominant in females; A_m dominant in males), all parameter conditions yield an ancestral balanced polymorphism [condition (1c)], and conditions for the invasion of gene duplicates are extremely permissive. For complete linkage and B expressed in both sexes, the strength of positive selection on repulsion haplotypes (A_fB_m and A_mB_f; coupling haplotypes are never favored) is

(8)

where and represent mean male and female fitness prior to duplication. Under loose linkage, B_f and B_m alleles are always favored, with the strength of positive selection given by

(9a)

and

(9b)

Equations 8 and 9 closely parallel results from overdominant selection models of gene duplication, to first order in the selection coefficients (see equations A3 and A4 of Otto and Yong 2002).

When the dominance reversal is incomplete (0 < h_m, h_f < 0.5), a substantial range of parameter space for s_m and s_f may not yield a balanced polymorphism, yet invasion from a boundary (i.e., or ) remains more permissive than under the constant dominance scenario. Consider again the boundary condition (1a) where the male-beneficial allele A_m will not invade at the ancestral locus. When the B locus is equally expressed in males and females, invasion of a male-beneficial duplicate will be favored when

(10)

This condition for B_m is broader than the ancestral invasion criteria for A_m [condition (1c)] when k > 1, which is necessarily true under a dominance reversal.

Recall that when A_f or A_m is initially fixed within the population, duplications are not immediately favored because they are completely redundant with resident alleles at the ancestral locus (e.g., A_fA_fB₀B_f genotypes are selectively equivalent to A_fA_fB₀B₀). Two mutation steps—the duplication step followed by a B_f → B_m or B_m → B_f mutation—are required to generate positive selection for a duplicate. This limitation is alleviated when the ancestral locus is polymorphic. When there is loose linkage between loci, both B_f and B_m alleles arise by duplication and are immediately favored by selection [for a duplication rate of u_dup, the B_f and B_m alleles are created at rate and , respectively]. For tandem duplicates, repulsion haplotypes can be created by nonhomologous crossover events in A_fA_m heterozygotes (Bailey et al. 2003) or by duplication followed by recombination within A_mB₀/A_fB_f or A_fB₀/A_mB_m heterozygotes.

To what degree can dominance reversals shorten the waiting time until gene duplication resolves the sexual antagonism? For the permissive case where h_f = h_m → 0 and duplicates are unlinked, there are two paths toward duplicate invasion. If B_m is the first to invade, then selection in males against A_f is relaxed and the two-locus system will eventually become fixed for A_f and B_m. If the invading duplicate is B_f, then selection in females against A_m relaxes and A_m and B_f will eventually become fixed. Assuming a relatively low duplication rate (u_dup ≪ 1/2N_e ≪ 2s(B_f), 2s(B_m) ≪ 1), that duplicates are expressed in both sexes, and that the probability of invasion for each is roughly twice the selective advantage [e.g., Haldane 1927; i.e., 2s(B_f) and 2s(B_m), using Equations 9a and 9b], the mean waiting time until sexual antagonism is resolved by gene duplication is

(11)

Under dominance reversal conditions, the resolution of sexual antagonism may involve the invasion of either a male- or a female-beneficial duplicate: Equation 11 reflects this by incorporating probabilities of B_m and B_f invasion (both alleles can invade a population fixed for B₀). Relative to the constant dominance model, dominance reversals drastically reduce the waiting time until duplicate invasion (Figure 2b).

Figure 2.—

Waiting times until duplicate gene invasion. (A) The mean waiting time until invasion of a male-beneficial duplicate allele (B_m) under the constant dominance model with autosomal inheritance. Results were obtained using Equations 6 and 7, with parameters k = 1 (h_m = h_f = 0.5), u_dup = 10⁻⁷, f_m = 0.01, u_n = 10⁻⁶, and s_f = 0.01. (B) The mean waiting time until invasion of a duplicate gene under the dominance reversal model with autosomal inheritance. For purposes of contrast, the shaded curves present the results from A. The solid curves represent the dominance reversal results for N_e = 10⁵ (top) and N_e = 10⁷ (bottom) and were obtained with Equation 11 and parameters u_dup = 10⁻⁷ and s_f = 0.01.

X-linkage and the invasion of male- and female-beneficial gene duplicates:

Two opposing asymmetries typically characterize X-linked inheritance. The X is transmitted through females and exposed to female-specific selection twice as often as it is exposed to male-specific selection. However, hemizygosity in males heightens allelic expression and thereby enhances male-specific selection on recessive X-linked alleles. While the twofold transmission bias toward females will always be true for the X, gene duplication and gene copy number variation should generally dampen the differential expression of X-linked alleles in males. Duplication dampens allelic expression differences between the sexes because males are no longer hemizygous—duplication of an X-linked locus results in a diploid-like state in males and a triploid-like state in females (for the case where the duplicate gene is rare).

The evolutionary dynamics of sexually antagonistic selection at single X-linked loci have been well described and suggest that enhanced expression of recessive male-beneficial alleles can override the twofold greater opportunity for selection on the X in females (Rice 1984; Patten and Haig 2009). As we show below, the tipping point that favors the invasion of male-beneficial alleles at single loci (i.e., h_f < 0.5) does not extend to multiple duplicate loci. Consequently, the X is systematically biased toward the evolutionary invasion of female-beneficial duplicates and against the invasion of male-beneficial duplicates, independent of dominance. We assume below that positive selection is sufficiently strong in heterozygotes [i.e., s_m(1 − h_m), s_f(1 − h_f) ≫ 1/N_e] that different X-linked and autosomal effective population sizes will marginally influence probabilities of duplicate invasion.

Duplicate invasion when both loci are X-linked:

Without dominance (h_f = h_m = 0.5), an ancestral X-linked locus will be fixed for the A_f allele when s_m < 2s_f/(2 + s_f) (i.e., ∼s_m < s_f). An X-linked, male-beneficial (B_m) duplicate that is expressed in both sexes will invade when

(12a)

which is considerably more restrictive than the single-locus invasion condition. When s_f < s_m and A_m is fixed at the ancestral locus, an X-linked female-beneficial duplicate can invade when

(12b)

which is more permissive. Thus, for the additive case, female-beneficial duplicates do not necessarily require sex-limited expression to invade on the X (which is a requirement for autosomal invasion); B_f alleles will be favored on the X within the parameter range ∼s_m > s_f > 3s_m/4.

The X-linked scenario with h_f → 0 is of considerable interest because it is conducive to the maintenance of polymorphism at a sexually antagonistic locus (polymorphisms are balanced when 2s_f > s_m). How might this ancestral condition influence the invasion of X-linked duplicates? Under the constant dominance scenario (A_f, B_f dominant to A_m, B_m), with A and B expressed in both sexes, gene duplicates are never favored on the X (|λ| < 1 for all s_f, s_m), as is the case for autosomal inheritance. With dominance reversals (A_f, B_f dominant in females; A_m, B_m dominant in males), the invasion of duplicate alleles is favored from any possible ancestral equilibrium state—as was the case for the autosomal model.

Paralogs dispersed across the X and autosomes:

Duplication events can potentially involve paralogs dispersed between the X and autosomes. Under the additive model (h_f = h_m = 0.5), dosage compensation will influence the range of parameters and strength of selection in favor of duplicates because the relative expression of individual X-linked and autosomal genes can systematically differ within and/or between the sexes. We therefore modeled the opportunity for X-ancestral/autosome-derived and autosome-ancestral/X-derived evolutionary transitions and considered two scenarios: (a) no dosage compensation, where genomic location does not alter the expression of individual gene copies, and (b) dosage compensation, where a typical X-linked copy in males has twice the expression of a single autosomal copy. For purposes of contrast with the previous results, we consider the invasion dynamics at a duplicate locus that is expressed in both sexes.

Two patterns emerge from the additive model (Table 2). First, the X is a hospitable location for the invasion of female-beneficial duplicates, with the bias toward the X strongest when there is no dosage compensation. The underlying cause for this bias stems from the female-biased inheritance of X-linked genes: the X is exposed to selection in females twice as often as to selection in males. This preferential retention of female-beneficial duplicates is slightly weakened by dosage compensation, which enhances duplicate expression in males (and consequently the strength of purifying selection in males). Second, if the ancestral locus is X-linked and there is no dosage compensation, autosomal linkage can facilitate the invasion of a male-beneficial duplicate. This result can be explained by an asymmetry of allelic expression that arises under this combination of conditions. For a population with A_f fixed on the X, an autosomal, male-beneficial duplicate (B_m) will account for 50% of a male's gene product, yet only ∼33% of a female's gene product. Thus, a single B_m allele provides a benefit to males of (1 − s_m/2)/(1 − s_m) − 1 ≈ s_m/2 and a cost to females of ∼s_f/3. Consequently, the parameter range of s_f > s_m > 2s_f/3 favors the invasion of B_m from an initial condition with A_f fixed.

TABLE 2.

Conditions of duplicate gene invasion from the autosomes to the X and from the X to the autosomes

	Dosage compensation	No dosage compensation
A → X transitions
Bm allele
Parameters favoring invasion	None	None
Strength of positive selection	NA	NA
Bf allele
Parameters favoring invasion	sm > sf >	sm > sf >
Strength of positive selection
X → A transitions
Bm allele
Parameters favoring invasion	Nonea	sf > sm >
Strength of positive selection	NA
Bf allele
Parameters favoring invasion	None	None
Strength of positive selection	NA	NA

Duplicates are expressed in both sexes, and alleles have additive fitness effects.

^aB_m will invade within the parameter interval 2s_f/(2 − s_f) > s_m > s_f/(1 + s_f), which is negligible for small s_m, s_f.

With dominance, the fitness functions for each sex primarily depend on the presence or absence of male-beneficial (A_m, B_m) vs. female-beneficial alleles (A_f, B_f): when both allele types are present within a genome, and there is strong dominance, variability in the actual proportion of A_f/B_f vs. A_m/B_m gene product makes little difference. Consequently, dosage compensation has less of an impact on the evolutionary dynamics of gene duplicate invasion (results not shown; refer to appendix a for model details). For the constant dominance model (h_f = 1 − h_m → 0), invasion conditions for gene duplicates are generally restrictive. However, when female-beneficial alleles are dominant to male-beneficial alleles, the derived locus is X-linked, and ∼s_m > s_f (A_m is fixed at the autosomal locus), and female beneficial duplicates are favored when ∼2s_f > s_m, which is more permissive than the pure autosomal model (i.e., in contrast to Equation 4).

Under dominance reversal conditions (h_f = h_m → 0), duplicates will readily invade from any ancestral/derived chromosome orientation, yet the strength of positive selection—and consequently, the probability that a single duplicate will invade—varies with linkage. For an ancestral autosomal locus, selection in favor of X-linked female-beneficial and male-beneficial duplicates (respectively) will be

(13a)

and

(13b)

In contrast to the pure autosomal model (Equation 9), X-linkage increases selection for female-beneficial duplicates (by a factor of ) and decreases selection for male-beneficial duplicates (by a factor of ). If the ancestral locus is X-linked, selection in favor of female-beneficial and male-beneficial duplicates on the autosomes will be

(14a)

and

(14b)

While dominance reversals on the autosomes will always maintain a polymorphism, conditions for a balanced X-linked polymorphism require that 2s_f > s_m; otherwise, the male-beneficial allele will be fixed [condition (2a) (Rice 1984; Patten and Haig 2009; Fry 2010)]. When 2s_f < s_m (A_m fixed), female- but not male-beneficial duplicates can potentially invade the autosomes. Nevertheless, this type of invasion from a boundary state is limited by the waiting time for a duplication event followed by mutation from B_m → B_f. Consequently, there will be a higher rate of duplication originating from autosomal, sexually antagonistic loci because these loci are polymorphic over the same parameter range and can give rise to B_f and B_m alleles—both of which can immediately invade by positive selection (e.g., Equation 9). When ancestral X-linked loci are polymorphic (2s_f > s_m), the strength of positive selection for an autosomal, female-beneficial duplicate is s(B_f) = s_m²/(8s_f); ancestral X-linkage enhances positive selection (i.e., in contrast to Equation 9a) when s_m > s_f > s_m/2 and reduces selection when s_f > s_m. Positive selection for autosomal, male-beneficial duplicates is greatly enhanced by ancestral X-linkage: Equation 14b is greater than Equation 9b across a broad parameter range, 0 < s_m < s_f√3.

DISCUSSION

When sexually antagonistic selection favors different protein isoforms in males and females, gene duplication and differentiation between copies of an ancestral singleton gene can initiate the resolution of the intersexual genetic conflict (e.g., Ellegren and Parsch 2007; Bonduriansky and Chenoweth 2009). We show that sexual antagonism can favor the invasion of gene duplications, but that opportunities for duplicate invasion and preservation are sensitive to a variety of factors, including the balance of male vs. female selection at the ancestral locus (s_m vs. s_f), sex-specific dominance (h_m, h_f), dosage compensation, sex-specific expression of duplicate loci, recombination, and sex linkage. These factors can be classified into categories on the basis of whether they tend to favor or constrain the evolution of gene duplicates (Table 3).

TABLE 3.

A classification of factors that promote and constrain the invasion of gene duplicates of a sexually antagonistic locus

Factors constraining duplicate invasion	Factors promoting duplicate invasion
Constant allelic dominance for fitness, including additive expression	Allelic dominance reversals for fitness
Af or Am fixed at the ancestral locus	Ancestral balanced polymorphism
Expression of a duplicate by both sexes	Sex-limited expression of a duplicate
X-linkage of male-beneficial duplicates	X-linkage of female-beneficial duplicates; autosomal and Y-linkage of male-beneficial duplicates
Dosage compensation	No dosage compensation
Small Neudup	Large Neudup
Strong stabilizing selection on dosage	Weak stabilizing selection on dosage

Allelic dominance, ancestral polymorphism, and the invasion of duplicates:

The resolution of sexual antagonism by gene duplication is strongly influenced by the dominance of sexually antagonistic alleles. If allelic dominance is constant between the sexes (i.e., the same allele is dominant, or each is additive), then the timescale for invasion and differentiation of duplicate genes will be extremely long. Constant dominance conditions rarely maintain an ancestral polymorphism (e.g., Kidwell et al. 1977; Pamilo 1979; Prout 1999), and consequently a population experiencing sexual antagonism will initially be at least two mutation steps away from a beneficial duplicate allele. Furthermore, even if such a duplicate arises within the population, it cannot invade unless its expression is limited to the sex in which it is beneficial. In short, the resolution of sexual antagonism by duplication is so slow under constant dominance that it represents an almost insurmountable constraint to the evolution of sexual dimorphism.

Dominance reversals (within the parameter range of h_m, h_f < 0.5; A_f, B_f dominant in females; A_m, B_m dominant in males) greatly expand the conditions favorable for gene duplicate invasion, which no longer require sex limitation of duplicate loci. Because dominance reversals maintain sexually antagonistic polymorphism (e.g., Kidwell et al. 1977), ancestral loci give rise to both male- and female-beneficial duplicates, which can immediately respond to positive selection. This waiting time until duplicate establishment is reduced to a biologically plausible timescale.

These results parallel those of previous gene duplication models that do and do not invoke balancing selection. Heterozygote advantage readily favors the invasion of duplicate genes (Spofford 1969; Otto and Yong 2002; Proulx and Phillips 2006). Similarly, sexual antagonism with a dominance reversal generates a form of overdominant selection (with fitness averaged across the sexes) that maintains ancestral genetic variation and favors the invasion of gene duplications. In models of neofunctionalization and sexual antagonism with constant dominance (including additive fitness effects), ancestral loci are not polymorphic, and the duplication dynamics involve multiple mutation steps prior to establishment of a duplicate gene (Walsh 1995, 2003).

A key empirical issue is whether sexually antagonistic selection parameters are typically within the constant dominance or dominance reversal domains. Several authors have pointed out that dominance reversals can hypothetically emerge when fitness is a nonlinear function of the number of individual mutations or substitutions affecting a trait (e.g., Rice and Chippindale 2001; Fry 2010). There is widespread support for genotype-by-sex interactions affecting quantitative trait loci in Drosophila (Mackay 2001) and genetic diseases in humans (Ober et al. 2008), which at least implies that sex-by-dominance interactions might be common. On the other hand, very little is known about sex-specific dominance for fitness, so the question of dominance remains completely open. This issue is amenable to an empirical test by applying methods that combine approaches of Innocenti and Morrow (2010) with those of Simmons et al. (1978).

A tangential point worth considering is whether the dominance of sexually antagonistic alleles is likely to be static over time, as is often assumed in evolutionary models of the process. Although selection on a modifier of dominance is often assumed to be weak (on the order of the deleterious mutation rate; e.g., Wright 1929), selection for dominance modification is expected to increase with the population frequency and fitness cost of deleterious alleles, including alleles that are conditionally deleterious within a subset of the population (Fisher 1958; Otto and Bourguet 1999). The evolution of dominance in response to sexual antagonism should parallel dominance modification in a heterogeneous environment (see the “small-scale patchiness” model of Otto and Bourguet 1999), with two equally prevalent environments and disassortative mating between migrants. Selection will favor modifiers that decrease h_m and h_f (causing a dominance reversal), which can increase balanced polymorphism and decrease the waiting time until duplicate invasion.

Effects of sex linkage and physical linkage:

Single-locus models associate X-linkage with asymmetric selection between males and females as a result of (1) female-biased inheritance of the X, which enhances female-specific selection, and (2) preferential expression of rare recessive alleles in males, which enhances male-specific selection. The degree to which female- vs. male-specific selection dominates X chromosome evolution is mediated by allelic dominance. Female-specific selection is strongest when rare alleles are dominant; male-specific selection is stronger when rare alleles are recessive. There is no net bias in male vs. female selection for alleles with additive effects (Haldane 1927, 1937; Pamilo 1979; Rice 1984; Charlesworth et al. 1987; Patten and Haig 2009; Fry 2010; but see Vicoso and Charlesworth 2009).

This general relationship between allelic dominance and male- vs. female-dominated X chromosome evolution does not extend to the invasion of duplicates, which dampen the expression of recessive alleles in males. Consequently, female-specific selection dominates, and female-beneficial duplicates are disproportionately favored on the X. Opportunities for male-specific duplicate invasion will be greater on the Y chromosome and autosomes because these chromosomes spend a greater proportion of generations evolving within males (these results confirm the intuition of Wu and Xu 2003).

Data from Drosophila and mammals indicate that there is a good deal of duplicate gene traffic between the sex chromosomes and autosomes, with male functions (male- and testis-biased expression) preferentially associated with derived Y-linked and autosomal duplicates (Betrán et al. 2002; Skaletsky et al. 2003; Emerson et al. 2004; Meisel et al. 2009; Vibranovski et al. 2009b; Gallach et al. 2010). These patterns are consistent with predictions of the theory of sexual antagonism. Furthermore, the process of gene duplication and sex-specific cooption of individual paralogs can potentially account for many empirical linkage patterns of sex-biased genes (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; Mořkovský et al. 2010; Bellott et al. 2010).

There is also evidence from these same species that other processes—including mutational and gene expression biases during copy number evolution—may also influence patterns of gene duplication on the X and autosomes. For example, meiotic X chromosome inactivation silences X-linked genes during spermatogenesis (Lifschytz and Lindsley 1972; Hense et al. 2007; Namekawa and Lee 2009; Vibranovski et al. 2009a) and should, if anything, make the X a less hospitable location for genes with male function (whether or not they are sexually antagonistic). X inactivation may also generate selection for increased dosage of X-linked genes and favor the fixation of autosomal duplicates.

The mutation process that generates new duplications may also be biased between different regions of the genome. Different mechanisms of duplication (e.g., DNA-mediated duplication vs. retroposition) can generate different patterns of physical linkage between ancestral and derived paralogs. Tight linkage influences the evolutionary dynamics of duplicate alleles (Otto and Yong 2002; see above), and may also influence the degree of gene expression similarity between ancestral and derived paralogs. Intuitively, position effects (including sex-specific expression differences) would seem more likely for dispersed rather than tandem paralogs. If sex-limited expression is required prior to duplicate invasion, then dispersed duplicate alleles might have a greater chance of meeting the minimum criteria for invasion. Recent work in Drosophila has also shown that copy number variation (CNV) is nonrandomly distributed across the genome [with respect to the X and autosomes (Dopman and Hartl 2007; Emerson et al. 2008) and with respect to the timing of DNA replication (Cardoso-Moreira and Long 2010)] and that duplicate fixation and preservation rates differ between the X and autosomes (Zhang et al. 2010). These patterns may be due, in part, to mutational biases between different regions of the genome, which should translate to vastly different opportunities for duplication of different genes. Additional CNV studies will be required to better understand the mutational properties and the physical distribution of novel gene duplicates. This subject is critical to theoretical predictions, which strongly depend on poorly known dynamics of mutation.

Implications for alternative splicing and speciation:

Although we present results within the context of whole gene duplications, they may also apply to the evolution of alternatively spliced exons. Recent studies using mammals and Drosophila indicate an association between alternative splicing and sex-biased transcription (McIntyre et al. 2006; Telonis-Scott et al. 2009; Blekhman et al. 2010). The evolution of male- and female-specific splice forms may evolve by tandem duplication of an exon under sexual antagonism, followed by substitutions causing alternative splicing of male- and female-beneficial exons (e.g., exon shuffling). The fate-determining step of the process may be similar to a scenario of tandem duplication of a sexually antagonistic gene. By analogy, the duplication/invasion step will be facilitated by dominance reversal at the ancestral locus. On the other hand, the evolution of alternative splicing does not necessarily require exon duplication as a first step and may instead incorporate previously intronic coding sequence.

The results also have implications for the evolution of postzygotic reproductive isolation. Duplication and differentiation between paralogs can alter the genetic architecture of male and female traits, and hybridization between species can potentially disrupt the transmission of sex-specific genes, leading to sex-specific hybrid inviability or sterility (see Lynch and Force 2000; Masly et al. 2006). Gene duplication between chromosomes, particularly the movement of male functions from the X to the autosomes, can disproportionately affect males and might partially account for the two rules of speciation (Moyle et al. 2010): “Haldane's rule” (Haldane 1922; Coyne and Orr 2004) and the “large-X effect” (Coyne and Orr 1989; True et al. 1996; Tao et al. 2003; Masly and Presgraves 2007; Presgraves 2008).

Conclusions:

The resolution of sexual antagonism may occur by gene duplication, by alternative splicing, or by the evolution of sexually dimorphic gene expression of single-copy genes (Bonduriansky and Chenoweth 2009; van Doorn 2009; Stewart et al. 2010). A combination of each process may contribute to two widely observed phenomena: the evolution of sexual dimorphism and the nonrandom chromosomal distribution of sex-biased genes (Parisi et al. 2003; Rogers et al. 2003; Wu and Xu 2003; Oliver and Parisi 2004; Connallon and Knowles 2005; Ellegren and Parsch 2007; Gurbich and Bachtrog 2008; Mank 2009a; Connallon and Clark 2010b; but see Bachtrog et al. 2010). The relative role of gene duplication during the evolution of sexual dimorphism is currently difficult to answer due to lack of available data on the genetics of sexually dimorphic traits, yet recent studies of species hybrids suggest a promising approach for addressing this question (Coyne et al. 2008; Loehlin et al. 2010a,b).

Our results show that gene duplication and sex-specific divergence between paralogs can resolve sexual antagonism by decoupling intersexual genetic correlations. Under some ancestral conditions (i.e., constant dominance of sexually antagonistic alleles), the duplication rate of sexually antagonistic loci is likely to be slow, and sexual antagonism may be resolved primarily by a change in the pattern of selection itself. On the other hand, the process of duplication and sex-specific cooption of paralogs may still be common if many genes are simultaneously evolving under sexually antagonistic selection or if ancestral loci are highly polymorphic.

The accumulation of gene duplicates is expected to be nonrandom, with male-beneficial duplicates preferentially accumulating on autosomes and female-beneficial duplicates favored on the X. In contrast to single-locus models of sexually antagonistic variation, whose predictions for X and autosomal variation depend on dominance parameters (e.g., Rice 1984; Patten and Haig 2009; Fry 2010), the fates of male- and female-beneficial duplicates are comparatively independent of dominance. This implies that the genetic linkage patterns of sexually antagonistic variation and intersexual divergence may be quite different, despite being shaped by similar processes of sex-specific selection. This may account for observations of inflated sexually antagonistic variation on the Drosophila X (e.g., Gibson et al. 2002; Long and Rice 2007; Innocenti and Morrow 2010), despite a deficit of X-linkage for male-biased genes (e.g., Parisi et al. 2003; Ranz et al. 2003).

APPENDIX A: RECURSIONS AND STABILITY FOR (I) AUTOSOME → AUTOSOME, (II) X → X, (III) X → AUTOSOME, AND (IV) AUTOSOME → X DUPLICATION MODELS (ANCESTRAL → DERIVED LOCUS LINKAGE RELATIONSHIPS)

General properties

Haplotypes and their frequencies are presented in Table A1. Fitness per genotype is presented in Tables A2 and A3.

TABLE A1.

Haplotypes and their frequencies

Haplotype	Frequency in sperm	Frequency in eggs
AfB0	y1	x1
AmB0	y2	x2
AfBf	y3	x3
AfBm	y4	x4
AmBf	y5	x5
AmBm	y6	x6

TABLE A2.

Fitness per genotype

Maternally inherited haplotype	Paternally inherited haplotype
	AfB0	AmB0	AfBf	AfBm	AmBf	AmBm
AfB0	f11, m11	f12, m12	f13, m13	f14, m14	f15, m15	f16, m16
AmB0	f21, m21	f22, m22	f23, m23	f24, m24	f25, m25	f26, m26
AfBf	f31, m31	f32, m32	f33, m33	f34, m34	f35, m35	f36, m36
AfBm	f41, m41	f42, m42	f43, m43	f44, m44	f45, m45	f46,m46
AmBf	f51, m51	f52, m52	f53, m53	f54, m54	f55, m55	f56, m56
AmBm	f61, m61	f62, m62	f63, m63	f64, m64	f65, m65	f66, m66

f_ij refers to female fitness; m_ij refers to male fitness.

TABLE A3.

Fitness per hemizygous genotype

Genotype	AfB0	AmB0	AfBf	AfBm	AmBf	AmBm
Fitness	m1	m2	m3	m4	m5	m6

When A and B are on the same chromosome, the probability of recombination between loci, per female meiosis, is r_f; the probability in males is r_m. When the loci are on different chromosomes, . Below, we present the details of the pure autosomal model and stability analysis. We use the same general approach and present the main results for analyses involving X-linkage of the ancestral and/or derived loci.

I. Autosomal inheritance:

Male recursions:

Female recursions:

Evolutionary stability at the duplicate (B) locus:

With B₀ fixed within the population, and the A locus at deterministic equilibrium (as defined in the main text), the Jacobian matrix for the system of recursions is a 12-by-12 block upper triangular matrix, with three 4 × 4 matrices along the diagonal. One of these refers to the resident haplotypes within the population (A_fB₀, A_mB₀), which under our equilibrium assumption will yield a leading eigenvalue |λ_L| < 1. Stability with respect to B₀ can be determined using the remaining 4 × 4 matrices, which are

To find the leading eigenvalue for each 4 × 4 matrix, we assume that selection coefficients are relatively small (s_m, s_f ≪ 1) such that the equilibrium frequencies of A₁ and A₂ are approximately equal between the sexes (). This yields a matrix of the form

where a–h refer to the terms in matrix J₁ or J₂. After some manipulation, the determinant for (J – λI) can be found,

which yields a leading eigenvalue:

(A1)

For our purposes, it is useful to separately consider the invasion condition for individual, derived haplotypes (i.e., A_fB_f, A_fB_m, A_mB_f, A_mB_m). We therefore take the approach of Otto and Bourguet (1999) and approximate haplotype-specific eigenvalues for tight and loose linkage. These approximations are accurate, given our assumptions, as the largest of these compares very well with the more exact Equation A1 (T. Connallon and A. G. Clark, unpublished results; for a similar analysis, see Otto and Bourguet 1999).

Under tight linkage (r_m, r_f ≈ 0), the four relevant eigenvalues are

Under loose linkage (such that the magnitude of recombination r ≫ s_m, s_f, with s_m², s_f² ≈ 0), eigenvalues associated with B_f and B_m alleles are

These eigenvalues, along with the fitness model presented in the main text, are used to calculate invasion conditions (|λ| > 1) and selection coefficients (≈λ − 1) for derived alleles and haplotypes.

II. X-linked inheritance:

Female fitness and recursions are the same as those from the autosomal model.

Male recursions:

Evolutionary stability at the duplicate (B) locus:

Under the assumption that selection is weak and the population is at equilibrium at locus A, with B₀ fixed (equilibrium conditions are defined in the main text), the eigenvalues for the X-linked model can be approximated for tight and loose linkage. Under tight linkage (r_f ≈ 0), these are

Under loose linkage, the relevant eigenvalues are

III. Ancestral autosomal linkage—X-linked duplicate:

Under weak selection and A at equilibrium with B₀ fixed, the relevant eigenvalues for the two derived alleles (B_f, B_m) are

where w_m and w_f refer to sex-specific mean fitness with respect to the A locus at [B₀] = 1.

IV. Ancestral X-linkage—autosomal duplicate:

Under weak selection and A at equilibrium with B₀ fixed, the relevant eigenvalues for the two derived alleles (B_f, B_m) are

where w_m and w_f refer to sex-specific mean fitness with respect to the A locus at [B₀] = 1.

APPENDIX B: GENE DUPLICATION BY SEQUENTIAL FIXATION OR STOCHASTIC TUNNELING

Several prior studies (e.g., Iwasa et al. 2004; Weissman et al. 2009; Lynch 2010; Lynch and Abegg 2010) have modeled adaptation in a two-locus system (A and B), where the population is currently two mutation steps away from a higher fitness state, and intermediate genotypes are neutral (i.e., the fixed wild-type haplotype is A₁B₁, the most fit haplotype is A₂B₂, and A₁B₂ and A₂B₁ have the same fitness as A₁B₁). Our gene duplication scenario is very similar, in that a transition from B₀ (no duplicate) to B_m (male-beneficial duplicate) requires two mutation steps: with A_fA_f ancestral, the first mutation is from B₀ to B_f, and the second one is from B_f to B_m. As described in the text, we assume that duplicate alleles are expressed only in males.

If the transition from B₀ to B_m is sequential, B_f is fixed first by genetic drift, followed by the eventual fixation of B_m by positive selection, as described in Equation 6. Alternatively, B₀ to B_m transitions may occur by the tunneling process, where individual B_f mutations will segregate in the population, some of which may mutate to B_m and then become fixed by selection. Prior theory for the two-locus case finds that each mutation to a neutral intermediate will eventually become fixed by tunneling with probability , where u represents the mutation rate from A₁B₂ (or A₂B₁) to A₂B₂, and is the probability that a single A₂B₂ haplotype is driven to fixation. For our gene duplication scenario, the probability that a single B_m allele invades a population fixed for A_fB₀ is . Thus, the previous result is modified to . Since the rate of mutation to B_f is 2N_eu_dupf_m, per generation, the rate of tunneling is , which is used in the formulation of Equation 7.

The mean waiting time until B_m becomes fixed is a function of the sequential and tunneling waiting times: T ∼ 1/(1/T_drift + 1/T_res). Stochastic simulations confirm that the analytical result represents a reasonable approximation for the timescale of male-beneficial duplicate fixation (Figure S1).