Inter-locus sexually antagonistic coevolution can create indirect selection for increased recombination

Abstract

The ubiquity of recombination in nature is a paradox because it breaks up combinations of alleles favored by natural selection. Theoretical work has shown that antagonistic coevolution between hosts and parasites can result in rapid fluctuations in epistasis, which can create a short-term advantage to recombination. Here we show that another kind of antagonistic coevolution, inter-locus sexually antagonistic coevolution (SAC), can also create indirect selection for modifiers that increase the rate of recombination, and that it can lead to very high levels of recombination at equilibrium. Recombination is favored because inter-locus SAC creates heterogeneity in the strength and direction of selection, both within and between generations, which maintains an excess of disadvantageous haplotypes in the population. This result is similar to and consistent with dynamics of fluctuating epistasis produced in models of host-parasite coevolution. However, the conditions under which inter-locus SAC provides an advantage to recombination are more permissive.

Introduction

Recombination allows gametes to possess unique combinations of alleles not present on the parental chromosomes, resulting in an increase in genetic diversity among offspring of doubly heterozygous parents. On the population level, recombination leads to a decrease in the magnitude of linkage disequilibrium (the non-random association between alleles at different loci), because it increases the frequency of under-represented genotypes within the population. In a related manner, recombination has the potential to create new allele combinations by introducing novel mutations onto genetic backgrounds different from the ones on which they arose (Muller 1964). However, recombination also breaks apart beneficial allele combinations favored by selection, and theoretical models show that, under many conditions, genetic modifiers of recombination spread only if they reduce the recombination rate (Feldman et al. 1980, Zhivotovsky et al. 1994, Barton 1995). The ubiquity of recombination in spite of such theoretical predictions is referred to as the paradox of recombination.

In order to explain this paradox, population genetics theory has focused on identifying conditions that result in the build-up of disadvantageous linkage disequilibrium. Notably, both selection and drift can generate linkage disequilibrium (Hill and Robertson 1966, Felsenstein 1974, Barton 1995, Otto and Feldman 1997). Here we focus only on linkage disequilibrium generated by selection, where two general conditions can favor increased recombination: directional selection and heterogeneous selection (Barton 1995, Peters and Lively 1999, Lenormand and Otto 2000). Directional selection favors increased recombination when epistasis is weakly negative. This occurs because recombination breaks down negative linkage disequilibrium, which hinders adaptation, and creates more extreme genotypes that can respond rapidly to selection. As such, recombination increases the genetic variance for fitness in the population (Feldman et al. 1980, Barton 1995, Otto and Feldman 1997). Empirical estimates of epistasis have not established a widespread pattern of negative epistasis, making the generality of this scenario unclear (de Visser et al. 2011).

Temporal variation in selection can favor recombination when past selection has produced an excess of haplotypes that are deleterious in the current environment (Sturtevant and Mather 1938, Charlesworth 1976, Maynard Smith 1978, Barton 1995). This scenario, also referred to as fluctuating epistasis, can create consistent indirect selection for increased recombination only if the fluctuations in epistasis are sufficiently rapid (Barton 1995). Slower fluctuations allow selection to break down the unfavorable linkage disequilibrium completely, removing an advantage for recombination. Thus, it is unlikely that temporal fluctuations in the abiotic environment are sufficiently rapid or predictable enough to allow this mechanism to act consistently over time (Charlesworth 1976, Maynard Smith 1977, Barton 1995). However, biotic interactions, more specifically host-parasite coevolution, have been shown to produce the conditions necessary to favor recombination if the parasite is sufficiently virulent (Peters and Lively 1999, 2007; Gandon and Otto 2007). Similarly, when selection varies in sign or magnitude across space, migration between demes can produce consistently disadvantageous linkage disequilibrium (Pylkov et al. 1998, Lenormand and Otto 2000). The prevalence of this scenario depends on whether heterogeneity in selection among natural populations occurs over a spatial scale relevant to migration (Lenormand and Otto 2000). Furthermore, it requires that the covariance in selection between demes is opposite of epistasis (Lenormand and Otto 2000). Spatial heterogeneity in selection has been shown to favor sexual reproduction in a controlled laboratory experiment (Becks and Agrawal 2010).

However, heterogeneity in selection can occur on a finer scale, within populations, if the strength or direction of selection differs between the sexes. Such differences can arise when selection favors different phenotypes in males and female (intra-locus sexual conflict) or when there is conflict over the outcome of an interaction between males and females that is determined by at least two loci (inter-locus sexual conflict) (Arnqvist and Rowe 2005). Intra-locus sexual conflict is capable of producing stable linkage disequilibrium, and thus it has the potential to create indirect selection for sex and recombination (Úbeda et al. 2011, Patten et al. 2013). However, Roze & Otto (2012) found that intra-locus sexual antagonism can maintain sex only when the strength, but not the direction, of selection varies between the sexes. Nonetheless, it is important to note that sex and recombination are not equivalent and under certain conditions, recombination can be favored even when sex is not (Agrawal 2009).

In the present study, we examine whether inter-locus sexually antagonistic coevolution (SAC), can produce indirect selection for recombination. Commonly explored examples of traits subject to inter-locus SAC include mating rate, relative parental effort, re-mating behavior, and female reproductive rate (Arnqvist and Rowe 2005). Inter-locus SAC shares a number of characteristics with host-parasite coevolution. Most obviously, the strategies for increasing the individual fitness of one group of individuals (i.e. males or parasites) are directly at odds with optimal strategies increasing the fitness of another group of individuals (i.e. females or hosts). As a result, both inter-locus SAC and host-parasite coevolution can produce frequency-dependent selection with the potential to cause long-term cycling of genotype frequencies (Nee 1989). Unlike host-parasite coevolution, however, inter-locus SAC occurs between loci within the same species, and thus, within a shared genome.

We used a three-locus model to observe indirect selection on a modifier locus that influences the recombination rate between two loci under inter-locus sexually antagonistic selection (Nei 1967). We found that inter-locus SAC is capable of producing stable, but fluctuating, linkage disequilibrium. Furthermore, inter-locus SAC consistently creates indirect selection for increased rates of recombination. To determine whether selection for increased recombination was produced due to directional or heterogeneous selection, we measured both the short-term and long-term effects on the recombination modifier. Our results suggest that heterogeneity in selection, both within and between generations, produces a short-term benefit that favors recombination. Interestingly, inter-locus SAC can maintain selection for recombination even when the fluctuations in linkage disequilibrium, and thus epistasis, are very slow, as well as when the rate of recombination in the population approaches the maximum.

Model

The model considers a diploid, sexually reproducing population with 50:50 sex ratio. The inter-locus sexually antagonistic coevolution occurs between two autosomal loci, A and B. Every individual, male or female, carries both loci, and each locus possesses two alleles (See Table 1 for a description of abbreviations and subscripts used). However, the A locus is expressed only in males, and the B locus is expressed only in females. The interaction framework between the antagonistic loci is analogous to the matching alleles models from host-parasite coevolution theory (Hamilton 1980, Frank 1996). Certain male genotypes are better able to manipulate certain female genotypes and, vice versa. As such, no genotype is universally better at manipulation or resistance (Table 2). More specifically, when the male genotype matches the female genotype (AA_♂ & BB_♀, Aa_♂ & Bb_♀, and aa_♂ & bb_♀), the male receives a fitness advantage from the interaction, and the female receives a fitness disadvantage from the interaction. Likewise, when the male genotype does not match the female genotype (AA_♂ & bb_♀ and aa_♂ & BB_♀), the female receives a fitness advantage from the interaction, and the male receives a fitness disadvantage from the interaction. The fitness cost of disadvantageous interactions is given by the selection coefficient, s (Table 2). We consider partial matches (such as: AA_♂ & Bb_♀ and aa_♂ & Bb_♀), to have intermediate fitness effects (such that the dominance coefficient, h, is equal to 0.5) (Table 2). These interactions change the sex-specific genotype frequencies, and mating is random among the post-selection genotypes. Additionally, we consider how the outcome of interactions between heterozygous adults (Aa_♂ & Bb_♀), determined by h_H, influences the evolutionary dynamics.

Table 1.

Description of abbreviations and subscripts.

Abbr.	Description
SAC	sexual antagonistic coevolution
G	genotype frequency
H	haplotype frequency
p	allele frequency
I	interaction frequency for a given combination of male and female genotypes
s	selection coefficient
h	dominance coefficient
hH	determines the fitness of interactions between two heterozygous adults
W	absolute fitness
w	relative fitness
rM, rm	recombination rate of alleles M and m
ε	epistasis
D	linkage disequilibrium
Subscript	Description
*	denotes an interaction
J,K,L	denotes a given genotype at locus A, B, and M, respectively
j,k,l	denotes a given allele at locus A, B, and M, respectively

Table 2.

Sex-specific fitness effects for each potential type of interactions between males with genotype J at locus A and females with genotype K at locus B. The fitness effects are determined by s, the selection coefficient, h, the dominance coefficient, and h_H, which determine the type of interaction between two heterozygous adults. (A) Fitness effects on males with genotype J based upon their interactions with females with genotype K. (B) Fitness effects on females with genotype K based upon their interactions with males with genotype J.

(A)
Male Genotype, J	Female Genotype, K
	BB	Bb	bb
AA	1	1 − (1 − h)s	1 − s
Aa	1 − (1 − h)s	1 − (1 − hH)s	1 − hs
aa	1 − s	1 − hs	1

(B)
Male Genotype, J	Female Genotype, K
	BB	Bb	bb
AA	1 − s	1 − hs	1
Aa	1 − hs	1 − hHs	1 − (1 − h)s
aa	1	1 − (1 − h)s	1 − s

A third autosomal locus, M, modifies the recombination rate between the sexually antagonistic loci, A and B, and is not under direct selection. Recombination occurs on the A–B interval and the recombination modifier locus, M, is completely unlinked to either A or B. The modifier allele, M, is dominant and confers a recombination rate, r_M, that differs from the baseline recombination rate, r_m. We assume a life cycle in which selection occurs at the adult diploid stage, followed by gamete production and reproduction.

Selection

Let G_JKL be the frequency of individuals with genotype J at locus A, genotype K at locus B, and genotype L at locus M. Genotype denotes one of three potential diploid combinations of the two alleles present at each locus. Individuals of one sex encounter individuals of the other sex at random such that the frequency of interactions between two given genotypes, I_JKL*JKL, is given by the product of the frequencies of the genotypes within the population, I_JKL*JKL = G_JKL × G_JKL. As a result of these interactions, selection acts on the antagonistic loci within the sex in which each locus is expressed. For example, within males, selection acts directly on locus A, and the fitness of a given genotype J is determined by the genotype frequencies at locus B among females in the population. The total frequency of a given genotype J or K is given by the sums G_J = Σ_KΣ_LG_JKL and G_K = Σ_JΣ_LG_JKL, respectively. Thus, by defining W_J*K as the relative fitness of males with genotype J when interacting (signified by the asterisk subscript) with females with genotype K, it follows that the average fitness of genotype J among males is given by:

$${\overline{W}}_{J♂}={\sum }_K{G}_K{W}_{J\ast K},$$ (1)

and average male fitness is given by:

$${\overline{W}}_{♂}={\sum }_J{G}_J{\sum }_K{G}_K{W}_{J\ast K}.$$ (2)

Likewise, with females, selection acts directly on locus B and the average fitness of a given genotype K is determined by the frequencies of genotypes at the A locus in the male population. Thus, the average fitness of genotype K among females is given by:

$${\overline{W}}_{K♀}={\sum }_J{G}_J{W}_{K\ast J},$$ (3)

and average female fitness is given by:

$${\overline{W}}_{♀}={\sum }_K{G}_K{\sum }_J{G}_J{W}_{K\ast J}.$$ (4)

The frequencies of genotypes at loci A and B after selection, $G_{J♂}^{\prime }$ and $G_{K♀}^{\prime }$, in males and females, respectively, are a function of the frequency of genotypes before selection and the relative fitness of that genotype in the sex in which it is expressed. Thus, it follows that $G_{J♂}^{\prime }=G_J{\overline{w}}_{J♂}$ and $G_{K♀}^{\prime }=G_K{\overline{w}}_{K♀}$, where the relative fitnesses, w̄_J♂ and w̄_K♀, are given by $\raisebox{1ex}{${\overline{W}}_{J♂}$}\!\left/ \!\raisebox{-1ex}{${\overline{W}}_{♂}$}\right.$ and $\raisebox{1ex}{${\overline{W}}_{K♀}$}\!\left/ \!\raisebox{-1ex}{${\overline{W}}_{♀}$}\right.$, respectively.

Gamete & Offspring Production

After selection has occurred, haploid gametes are produced in proportion to the genotype frequencies in the adult population after selection. Recombination occurs during gamete production at a rate determined by the alleles present at the recombination modifier locus. Production of the male and female gametes is governed by the standard rules of segregation and recombination. Offspring are created through the random union of male and female gametes after selection and recombination. Thus, offspring genotype frequencies, $G_{\mathit{JKL}}^{″}$, are determined by the product of the haplotype frequencies in the female gamete pool and the haplotype frequencies in the male gamete pool, $G_{\mathit{JKL}}^{″}=H_{\mathit{jkl}♂}\times H_{\mathit{jkl}♀}$, where H_jkl is the frequency of haplotypes with allele j at locus A, allele k at locus B and allele l at locus M. A 50:50 sex ratio is assumed, such that genotype frequency is equal in male and female offspring at the start of each generation. Note that locus A is neutral in females and locus B is neutral in males. Thus, when s ≠ 0, the frequency of genotypes among males and females is equivalent before, but not after, selection, such that G_J♂ = G_J♀ and G_K♂ = G_K♀, but $G_{J♂}^{\prime }\ne G_{J♀}^{\prime }$ and $G_{K♂}^{\prime }\ne G_{K♀}^{\prime }$.

Analyses

Numerical iterations of the model outlined above were performed in Matlab (version R2012a). By tracking the change in frequency of the modifier allele, M, over time, we determined whether the dynamics of sexually antagonistic coevolution between A and B favor change in the recombination rate between these loci (Nei 1967). For example, if the modifier allele increases the recombination rate, then a corresponding increase in the frequency of the modifier allele indicates that SAC favors increased recombination. Similarly, a decrease in the frequency of the same modifier allele indicates that SAC favors reduced recombination. We ran simulations both with and without a “burn-in” period. The presence or absence of a burn-in period did not affect the outcome and the simulations presented here were all run without a burn-in period.

In addition to keeping track of the frequency of the modifier allele over time, we also measured linkage disequilibrium and epistasis. The initial genotype and haplotype frequencies in the population were determined only by the starting allele frequencies; hence, the initial linkage disequilibrium in the population was equal to zero. As such, all linkage disequilibrium measured in the population must have arisen due to the effects of sexually antagonistic coevolution between locus A and B. Linkage disequilibrium and additive epistasis were calculated in the offspring for loci A and B using the standard equations, given by:

$$D=H_{AB}{H}_{ab}-H_{AB}{H}_{ab},$$ (5)

and

$$\epsilon ={\overline{W}}_{AB}-{\overline{W}}_{Ab}-{\overline{W}}_{aB}+{\overline{W}}_{ab},$$ (6)

where H_jk and W̄_jk are haplotype frequency and average haplotype fitness, respectively, and the subscripts refer to the alleles at each locus (Crow & Kimura 1970). As haplotype fitnesses differ between the sexes, epistasis was calculated separately for males and females. Direct fitness benefits for recombinant offspring occur when the sign of linkage disequilibrium and epistasis are opposite, such that recombinant offspring receive a direct fitness advantage. We used the following equation to measure the opportunity for direct fitness benefits for recombination:

$$S_{\mathit{direct}}=-D_{AB}\epsilon .$$ (7)

As epistasis differs for males and females, S_direct was also calculated in a sex-specific manner.

A recombination modifier allele may spread to due its association with beneficial allele combinations (short-term effects) or beneficial alleles (long-term effects). In models of host-parasite coevolution, short-term effects and long-term effects have both been associated with selection on the recombination modifier (Otto and Nuismer 2004, Peters and Lively 2007, Salathé et al. 2009). To address how these effects drive the changes observed at the modifier allele, we measured the associations between the recombination modifier allele and the alleles at the sexually antagonistic loci, and determined whether these associations could contribute to positive or negative changes in frequency of the modifier allele. Short-term effects are the result of associations between the modifier allele and certain allele combinations at the loci under direct selection (Salathé et al. 2009). As such, we measured the short-term effects in each sex as follows:

$$S_{\mathit{short}}=D_{\mathit{ABM}}\epsilon$$ (8)

where D_ABM is the three-way linkage disequilibrium between loci A, B, and M. Thus, if the modifier allele is associated with favored haplotypes, we should observe that S_short > 0.

Long-term effects arise due to the association between the modifier allele and certain alleles at the loci under direct selection (Salathé et al. 2008, 2009). Likewise, we measured long-term effects in each sex as follows:

$$S_{\mathit{long}}=D_{AM}{s}_A+D_{BM}{s}_B$$ (9)

where S_A measures selection at locus A (given by: S_A = (w_Ab − w_ab)p_b + (w_AB − w_aB)p_B) and S_B measures selection at locus B (given by: S_B = (w_aB − w_ab)p_a + (w_AB − w_Ab)p_A). When S_long > 0, it indicates that the modifier allele is associated with advantageous alleles at the sexually antagonistic loci.

Results

The sexually antagonistic interaction between locus A and locus B resulted in frequency-dependent selection, as well as long-term stable cycles of genotype frequencies within the population (See Figure S1). The sexually antagonistic interactions between A and B also produced linkage disequilibrium that fluctuated in magnitude over time (Figure 1A). Recombination reduced the magnitude of linkage disequilibrium between A and B, and allowed linkage disequilibrium to fluctuate in both sign and magnitude (Figure 1A). Epistasis also emerged between alleles at the two loci (Figure 1B). As the two loci are never simultaneously expressed within the same sex, epistasis must arise because certain haplotypes are more or less likely to be contained in certain genotypes. The sign and magnitude of epistasis fluctuated over time, as genotype frequencies within the population changed (Figure 1B). These results demonstrate that SAC is capable of producing linkage disequilibrium and fluctuating epistasis.

Figure 1.

The dynamics of linkage disequilibrium and epistasis produced by inter-locus sexually antagonistic coevolution. (A) SAC leads to the build-up of fluctuating linkage disequilibrium between locus A and locus B. Increasing the rate of recombination conferred by the modifier allele (r_M) diminishes the magnitude of linkage disequilibrium. Starting parameters: s = 0.1, h = 0.5, h_H = 1, p_A = 0.8, p_B = 0.2, p_M = 0.1, r_m = 0. (B) Epistasis between locus A and locus B fluctuates over time. The sign of epistasis is opposite in males (dashed) and females (solid). Starting parameters: s = 0.1, h = 0.5, h_H = 1, p_A = 0.8, p_B = 0.2, p_M = 0.1, r_m = 0, r_M = 0.1. (C) The modifier allele increases in frequency (non-shaded areas) when the short-term effects favor recombination in the sex under slightly stronger selection (here females). Conversely, the modifier allele decreases in frequency (shaded areas) when the short-term effects in females do not favor recombination. Note that it is the fitness interaction between double heterozygotes (h_H) that causes one sex to be under slightly stronger selection, not the selection coefficient (s), which is the same for both sexes. Starting parameters: s = 0.1, h = 0.5, h_H = 1, p_A = 0.8, p_B = 0.2, p_M = 0.1, r_m = 0, r_M = 0.1.

Across a wide range of selection coefficients and starting allele frequencies, the simulations consistently showed that the sexually antagonistic dynamics between locus A and locus B imposed indirect selection for increased recombination rate within populations (Figure 2). If the recombination rate conferred by the modifier allele was higher than the baseline recombination rate (r_M > r_m), the modifier allele increased the recombination rate in the population, and it experienced a net increase in frequency over time. Alternatively, if the recombination rate conferred by the modifier allele was lower than the baseline recombination rate (r_M < r_m), the recombination modifying allele decreased the recombination rate in the population and exhibited negative net change. If the recombination modifier allele had no effect (r_m = r_M), the modifier allele did not change in frequency in the population.

Figure 2.

The net change in the frequency of the recombination modifier allele (M) after 1000 generations for all possible combinations of the baseline recombination rate (r_m) and the modified recombination rate (r_M). The recombination modifier allele exhibits positive net change when it increases the rate of recombination in the population (r_M > r_m), which occurs above the 1:1 line. Below the 1:1 line, the modifier allele decreases the rate of recombination in the population (r_M < r_m) and decreases in frequency. The starting frequency of the modifier allele (p_M = 0.1) causes the asymmetries in net change around the 1:1 line. Starting parameters: s = 0.2, h = 0.5, h_H = 1, p_A = 0.8, p_B = 0.2, p_M = 0.1.

Furthermore, when r_M > r_m, the recombination modifier allele exhibited a net increase in frequency for all starting frequencies between 0 and 1 (See Figure S2). This indicates that the modifier allele can invade when rare and will approach fixation given sufficient time. Moreover, this result was not sensitive to the magnitude by which the modifier increased the recombination rate. For example, a net increase in the frequency of the recombination modifier was observed even when it confers the maximum recombination rate (r_M = 0.5) and was very prevalent in the population (p_M > 0.95) (See Figure S2). This result suggests that SAC favors the maximum recombination rate between the antagonistic loci, rather than approaching an intermediate optimal rate of recombination as is seen in models of host-parasite coevolution (Peters and Lively 2007).

Altering starting conditions of the model influenced the rate of change in frequency of the modifier allele, but did not modify the qualitative results. When the modifier allele increased recombination, it exhibited a net increase in frequency over time under all starting allele frequencies with the exception of the trivial, stable equilibria that occur when one allele is fixed at any of the three loci, and the unstable interior equilibrium that occurs when allele frequencies at the sexually antagonistic loci are exactly equal (A = 0.5 and B = 0.5). This unstable interior equilibrium is consistent with two-loci models of epistasis (Karlin 1979). Greater initial change in the frequency of the recombination modifier allele occurred as the starting allele frequencies at the SAC loci moved away from this interior equilibrium. Furthermore, the increase in frequency of the recombination modifier was not dependent on strong selection, and occurred for all values of the selection coefficient. Stronger selection led to more rapid change in the frequency of the modifier allele, and changing the value of the selection coefficient (s) had a much stronger effect on the rate of increase of the recombination modifier allele (M) than modifying the starting allele frequencies. When the magnitude of the selection coefficient differed in males and females (s_♂ ≠ s_♀), recombination was favored provided that strong selection in one sex did not cause polymorphism at either of the sexually antagonistic loci to be lost (See Figure S3).

While the net change in the frequency of the modifier was consistently positive when it increased recombination, the sign and magnitude of the change in the modifier allele frequency varied over time as the genotype frequencies within the population fluctuated (See Figure S2). This allowed us to further dissect the mechanism promoting the increase in frequency of the recombination modifier allele by asking what evolutionary dynamics were associated with those periods in which the recombination modifier increased in frequency and were absent during the brief periods in which the modifier allele decreased. We found that the recombination modifier increased when linkage disequilibrium and epistasis had opposite signs in the sex under stronger selection (S_direct > 0) (See Figure S4). Conversely, during the brief periods of time in which the common haplotypes were also the most favored by selection, and thus linkage disequilibrium and epistasis had the same sign, the recombination modifier decreased in frequency (S_direct < 0). In other words, the recombination modifier increased in frequency when there was an immediate fitness benefit to breaking up common haplotypes and producing rare haplotypes in one sex. We also observed short-term benefits, which were identical to the direct fitness benefits (See Figure S5). Thus, when the modifier allele created a direct fitness benefit, it also became associated with the beneficial haplotypes that it created. While potential long-term effects were observed, they were not correlated with the observed change in frequency of the recombination modifier allele (See Figure S5). For example, during periods in which the modifier allele was associated with beneficial alleles at the SAC loci, it was observed to decrease in frequency.

While the degree of linkage disequilibrium was the same among males and females, the sign of epistasis was opposite (Figure 1B). As a consequence, the short-term effects of recombination were also opposite for males and females (See Figure S5). When interactions between two heterozygous adults were considered to favor male fitness (h_H > 0.5), male genotype frequency closely tracked female genotype (See Figure S1), imposing slightly stronger selection on females (See Supplemental Figure 4). Correspondingly, the recombination modifier allele increased when it provided short-term benefits to females (S_direct♀ > 0) (See Figure S5). When this assumption was reversed, such that interactions between heterozygotes were considered to favor female fitness (h_H < 0.5), female genotype frequency closely tracked the male genotype and males became the sex under slightly stronger selection. As expected, the changes in the frequency recombination modifier occurred in accordance to the male’s short-term benefits (S_direct♂ > 0). When male and female fitness outcomes were exactly equal during interactions between two heterozygotes (h_H = 0.5), neither sex had a consistent advantage. Rather, the dynamics alternated and, likewise, the sex whose short-term benefits favor increased recombination also alternates. However, the simulations still produced consistent increases in the frequency of the modifier allele. This special case occurred only when h_H was exactly 0.5.

Discussion

This model demonstrates that, similar to host-parasite coevolution, inter-locus sexually antagonistic coevolution can create indirect selection for increased recombination. Inter-locus SAC led to the build-up of linkage disequilibrium and epistasis that consistently fluctuated in sign over time. Furthermore, linkage disequilibrium and epistasis were often of opposite signs, such that the favored haplotypes were more often rare than common. Thus, recombination is favored, because it breaks up disfavored haplotypes, and the modifier allele becomes associated with the favored, but rare, haplotypes in the sex under slightly stronger selection. This is largely consistent with the dynamics expected when fluctuating epistasis favors recombination, and it is similar to dynamics observed in models of host-parasite coevolution. Furthermore, the long-term effects of recombination were not correlated with the observed changes in frequency of the modifier allele. Thus, directional selection is not likely to strongly contribute to the observed indirect selection for increased recombination. We cannot rule out that more complex dynamics, such as those described in Lenormand (2003), could also be contributing to the change in frequency of the recombination modifier.

Unlike models of host-parasite coevolution, however, linkage disequilibrium arises due the admixture of two gene pools, male and female, that are exposed to different selection pressures (Úbeda et al. 2011, Patten et al. 2013). As such, sexually antagonistic coevolution shares many of the characteristics found in models of spatially heterogeneous selection (Lenormand & Otto 2000). Males and females represent two demes that are separated not by space, but rather by their reproductive role. It follows that the production of offspring is analogous to an extreme from of migration between the two demes (males and females), in which all offspring possess a parent from each deme. This means that, rather than being a product of past selection, LD is being produced every generation during syngamy. As a result, inter-locus SAC produces persistent LD, which is maintained for all frequencies of recombination (Úbeda et al. 2011). Consequently, inter-locus SAC ultimately favors maximum recombination. Thus, we may expect to find that sexually antagonistic loci are located on separate chromosomes.

Furthermore, inter-locus SAC does not require that that selection is strong, such that the sign of epistasis fluctuates rapidly, in order to favor recombination (Barton 1995, Peters and Lively 1999). In this model, even weak selection is capable of producing indirect selection for increased recombination. In such cases, the sign of epistasis fluctuates on the order of hundreds of generations, which is outside of the ‘Barton zone’ of 2–5 generations (Barton 1995, Peters and Lively 1999). Like models of recombination in spatially heterogeneous environments, strong differences in the sign of linkage disequilibrium and epistasis can be maintained over long periods of time, creating indirect selection for high rates of recombination (Lenormand & Otto 2000). However, unlike models of spatially heterogeneous environments, SAC also drives consistent fluctuations in the strength and sign of linkage disequilibrium and epistasis over time. As a result, SAC overcomes a major constraint for the applicability of fluctuating epistasis in understanding the evolution of recombination rate and provides a general solution to the question proposed by Barton (1995): “How can such a perverse combination of linkage disequilibrium and epistasis be stable?”

Interestingly, the epistasis observed in these simulations is not produced in the traditional sense. As the two loci are sex-specific, it is not possible that they simultaneously expressed in the same individual. Thus, differences in haplotype fitness must arise because certain haplotypes are associated with certain genotypes. For example, for AB haplotypes to be favored it must be the case that they are more likely to be associated with either the AA or Aa genotype than Ab haplotypes. This may occur, as for linkage disequilibrium, due the admixture of gamete pools that underwent differential selection in the parental generation. Further analytical investigations of the dynamics produced in the simulations presented here may further elucidate the consequences of inter-locus SAC on linkage disequilibrium and epistasis. Nevertheless, it is still the case that recombination is favored because it breaks down disadvantageous associations between alleles produced by inter-locus SAC.

The results of this model differ substantially from previous theory examining how sexually antagonistic coevolution a single locus (intra-locus) influences the evolution of sex. Roze & Otto (2012) demonstrated that intra-locus sexual conflict can maintain sex only when the strength of selection is stronger in males, but in the same direction in both sexes. In contrast, inter-locus sexual conflict can selection for recombination when selection differs in direction between the sexes, as well as when it is slightly stronger in either sex. However, the evolutionary dynamics that promote sex and recombination are not identical, and thus, we cannot directly compare the results of this model to that of Roze & Otto (2012), nor can we speculate as to whether inter-locus sexual antagonism could maintain sex (Agrawal 2009). Interestingly, Wyman & Wyman (2013) found that an antagonistic allele is more likely to invade a population if it reduces the recombination rate between two intra-locus sexually antagonistic loci. This indicates that unlike inter-locus sexual antagonism, intra-locus sexual antagonism likely selects against both sex and recombination.

The model presented here simulates the interactions between a single pair of sexually antagonistic loci. However, it is possible that there are many pairs of such loci within one genome. The genes involved in inter-locus SAC are likely both prevalent and diverse and, as such, have the potential to be ubiquitous contributors to the evolution of recombination. The coevolution between male seminal fluid proteins and the female proteins native to the reproductive tract is a well-cited example (Wolfner 1997, Chapman 2001, Swanson and Vacquier 2002, McGraw et al. 2004, Wigby and Chapman 2005). However, inter-locus SAC can also arise between genes without a direct interaction. For example, the genes that produce the elaborate griping structures that male water striders use to capture females and force copulations can coevolve antagonistically with genes that underlie female resistance traits (Arnqvist 1992, 1997, Westlake et al. 2000). The presence of multiple pairs of sexually antagonistic loci may act synergistically to favor recombination. For example, there are over 100 known Drosophila melanogaster seminal fluid proteins (Findlay et al. 2008). Thus, while the indirect selection for recombination produced by one pair of sexually antagonistic loci may be weak, the effect may be greatly amplified due to the large number of sexually antagonistic loci. This outcome is possible, because the modifier in our model is not linked to the loci under sexually antagonistic selection. Hence, one modifier could influence the rate of recombination between multiple of pairs of sexually antagonistic loci. However, recombination modifier models that consider the effects of multiple pairs of loci indicate that the results may not be so straightforward (Lenormand and Otto 2000). Future work should investigate how the presence of multiple pairs of sexually antagonistic loci influences the evolution of recombination.

Conclusion

Inter-locus sexually antagonistic coevolution presents a novel solution to the paradox of recombination. Heterogeneity in selection, both across generations, due to negative frequency dependent selection, and within generations, due differences between the sexes, produces an excess of disadvantageous haplotypes within a single population. Recombination is favored because it breaks apart common, but unfavorable, haplotypes and produces more rare, but advantageous, haplotypes. Furthermore, the conditions necessary for SAC to favor recombination are more permissive than those observed in models of host-parasite coevolution and ultimately favor higher rates of recombination. This finding, coupled with the prevalence of inter-locus sexual conflict, indicates that SAC may be a widespread factor contributing to variation in recombination rates.