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Keywords:

  • heterochiasmy;
  • sexual antagonism;
  • sexual conflict;
  • sexual dimorphism

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model
  5. Results and discussion
  6. Conclusions
  7. Acknowledgments
  8. References
  9. Appendix A: Model of selection and dominance
  10. Appendix B: Recursion equations and stability analysis
  11. Supporting Information

The introduction and persistence of novel, sexually antagonistic alleles can depend upon factors that differ between males and females. Understanding the conditions for invasion in a two-locus model can elucidate these processes. For instance, selection can act differently upon the sexes, or sex linkage can facilitate the invasion of genetic variation with opposing fitness effects between the sexes. Two factors that deserve further attention are recombination rates and allele frequencies – both of which can vary substantially between the sexes. We find that sex-specific recombination rates in a two-locus diploid model can affect the invasion outcome of sexually antagonistic alleles and that the sex-averaged recombination rate is not necessarily sufficient to predict invasion. We confirm that the range of permissible recombination rates is smaller in the sex benefitting from invasion and larger in the sex harmed by invasion. However, within the invasion space, male recombination rate can be greater than, equal to or less than female recombination rate in order for a male-benefit, female-detriment allele to invade (and similarly for a female-benefit, male-detriment allele). We further show that a novel, sexually antagonistic allele that is also associated with a lowered recombination rate can invade more easily when present in the double heterozygote genotype. Finally, we find that sexual dimorphism in resident allele frequencies can impact the invasion of new sexually antagonistic alleles at a second locus. Our results suggest that accounting for sex-specific recombination rates and allele frequencies can determine the difference between invasion and non-invasion of novel, sexually antagonistic alleles in a two-locus model.


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model
  5. Results and discussion
  6. Conclusions
  7. Acknowledgments
  8. References
  9. Appendix A: Model of selection and dominance
  10. Appendix B: Recursion equations and stability analysis
  11. Supporting Information

Selection can act differently in males vs. females (Arnqvist & Rowe et al., 2005), resulting in antagonism over the expression of shared traits. Fitness itself can be under sexual conflict so that reproductive success has a negative genetic correlation between the sexes (e.g. Chippindale et al., 2001; Foerster et al., 2007). Alternatively, individual traits may also be under sexual conflict because trait expression in one sex has opposing fitness effects in the other sex. Among sexually dimorphic traits, ∼17% of selection estimates were sexually antagonistic (Cox & Calsbeek, 2009). Sexual conflict can operate at even finer scale levels: the expression of ∼8% of genes is beneficial for one sex, but detrimental in the other in Drosophila melanogaster (Innocenti & Morrow, 2010). Understanding sexual antagonism requires experimental (e.g. Long et al., 2012) as well as theoretical approaches. In a one-locus model, Rice (1984) suggested that sex linkage can facilitate the initial spread of sexually antagonistic alleles. His model demonstrated that recessivity and hemizygosity (possessing one major sex chromosome) can shield novel antagonistic alleles from selection in the nonbenefitting sex. However, Fry (2010) has shown that differences in sex-specific dominance may affect the invasion advantage of sex linkage.

Recent theoretical work on the invasion of sexually antagonistic alleles has focused on two-locus models (Connallon & Clark, 2010; Patten et al., 2010; Úbeda et al., 2011) which introduce realistic factors such as epistasis, linkage disequilibrium and recombination. These factors are known to alter the invasion outcomes for novel, sexually antagonistic alleles. For instance, Connallon & Clark (2010) employed a model whereby sexual antagonism can be introduced through epistatic interactions between two different loci. When epistasis is present, the recombination rate difference between males and females affects the invasion of sexually antagonistic allelic combinations. In a related model, Patten et al. (2010) showed that a second locus can both increase the opportunities for polymorphic equilibria relative to a one-locus model and preserve allelic combinations that increase fitness variation through linkage disequilibrium.

Differences in sex-specific recombination rates are often dramatic (e.g. Lenormand & Dutheil, 2005; Mank, 2009; Brandvain & Coop, 2012), suggesting the need for further study into its implications. Up to 75% of recombining species demonstrate >5% overall rate difference between the sexes (Burt et al., 1991; Lenormand, 2003). Recombination rates on a local scale can also vary between individuals (Coop et al., 2008; Baudat et al., 2010; Fledel-Alon et al., 2011) and between the sexes (e.g. Hansson et al., 2005; Berset-Brändli et al., 2008; Kong et al., 2010). In fact, recombination rates have very little intersexual covariance (Coop et al., 2008), so that tracking the rates in males and females separately may be necessary. It is important to understand what, if any, implications sexual dimorphism in recombination might have on the introduction of sexually antagonistic phenotypes. For instance, it has been suggested that lower recombination can maintain combinations of genes beneficial to males because those very genes have successfully undergone sexual selection (Trivers, 1988). Although sex-specific selection at the diploid stage is unlikely to facilitate the evolution of heterochiasmy (Lenormand, 2003), once heterochiasmy is established, it may subsequently affect the formation of male- or female-benefit regions in the genome. Indeed, lower sex-specific recombination makes it easier for alleles benefitting that same sex to invade on sex chromosomes (Connallon & Clark, 2010).

Here, we expand upon previous models and investigate the effect of heterochiasmy in three ways. First, we study the effect of sexually dimorphic recombination rates on the invasion of sexually antagonistic alleles. Specifically, we examine whether dimorphism in recombination can hinder or facilitate the spread of an autosomal male-benefit, female-detriment allele (and similarly for female-benefit, male-detriment alleles). Second, to capture the reality of sex-specific and individual variation in recombination rates, we study how sex- and genotype-specific recombination rates affect invasion outcomes. Finally, we analyse the impact of sexually dimorphic allele frequencies at one locus for invasion outcome at a second locus. Considered together, these results suggest that novel, sexually antagonistic variation can potentially spread more easily due to pre-existing sexual dimorphism in recombination and allele frequencies.

Model

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model
  5. Results and discussion
  6. Conclusions
  7. Acknowledgments
  8. References
  9. Appendix A: Model of selection and dominance
  10. Appendix B: Recursion equations and stability analysis
  11. Supporting Information

We constructed an autosomal two-locus diploid model. The two-locus diploid model has been studied extensively in other contexts (e.g. Curtsinger & Heisler, 1988; Otto, 1991; Albert & Otto, 2005; Patten et al., 2010). Much of our notation is adopted from Connallon & Clark (2010) to facilitate comparison. The A and B loci both have two alleles so that there are 10 unique male and 10 unique female genotypes and associated fitnesses (see Table 1). Recombination between the A and B locus occurs at a rate inline image in males and inline image in females. Discrete time recursion equations for the AB genotypes are reproduced in the Appendix B [and in a Mathematica notebook (Supplement S1)]. The genotype frequencies in the next generation depend upon the maternal and paternal allele frequencies at the A and B loci, as well as upon recombination and linkage disquilibrium between the two loci.

Table 1. Shorthand for genotype fitnesses
Diploid genotypeFemale fitnessMale fitness
inline image inline image inline image inline image
inline image inline image inline image inline image
inline image inline image inline image inline image
inline image inline image inline image inline image
inline image inline image inline image inline image
inline image inline image inline image inline image
inline image inline image inline image inline image
inline image inline image inline image inline image
inline image inline image inline image inline image
inline image inline image inline image inline image

First, we analyse recombination rates that are nonzero (i.e. no achiasmy) and different in each sex for the polymorphic equilibrium case (e.g. inline image is not fixed upon the invasion of inline image). We first work with a general model to derive our expressions and then plug in a specific fitness parameterization to investigate the effects of dominance and epistasis on our results (parameterization in Appendix A). Because we only specialize at the last step, any fitness parameterization may be used, and our general results are not contingent upon them. Although we are using a two-locus diploid model in the context of sexual antagonism, our model is not to be conflated with models of interlocus sexual conflict (i.e. different loci in each sex interact to affect the expression of a shared trait). Second, we treat recombination rate itself as a novel, sexually antagonistic trait in our model and study its effect on invasion. Lastly, we study the model's behaviour on the invasion of new alleles when resident allele frequencies are different between the sexes.

Results and discussion

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model
  5. Results and discussion
  6. Conclusions
  7. Acknowledgments
  8. References
  9. Appendix A: Model of selection and dominance
  10. Appendix B: Recursion equations and stability analysis
  11. Supporting Information

Sex-averaged vs. sex-specific recombination rates

Upon first consideration, it may appear that only the sex-averaged recombination rate can matter for invasion (e.g. Hedrick, 2007). After all, recombination shuffles loci in males and females, so that any particular allelic combination has approximately the same likelihood of appearing in both sons and daughters. These combinations experience sex-specific selection, but then get reshuffled in males and females of the second generation. As reshuffling is inevitable, it may seem that the effects of sex-specific recombination are averaged out between the sexes over time. However, a few lines of evidence suggest that this view is too simplistic. First, sexual antagonism can preserve stable linkage disequilibrium in the face of recombination so that sex-specific patterns of variation persist as polymorphic equilibria (Patten et al., 2010; Úbeda et al., 2011). Second, when sex-specific recombination rates are explicitly modelled, their particular values appear to affect the conditions for invasion (Connallon & Clark, 2010). Finally, whether a novel allele spreads or not depends only upon the present conditions, not the long-term conditions. Selection precedes fertilization, so that recombination occurs in a subset of individuals different from the total set initially preset at birth. Selection may occur at the adult stage, but may also occur at the gamete stage via fertility selection. Thus, the interplay of sexually antagonistic fitness and sex-specific recombination rates may lead to small but potentially consequential sex differences in the frequency of novel alleles, affecting their invasion.

We can study these dynamics by conducting a local stability analysis of the recursion equations governing a two-locus model with sex-specific recombination (Appendix B). We calculated the eigenvalues of the Jacobian matrix for the recursion equations; whenever the dominant eigenvalue describing an invasive allele is greater than one, the population is subject to invasion by that novel allele. We conducted this analysis under the condition that the first A locus is polymorphic for inline image (frequency q) and inline image (frequency p) and the second B locus is fixed for inline image upon the introduction of inline image. The polymorphic case may be more interesting under the assumption that a novel mutation may appear on a background that is genetically variable throughout a population (NB: The eigenvalue for the equilibrium whereby inline image is fixed may be recovered by setting p = 0 and q = 1). The full results from our analysis is reproduced in Supplement S2. Since these results are lengthy, we provide approximate and abbreviated formulae below.

The dominant eigenvalue (inline image) for the invasion of a novel inline image allele when the male and female frequencies are equal (inline image) at the A locus is given by equation 6a in Connallon & Clark (2010):

  • display math(1)

As Connallon & Clark (2010), we have used inline image and inline image to designate female and male fitnesses, respectively (Table 1). For instance, genotype inline image has the fitness type 32. In addition, the 22 types are either 22C or 22R, because 22 is heterozygous at both loci, but may occur as coupling or repulsion (Table 1). inline image and inline image represent the mean female and male fitnesses, respectively.

inline image can be rewritten by expressing fitness values as a deviation from 1. For instance, for the inline image genotype, the female fitness can be re-expressed as inline image, where inline image is the deviation of inline image from 1. The u term is a counting parameter for doing a series expansion when u is small. Terms with no u (i.e. inline image) are larger than terms with inline image, and terms with inline image are larger than terms with inline image, etc. This expansion is made assuming weak selection; we also assume that mutation is weak. Finally, we assume that recombination rates are small and also multiply them by u. Eqn (1) can now be rewritten as an expansion in u:

  • display math(2)

This approximation has a few nice features that aid our understanding. The first line contains just the mean fitnesses and is approximately 1. The second line contains terms that have one power of u; the third line contains even smaller pieces (where inline image indicates the order in u). Taking inline image simplifies things further:

  • display math(3)

We observe that the main recombination effect is indeed the sex-averaged recombination rate (the last term in the second line of eqn (3)), as is typically assumed (Hedrick, 2007). However, when males and females differ in fitness for the genotype inline image – that is, when inline image (as is expected when inline image is sexually antagonistic) – the sex-averaged rate is adjusted by the third line (i.e. inline image). The strength of the third line of eqn 3 is proportional to 1−p and affects genotypes bearing inline image. As the strength of the sexual antagonism increases, the effect of heterochiasmy also increases.

In general, the inline image terms will be smaller than the inline image terms. However, this assumption is not met in two cases. Most obviously, the approximation is not valid under strong selection or strong sexual antagonism (i.e. genotype fitnesses are not close to 1) so that the approximation inline image breaks down. A more subtle scenario (that does not violate the approximation) is when the terms on the second line almost cancel out, allowing order 1 terms to be order 2 in eqn 3:

  • display math(4)

For example, for the stable polymorphic equilibrium p = 0.25 [i.e. polymorphic equilibrium is determined by inline image)/(inline image, in the notation of Otto & Day (2007)], the δm terms are all slightly negative (male-benefit) and the δf terms are all slightly positive (female-detriment). As a result, the magnitude of inline image the magnitude of inline image and the magnitude of inline image magnitude of inline image, but with the overall sum of the four terms having a slightly negative effect. In other words, the inline image and inline image alleles are only slightly male-benefit and slightly female-detriment. Such alleles may be important to study under the assumption that new mutations may have minor phenotypic effects in relation to the current phenotype, as we assume under an infinitesimal model. Then, the sum within the brackets of eqn (4) may be very small, that is to say of inline image. Since inline image and inline image are of inline image, the ‘first’-order term is the same or smaller magnitude as the second-order term. In such a case, the second-order term makes a substantial difference to the outcome of invasion and must be included for consistency; so the sex-averaged rates and the sex-specific rates may contribute equally when eqn (4) holds. Even though this special case may only persist for a short time in a population, it can affect the invasion of new alleles during that time. (NB: We emphasize that when inline image due to an accidental cancellation, the approximation is still valid because we still expect inline image). We have demonstrated qualitatively how eqn (4) may be relevant. Below we also present an example using a particular fitness model to highlight both the qualitative and quantitative implications of eqn (4) to invasion.

Implications of sex-specific recombination

The analysis above shows that the sex-specific recombination rates matter for invasion, in addition to the sex-averaged rate. A specific fitness model (see Appendix B) can be used for further investigation. We determine whether inline image can invade on a polymorphic A background, given that inline image decreases fitness in both males and females, but also interacts epistatically with inline image to increase only male fitness and not alter female fitness (i.e. the inline image combination is sexually antagonistic overall). The fitness model assumes linear dominance (dominance coefficients h and g for the A and B loci, respectively). The selection coefficients are s and t (for the A and B loci, respectively); positive values indicate a fitness detriment, and negative values indicate a fitness benefit. Epistasis (ε) is additive; positive values indicate a fitness detriment, and negative values indicate a fitness benefit.

For the A locus, we assumed equal dominance and selection coefficients between the sexes: inline image, inline image, inline image and inline image. For the B locus, we also assumed equal dominance and selection coefficients: inline image, inline image, inline image and inline image. For these set of parameters, the inline image allele is detrimental to both males and females. However, we assumed epistasis to be inline image and inline image. As a result, when inline image and inline image are together, the allele combination confers an overall benefit to males but not females (since inline image but inline image). It would be easy enough to make the alleles themselves sexually antagonistic by setting selection to have opposite signs between the sexes (i.e. inline image, inline image0 and inline image, inline image0). However, doing so at the A locus would violate the assumption of equal allele frequencies in males and females (see more below). Meanwhile, making the B locus sexually antagonistic independently of the A locus would reduce the dynamics to a one-locus model.

Figure 1a demonstrates a few features of the model. The solid line is the critical invasionary eigenvalue of one; invasion occurs to the left, and not to the right, of this line. First, we observe that the range of recombination rates required for the new inline image allele to invade due to its epistatic interaction with inline image differs between male and females. Males have a smaller range of recombination rates conducive to invasion than females. So, overall, inline image. Connallon & Clark (2010) also show this result but present it in terms of the wait times for successful co-invasion of epistatically beneficial alleles. An intuitive explanation for this theoretical result rests in the fact that keeping inline image and inline image together in males requires suppressing recombination to some degree. By contrast, because inline image does not benefit females with an inline image background, a wider range of recombination values that can potentially break up the detrimental combinations benefits females. In support of this interpretation, when inline image hurts females (inline image and inline image), as in the case of sexual antagonism, this range of female recombination rates increases.

A second, and key, observation is that within the overall invasion space (left of the solid line), inline image, inline image and inline image all yield valid conditions for the successful invasion of inline image. This may explain why we might not expect to see a consistent fine-scale correlation between sex-specific recombination rates and the concentration of male- or female-benefit alleles – even though the range of invasionary recombination rates differs between the sexes. So while the male recombination rate should be lower than the female recombination rate overall in regions enriched with genes beneficial to males, in particular cases, they may not be. Conversely, the female recombination rate may not necessarily be lower in regions of female-benefit genes. Positive correlations between the concentration of sex-specific beneficial alleles and lower recombination rates may only be apparent in the most extreme circumstances: for example, when male and female recombination values are near the invasion border and when males do not recombine (i.e. along the x-axis of Fig. 1a). As Connallon & Clark (2010) have shown, the lack of male recombination and the fact that the X chromosome experiences selection more often in females than in males may explain empirically why the concentration of male-biased genes is lower on the X chromosome of a male non-recombining species such as Drosophila. This must be the case if male-biased gene expression can be roughly equated with male-benefit, as the current evidence suggests (Pröschel et al., 2006; Andolfatto et al., 2010; Innocenti & Morrow, 2010; Wyman et al., 2010).

Finally, Fig. 1a demonstrates that the correction (solid line) to the sex-averaged (dotted line) recombination rate makes the greatest impact near the border of invasion. This borderline region is where the inline image effects and the inline image effects compete with each other to determine invasion. Both lines indicate the slope of the critical invasionary eigenvalue. However, in region II, the sex-averaged rate will incorrectly predict the lack of invasion for this given set of parameters; conversely, in region I, the sex-averaged rate will incorrectly predict successful invasion. The numerical results support this mismatch between the two critical eigenvalues (Fig. 1b). Reliably predicting the persistence or disappearance of novel, sexually antagonistic alleles based upon the sex-averaged recombination rate will require that the rate is well within or beyond this borderline.

Many factors lead to variation in the rate of recombination among individuals – such as age, genetic background and environmental stress (see Tedman-Aucoin & Agrawal, 2011 for review). Sources of variation relevant to this study include remating rate (Priest et al., 2007) and male genotype (Stevison, 2011). Sexually antagonistic allele combinations may invade more easily in populations in which males can lower their own recombination rates or can induce lower average recombination rates in females, relative to the critical invasionary values. Conversely, male-benefit, female-detriment alleles may invade less easily in populations where males do not decrease (or rather, increase) their own recombination rate or their partners.

Individual variation in sex-specific recombination rates

The analyses above support that the sex-averaged recombination rate is not the only quantity of interest. The sex-specific rates can matter through sexually antagonistic effects on fitness from epistasis. However, because recombination itself is a phenotype, it is also important to consider its variation among individuals (Coop et al., 2008; Baudat et al., 2010; Fledel-Alon et al., 2011) as well as between the sexes. In fact, variation in recombination rate is known to affect variation in fitness; mothers with higher average recombination rate have more children (Kong et al., 2004; Fledel-Alon et al., 2011).

To capture these biological realities, our model can be modified so that each sex by genotype combination has a distinct recombination rate, for example, an adult inline image male has a recombination rate, inline image (adopting the notation in Table 1). Thus, A and B affect both fitness and recombination rate. In the new eigenvalues of the modified model, recombination only affects terms that carry the double heterozygote genotype (22 = inline image; see Table 1):

  • display math(5)

or

  • display math(6)

This result comports nicely with intuition. Recombination in the double heterozygotes matters because recombination destroys inline image (from the 22C genotype) or creates inline image (from the 22R genotype). Keeping track of these processes will determine the size of the respective eigenvalues. The other genotypes with the inline image allele are either extremely rare (i.e. inline image) or have no effective recombination (i.e. recombination occurs but does not change the haplotypes).

More generally, eqns (5) and (6) have the form:

  • display math(7)

If fitness is always a positive number and the new allele is male-benefit and female-detriment, the fourth term is the largest negative term. As a result, invasion becomes more likely (i.e. λ > 1) as inline image becomes smaller. In particular, inline image or inline image must be smaller, and not any other genotype-specific male recombination rate (e.g. inline image, inline image, inline image, etc.). Furthermore, in order for a sexually antagonistic allele to invade, the recombination rate must be low in the sex benefitting from the new allele, and in particular, recombination must be low in the double heterozygote genotype, which carries one copy of the new allele. Interestingly, it is also true that inline image or inline image, and not any other genotype-specific female recombination rate, should be smaller for a male-benefit allele to invade.

Sexually dimorphic allele frequencies

In the previous sections, we allowed for a polymorphic equilibrium at the A locus, but made the assumption that allele frequencies in males and females were equivalent, inline image. While assuming equal allele frequencies in the sexes simplifies the math and is a fair approximation, as a precise statement, it may have limited biological relevance. After all, when an allele is sexually antagonistic, allele frequencies should be slightly higher in the helped sex and slightly lower in the harmed sex. Although male and female offspring are equally likely to inherit their alleles from their fathers and mothers, sexually antagonistic selection and fertility selection can alter allele frequencies prior to mating and fertilization (e.g. Patten et al., 2010; Úbeda et al., 2011).

To assess and estimate the importance of this mismatch between male and female allele frequencies, we used the recursion equations and found polymorphic equilibria for inline image. The equilibrium allele frequencies can be rewritten in terms of two numbers inline image and inline image:

  • display math(8)

where inline image is the proportion of the ova carrying the inline image allele, inline image the proportion carrying inline image. inline image is the proportion of sperm carrying inline image, inline image carrying inline image. inline image is the difference in allele frequency between males and females, inline image = inline imageinline image. Plugging eqn (8) into the recursion formulas, using f = 1−u δf, expanding out the mean fitnesses inline image and inline image, and keeping only inline image terms, we found the following approximate equilibrium solutions:

  • display math(9)
  • display math(10)

Although the full expression (to higher order in u) is necessary to find accurate equilibria (see Mathematica notebook, Supplement S1), this first-order approximation has a few instructive properties. inline image appears with no factors of u, which makes sense as allele frequencies can be large relative to recombination when A is polymorphic. However, inline image appears as a multiple of inline image (four factors of δf or δm in the numerator and three in the denominator of inline image). Thus, the difference between inline image and inline image has the same size effect on the eigenvalue as the sex-specific fitnesses do. The assumption inline image, in addition to its limited biological appeal, is not necessarily consistent.

We can calculate how much the eigenvalue is changed relative to inline image when inline image by determining the value of Δλ, defined as inline image (where λ is the full eigenvalue). We can rewrite inline image as inline image to make its size obvious in the counting parameter u. Δλ is a long expression found by solving for the characteristic polynomial; the most important piece is the new first-order contribution to the eigenvalue from inline image:

  • display math(11)

Invasion is substantially impacted by including sexually dimorphic allele frequencies, as represented by inline image. Moving onto the next smaller set of terms, we find additional inline image contributions to the eigenvalue that affect inline image. Since our main focus is on sexually dimorphic recombination rates, we show only terms with inline image or inline image:

  • display math(12)

Hence, whenever the sex-specific recombination rates are important (i.e. borderline of Fig. 1), we should also keep track of the new parts of the eigenvalue that are proportional to inline image.

image

Figure 1. Sex-specific recombination rates. See text for parameter selection. a) The dashed line represents the sex-averaged recombination dependent term; it connects the same recombination value on the inline image and inline image axes. The solid line represents inline image that provides the correction to the sex-averaged estimate. Because the individual recombination rates matter, the solid line does not connect the same values on both axes. Both lines represent the critical eigenvalue of one. Invasion occurs to the right of the each line; invasion does not occur to the left. In order for recombination to facilitate the invasion of the second male-benefit, female-detriment allele, the range of male recombination rates has to be smaller than the range of female recombination rates. However, within the invasion space, inline image and inline image can all permit invasion. b) In region II, the recombination rates (inline image) allow invasion of inline image even though the sex-averaged rate would not predict invasion. In region I, the recombination rates (inline image) do not allow invasion of inline image, even though the sex-averaged rate would predict invasion.

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To interpret inline image, we can again plug in our fitness model (see Appendix A):

  • display math(13)
  • display math(14)

The equation for inline image shows that when male and female dominances are equal, inline image, so that male and female allele frequencies are identical. Thus, sexual dimorphism in dominance can affect the invasion probabilities, in addition to the effect from sexual dimorphism in recombination rates.

In Fig. 2, we numerically confirm that inline image has the potential to impact whether a sexually antagonistic allele inline image can invade in the population. For the same averaged male and female allele frequencies at locus A, inline image can eventually invade in one population (black points: inline image), but not in the other population (grey points: inline image). This occurs despite the fact that both populations have the same recombination parameters (inline image) and the same sex-averaged p. We note that invasion depends on the size and sign of inline image (larger inline image has a greater impact). That is, when inline image and a new allele is male-benefit, invasion is easier. On the other hand, parameter selection that results in inline image will hinder the invasion of a male-benefit sexually antagonistic combination (or favour female-benefit alleles). In summary, sex differences in allele frequencies can have definite effects on the possibility of invasion.

Limitations of the model

As with any model, the results are subject to limitations of the assumptions. Two factors may mitigate the applicability of these results. First, the role of sexually dimorphic recombination was only investigated for weak selection. When the strength of selection increases in either sex, the approximation (eqn (3)) breaks down, and the sex-averaged recombination rate determines invasion outcome over the sex-specific recombination rates. Second, since populations are finite, it is important to assess the effect of the modified eigenvalues for realistic conditions. When the eigenvalue is near one (i.e. sex-specific recombination rates make a difference to invasion), we must worry about genetic drift. Drift dominates when N(λ−1) ≪ 1, where N is the population size and λ−1 is the deviation of the eigenvalue for invasion from one. 1/(λ−1) measures the time to invasion for a novel allele. Hence, in a case when the fitnesses and recombination rates are on the order of 1%, inline image. Drift becomes less important when inline image, a modest population size.

Conclusions

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model
  5. Results and discussion
  6. Conclusions
  7. Acknowledgments
  8. References
  9. Appendix A: Model of selection and dominance
  10. Appendix B: Recursion equations and stability analysis
  11. Supporting Information

Recombination is important for adaptation because it shuffles loci and places favourable alleles together. However, recombination also hinders adaptation by breaking up the very same favourable combinations. The model studied here demonstrates that the advantages and drawbacks of recombination occur alongside each other, albeit in the different sexes. In fact, sex-specific recombination affects invasion in a manner distinct from simply the sex-averaged recombination rate, especially at the border of invasion. Low recombination permits the helped sex to keep favourable sexually antagonistic combinations together. Low recombination also permits stable linkage disequilibria to maintain sexually antagonistic variation (Patten et al., 2010; Úbeda et al., 2011). Both effects occur because low sex-specific recombination increases allele frequencies in the benefitting sex over the harmed sex without the differences washing out over time.

However, within the invasion space, we find that male recombination can be greater than, less than or equal to female recombination and still allow a male-benefit allele combination to invade. In other words, successful invasion does not always require that the benefitting sex has a lower recombination rate than the harmed sex. Empirically, the model suggests that low male recombination rates may not always be correlated with a high concentration of male-benefit genes on chromosomes.

When recombination is treated as the phenotype of interest and allowed to vary in a genotype-specific manner, only the recombination rate of the double heterozygotes matters for invasion outcome. The double heterozygotes are the only genotype in which recombination will make a difference to increase or decrease the frequency of the new sexually antagonistic allele. This is a sex-specific effect, so that a male-benefit, female-detriment allele that simultaneously increases male fitness and decreases male recombination rate invades more easily than those alleles that do only one or the other.

We also find that sexual dimorphism in allele frequencies affects invasion outcomes. As sexually antagonistic alleles benefit only one sex, allele frequencies may be higher in the helped sex (Patten et al., 2010; Úbeda et al., 2011). Here, sex differences in allele frequencies at one locus impact the invasion of novel alleles at a different locus (Fig. 2). The magnitude of this effect can be as important as that of sex-specific recombination. These results suggest that maintaining sexually dimorphic allele frequencies at one locus would be a way to both facilitate or hinder the spread of sexually antagonistic alleles at a second locus.

image

Figure 2. Differences in allele frequencies of inline image between the sexes. Invasion of the inline image allele (males, upper black points; females, lower black points) can occur in a population which has sexually dimorphic frequencies of inline image, with inline image). However, invasion of inline image does not occur (both sexes, grey points) when inline image frequency is monomorphic, inline image (inline image, inline image, inline image, inline image, inline image, inline image, inline image, inline image, inline image). This difference in invasion outcome exists even though the sex-averaged allele frequencies were initially identical and even though recombination was the same in both populations (inline image).

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In sum, there is a delicate interplay between sex-specific selection, sex-specific recombination rates and sex-specific allele frequencies. The impact of sexual dimorphism in recombination rates and allele frequencies became more apparent by doing the calculations than by relying upon intuition alone. Two-locus models of sexual antagonism have a great deal of complexity unavailable to one-locus models that might explain the presence of widespread sexual antagonism in nature.

Acknowledgments

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model
  5. Results and discussion
  6. Conclusions
  7. Acknowledgments
  8. References
  9. Appendix A: Model of selection and dominance
  10. Appendix B: Recursion equations and stability analysis
  11. Supporting Information

We thank L. Rowe, A.F. Agrawal, J.C. Perry and 2 anonymous referees for comments. MJW was supported by the University of Toronto Connaught Scholarship and Doctoral Completion Award and by NSERC grants to L. Rowe. MCW was supported by US Department of Energy contract DE-FG02-90ER-40560 and by the Kavli Institute for Cosmological Physics at the University of Chicago through grants NSF PHY-0114422 and NSF PHY-0551142 and an endowment from the Kavli Foundation and its founder Fred Kavli.

References

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model
  5. Results and discussion
  6. Conclusions
  7. Acknowledgments
  8. References
  9. Appendix A: Model of selection and dominance
  10. Appendix B: Recursion equations and stability analysis
  11. Supporting Information

Appendix A: Model of selection and dominance

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model
  5. Results and discussion
  6. Conclusions
  7. Acknowledgments
  8. References
  9. Appendix A: Model of selection and dominance
  10. Appendix B: Recursion equations and stability analysis
  11. Supporting Information

The m and f prefix or subscript indicate male and female, respectively. The h and s terms describe dominance and selection at the A locus. The g and t terms describe dominance and selection at the B locus. Epistasis, ε, is additive; having two copies of inline image has twice the effect of having one copy. This model assumes cis-epistasis since haploid gametes are being modelled; recombination occurs between the A and B locus within a chromosome.

inline imageinline image

Appendix B: Recursion equations and stability analysis

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model
  5. Results and discussion
  6. Conclusions
  7. Acknowledgments
  8. References
  9. Appendix A: Model of selection and dominance
  10. Appendix B: Recursion equations and stability analysis
  11. Supporting Information

We write the frequency of haplotype inline image in generation n abstractly as inline image (for eggs) and inline image (for sperm). The recursion equations for these frequencies are given by the general equations

  • display math(15)
  • display math(16)

where {i,j,ℓ,m} ∈ {1,2}, inline image is the fitness of a female (male) with a maternal inline image and paternal inline image genotype, inline image is the recombination rate of females (males), inline image is the mean female (male) fitness and inline image is the female (male) linkage disequilibrium rate. Mean fitness is defined as

  • display math(17)
  • display math(18)

We define the sex-specific linkage disequilibrium factors by

  • display math(19)
  • display math(20)

If we put the haplotype frequencies in a vector:

  • display math(21)

then the i,j entry of the Jacobian stability matrix J associated with these recursion equations is given by

  • display math(22)

The eigenvalues associated with the lower right 4 × 4 submatrix of J will be the ones relevant to the invasion of the inline image allele, the one relevant to our study. These are reproduced in the main text. We derive them by first finding the characteristic in full generality and then solve it order by order in an expansion in a small parameter. In our calculations, we do not need to assume weak selection, but always assume weak recombination and a small intersexual fitness difference. That is, we assume all recombination rates are ≪1 and that the difference between the male and female equilibrium allele frequencies is also small – that is, inline image. In the main text of the article, we have also assumed weak selection when we write most of our equations. These calculations, including equations without the assumption of weak selection, are all reproduced in the associated Mathematica notebook (Supplement S1).

Supporting Information

  1. Top of page
  2. Abstract
  3. Introduction
  4. Model
  5. Results and discussion
  6. Conclusions
  7. Acknowledgments
  8. References
  9. Appendix A: Model of selection and dominance
  10. Appendix B: Recursion equations and stability analysis
  11. Supporting Information
FilenameFormatSizeDescription
jeb12236-sup-0001-Supplement1.pdfapplication/PDF113KSupplement S1 Mathematica notebook with full details of the calculation.
jeb12236-sup-0002-Supplement2.nbapplication/nb12413KSupplement S2 Eigenvalues to higher order in inline image and inline image.

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