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.
For example, for the stable polymorphic equilibrium p = 0.25 [i.e. polymorphic equilibrium is determined by )/(, 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 the magnitude of and the magnitude of magnitude of , but with the overall sum of the four terms having a slightly negative effect. In other words, the and 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 . Since and are of , 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 due to an accidental cancellation, the approximation is still valid because we still expect ). 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
For the A locus, we assumed equal dominance and selection coefficients between the sexes: , , and . For the B locus, we also assumed equal dominance and selection coefficients: , , and . For these set of parameters, the allele is detrimental to both males and females. However, we assumed epistasis to be and . As a result, when and are together, the allele combination confers an overall benefit to males but not females (since but ). It would be easy enough to make the alleles themselves sexually antagonistic by setting selection to have opposite signs between the sexes (i.e. , 0 and , 0). 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 allele to invade due to its epistatic interaction with differs between male and females. Males have a smaller range of recombination rates conducive to invasion than females. So, overall, . 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 and together in males requires suppressing recombination to some degree. By contrast, because does not benefit females with an background, a wider range of recombination values that can potentially break up the detrimental combinations benefits females. In support of this interpretation, when hurts females ( and ), 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), , and all yield valid conditions for the successful invasion of . 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 effects and the 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).
More generally, eqns (5) and (6) have the form:
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 becomes smaller. In particular, or must be smaller, and not any other genotype-specific male recombination rate (e.g. , , , 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 or , 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, . 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).
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 .
The equation for shows that when male and female dominances are equal, , 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.
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%, . Drift becomes less important when , a modest population size.