The Evolution of Sex Dimorphism in Recombination

Thomas Lenormand1 CEFE-Centre National de la Recherche Scientiﬁque, 34293 Montpellier, France Manuscript received May 13, 2002 Accepted for publication November 14, 2002

ABSTRACT Sex dimorphism in recombination is widespread on both sex chromosomes and autosomes. Various hypotheses have been proposed to explain these dimorphisms. Yet no theoretical model has been explored to determine how heterochiasmy—the autosomal dimorphism—could evolve. The model presented here shows three circumstances in which heterochiasmy is likely to evolve: (i) a male-female difference in haploid epistasis, (ii) a male-female difference in cis-epistasis minus trans-epistasis in diploids, or (iii) a difference in epistasis between combinations of genes inherited maternally or paternally. These results hold even if sources of linkage disequilibria besides epistasis, such as migration or Hill-Robertson interference, are considered and shed light on previous verbal models of sex dimorphism in recombination rates. Intriguingly, these results may also explain why imprinted regions on the autosomes of humans or sheep are particularly heterochiasmate.

EIOSIS in males and females differs in several as achiasmy, or it may vary quantitatively between sexes, M important respects. A female produces only one a phenomenon that I term “heterochiasmy.” large gamete (ovule) from the four meiotic products Recombination dimorphism on sex chromosomes: In whereas a male produces four small motile gametes species with a large sex-chromosome heteromorphism (spermatozoa) from the four meiotic products. Often, (X vs. YorZvs. W), the sex chromosomes in the hetero- the timing of male and female meiosis is different: in gametic sex do not exchange genetic material along animals, for example, male meiosis tends to be continu- much of their length. This is the most conspicuous and ous whereas female meiosis generally stops twice, just widespread recombination dimorphism between the after meiosis begins and just before it ends. And at sexes. Two related theories have been advanced to ex- least sometimes, the amount of genetic recombination plain selection for reduced recombination around sex- during meiosis differs between males and females be- determining loci: (i) recombination is disadvantageous cause of differences in crossing over number and/or for sex-linked alleles with opposite effects in the two position. How did these meiotic differences evolve, and sexes (Charlesworth and Charlesworth 1980), and how are they maintained? (ii) recombinant genotypes have an “intersex” unfit ge- The first aspect—the evolution of anisogamy—has notype because of the accumulation of linked sex factors received considerable theoretical treatment (see, for (Nei 1969). example, Randerson and Hurst 2001); but aspects Achiasmy: Although it has received less attention, re- such as the evolution of dimorphism in recombination combination dimorphism on autosomes is also com- have received much less attention, especially with re- mon. In the most conspicuous cases, achiasmy, one sex spect to formal models. The aim of this article is to apparently lacks recombination completely. This is not determine the conditions under which a recombination related to the loss of meiosis that often occurs with dimorphism can evolve. I begin by reviewing the facts parthenogenesis: achiasmy occurs in taxa where parthe- about recombination dimorphism and the different nogenesis is rare or unknown, e.g., Lepidoptera, Tri- hypotheses that have been proposed to account for it. choptera, Diptera, and isolated species of molluscs, wa- I then present a formal model for the evolution of re- ter-mites, copepods, grasshoppers, and alder-flies (Bell combination dimorphism on autosomes and on sex 1982). When achiasmy occurs in dioecious species, it is chromosomes. almost always the heterogametic sex that is achiasmate, A recombination dimorphism can occur on sex chro- a phenomenon known as the “Haldane-Huxley rule” mosomes (or close to a sex-determining locus) or on [Haldane 1922; Huxley 1928; for examples, see Bell autosomes. In the autosomal case, recombination may 1982, Table 5.3; however, some species in Musca and be completely absent in one sex, a phenomenon known Culex genera are possible exceptions (Bull 1983)]. The Haldane-Huxley rule has been explained either as a pleiotropic consequence of selection against recombi- 1Address for correspondence: CEFE-CNRS, 1919 rte. de Mende, 34293 nation between X and Y (or Z and W)—the pleiotropy Montpellier, France. E-mail: [email protected] hypothesis—or as the consequence of the evolution of

Genetics 163: 811–822 ( February 2003) 812 T. Lenormand the Y (or W) in the sex that initially had no recombina- Trivers (1988) and Burt et al. (1991) reviewed chi- tion—the no-recombination hypothesis. asma count data and found that differences between Both hypotheses are plausible in principle. However, the sexes are often large. Recent map data tend to con- both can be criticized in several ways. For example, the firm their analyses, although these data have yet to be of chiasmate %75ف pleiotropy hypothesis provides no explanation for why rigorously examined. Moreover, in the pleiotropic effect should be so extreme: after all, species (whether dioecious or hermaphrodite), recom- there is ample within-species genetic variation for re- bination rate, measured using either chiasma count combination rates on autosomes. The hypothesis does (Burt et al. 1991) or map length (my personal observa- not explain why achiasmate meiosis occurs in only one tions), differs by Ͼ5% between male and female meiosis. gender within hermaphrodites (e.g., some Liliaceae in In extreme cases, recombination in female meiosis can the genus Fritillaria; Noda 1975). It cannot explain why be as much as 3 times higher [for instance, in the zebra- achiasmy has evolved in the heterogametic sex in species fish Danio rerio, which has no heteromorphic sex chro- with a XX/XO sex-determination system (e.g., accord- mosome (Singer et al. 2002), and in the planarian ing to White 1976 this has occurred eight separate Schmidtea polychroa, a hermaphrodite (Pongratz et al. -times lower (e.g., in the monecious spe 1.5ف times in Mantodea), unless one supposes that each XX/ 2001)] or XO system has passed through an XX/XY system fol- cies Pinus taeda; Sewell et al. 1999). Nonetheless, Triv- lowed by the complete degeneration of the Y. Finally, ers (1988) argued that in dioecious species the direc- the pleiotropy hypothesis does not explain why achias- tion of heterochiasmy tends to be biased toward less mate meiosis in the heterogametic sex is maintained recombination in male meiosis and is not affected much once sex-chromosome heteromorphism has evolved by heterogamety. such that the sex chromosomes would no longer be Heterochiasmy theories: Several ideas have been put homologous and able to recombine (e.g., in the extreme forward to explain the occurrence of heterochiasmy. case recombination on the sex chromosome cannot oc- They fall into several groups that I briefly review before cur in XO individuals). turning to the model. The no-recombination hypothesis provides no expla- Mechanistic explanations: A different internal environ- nation for why sex differences in recombination should ment between male and female tissue, due to physiologi- preexist the formation of Y or W. Furthermore, the cal or molecular processes, is a potential cause of hetero- assumption on which this hypothesis rests—that heterochiasmy. For instance, Bernstein et al. (1988) argued gamety will always gradually evolve in the sex with low that higher recombination rates in females could be due recombination—will not always hold: if the sex-deter- to higher metabolic rates in females. This hypothesis is mining mechanism depends on a single locus (even if weak since, in hermaphrodites, both male and female this is considered improbable; Charlesworth 2002), meiosis occurs in the same individual. However, even it does not depend on recombination and therefore in hermaphrodites, male and female meiosis may not the “heterozygote” sex, which is likely to become the occur at the same time—and therefore may occur under heterogametic sex, should be equally likely to be the different conditions. For instance, in Pinus, male and sex with or without recombination. female meiosis occurs in different seasons. Differences Heterochiasmy data: Measuring heterochiasmy is dif- in temperature, which have been shown to influence ficult. Most data that have been collected consist of recombination rates, could thus explain the observed chiasma counts. This method does not often take into heterochiasmy with more recombination in males in this account the position of crossing over along chromo- genus (see Plomion and Omalley 1996). But without somes, which in general varies between males and fe- more data on the timing of meiosis in hermaphrodites, males, resulting in a strong bias (male chiasmata are the extent to which timing may explain heterochiasmy often either proterminal or procentric; e.g., Fletcher cannot be evaluated. Another possibility is that hetero- and Hewitt 1980). This difference of chiasma position chiasmy itself may be a way to control the timing of between the sexes is also a problem in studies using map meiosis. Crossing over has been hypothesized to regu- distance between few markers: depending on where the late segregation and DNA repair; chiasma number markers are along the chromosomes more recombina- could also modulate the speed of meiosis. For example, tion may appear to be in one sex or the other when in as mentioned above, in gonochoric animals, male meio- fact it is not. Molecular techniques have also revealed sis tends to be continuous whereas female meiosis gener- a large source of variation in recombination rates from ally stops twice; numerous chiasmata could stabilize the chromosome to chromosome and from genotype to female meiosis when it stops (sometimes for long peri- genotype (see Korol et al. 1994, p. 280, for examples). ods of time), whereas few chiasmata could allow males Analyzing heterochiasmy data can present further diffi- a fast gametogenesis. However, this hypothesis may not culties. For example, not knowing the rate of evolution apply to most plants for which the timing of male and of heterochiasmy can make it difficult to judge whether female meiosis does not seem to differ much. there is phylogenetic inertia and thus how many species Pleiotropic effect of sex-chromosome heteromorphism: The or groups of species must be contrasted when attemp- Haldane-Huxley rule could explain heterochiasmy as ting to test hypotheses. well as achiasmy (for instance Huxley 1928 invoked it Sex Dimorphism in Recombination 813 for marked heterochiasmy). However, at the very least, Genetic setting: Consider a sexual dioecious popula- this is not a general explanation: counter examples are tion with three autosomal loci {i, j, k}. Suppose that locus numerous, and a pleiotropic effect of the evolution of i is a sex-specific recombination modifier locus and that sex-chromosome heteromorphism cannot account for loci j and k are under viability selection. The aim is to heterochiasmy in hermaphrodites (see Burt et al. 1991). compute the frequency change at the modifier locus The neutral hypothesis: The evolution of the average over one generation to determine under which condi- recombination rate has been well studied theoretically tions a recombination dimorphism can evolve. I follow (see, for review, Barton and Charlesworth 1998; notation used in Barton and Turelli (1991) and Bar- Otto and Lenormand 2002) and some experiments ton (1995) for variables (see Table 1). Each locus l has have shown that it can evolve (e.g., Korol and Iliadi two alleles and is modeled using a random variable Xl, 1994). On the other hand the evolution of the differ- which takes the value of 0 or 1 for the first and second ϭ ence in recombination rates between males and females allele, respectively. Let x(s) {Xi(s), Xj(s), Xk(s)} and x(s)* has neither theoretical nor empirical support. When represent a haploid set of alleles inherited from the Burt et al. (1991) failed to find support for a correlation father and the mother, respectively, in an individual of between the magnitude of heterochiasmy and the op- sex s (s ϭ m, f for male or female) and let the couple portunity for sex difference in selection (which, they (x(s), x(s)*) be a diploid genotype (which is either a male argued, should be high in dioecious animals, intermedi- or a female). The subscript (s) denotes a sex-specific ϭ Ϫ ate in hermaphroditic plants, and low in hermaphro- value throughout. The average frequency pl (pl 1 ditic animals), they suggested that heterochiasmy might ql) of the allele coded by 1 in the whole diploid popula- ϩ ϩ ϩ be neutral. A failure to find support for one hypothesis, tion is the expectation of Xlm Xlm* Xlf Xlf*. The however, does not mean that a trait is neutral—especially linkage disequilibria between loci are measured by when, as in this case, the hypothesis under consideration ϵ Ϫ Ϫ had no clear theoretical foundation and no empirical CU,V E΄͟(Xl pl)͟(X*l pl)΅, (1) l ʦU l ʦV justification. Evolutionary explanations, sexual selection: Trivers (1988) where U, V represents the different possible sets of loci proposed that heterochiasmy could be due to sexual (i.e., U, V ʦ {л, i, j, k, ij, ik, jk, ijk}) distributed on selection. He supposed that because of the higher vari- maternal and paternal chromosome and by convention ance of reproductive success in males, “the genes and Xл ϭ X *л ϭ 1 and pл ϭ 0. In haploids, only the associa- combinations of genes being passed in males would tions between loci on a single chromosome are needed be superior on average, compared to genes passed in (U or V is empty). I also assume for simplicity (and females” (p. 277). He concluded that “insofar as the because I am not aware of any corresponding genetic actual combinations in which a male’s gene appear are mechanism) that sex-of-origin effects do not extend back important to their success, then he will be selected to more than one generation (i.e., like with imprinting, reduce rates of recombination (compared to females) meiosis resets eventual sex-of-origin marks of the previ- ϭ in order to preserve these beneficial combinations.” ous generation). As a consequence, in haploids CU, л(s) Trivers argued that this explains why males recombine Cл, U(s) and we simply note the disequilibria CU(s) . less—and that exceptions can be accounted for by Life cycle: The model describes a species undergoing changes in the regime of sexual selection. However, this the following events during its life cycle: diploid selec- theory is largely inspired by models of evolution of sex tion (D), meiosis (M), haploid selection (H), and syn- chromosomes (Nei 1969) and has received as yet no gamy (S). The superscripts D, M, H, and S denote these theoretical foundation for autosomes. Worse, the oppo- different events. By construction of the life cycle, male site verbal model has been made: “As two potentially and female populations are strictly identical just after important sources of linkage disequilibrium are selec- syngamy on autosomes because both male and female tion and drift, one might expect that the sex experienc- individuals are made from the fusion of a male and a ing the more intense selection, or otherwise having the female gamete and because I suppose for now that sex higher variance in reproductive system, should have is determined at unlinked loci (I consider linkage to a more recombination” (Burt et al. 1991). sex-determining locus at the end of the model section). Therefore, I consider the start of a generation just after syngamy when male and female populations have ex- MODEL actly the same frequency and combinations of autoso- Here, I present a three-locus model to determine mal genes. The linkage disequilibria are measured the selection coefficient on a recombination modifier within a generation relative to the gene frequency at having different effects in males and females. Alleles this moment. Denote CU,V any value of linkage disequi- at this modifier locus change the recombination rate librium measured just after syngamy. Within one gener- between two loci subject to both haploid and diploid ation during the life cycle, the CU,V will vary around this selection. A Mathematica notebook (Wolfram 1999) value and these variations will be sex specific until the Ͻ D with the full recursions can be obtained at http:// next syngamy event. I therefore denote CU,V, C U,V(s), Ͼ DM DMH DMHS www.cefe.cnrs-mop.fr/wwwgdyn . C U(s), C U(s) , C U,V the linkage disequilibria values mea- 814 T. Lenormand

TABLE 1 Table of notations

Parameters a Selection coefficient ␣, ␤ Selection coefficient when partitioned in average effect (␣), sex effect (␣ˆ), sex-of-origin effect (␣˜), and sex-by- sex-of-origin interaction (␤), [see (17)] ε Multiplicative epistasis [see (20–21)] ␦ Unspecified source of linkage disequilibrium between j and k loci r Recombination rate ␳ Effect of allele 1 at the modifier locus on the recombination rate between loci j and k Variables X Haploid genotype at a locus taking value 0 or 1 for the first and second allele, respectively x and x* Set of loci inherited from the father and the mother, respectively p and q Allele 1 frequency and allele 0 frequency C Measure of association of alleles Subscripts (s) Refers to the value of a parameter or variable taken in an individual of sex s; s can be male/female (m or f ) or homogametic/heterogametic (00/01) U Refers to a haploid set of loci U, V Refers to a diploid set of loci made up of two haploid sets: U inherited from the father and V inherited from the mother l Refers to a single locus i, j, and k Refer to the loci labeled i (the modifier locus), j (a selected locus or a sex-determining locus), k (a selected locus) Superscripts D, M, H, S Refer to the different events in the life cycle (diploid selection, meiosis, haploid selection, syngamy): when applied to a variable, it indicates its value after the last event listed [e.g., V DM is the value of the variable V after (1) diploid selection and (2) meiosis, in this order]; when applied to a parameter it indicates its value summed over the events listed (e.g., P DH is the sum of value of P in the haploid and diploid phases) V or P A bar on a parameter or a variable indicates the average over male and female values Vˆ or Pˆ A circumflex on a parameter or a variable indicate the difference between male and female values Њ Refers to a QLE value for an association C B Refers to recombination rates in the absence of the modifier allele

sured along the life cycle (Figure 1), after syngamy, equilibria deﬁned on haploids are needed to describe diploid selection, meiosis, haploid selection, and syn- the population. I follow these events in this order in gamy, respectively. Note that after meiosis, only the dis- the next sections.

Figure 1.—Life cycle. Thick and thin lines represent diploid and haploid phases, respectively. Dashed and non- dashed lines represent male and female life cycles, respectively. The notation for the linkage disequilibria is described in the text. Sex Dimorphism in Recombination 815

D D D D Diploid selection: I use a sex-specific diploid fitness Symmetrical moments (C л,jk, C k,j, C л,ij, C л,ijk) can be ob- function that allows for dominance, cis-, and trans-epista- tained by permuting all U and V indices in (3) and (4). D D sis terms (i.e., a combination of genes may have different The recursions for C ik,л and C л,ik are equivalent to re- D D fitness effects if the genes are on the same or different cursions for C ij,л and C л,ij, respectively, with subscript j chromosomes) and sex-of-origin effects (i.e., a gene or replaced by k throughout. Note that only the variations combination of genes in a diploid individual is not con- in the disequilibria that are useful below are given in sidered to have the same fitness effect if it is contributed (3) (for instance, diploid selection changes Cj,j, but this from the mother or the father). Selective interactions moment does not influence frequency change at the between more than two loci are ignored. Specifically, recombination modifier locus). the fitness function is The assumption that selective interactions between D ϭ D alleles are smaller than directional selection allows the W (x(s), x*(s)) ͚aU,V(s) ͟Xl(s) ͟X*l (s) , (2) U,V l ʦU l ʦV analysis of a case more general than the situation in which all selection coefficients have the same order where U and V represent the set of selected alleles inher- (Barton 1995, see results section for details). For the ited from the father and the mother, respectively (i.e., ϭ л sake of discussion, I also introduce in (3) an unspecified one of the following set of indices U, V { , j, k, jk}, ␦D sex-specific source of linkage disequilibrium, jk(s), be- with the convention that a лD л ϭ 1), (s) indicates the , (s) tween the selected loci j and k during the diploid phase, gender of the individual carrying the alleles (whether it which could be created by forces other than those con- is a male or a female), and the superscript D indicates that sidered in this model, such as migration (Lenormand these parameters represent selection during the diploid and Otto 2000) or drift (Otto and Barton 1997). phase. For instance a Dл is the additive effect of the se- j, (s) Meiosis: Meiosis occurs after diploid selection in a lected allele at locus j during the diploid phase (D) in an B individual of sex (s) when this allele is inherited from the sexual life cycle. Let r U(s) equal the basal recombination father. Overall, for two loci, diploid selection is described rate between the set of loci U, i.e., when the 0 allele at using 30 independent parameters: there is no constraint the modifier locus is fixed in the population. Each copy on the fitness matrix (16 selected genotypes are in each of the modifier allele at locus i modifies the recombina- sex and hence 15 relative fitness). tion rate between the viability loci j and k by a small ␳ ϭ ␰ Assuming that the directional selection coefficients amount (s) O( ) in an individual of gender s. I simply D D D D ␰ denote rU(s) the average recombination between the set a j,л(s), a k,л(s), a л,j(s), a л,k(s) are small, of order , a small parameter, and that all other selection coefficients— of loci U over the different genotypes at locus i in the interactions between alleles—are smaller, of order ␰2, population of gender s. Assuming that the loci are in the different disequilibria measured between loci i, j the order i-j-k, and k change after selection on diploids as ϭ B ϩ␳ ϩ ϭ B ϩ ␳ rjk(s) r jk(s) (s)E[Xi X*i ] r jk(s) 2 (s)pi D ϭ ϩ C jk,л(s) Cjk,л A ϭ B rij(s) r ij(s)

ϩ D Ϫ D D ϩ D ϩ D ϩ D ϩ␦D (a jk,л(s) a j,л(s)a k,л(s) a jk,j(s)pj a jk,k(s)pk a jk,jk(s)pjpk jk(s)) ϭ Ϫ B B ϩ␳ rik(s) rijk(s) r ij(s)(r jk(s) (s))

ϫ ϩ ␰2 ϭ B ϩ B ϩ␳ Ϫ B B ϩ␳ pqjk o( ) rijk(s) r ij(s) r jk(s) (s) r ij(s)(r jk(s) (s)), (6)

D ϭ Ϫ C j,k(s) Cj,k A where rijk is the chance that the trilocus genotype is broken apart by recombination. Note that when locus ϩ D Ϫ D D ϩ D ϩ D ϩ D ϩ␦D (a j,k(s) a j,л(s)a л,k(s) a j,jk(s)pj a jk,k(s)pk a jk,jk(s)pjpk jk(s)) i is involved, the recombination rate does not depend

ϫ ϩ ␰2 on the frequency of the modifier allele because recombi- pqjk o( ) nation matters only when locus i is heterozygous. After D ϭ ϩ D ϩ ␰ C ij,л(s) Cij,л a k,л(s)C ijk,л o(Cijk,л ) meiosis the different disequilibria measured between loci i, j, and k change as follows: D ϭ ϩ C ijk,л Cijk,л o(Cijk,л), (3) 2C DM ϭ (1 Ϫ r )(C D л ϩ C лD ) ϩ r (C D ϩ C D ) where jk(s) jk(s) jk, (s) ,jk(s) jk(s) j,k(s) k,j(s) DM ϭ Ϫ D ϩ D 2C ijk(s) (1 rijk(s))(C ijk,л(s) C л,ijk(s)) D Ϫ D D Ϫ D ϭ a j,л(s) a л,j(s) ϩ a k,л(s) a л,k(s) A pq j Ck,л pqk Cj,л ϩ␳ D ϩ D Ϫ D Ϫ D ϩ DM 2 2 (s)pqi(C j,k(s) C k,j(s) C jk, л(s) C л,jk(s)) o(C ijk(s)). (7) D Ϫ D D Ϫ D ϩ (a j,л(s) a л,j(s))(a k,л(s) a л,k(s)) pq jk (4) DM DM 4 The recursion for C ij(s) (respectively C ik(s)) is equivalent DM to recursion for C jk(s) with subscript k (respectively j) (with the value of Cj,л and Ck,л given in the syngamy replaced by i. section) and Haploid selection: Haploid fitness is defined in the ϭ Ϫ same way as diploid fitness except that there are no pqU ͟pl(1 pl). (5) l ʦU trans-effects and no sex-of-origin effects of alleles. A 816 T. Lenormand ϭ superscript H indicates selection occurring during the where pjk pjpk . To simplify this expression, I use a haploid phase, with fitness equal to quasi-linkage equilibrium (QLE) approximation (see H ϭ H Nagylaki 1976; Barton and Turelli 1991; Barton W (x(s)) ͚a U(s) ͟Xl(s) . (8) U l ʦU 1995) to determine the value of the different disequilibria after syngamy (C ) or meiosis (C DM ). Overall, for two loci, haploid selection is described using U,V U(s) QLE assumption: Assuming that recombination rates six parameters (three relative fitnesses in each sex). are of higher order than epistasis, the different disequi- Assuming, as for diploid selection, that ␣H , ␣H are j(s) k(s) libria quickly reach “quasi-linkage” equilibrium, at O(␰) and that ␣H is O(␰2) the different linkage disequi- jk(s) which point their values, denoted with a circle super- libria change after selection on haploids as script, can be obtained by solving to leading order in ␰ DMH ϭ DM ϩ H ϩ␦H ϩ ␰2 C jk(s) C jk(s) (a jk(s) jk(s))pqjk o( ) the difference equation, DMH ϭ H DM ϩ DM ␰ DMHS Ϫ ϭ C ij(s) a k(s)C ijk(s) o(C ijk(s) ) C U CU 0, (13) DMH ϭ DM ϩ DM DMHS C ijk(s) C ijk(s) o(C ijk(s)), (9) where C U is rewritten in terms of CU using Equations 10, 9, 7, and 3. To simplify the result, it is much simpler where again, I introduce an unspecified, sex-specific to partition the selection coefficients into four terms: source of linkage disequilibrium, ␦H , between the se- jk(s) the average effect of a gene or gene combination over lected loci j and k during the haploid phase. sex and sex-of-origin, Syngamy: I assume that each new diploid individual ␣D ϭ D ϩ D ϩ D ϩ D results from the random fusion of a male and a female U,V (a U,V(m) a U,V( f ) a V,U(m) a V,U( f ))/4, (14) gamete and that its gender is independent of its autosomal genes. After syngamy the different disequilibria the sex-effect averaged over sex-of-origin, measured between loci i, j, and k change as ␣D ϭ D ϩ D Ϫ D Ϫ D ˆ U,V (a U,V(m) a V,U(m) a U,V( f ) a V,U( f ))/2, (15) DMHS ϭ DMH ϩ DMH DMH ϩ DMH DMH DMH C U,V ΂C U(m) ͚ C S(m) C T(m) ͚ C S(m) C T(m) C W(m)΃ SϩTϭU SϩTϩWϭU the sex-of-origin effect averaged over sex,

ϫ DMH ϩ DMH DMH ϩ DMH DMH DMH ␣D ϭ D ϩ D Ϫ D Ϫ D ΂C V( f ) ͚ C S( f ) C T( f ) ͚ C S( f ) C T( f ) C W( f )΃, (10) ˜ U,V (a U,V(m) a U,V( f ) a V,U(m) a V,U( f ))/2, (16) SϩTϭV SϩTϩWϭV and the interaction between the sex and the sex-of- where the sums are over disjoint partitions of U or V origin effects, with the convention S, T, W ϶ л if these partitions exist ␤D ϭ D ϩ D Ϫ D Ϫ D [because U and V may contain less than three (two) U,V a U,V(m) a V,U( f ) a V,U(m) a U,V( f ) , (17) loci, the triple (double) partition may not be possible] which gives and ␣D ␣D ␤D ϭϪ ϭ ϭϪ ϭ DMH Ϫ DMH D ϭ ␣ D ϩ ˆ U,V ϩ ˜ U,V ϩ U,V Cl(m) Cl(f ) Cl,л C л,l (p l(m) p l( f ) )/2 a U,V(m) U,V 2 2 4 ϭ DH ϩ DH Ϫ DH Ϫ DH pql(a л a л a л a л )/4 (11) l, (m) ,l(m) l, ( f ) ,l( f ) ␣D ␣D ␤D D ϭ ␣ D Ϫ ˆ U,V ϩ ˜ U,V Ϫ U,V DH D ϩ H a U,V( f ) U,V in which the notation x is shorthand for x x . 2 2 4 DMHS ϭ DMH DMH Under random mating C U,V C U C V (Barton ␣D ␣D ␤D and Turelli 1991). In this model, however, mating is D ϭ ␣ D ϩ ˆ U,V Ϫ ˜ U,V Ϫ U,V a V,U(m) U,V not random (female gametes fuse with male gametes). 2 2 4 Extra terms in C DMHS arise because the frequencies of U,V ␣D ␣D ␤D D ϭ ␣ D Ϫ ˆ U,V Ϫ ˜ U,V ϩ U,V the selected alleles are different between male and fe- a V,U( f ) U,V . (18) male gametes before syngamy. These frequency differ- 2 2 4 ences are caused by differences in diploid and haploid H The haploid selection coefficients a U(s) and the modifier selection between males and females. This extra source ␳ effects (s) are rewritten in the same way (but without sex- of linkage disequilibrium is analogous to the linkage of-origin and sex-by-sex-of-origin effect); i.e., the overbar disequilibria created by migration between populations indicates an average value over males and females while with different gene frequencies (Lenormand and Otto a hat indicates a difference between male and female 2000). values. Frequency change at the modifier locus: The frequency The linkage disequilibrium between the selected loci change at the modifier locus over one generation is Cjk does not enter directly in (12), but its average value found by linearizing the exact recursions to order ␰5, produces Cijk [see (7)], which in turn produces Cij and ⌬ ϭ DMHS Ϫ pi p i pi Cik [see (3) and (9)]. Its average QLE value measured after syngamy is 1 ϭ D ϩ D Њ Њ ͚ ͚ (a U,л(s)CUi,л a л C л,Ui ϩ ,U(s) C jk,л C л,jk (E ϩ D) 4 sϭm, f Uϭj,k,jk ϭ Ϫ D pq jk ΂ E ΃, (19) ϩ D ϩ D ϩ H DM ϩ ␰5 2 rjk a jk,U(s)pUCijk,л a U,jk(s)pUC л,ijk 2a U(s)C Ui(s)) o( ), (12) where Sex Dimorphism in Recombination 817

ϵ ε DH ϩ ␣H␣D ϩ␣D␣H ϩ␣D␣DH ϩ␣D␣DH ␰ E jk (ˆ j ˆ k ˆ j ˆ k ˜ j ˆ k ˜ k ˆ j )/4 are small (of order ). The more general result with a D ϵ ␣ D Ϫ ␣ D ϩ ␣ D␣ DH ϩ␣D␣ DH ϩ␣DH␣ DH large difference in basal recombination rates between E jk j,k (˜ j ˆ k ˜ k ˆ j )/2 ˆ j ˆ k /4 Њ ϩ males and females is simple for the value of (C jk,л ϭ ␦ D ϩ ␦ H Њ Ϫ D D jk jk . (20) C л,jk)/2 [it adds a term equal to pq jk Eˆ ˆrjk/4rjk to its Њ ϩ Њ value in (19)] and for the values of (C ijk,л C л,ijk)/2 Equation 20 summarizes the different forces producing Њ Ϫ Њ Ϫ␳ and (C ijk,л C л,ijk) (it adds the same term times the linkage disequilibrium between the selected loci: Ϫ␳ pqi/rijk and ˆ pqi, respectively). However, with a large diploid multiplicative epistasis, difference in basal recombination rates between males ε D ϵ ␣ D Ϫ ␣ D␣ D ϩ ␣ D ϩ ␣ D ϩ ␣ D jk jk j k jk,jpj jk,kpk jk,jkpj pk , (21) and females, the QLE values of Cij and Cik are complicated (see results below). haploid multiplicative epistasis, General result for autosomes: The frequency change ε H ϵ ␣ H Ϫ ␣ H␣ H at the modifier is obtained using Equation 12. The QLE jk jk j k , (22) values of the disequilibria measured after syngamy, Њ mixing at syngamy of male and female gametes with C U, are given by Equation 23, and the QLE values of ␣ DMЊ different frequency (all terms including ˆ); difference the disequilibria measured after meiosis, C U(s) , are gen- ␣ D Ϫ ␣ D between average cis- and trans-epistasis ( jk j,k), and der specific and were computed using Equations 7 and Њ finally, the unspecified source of linkage disequilibrium 3 and the C U values. This gives ␦ DH that I introduced for generality ( jk ). However, and ␳ ϩ ␳ˆ D ⌬ ϭϪ ϩ⌳ (E D) ϩ Ê ϩ H ϩ D more importantly, (19) shows that Cjk is made up of two pi pqijk ΄(E )΂ ΃ (Eˆ E˜ ) terms: one depending on recombination (proportional rijkrjk 4rijk ϩ to E D) and another independent of recombination ␳Eˆ D ␳ˆ(E ϩ D) (proportional to E D). As a consequence, the different ϫ ΂ ϩ ΃΅ ϩ o(␰5) (25) 4 4r forces summarized in E D play no role for the evolution jk of recombination (for instance, average trans-epistasis with E, Eˆ D, Eˆ H, ⌳, D: ␣ D j,k has no effect at all on the evolution of recombination ϵ εDH ϩ ␣H␣D ϩ␣D␣H ϩ␣D␣DH ϩ␣D␣ DH rate, a result already mentioned by Pylkov et al. 1998 E jk (ˆ j ˆ k ˆ j ˆ k ˜ j ˆ k ˜ k ˆ j )/4 for the evolution of average recombination rates). D ϵ ␣D Ϫ␣D ϩ ␤D␣DH ϩ␤D␣DH Eˆ ˆ jk ˆ j,k ( j ˆ k k ˆ j )/2 The QLE values of C ij, л, C л,ij, C ik, л, C л,ik, C ijk, л, C л,ijk D ϵ ␣D ϩ␣D ϩ␣D ϩ ␣D ␣D ϩ ␣DH ϩ␤D Ϫ are more complicated but can be deduced from the E˜ ˜ jk ˜ jk,j pj ˜ jk,k pk ͚˜ j ( k k /rik k (1 rij)/4) ↔ values of j k ˆ H ϵ ␣ H ϩ ͚␣ H ␣ H ϩ ␣ DH ϩ␤D Ϫ pq E ˆ jk ˆ j ( k k /rik k (1 rij)/4) Њ ϭϪ ijk ␣ H ϩ ␣H ϩ Ϫ ␣D ϩ␤D j↔k C ij,л ΂[4 k rij 2ˆ k (1 rij)(2˜ k k rij)] 2 1 1 ⌳ ϵ ␣ DH␣ DH΂ ϩ Ϫ 1΃ ϩ ͚r ␣D(␣H ϩ␣D)/4 D j k ij ˆ k ˆ j ˜ j ␳Eˆ ␳ˆ(E ϩ D) ↔ ϫ ΄ ϩ ΅ rij rik j k 4rij 4rijrjk ϵ ␦ D ϩ ␦ H D jk jk . (26) ϩ ␣ H ϩ␣H ϩ Ϫ ␣ D ϩ␣D [2 k ˆ k rij (1 rij)(2 k ˆ k rij)] E, D, and Eˆ D are already defined above but are repeated ␳(E ϩ D) ␳Êˆ D ↔ ϫ ΄ ϩ ΅΃ ϩ o(␰4) here for clarity. Sums over j k indicate that a term rijrjkrijk 4rijrijk with indices j and k switched must be added to the term ϩ ˆ D ˆ D ϩ in the sum. The same computations can be made by Њ ϭϪ ␳ (E D) ϩ E ϩ␳ E ϩ (E D) ϩ ␰3 C ijk,л pqijk ΂ ΂ ΃ ˆ΂ ΃΃ o( ),(23) ␰ rjkrijk 2 4rijk 2rjk assuming that epistasis terms are of order instead of ␰2 (in which case recombination rates are assumed to where be of order 1). In that case, the frequency change at D ϵ ␣D Ϫ␣D ϩ ␤D␣DH ϩ␤D␣DH the modifier locus is stronger (of order ␰3) and is also Eˆ ˆ jk ˆ j,k ( j ˆ k k ˆ j )/2. (24) given by (25) where only terms of order ␰ are kept in Њ Њ C ik,л is obtained from C ij,л by switching j and k subscripts E, Eˆ D, Eˆ H, E˜ D, ⌳, that is, with Њ Њ and the C л,U are obtained from the C U,л by switching ϭ ε DH the sign of all ␣ˆ, ␣˜ and the sign of ␳ˆ (the male-female E jk difference of the modifier effect). The three-way associa- Eˆ D ϭ␣ˆ D Ϫ␣ˆ D Њ jk j,k tion C ijk is built at meiosis if cis- and trans-pairwise dis- ˜ D ϭ␣D ϩ␣D ϩ␣D equilibria between loci j and k do not equal one another E ˜ jk ˜ jk,j pj ˜ jk,k pk D Ϫ D ϶ (C jk(s) C j,k(s) 0) (note that Cijk is only added to in Eˆ H ϭ␣H D ˆ jk Equation 7). This condition is easily met because C jk D ⌳ϭ accumulates through time whereas C j,k is strongly re- 0 duced by segregation each generation [see (10)]. These ϭ ␦ D ϩ ␦ H D jk jk . (27) QLE solutions are valid when there are differences between males and females in basal recombination rates As a consequence, Equation 25 covers both cases and B ϶ B (i.e., when r U(male) r U(female)) provided these differences allows one to discuss situations where epistasis terms are 818 T. Lenormand closer to zero (which is not possible with the strong ence in recombination rate between males and females epistasis approximation). A modifier allele will change (i.e., with ␳ϭ0, call it a “symmetric” modifier) captures in frequency because it causes the average recombina- the selection pressure acting on heterochiasmy, al- tion rate (when ␳ ϶ 0) to evolve, because it causes a though a sexual dimorphism in recombination rate may dimorphism in recombination rates between males and evolve by selection on a modifier that also changes the females to evolve (when ␳ˆ ϶ 0), or because of both. A average recombination rate. A symmetric modifier will modifier allele with an effect restricted to one sex will change in frequency if fall into this last category. ˆ D ϩ⌳ ˆ H ϩ ˜ D ϩ ⌬ ϭϪ ␳ E (E ) ϩ (E E )(E D) ϩ ␰5 Evolution of the average recombination rate: The evolution pi pqijk ˆ΂ ΃ o( ) of the average recombination rate has been determined 4rijk 4rjk several times in the literature. It has been found that (32) increased recombination evolves when there is weak is nonzero [where E, Eˆ D, Eˆ H, ⌳, D are given by (26)]. Ϫ⌳ Ͻ ␣ Ϫ negative multiplicative epistasis (i.e., when jk This result holds if the recombination rate does not ␣ ␣ Ͻ ⌳ j k 0, where is a threshold value; Barton 1995, differ much between male and female. When epistasis see below). This result has been shown to hold for both and directional selection are of the same order (␰), only haploid and diploid selection (Barton 1995, Equation Њ Њ the QLE values of C ijk,л and C л,ijk are needed to compute A1.5f). It can be obtained from Equation 25 assuming the frequency change at the modifier locus to leading that there are no selective differences between males order. In that case, the more general result with a large and females during both diploid and haploid phase, male-female difference in basal recombination rates no selective differences associated with sex-of-origin of (noted ˆr) is not a lot more complicated than (32). The alleles, and no other source of linkage disequilibrium following term must be added to the expression in (32) besides epistasis (i.e.,if in which the values of E, Eˆ D, Eˆ H, D are given by (27): ␣ˆ ϭ 0, ␣ ϭ 0, D ϭ 0), (28) U ˜ U pq ␳ˆ Eˆ DEˆ Hˆr Eˆ D(Eˆ H ϩ E˜ D)ˆr ijk ΂ ijk ϩ jk΃ ϩ o(␰3). (33) which leads to 16 rijk rjk ␳ ϩ⌳ ⌬ ϭϪ E(E ) ϩ ␰5 A necessary condition for the evolution of dimorphism pi pqijk ΄ ΅ o( ), (29) rijk rjk is therefore where the expressions for E and ⌳ simplify to Eˆ D ϶ 0orEˆ H ϩ E˜ D ϶ 0, (34) ϭ ε DH E jk in both cases [i.e., in (32) and (33)] even if other unspecified forces generate nonrandom associations between ⌳ϭ␣ DH␣ DH 1 ϩ 1 Ϫ j k ΂ 1΃. (30) the selected alleles (i.e., even if D ϶ 0). Large differ- rij rik ences in recombination rate between male and female Thus, the weak negative epistasis result holds provided matter in (33) if both conditions are met, which is very that assumptions (28) are fulfilled. However, Equation unlikely to be a common situation and is not discussed 25 shows that the average recombination rate may evolve further here. by other means: it can evolve if First, heterochiasmy can evolve if Eˆ D ϶ 0, i.e., if the (E ϩ D)(E ϩ⌳) Eˆ D(Eˆ H ϩ E˜ D) difference between cis- and trans-epistasis is different in ⌬ ϭϪ ␳ ϩ ϩ ␰5 D D pi pqijk ΂ ΃ o( ) males and females during diploid selection (␣ˆ Ϫ␣ˆ ϶ r r 4 jk j,k ijk jk 0) or if there are sex and sex-by-sex-of-origin effects on (31) ␤D␣DH ϩ␤D␣DH ϶ directional selection coefficients ( j ˆk k ˆj 0). ␣D ␣D is nonzero, which suggests three other possibilities: it There is no simple reason why ˆ jk and ˆ j,k should differ. can evolve if Eˆ D Eˆ H or Eˆ D E˜ D is nonzero, which means For instance, if epistasis is the consequence of the bio- that the difference between cis- and trans-epistasis is chemical properties of two proteins coded by loci j and different in males and females during diploid selection k, it should not matter whether these proteins are coded or that there are sex-by-sex-of-origin selective interac- by alleles on the same or different chromosomes. As a tions [i.e., Eˆ D ϶ 0; see definition of Eˆ D in (26)] and consequence, a sex difference in diploid epistasis is not haploid selection differences between males and fe- very likely to produce a selection pressure on heterochi- males (Eˆ H ϶ 0) or sex-of-origin effects during diploid asmy. selection (E˜ D ϶ 0). The average rate of recombination Second, sex dimorphism in recombination can also can also evolve if there is another source of linkage evolve if Eˆ H ϶ 0orE˜ D ϶ 0. Sex effects during haploid ␣H disequilibrium (D ϶ 0), for instance, that produced by phase (ˆ U ) and sex-of-origin effects during the diploid ␣D migration (Lenormand and Otto 2000) or drift (Otto phase ˜ U play a very similar role on the evolution of and Barton 1997, 2001). heterochiasmy [compare the expression of Eˆ H and E˜ D Evolution of a dimorphism in recombination rate: The fre- in (26)]. Eˆ H ϶ 0 can be due to any selective difference quency change of a modifier that affects only the differ- between males and females during the haploid phase Sex Dimorphism in Recombination 819

␣ H ϶ ␣ H ϶ ␣ H ϶ D ϶ (ˆ jk 0orˆ j 0orˆ k 0) and similarly, E˜ 0 respectively. Note that Cjk and Cijk are zero in the homo- can be due to any sex-of-origin effect during diploid gametic sex because locus j cannot be heterozygous in selection. However, when epistasis and directional selec- this subpopulation. Frequency change at locus i is given tion have the same order, Eˆ H and E˜ D are both dominated by by epistasis terms [see (27)]. In this last situation, epista- ⌬ ϭ 1 ␣ D Њ ϩ 1␣D Њ ϩ Ϫ ␣H Њ ϩ␣H Њ sis is also an important source of linkage disequilibrium pi ΂ k C ik ˆ k Cˆ ik (1 rik)( k(00) C ik(00) k(01)C ik(01))΃ 2 2 (it is the main term in E). As a consequence, having ␣ DH ϭ ␰ 3 strong epistasis, jk O( ), with a haploid sex effect ϩ o(␰ ), (36) ␣ H ␣ D ϭ ␰ or a diploid sex-of-origin effect, ˆ jk or ˜ jk O( ), is the which simplifies to simplest sufficient condition to have an important selection pressure for heterochiasmy. For instance, in the ␳ ␣ˆ DH[␣ˆ D ϩ␣ˆ H (1 Ϫ r )]pq ⌬ ϭϪ (01) k k k ik ik ϩ ␰3 simple situation where selection occurs only during the pi ϩ Ϫ ϩ o( ). (37) 4rjk[rik (2 rik)(rij rjk)] haploid phase and where haploid epistasis is the main ␳ source of linkage disequilibrium, D ϭ o(␰), a lower re- The effect of the modifier in the homogametic sex (00) combination rate is expected to evolve in the sex with plays no role because recombination between loci j and the strongest absolute value of epistasis. k is irrelevant when the j locus is homozygous. This Nevertheless, the condition (34) may be met even result indicates that a modifier allele that decreases the in the absence of epistasis, although this will tend to recombination rate between a sex-determining locus generate a weak selection pressure on heterochiasmy. and a selected locus will always increase in frequency ␣DH ϭ ␣D Ϫ␣H In this situation, a mechanism other than epistasis gen- except if ˆ k 0orifˆ k lies between ˆ k and Ϫ␣H Ϫ ˆ k (1 rik). The first condition indicates that there erating the linkage disequilibrium Cjk is necessary in most cases for heterochiasmy to evolve. This mechanism must be a sex difference in selection in either the hap- may be a covariance in directional selection coefficients loid or the diploid phase for recombination to evolve between sexes across loci across both haploid and dip- on sex chromosomes. The second condition is unlikely ␣ H␣ D ϩ␣D␣ H ϶ as it requires that locus k be under both diploid and loid phases (ˆ j ˆ k ˆ j ˆ k 0) or the presence of both sex-of-origin effects and sex effects within or across haploid selection, that within each phase selection dif- ␣ D␣ DH ϩ␣D␣ DH ϶ fers between males and females, and that the male- phase (˜ j ˆ k ˜ k ˆ j 0). This mechanism could also be unspecified in this model (D ϶ 0). In these last female difference has opposite signs in the haploid and cases, the direction of evolution of heterochiasmy de- diploid phases and falls in a narrow interval. Note that pends on the sign of these forces generating the linkage the selection on the modifier is important even if the disequilibrium. three loci recombine freely. These results are qualita- Evolution of recombination around a sex-determining tively consistent with results obtained by Nei (1969), locus: Consider the same model as above except that who studied the case where allele 0 at locus k causes locus j is a sex-determining locus (i.e., one sex is homozy- sterility in females (and is dominant in females) while gous 00 and the other is heterozygous 01, and the geno- allele 1 causes sterility in males (and is recessive in type 11 does not exist). Locus k is a selected locus and males). locus i a recombination modifier locus, with alleles changing the recombination rate between j and k.As DISCUSSION before, suppose that locus k is exposed to viability selection during both haploid and diploid phase. There is The evolution of heterochiasmy: The model pre- no epistasis term because there is only one selected sented in this article indicates that heterochiasmy can locus. However, when selection at locus k differs be- evolve for three different reasons associated with sex tween males and females, the situation is somewhat anal- differences in selection: (i) because of a difference in ogous to epistasis between the sex-determining locus epistasis during the haploid phase between the gametes and locus k. At QLE, the values for the different linkage of males and females; (ii) because of a male-female disequilibria are difference in cis-epistasis minus trans-epistasis during the diploid phase; and (iii) because of a difference in direc- ␣ DH Ϫ Њ ϭϪˆ k pqk(1 2rjk) ϩ ␰ tional selection during the haploid phase between the C jk(01) o( ) 8rjk gametes of males and females if some linkage disequilibrium between the selected loci is produced by some Њ ϭϪ Њ ϭ Ϫ Њ ϩ ␰2 C jk(00) C ik(01) (1 rik)C ijk(01) o( ) mechanism. In parallel, heterochiasmy can evolve for ␳ ␣ˆ DHpq three similar reasons associated with sex-of-origin differ- Њ ϭϪ (01) k ik ϩ ␰2 C ijk(01) ϩ ϩ Ϫ ϩ o( ), ences in selection: (a) because of a difference in epistasis 2rjk[rik(1 rij rjk) 2(rij rjk)] (35) during the diploid phase between the chromosomes inherited from the father and the mother; (b) because where the subscripts 00 and 01 indicate genotypic values of a sex effect and sex-by-sex-of-origin effects on diploid at locus j in the homogametic and heterogametic sex, directional selection coefficients; and (c) because of a 820 T. Lenormand difference in directional selection during the diploid monoallelic expression with random parental allele ex- phase between alleles inherited from mother and father pression (such autosomal genes, like genes coding for if some linkage disequilibrium between the selected loci immunoglobulin, T-cell receptor, and olfactory recep- is produced by some mechanism. tor have been described; Burns et al. 2001). In addition, It is difficult to judge the likelihood of these different conditions ii and b require a sex difference in selection conditions. Conditions i and a are the most straightfor- during the diploid phase and therefore do not apply to ward as they require only that epistasis varies with the hermaphroditic species. sex during haploid phase or with the sex-of-origin dur- Conditions iii and c require sex effects on the direc- ing the diploid phase (assuming that the average epista- tional selection coefficient during haploid phase or sex- sis is not exactly zero, such that epistasis produces also of-origin effect on the directional selection coefficient the linkage disequilibrium); both mechanisms apply to during the diploid phase, respectively. Both conditions hermaphroditic and gonochoric species. Condition i apply to gonochoric and hermaphroditic species. Con- will arise, for instance, whenever genes are expressed dition iii, in particular, which requires haploid expres- and selected in combinations during the haploid phase sion of genes but no epistasis, may be quite common. in just one sex. It is probably the most general and likely Condition c, like other conditions involving sex-of-ori- condition. The general trend noted by Trivers (1988) gin effect, may not be very likely as it requires a mecha- toward more recombination in females could then be nism like imprinting and may concern very few genes. due to the fact that the haploid phase is often shorter in However, both conditions require, in addition, a general females, as there may be little opportunity for selection mechanism generating the linkage disequilibrium be- because female meiosis is often completed at fertiliza- tween the selected loci (E ϩ D ϶ 0). Among the various tion (Cohen 1977). However, to work, this hypothesis possibilities, some are not very likely [when genes must also requires genes to be expressed during the haploid be imprinted or selected in both haploid and diploid stage, a phenomenon that is common in plants (where phases, see the expression of E in (26)] and others may a large fraction of genes are expressed in pollen and be more common (haploid or diploid epistasis or some ovules), but may be less common in animals (where mechanism unspecified in the model such as migration fewer genes are expressed at the haploid stage; McCor- or drift). In these last cases, conditions iii or c may occur mick 1991; Treier and Beck 1991; Kramer and Kra- but will tend to produce a weak selection pressure on wetz 1997; Taylor and Hepler 1997; Christians et a modifier of heterochiasmy. al. 1999; Steger 1999; Xu et al. 1999). Condition a (like Interference in the evolution of heterochiasmy: The differ- conditions b and c) requires a mechanism producing ent sets of conditions for the evolution of a dimorphism sex-of-origin effects in diploids such as imprinting. Even outlined above are valid for modifier alleles that have if this kind of mechanism has been described in plants exactly opposite effects in males and females on autoso- and many groups of animals (see Lloyd et al. 1999), it mal recombination rates. However, any particular allele may concern few genes in each case. For instance, that modifies recombination can also change the aver- Burns et al. (2001) estimated that imprinted genes may age recombination rate over males and females and/ of genes in mammals. However, these or the recombination rate on sex chromosomes. Since %0.1ف represent conditions might explain an intriguing pattern that has the conditions for the evolution of recombination are been found recently in sheep and humans: imprinted different for each of these cases, the outcome may be regions of the genome (imprinted genes tend to be quite complicated (see Equation 25). For instance, a clustered and in most of these clusters, both maternally modifier with a sex-limited effect (i.e., acting only in one and paternally imprinted genes are found; Bartolomei sex) on all chromosomes may be selected for because it and Tilghman 1997) seem to exhibit particularly changes the average recombination rate on autosomes, high recombination dimorphism (Paldi et al. 1995; because it changes the difference in recombination rate McLaren and Montgomery 1999). This pattern has between males and females on autosomes, and, if it been interpreted the other way around, with heterochi- acts in the heterogametic sex, because it changes the asmy the consequence rather than the cause of im- recombination rate between sex chromosomes. Genetic printing (Paldi et al. 1995), although this hypothesis variation in recombination rates within species has been may not work well because both maternal and paternal extensively demonstrated (Korol et al. 1994). However, imprinted genes are often present in the same cluster. whether there is ample and independent genetic varia- Conditions ii and b are more difficult to meet as they tion for each of these “components” is less clear. The require sex differences in cis- minus trans-epistasis in frequent association of sex-chromosome heteromor- diploids or sex and sex-by-sex-of-origin effects in dip- phism and achiasmy (perhaps also heterochiasmy) loids, respectively, which may concern a very small frac- could suggest that there is not. On the other hand, tion of genes within genomes. As with condition a, these the association within chromosomes of heterochiasmate conditions require a mechanism to produce cis-trans or and imprinted regions may indicate the opposite. sex-of-origin effects. Imprinting may cause both, but cis- What to conclude about existing theories: The conditions trans effects may also be produced when genes undergo for the evolution of (i) autosome average recombina- Sex Dimorphism in Recombination 821 tion rates, (ii) autosome heterochiasmy, and (iii) sex- rate, because it is hard to quantify the different sources chromosome heterochiasmy are very different. The au- of linkage disequilibrium. The evolution of heterochi- tosome average recombination rate can be selected for asmy is a simpler problem since variation in epistasis is or against at either the haploid or the diploid stage, its most likely selection-variation explanation. Studying depending on the relative values of the linkage disequi- heterochiasmy may thus provide a way to investigate the bria and epistasis averaged over sexes. The autosome importance of epistasis. Empirical findings consistent heterochiasmy can be selected in either direction but with the predictions made here would shed light on the for different reasons in haploid and diploid phases (see role of epistasis in molding recombination rates and above). Sex-chromosome heterochiasmy almost always would therefore strongly corroborate theories for the evolves in the same direction, due to either haploid or evolution of sex based on selection and variation, as diploid selection (i.e., reduced recombination in the opposed to mechanistic theories. heterogametic sex), does not necessarily involve epista- I particularly thank Mark Kirkpatrick for his help and stimulation sis, and depends only on the effect of modifiers in the throughout this project. I also thank O. Judson and S. P. Otto for heterogametic sex. Furthermore, arguments based on greatly improving the manuscript and stimulating discussions and P. models for the evolution of sex-chromosomes heterochi- Jarne for helpful comments. This study was supported by the Centre asmy do not extend to autosomes. National de la Recherche Scientifique and French Ministry of Re- search. This model offers a set of predictions for when heterochiasmy will and will not evolve. 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