Abstract
The sterility and inviability of species hybrids can be explained by betweenlocus “DobzhanskyMuller” incompatibilities: alleles that are fit on their “normal” genetic backgrounds sometimes lower fitness when brought together in hybrids. We present a model of twolocus incompatibilities that distinguishes among three types of hybrid interactions: those between heterozygous loci (H_{0}), those between a heterozygous and a homozygous (or hemizygous) locus (H_{1}), and those between homozygous loci (H_{2}). We predict the relative fitnesses of hybrid genotypes by calculating the expected numbers of each type of incompatibility. We use this model to study Haldane's rule and the large effect of X chromosomes on postzygotic isolation. We show that the severity of H_{0} vs. H_{1} incompatibilities is key to understanding Haldane's rule, while the severity of H_{1} vs. H_{2} incompatibilities must also be considered to explain large X effects. Large X effects are not inevitable in backcross analyses but rather—like Haldane's rule—may often reflect the recessivity of alleles causing postzygotic isolation. We also consider incompatibilities involving the Y (or W) chromosome and maternal effects. Such incompatibilities are common in Drosophila species crosses, and their consequences in male vs. femaleheterogametic taxa may explain the pattern of exceptions to Haldane's rule.
IN a landmark article, Dobzhansky (1936) reported the first thorough genetic analysis of any form of reproductive isolation: the sterility of F_{1} male hybrids between Drosophila pseudoobscura and D. persimilis. Using stocks carrying visible mutations on each major chromosome arm, he backcrossed fertile F_{1} females to both pure species and assessed the testis size (a proxy for fertility) of the resulting backcross males. He found that certain hybrid genotypes were consistently sterile, while others were consistently fertile. He posited that hybrid sterility and inviability often arise as a pleiotropic byproduct of independent evolution in geographically separate lineages: alleles that enhance fitness on the “normal” genetic background may occasionally lower fitness when brought together in hybrids with alleles from another species. In this way, two taxa can become separated by an adaptive valley (corresponding to the unfit hybrids) without either lineage ever passing through such a valley. This resolves a paradox that plagued Darwin's (1859, Chap. 8) account of the origin of species: how can natural selection allow for the routine evolution of hybrid sterility and inviability, phenotypes that appear patently maladaptive?
This model of the evolution of postzygotic isolation was later elaborated by Dobzhansky (1937, p. 256) and Muller (1940, 1942), and it remains central to understanding speciation in allopatry. The model obviously focuses on developmentally mediated (“intrinsic”) fitness loss in hybrids, not ecologically mediated selection against intermediate phenotypes (Rice and Hostert 1993). The socalled DobzhanskyMuller model underlies many recent advances in our understanding of the evolution of postzygotic isolation (Orr 1995; Orr and Orr 1996; Gavrilets and Hastings 1996; Gavrilets 1997) and of the causes of Haldane's rule—the pattern that sexspecific hybrid problems typically afflict the heterogametic sex, a rule that holds across Drosophila, mammals, birds, and Lepidoptera, among other groups (Haldane 1922; reviewed by Laurie 1997; Orr 1997; Turelli 1998).
This article has two purposes. First, we hope to fill a vacuum in the theory of speciation. Although Dobzhansky's experimental approach has been followed in many studies of postzygotic isolation, there has been no attempt to predict the pattern of fitness differences seen across the hybrid genotypes produced. Are certain genotypes predictably less fit than others? Here, starting from the DobzhanskyMuller model, we present a theoretical analysis of the expected fitness of different hybrid backcross and F_{2} genotypes. This analysis differs from our previous ones (Orr 1993a; Turelli and Orr 1995) in several ways. For one, we consider types of incompatibilities that we previously ignored (e.g., XX, XY, and Yautosomal, and maternal effects). For another, we consider classes of incompatibilities that appear only in backcross and F_{2} hybrids, namely those between loci that are both homozygous or hemizygous for incompatible alleles. As Muller (1940, pp. 204–205) recognized, such incompatibilities should have more severe effects than those acting in F_{1} hybrids and will thus contribute to “hybrid breakdown,” a form of postzygotic isolation that may often arise before F_{1} problems.
This fuller analysis forces us to deal with several subtleties of the DobzhanskyMuller model. Analysis of DobzhanskyMuller interactions is more difficult than it first appears, as these interactions involve both dominance and epistasis: the severity of the interactions between two loci depends on the dominance of each incompatible allele. In addition, the dominance at one locus might well depend on the genotype at the other locus. In our previous work, we simplified our picture of DobzhanskyMuller incompatibilities to render the analysis tractable and intuitive (e.g., when studying Haldane's rule, we assigned all dominance effects to the Xlinked locus involved in an Xautosomal incompatibility). Here we present a complete model of twolocus DobzhanskyMuller incompatibilities.
We use this model to address the only two known patterns in the genetics of speciation: Haldane's rule and the “large X effect” (Coyne and Orr 1989a). Our previous conclusions about the central role of dominance in both patterns are supported but generalized. In particular, we show that a single dominance coefficient no longer suffices to characterize the hybrid incompatibilities experienced by backcross and F_{2} genotypes.
Our second aim is to determine if the patterns of postzygotic isolation seen across hybrid genotypes in genetic studies of speciation are compatible with our model and to suggest new experimental tests.
GENETIC MODELS OF POSTZYGOTIC ISOLATION
We follow Dobzhansky (1937, p. 256) and Muller (1940, 1942) in assuming that the alleles causing inviability and sterility in hybrids have no such effects within lineages during their divergence from a common ancestor (see also Orr 1995; Orr and Orr 1996). For simplicity, we also assume that each species is fixed at all loci. Following Turelli and Orr (1995), we further assume that separate hybrid incompatibilities contribute additively to an underlying “breakdown score” that is translated into fitness by a monotone decreasing function, denoted V. (The additivity assumption is made for convenience and will be discussed below.) Hybrids become inviable/sterile when their breakdown score reaches a threshold value, C. Finally, we assume that loss of fitness in hybrids is caused by many incompatibilities between loci scattered throughout the genome. (Wu and Palopoli 1994 and Naveira and Maside 1998 review the evidence for this assumption.) This allows us to compare the expected breakdown scores of different genotypes.
Suppose that a hybrid genotype suffers n distinct incompatibilities and that the ith incompatibility contributes e_{i} > 0 to the breakdown score. (Note that specific loci may contribute to more than one incompatibility.) The fitness of this genotype is
Formally, the assumption that many incompatibilities contribute to hybrid sterility/inviability is central to our analysis. Without loss of generality, we can assume that the critical threshold is C = 1. When a large number, n, of incompatibilities is required to reach this threshold, the effect of each must be roughly proportional to 1/n. Hence, the variance of the breakdown scores must also be at most proportional to 1/n as the breakdown score is simply the sum of n random variables, each having mean on the order of 1/n and variance on the order of 1/n^{2}. If we assume that n is large, we can approximate the fitness of a genotype by evaluating the function V in (1) at the expected breakdown score. This simplification is used throughout.
A model of twolocus incompatibilities: The central idea of our analysis is that hybrids can experience three different types of twolocus DobzhanskyMuller incompatibilities, depending on whether the incompatible alleles at the interacting loci are homozygous or heterozygous. Imagine that one species' genotype is AAbb and the other aaBB, where a and b are incompatible in hybrids and A and B denote ancestral (and hence compatible) alleles. Further imagine that there are many such pairs of loci. We label the incompatibilities according to the number of loci that are homozygous for incompatible alleles. In an F_{2} hybrid genotype, certain incompatibilities might be homozygoushomozygous (e.g., aabb), which we denote H_{2}; each such incompatibility contributes, on average, h_{2} to the hybrid breakdown score. Other incompatibilities might be homozygousheterozygous (aaBb or Aabb), denoted H_{1}; each contributes, on average, h_{1} to the breakdown score. All remaining incompatibilities are heterozygousheterozygous (AaBb), denoted H_{0}; each contributes, on average, h_{0} to the breakdown score. Note that any locus homozygous for a compatible allele (AA or BB) cannot contribute to hybrid breakdown.
This last assumption merits comment. If the substitutions causing hybrid problems are driven by natural selection within species, one might expect that genotypes combining homozygous ancestral alleles at one locus with selectively favored alleles at another would enjoy enhanced fitness relative to the ancestral genotype, AABB. We assume, however, that any such effects are negligible compared with the deleterious DobzhanskyMuller effects. Indeed these positive effects must, on average, be small compared to the DobzhanskyMuller effects, or hybrids would not suffer large net fitness losses (relative to the parental means).
Initially, we consider autosomalautosomal, Xautosomal, and XX interactions, but ignore maternal effects and the Y. We also assume dosage compensation such that hemizygosity at Xlinked loci is equivalent in fitness effect to homozygosity. (This assumption is not needed for our analysis of Haldane's rule, which centers on incompatibilities that either do or do not involve a hemizygous X chromosome; these correspond to H_{1} vs. H_{0} incompatibilities, irrespective of dosage compensation.) To compare the breakdown scores of different genotypes, we assume that there are n twolocus incompatibilities in a reference genotype that carries one set of autosomes and one X from each parental species, as in F_{1} females. These incompatibilities are assumed to be randomly scattered throughout the genome. For normalization purposes, we use this reference genotype even when considering hybrid males.
To find the expected breakdown score for any particular hybrid genotype, we merely need to know the proportion of the genome that is homozygous (or hemizygous) from species 1 (p_{1}), the proportion that is homozygous (or hemizygous) from species 2 (p_{2}), and the proportion that is heterozygous for material from the two species (p_{H}, where p_{H} = 1 − p_{1} − p_{2}). A simple combinatoric argument shows that, on average, a fraction
Haldane's rule for inviability: The case of hybrid inviability is simpler than that of sterility as the same genes likely affect the viability of both hybrid males and females (Orr 1989a,b; Wu and Davis 1993). For F_{1} females, all loci are heterozygous (p_{H} = 1), and (2) gives
In the limiting case in which none of the genome is Xlinked (g_{X} = 0), (3) and (4) show that hybrid males and females have the same expected breakdown scores and Haldane's rule for inviability cannot arise (by this mechanism). As noted by Presgraves and Orr (1998), the case g_{X} = 0 arises in the genus Aedes in which the X and Y chromosomes are largely homologous and freely recombine. And, in fact, Haldane's rule for inviability does not occur in this genus.
More generally, (3) and (4) show that if g_{X} > 0, Haldane's rule is expected, i.e., E(S_{m}) > E(S_{f}), whenever
When there are unequal rates of evolution between the X and autosomes, g_{X} no longer equals the proportion of the genome that is Xlinked. If, for instance, the X evolves faster than the autosomes, as suggested by Charlesworth et al. (1987), g_{X} will exceed the fraction of the genome that is Xlinked. The above results still hold, however.
Haldane's rule for sterility: There are good reasons for thinking that different loci will affect hybrid male vs. female fertility. First, mutagenesis experiments within D. melanogaster show that the overwhelming majority of steriles afflict one sex only, while most lethals affect both sexes (reviewed in Ashburner 1989). Second, backcross and introgression data show that chromosome regions that cause the sterility of one hybrid sex typically do not affect the other (Orr and Coyne 1989; Orr 1992; Hollocher and Wu 1996; Trueet al. 1996).
This opens the door to a possibility that does not arise with hybrid inviability: when different loci affect the two sexes, we have no guarantee that substitutions ultimately afflicting males will occur at the same rate as those afflicting females. Indeed there is evidence that alleles causing sterility in hybrid males accumulate faster than those causing sterility in females, at least in Drosophila and mosquitoes (Hollocher and Wu 1996; Trueet al. 1996; Presgraves and Orr 1998). Such “fastermale” evolution might well contribute to Haldane's rule for sterility (Orr 1989a; Wu and Davis 1993). We consider this effect below and show how dominance and fastermale effects can be taken into account simultaneously.
Incompatibilities affecting hybrid female vs. hybrid male sterility may differ in both number and average effects. Let n_{f} (n_{m}) denote the number of incompatibilities in our reference genotype that affect hybrid females (males). We assume that the average effects of H_{0} and H_{1} incompatibilities are h_{0} and h_{1} for females and
Under this model, Haldane's rule for sterility is expected [E(S_{m}) > E(S_{f})] whenever
In general, both fastermale evolution and dominance might contribute to Haldane's rule for sterility in maleheterogametic species. To assess their relative contributions, one can consider the ratio of male to female breakdown scores. This ratio, which must exceed 1 for Haldane's rule in maleheterogametic species, is
Comparative analyses support the idea that fastermale evolution acts against Haldane's rule for sterility in femaleheterogametic species: Haldane's rule is more common for sterility than for inviability in maleheterogametic species, while the reverse appears true in femaleheterogametic species, as expected if fastermale evolution acts in both groups (Wu and Davis 1993; Turelli 1998). This pattern should not, however, obscure an even more striking one: Haldane's rule holds for 42 of 43 cases of unisexual hybrid sterility in birds and Lepidoptera (Laurie 1997); i.e., females are overwhelmingly sterile in these taxa, despite presumed fastermale evolution. Put differently, when dominance and fastermale effects are pitted against one another, dominance predominates, yielding virtually no exceptions to Haldane's rule. This suggests two possibilities that are not mutually exclusive: H_{0} incompatibilities have much smaller effects than H_{1} incompatibilities; or, as discussed below, Yassociated incompatibilities play a large role in postzygotic isolation. We address another important aspect of fastermale evolution in the discussion.
Our analyses of Haldane's rule did not require us to consider homozygoushomozygous incompatibilities, which cannot appear in F_{1} hybrids. Thus, our previous simple “dominance” analyses (Orr 1993a; Turelli and Orr 1995) sufficed to correctly identify the conditions for Haldane's rule. But as we now show, a more detailed model is required to understand the full consequences of DobzhanskyMuller interactions.
The large X effect: As noted by Coyne and Orr (1989a), one of the most striking features of genetic studies of postzygotic isolation is that in backcrosses the X chromosome appears to have a disproportionately large effect on hybrid male fitness relative to comparably sized autosomes. There has, however, been a great deal of confusion about the biological significance, if any, of this socalled large X effect. Wu and Davis (1993) and Hollocher and Wu (1996) have argued that the effect is an artifact of backcross analysis: in any backcross, X substitutions in males replace a hemizygous X from one species with a hemizygous X from the other. Autosomal substitutions, on the other hand, replace a single autosome from one species with a single one from the other. Thus, they argue, hemizygous X substitutions should have larger effects on hybrid fitness than heterozygous autosomal ones. Orr (1997) and others, however, suggest that the large X effect may not be an inevitable artifact of backcross analysis but evidence for the partial recessivity of the genes causing postzygotic isolation. A quantitative treatment is clearly required to find the conditions under which large X effects are expected.
Consider a backcross analysis of male sterility in which fertile F_{1} females are crossed to males from taxon 1. When assessing the effect of “foreign” loci, note that the introgressed segments of X_{2} are necessarily hemizygous, while the introgressed autosomal segments are heterozygous. To quantify the X effect, we require an explicit comparison between Xlinked and autosomal introgressions. Although discussions of the large X effect have been mostly qualitative, at least two quantitative criteria can be used. A large X effect might be declared if a hemizygous X introgression has (1) a greater effect than a heterozygous autosomal introgression that is twice as large; or (2) more than twice the effect of an equalsized heterozygous autosomal introgression. The first criterion requires fewer assumptions, but the second (used by Coyne and Orr 1989a, among others) leads to similar results.
We first use criterion 1, comparing Xlinked and autosomal introgressions from species 2 into a background homozygous for species 1. For the X introgression, let p_{2} = q, where q is the fraction of the haploid genome that is introgressed, and p_{1} = 1 − q. For the autosomal introgression, let p_{H} = 2q (i.e., it is twice as large as the X introgression) and p_{1} = 1 − 2q. The expected breakdown scores are
The analysis above is idealized. Real backcross analyses involve comparing many genotypes, some of which carry the “foreign” X (or portions of it) and some of which do not. We thus extend our analysis to more realistic situations in which the “background” for an introgression includes heterozygous material (i.e., p_{H} > 0). In this case, we compare E(S_{X}) = E(Sp_{2} = q, p_{H} = r, p_{1} = 1 − r − q) with E(S_{auto}) = E(Sp_{H} = r + 2q, p_{1} = 1 − r − 2q). Again, (12) and (13) suffice to give E(S_{X}) > E(S_{auto}). Thus, no matter what the rest of the hybrid genome looks like, introgressing part of the X will lower hybrid fitness more than introgressing twice as much autosomal material when (12) and (13) hold.
Criterion 2 is trickier to apply: while it is easy to compare the average breakdown scores of X vs. autosomal introgressions, we must actually compare the fitness effects of X vs. autosomal introgressions. But these are known only if we know the function V that maps breakdown score onto fitness. For simplicity, then, we assume that the fitness function is nearly linear over the relevant range of values, so that comparing breakdown scores themselves suffices. If fitness declines faster than linearly (e.g., Kondrashov 1988; Charlesworth 1990), our conditions will be too restrictive.
We again first compare Xlinked vs. autosomal introgressions from taxon 2 into a background homozygous for taxon 1. The breakdown score is initially zero in both cases, and we want to compare E(S_{X}) = E(Sp_{2} = q, p_{1}= 1 − q) with 2E(S_{auto}) = 2E(Sp_{H} = q, p_{1} q). With (2), we see that E(S_{X}) > 2E(S_{auto}) if
Next we consider introgressions into backgrounds in which some of the autosomal genome is already heterozygous. We want to compare E(ΔS_{X}) ≡ E(Sp_{2} = q, p_{H} = r, p_{1} = 1 − r − q) − E(Sp_{H} = r, p_{1} = 1 − r) with E(ΔS_{auto}) ≡, E(Sp_{H} = r + q, p_{1} = 1 − r − q) − E(Sp_{H} = r, p_{1} = 1 − r). Using (2), we see that E(ΔS_{X}) > 2E(ΔS_{auto}) if
Our key biological conclusion is that large X effects, by either criterion 1 or 2, are not automatic consequences of comparing hemizygous X with heterozygous autosomal substitutions. Instead, large X effects imply that the alleles causing hybrid sterility and inviability are fairly recessive. Indeed, large X effects are essentially guaranteed if condition (17) is met (in Drosophila). The more extreme the recessivity of the incompatible alleles, the more pronounced the large X effect. The “faster X” mechanism of Charlesworth et al. (1987) may also, of course, contribute to the large X effect.
Comparisons among other backcross genotypes: There are two different classes of backcross to consider, depending on whether one crosses F_{1} hybrid males or females to a parental species. When F_{1} males are backcrossed, the resulting progeny all inherit an X and a complete set of autosomes maternally. In this case, H_{2} incompatibilities are impossible, and (2) reduces to
When F_{1} females are backcrossed, the resulting male progeny can inherit an X that is incompatible with the haploid set of autosomes inherited paternally. This can produce H_{2} incompatibilities so that all three terms in (2) may be nonzero, precluding simple predictions analogous to (19).
Other incompatibilities affecting hybrids: Data from several wellknown hybridizations have shown important effects of YX (e.g., Orr 1987), Yautosome (e.g., Zouroset al. 1988; Heikkinen and Lumme 1991), and maternalzygotic incompatibilities (Sawamura 1996; Hutter 1997). Indeed our review of the literature suggests that both Y and maternal effects on postzygotic isolation are extraordinarily common. We surveyed all known genetic analyses of both hybrid sterility and inviability in Drosophila; the results are shown in Table 1. We found that 10 out of 11 species crosses involving hybrid male sterility show Y chromosome effects in at least one direction of the cross. Similarly, 4 out of 18 hybridizations show maternal effects; excluding male sterility, this figure rises to 4 out of 7. Moreover, the observation of nonreciprocal hybrid female inviability—or even sterility—in Drosophila (Coyne and Orr 1989b) suggests prima facie that maternal effects on hybrid fitness are common, as reciprocal F_{1} females have identical nuclear genotypes. [In the D. melanogasterD. simulans and D. montanaD. texana hybridizations, for instance, females are lethal in one direction of the cross only. (These taxa belong to different subgenera, so their hybrid outcomes are phylogenetically independent.) Analysis of each has proven the role of maternal factors (reviewed in Wu and Davis 1993 and Sawamuraet al. 1993).]
Given their frequent roles, it is important to incorporate Y and maternal effects into our analysis. This can be accomplished via Equation 1, by assuming that Y and maternal incompatibilities simply add to the total breakdown score, which we now denote S_{T}. Keeping with our earlier notation, we denote the cumulative breakdown score attributable to XX, Xautosome, and autosomeautosome incompatibilities by S. Additional contributions from Yassociated and maternalzygotic incompatibilities are denoted S_{Y} and S_{MZ}. We discuss each in turn.
Ylinked incompatibilities: We consider both maleheterogametic (XY) and femaleheterogametic (ZW) taxa, but, for ease of discussion, refer to the sexlimited sex chromosome as the Y and its partner as the X. For simplicity, we focus on E(S_{Y}). Our qualitative predictions do not require that the number of Yassociated incompatibilities is large. Indeed the “large n” assumption is clearly implausible for Yassociated incompatibilities, as (1) the Y carries few complementation groups, at least in D. melanogaster and D. hydei (Ashburner 1989, Chap. 20); and (2) in at least one hybridization (D. mojavensisD. arizonensis), the Y interacts with a single autosome (Zouroset al. 1988). Because the Y has no essential somatic function, at least in D. melanogaster (Ashburner 1989), we might expect the Y to play a role in hybrid sterility but not inviability.
Because the Y is hemizygous, we treat the Ylinked partner in any incompatibility as effectively homozygous. We do not distinguish between YX and Yautosome incompatibilities and assume that each occurs in proportion to the fraction of the genome that is Xlinked vs. autosomal. Let n_{Y} denote the number of incompatibilities between Ylinked loci from taxon 1 and loci in a complete haploid set, including an X, from taxon 2. In F_{1} and backcross genotypes, these incompatibilities can occur in two forms, depending on whether the nonY partner is homozygous or heterozygous. Incompatibilities involving homozygous partners have average effect y_{2}, whereas those involving heterozygous partners have average effect y_{1}.
Because the Y is largely heterochromatic, we have no a priori basis for estimating the fraction of all incompatibilities that involve this chromosome and therefore no basis for drawing quantitative conclusions about S_{Y} vs. S. But, by considering E(S_{Y}) alone, we can still draw interesting, albeit qualitative, conclusions. If the Y is inherited from taxon 1 and the source of the rest of the nuclear genome is described by the fractions p_{1}, p_{2}, and p_{H}
Yassociated incompatibilities also relax the constraints on the dominance, h_{0}/h_{1}, required for Haldane's rule for sterility, particularly in femaleheterogametic species. For two reasons, this effect is likely to be especially important in taxa having small sex chromosomes. First, in the absence of Y effects, the upper bound on h_{0}/h_{1} needed for Haldane's rule when fastermale evolution acts is proportional to g_{X} when g_{X} is small (see 8b). Thus, in femaleheterogametic species with relatively small X's (on the order of 10% of the genome or less), such as birds (Abbott and Yee 1975) and lepidoptera (Robinson 1971, Chap. VIII), extreme recessivity would ordinarily be required to obtain Haldane's rule. Second, even if rare, Ylinked incompatibilities involve the potentially stronger H_{1} and H_{2} incompatibilities, whereas interactions not associated with the Y involve H_{0} and H_{1} incompatibilities.
Given this expected disproportionate effect of the Y in smallX taxa, it is interesting to note that Haldane's rule for sterility shows only a single exception in Lepidoptera and birds (Laurie 1997)—despite the presumption of fastermale evolution, which opposes Haldane's rule, in these groups (Wu and Davis 1993; Turelli 1998). This suggests that the combination of dominance and Y chromosome effects much more than compensates for any fastermale evolution.
Next, we consider the role of the Y in the large X effect. Consider a study of F_{1} male sterility in which F_{1} females are backcrossed to taxon 1. As above, we can compare the values of S_{Y} produced by an Xlinked introgression of size q vs. an autosomal introgression of size 2q. These are
Maternalzygotic incompatibilities: Again we focus on E(S_{MZ}) but recognize that the data suggest that maternal incompatibilities may involve few factors. These incompatibilities arise from interactions between loci in two different diploid genomes—that of the mother and her offspring—and they may generally affect viability only as maternal control of development is surrendered fairly early in development (Sawamura 1996). Such interactions are not surprising given that, in Drosophila, many genes deposit maternal transcripts in the unfertilized egg, and the transfer from maternal to zygotic control of development is gradual (Lawrence 1992). Maternal effects are distinct from cytoplasmic effects involving interactions between nuclear genes and maternally inherited cytoplasmic factors, such as microbes and mitochondria. Although cytoplasmic effects, such as those associated with Wolbachia, may contribute to reproductive isolation between some taxa (Hoffmann and Turelli 1997; Werren 1997), they seem to be relatively unimportant in producing the sexlimited hybrid viability differences on which we focus (Hurst 1993).
Maternalzygotic incompatibilities may occur in three forms depending on whether the incompatible alleles are homozygous or heterozygous. To distinguish these incompatibilities from those acting within the offspring genome, we denote them by M_{0}, M_{1}, and M_{2}. There may be a qualitative distinction between the two types of M_{1} incompatibilities (depending on whether the mother or offspring is homozygous for an incompatible allele), but we assume they have equal average effects. We assume that the average effect of M_{i} incompatibilities is m_{i} for i = 0, 1, 2 and that there are n_{MZ} maternalzygotic incompatibilities between a cytoplasm produced by a taxon 1 mother and her F_{1} hybrid daughters. Obviously, these would all be M_{1} incompatibilities involving homozygous maternal loci and heterozygous loci in the offspring. We assume that a fraction g_{X} of these incompatibilities involve loci on the zygotic X. With backcross analyses involving hybrid mothers, a wide range of incompatibilities can appear. We focus on only those that occur in hybrids with taxon 1 mothers.
If the offspring genome is characterized by the fractions p_{1}, p_{2}, and p_{H}, as before, the expected contribution to their breakdown score from maternalzygotic interactions is
In maleheterogametic species, the only relevant difference between F_{1} females and males is that females suffer from M_{1} incompatibilities between the paternal X and the maternal cytoplasm, whereas males do not (Patterson and Stone 1952, pp. 435–436, 489; Wu and Davis 1993). This appears in the breakdown scores as
In contrast, in femaleheterogametic species, F_{1} males have p_{H} = 1, but F_{1} females have p_{H} = 1 − g_{X} and p_{2} = g_{X}. Thus,
Thus, in femaleheterogametic species, dominance and maternal effects act in concert to promote Haldane's rule, whereas they act in opposition in maleheterogametic species. Maternal effects may, therefore, explain both the prevalence of exceptions to Haldane's rule for viability in Drosophila (and many of these exceptions appear evolutionarily independent; Sawamura 1996) and the virtual absence of exceptions in birds and Lepidoptera (Laurie 1997).
Complex genetic interactions: We have focused on twolocus incompatibilities as they capture the essence of the DobzhanskyMuller mechanism and are easily modeled. But hybrid inviability and sterility may be produced by more complex interactions involving three or more loci (e.g., Carvajalet al. 1996). In the appendix, we present an alternative analysis based on threelocus interactions. Although the threelocus results are much more complex than those of our twolocus analysis, our central conclusion remains clear: dominance can explain both Haldane's rule and the large X effect if homozygosity for an incompatible allele has more than twice the effect of heterozygosity on the incompatibility score (see A5).
HYBRIDIZATION DATA
While the above theory makes several predictions, existing data do not yet allow critical tests. Our purpose here therefore is merely to show (1) what predictions can be made; and (2) how these predictions can be tested with hybrid backcross data.
Data from introgressions: A large body of introgression data supports our assumption that H_{2} incompatibilities are more severe than H_{1}. True et al. (1996), for instance—in a study of 87 chromosome regions introgressed from D. mauritiana into D. simulans—found that many introgressions cause complete male or female sterility or inviability when homozygous. Although they made no quantitative measures of heterozygous fitness, the fact that introgressions proceeded through (obviously viable and fertile) heterozygous hybrids suggests that introgression heterozygotes were reasonably fit. (True et al. did not lose a large number of introgression lines, which would have indicated frequent severe heterozygous problems.) Similarly, Hollocher and Wu (1996), in a study of 18 second chromosome introgressions from D. mauritiana into D. simulans, report that none significantly reduces sterility or inviability when heterozygous, although “ … individuals homozygous for these same regions show dramatic increases in both.” More quantitative conclusions can be drawn from the D. buzzatiiD. koepferae (formerly D. serido) work of Naveira and collaborators.
D. buzzatiiD. koepferae: These species obey Haldane's rule for sterility. They have N = 6 chromosomes, with four autosomes and an X, all of roughly equal size, and a tiny sixth. Naveira and Fontdevila (1986) studied male fertility in a large set of X and heterozygous autosomal introgressions, using homologydependent polytene pairing to assay the size of introgressions. Their chief result is that male fertility is a function of the size of heterozygous autosomal introgressions (Naveira and Maside 1998; see also Marín 1996). Heterozygous introgressions of up to 30% of a large autosome (p_{H} ≈ 0.3/5 = 0.06) essentially never cause male sterility, whereas introgressions of >40% of a large autosome (p_{H} ≈ 0.4/5 = 0.08) almost always do. These introgressions correspond to p_{1} = 1 − p_{H} in formula (2) and the expected breakdown scores are given by (18). These autosomal data imply that
In contrast, Naveira and Fontdevila (1986) found that Xlinked introgressions of as little as 4% of the X always produce male sterility (corresponding to p_{1} ≈ 0.04/5 = 0.008 and p_{2} = 1 − p_{1} ≈ 0.992). Indeed, they argue that introgressions as small as 1% of the X cause sterility (i.e., p_{1} ≈ 0.002). These hemizygous introgressions cause all H_{2} incompatibilities, implying that
H_{0} vs. H_{1} incompatibilities: We have less direct evidence about the magnitudes of h_{0} and h_{1}. However, Coyne et al. (1998) recently found several regions from D. simulans that—when made hemizygous with D. melanogaster deficiencies—cause temperaturedependent inviability of otherwise heterozygous D. melanogasterD. simulans hybrid females. Though few such regions were found, their existence shows they individually satisfy h_{0} ⪡ h_{1}.
Hybrid backcross analyses: We now turn to traditional backcross/F_{2} analyses. Our approach is statistical: we pool all hybrid backcross or F_{2} genotypes that have the same p_{1}, p_{2}, and p_{H}. Consider, for instance, a backcross between two Drosophila species possessing five roughly equalsized chromosome arms, one of which is the X. We pool all data obtained when a single autosomal arm is introgressed, or two autosomal arms are introgressed, etc. We appreciate that experiments repeatedly show that particular chromosomes have large effects on backcross fitness while others do not, but our primary goal is not to explain the detailed outcomes of particular species crosses but to search for statistical regularities.
In a species cross, any of the incompatibilities discussed above might act. While we can make some predictions about the relative roles of Xautosomal vs. XX incompatibilities, we have no theory allowing us to predict how often, for instance, Ylinked or maternally acting genes might contribute to—or even dominate—postzygotic isolation. Thus we concentrate on cases in which Y and maternal effects are absent or small.
We make one further simplification. The above theory requires that we know p_{1}, p_{2}, and p_{H}. Unfortunately, most backcross studies in Drosophila involve taxa obeying Haldane's rule, so that backcrosses must proceed through F_{1} females. Because females recombine, markers remain associated with only some (inexactly known) chromosome region. We do not, therefore, know p_{1}, p_{2}, and p_{H}. This difficulty does not arise when backcrosses are performed through F_{1} males, as there is no recombination in Drosophila males. Thus we consider only backcrosses that proceed through F_{1} males. We know of two relevant studies: D. mojavensisD. arizonae (formerly arizonensis) and D. hydeiD. neohydei. We focus on the first for purposes of illustration.
D. mojavensisD. arizonae: Backcross hybrid males are sterile if they have a Y from D. arizonae and are homozygous for the fourth chromosome of D. mojavensis. These species have five chromosomes of roughly equal size (including the X), and a dot sixth chromosome that we will ignore. To avoid the complications of Ylinked incompatibilities, we focus on the backcross of F_{1} males (from D. mojavensis mothers) to D. arizonae females. Table 2 of Zouros et al. (1988) reports sperm motility for these males, all of whom carry an X and Y from D. arizonae. In their Figure 1, Zouros et al. (1988) pool their data into five classes, roughly corresponding to the fraction of the genome that is heterozygous (and ignoring the identity of the heterozygous chromosomes). Their categories correspond to p_{H} = 0, 0.2, 0.4, 0.6, and 0.8. For all of these males, p_{1} = 1 − p_{H}, so the expected breakdown scores are given by (18). Using their pooled data, we constructed exact 95% confidence intervals, based on the binomial distribution, for the fraction of males with immotile sperm. For two of the five classes (p_{H} = 0.6, 0.8), all of the males have immotile sperm. In these cases, we constructed an exact 95% lower bound.
The data are given in Table 2 along with the predicted breakdown scores from our twolocus analysis, Equation 18, and the threelocus analysis from the appendix. Because of small samples, there is only one statistically significant jump between adjacent fractions in the table—that from p_{H} = 0.2 (6/24) to p_{H} = 0.4 (27/29) (P < 10^{−5}).
It is worth noting that our large n assumption likely does not hold here. This is suggested by several lines of evidence. First, X_{m}Y_{a} A_{m}A_{a} F_{1} males (where A denotes a haploid set of autosomes) are fertile, whereas X_{a}Y_{m} A_{a}A_{m} males are all sterile. As both F_{1} males have the same expected breakdown scores, this difference reveals heterogeneity in the numbers of incompatibilities between “replicate” X chromosomes. Second, Zouros and collaborators have shown that hybrid male sterility is caused by Xautosome and autosomeautosome incompatibilities, not Y effects. But the data in Table 2 of Zouros et al. (1988) show statistically significant heterogeneity among the individual chromosomal classes combined in the p_{H} = 0.2 class—a heterogeneity that is inconsistent with large n.
Despite this, it seems worth asking if the pooled data in the first column of Table 2 agree with our predictions (given in the last two columns), where we assume that higher expected breakdown scores will be associated with greater sperm immotility.
First consider the twolocus predictions. From (18), as p_{H} increases, the expected breakdown score, E(S), rises to a maximum of nh_{1}/[4(1 − d_{0})] at p_{H} = 1/[2(1 − d_{0})] (see Equation 19), then falls to nh_{1}(0.16 + 0.64d_{0}) at p_{H} = 0.8. Because we ignore Y effects, the expected breakdown score for males with p_{H} = 0.8 in Table 2 is the same as for the sterile X_{a}Y_{m} A_{a}A_{m} F_{1} males. Indeed, both are sterile, as they suffer the same incompatibilities. Note that if d_{0} < 2/7, E(S) for p_{H} = 0.6 is larger than E(S) for p_{H} = 0.8. As expected, the p_{H} = 0.6 males are also all sterile. The lack of statistical power precludes more detailed tests, but the data suggest the kind of inferences possible. For instance, if the difference between the fractions of males with motile sperm for p_{H} = 0.4 and p_{H} = 0.8 were statistically significant, we could conclude that the corresponding breakdown scores must satisfy 0.24 + 0.16d_{0} < 0.16 + 0.64d_{0}, so that d_{0} > ⅙.
Now consider the threelocus predictions. The most interesting difference between the two and threelocus predictions is that the latter implies that the largest breakdown score occurs for smaller p_{H} than in the twolocus case. This can be seen in the breakdown scores for p_{H} = 0.2 vs. p_{H} = 0.8. In the twolocus model, E(S) is always larger for p_{H} = 0.8. In contrast, assuming that d_{0} = d_{1} = d, the threelocus model implies that E(Sp_{H} = 0.8) < E(Sp_{H} = 0.2) if d < 0.197. Thus the observation that males from this cross with p_{H} = 0.8 are less fit than those with p_{H} = 0.2 suggests that most of the D. mojavensis/arizonae incompatibilities act more like two than threelocus ones (or that these incompatibilities are less recessive than suggested by our inferences from other data).
DISCUSSION
Our understanding of speciation has been characterized by several steps in which large but nebulous problems have been reduced to smaller but sharper ones. During the modern synthesis, for instance, “the origin of species” was largely reduced to “the origin of reproductive isolation” (Dobzhansky 1937; Mayr 1942). Over the last decade, it has become clear that the origin of developmentally mediated (though not ecologically mediated; see Hatfield and Schluter 1999) postzygotic reproductive isolation can often be reduced to the origin of DobzhanskyMuller incompatibilities (Hutteret al. 1990; Orr 1995, 1997; Hutter 1997). A clear understanding of speciation thus requires a clear understanding of DobzhanskyMuller incompatibilities.
Fortunately, the DobzhanskyMuller mechanism is simple enough that it can be captured in mathematical models. Here we have presented a complete model of twolocus DobzhanskyMuller interactions. To simplify our analysis, we assumed that the number of incompatibilities is large and that individual incompatibilities contribute linearly to a “breakdown score,” such that higher scores lead to lower fitness (see Equation 2). Either assumption may be incorrect and more general models could be constructed. Increased generality would, however, lead to more ambiguous predictions, dependent on a proliferation of parameters whose values are unknown. We have also assumed that dosage compensation renders the effects of hemizygotes equivalent to those of homozygotes. This assumption is irrelevant to our analysis of Haldane's rule, as F_{1} individuals do not experience H_{2} incompatibilities and all of their H_{1} incompatibilities involve the hemizygous X (or Z) chromosome. Our assumption is, however, critical to our analyses of backcrosses. Fortunately, essentially all of the relevant data come from Drosophila in which dosage compensation occurs. For taxa like birds and lepidoptera, in which dosage compensation appears absent (Chandra 1994; Suzukiet al. 1999), one could add parameters that distinguish hemizygous from homozygous interactions. At present, however, this does not seem worthwhile as we have no data with which to estimate the required parameters.
We have used our model to address several questions in the genetics of speciation, including Haldane's rule and the large X effect.
Haldane's rule: Table 3 summarizes the forces hypothesized to contribute to Haldane's rule. We indicate whether each might act for hybrid sterility and/or inviability and in male and/or female heterogametic taxa. We first consider dominance alone.
Roughly speaking, our analysis shows that Haldane's rule arises if the factors causing postzygotic isolation act as partial recessives, i.e., if H_{1} incompatibilities are somewhat more than twice as severe as H_{0} ones. This condition emerges if either the same genes affect males and females or if male and female incompatibilities evolve at the same rate.
Fortunately, we possess data allowing rough inferences about the magnitude of h_{0}/h_{1}. First, in two Drosophila hybridizations, “normal” F_{1} females (who suffer H_{0} incompatibilities only) are viable, while “unbalanced” females carrying an attachedX stock (and who thus suffer some H_{1} incompatibilities) are inviable (Orr 1993b; Wu and Davis 1993). Similarly, Presgraves and Orr (1998) showed that mosquitoes that lack a hemizygous X (and so suffer H_{0} incompatibilities only) do not show Haldane's rule for viability, while other mosquitoes that possess a hemizygous X (and thus suffer H_{1} as well as H_{0} incompatibilities) do show Haldane's rule for inviability. Together, these findings suggest that h_{0}/h_{1} is small, at least for inviability. Additional indirect support comes from a comparative analysis of the time course of increasing postzygotic isolation between pairs of Drosophila with “small” vs. “large” X chromosomes (corresponding to roughly 20% vs. 40% of the genome). As expected if h_{0}/h_{1} is small, Haldane's rule occurs at a smaller average genetic distance between largeX than smallX pairs (Turelli and Begun 1997).
With hybrid sterility, our analysis is more complex, as different loci appear to affect males vs. females, allowing for the possibility that male and femaleexpressed genes evolve at different rates (Orr 1989a; Wu and Davis 1993). If so, the conditions under which Haldane's rule arises are set both by dominance and by any difference in the numbers/effects of incompatibilities affecting male vs. female fertility (see Figure 1). As expected, fastermale evolution (Wu and Davis 1993) always promotes Haldane's rule in maleheterogametic species but acts against it in femaleheterogametic ones. Greater recessivity, on the other hand, always facilitates Haldane's rule.
There is now considerable evidence that fastermale evolution occurs for genes causing postzygotic isolation (reviewed in Wuet al. 1996; Laurie 1997; Orr 1997; Turelli 1998). Nonetheless, our analysis—together with several other lines of evidence—suggests that the extent of fastermale evolution may have been overestimated. There are four reasons for thinking this. First, in taxa in which fastermale evolution and dominance are opposed—the former working against Haldane's rule and the latter for it—comparative work shows nearly perfect conformity to Haldane's rule. In birds and Lepidoptera, Laurie (1997) showed that 42 of 43 species crosses obeyed Haldane's rule for sterility, while Wu and Davis's (1993) review found no exceptions; i.e., females are nearly universally sterile despite phenotypic evidence for fastermale evolution based on sexual selection. This suggests that fastermale evolution is moderate enough to be overcome by dominance and Yassociated incompatibilities.
Second, the relative rates of accumulation of malesterilizing vs. femalesterilizing substitutions can be easily overestimated from the observed excess of hybrid male over female steriles. Hollocher and Wu (1996), for instance, found that male steriles were 4fold more common than female steriles in D. sechelliaD. simulans hybrids and 23fold more common in D. mauritianaD. simulans. While these ratios are subject to large error—True et al. (1996), for instance, found a 9fold (not 23fold) excess in D. mauritianaD. simulans—it would seem safe to conclude that hybrid male steriles are, say, six times more numerous than female. It does not follow, however, that maleexpressed genes evolve six times faster than femaleexpressed genes.
The reason is simple. If there have been K_{f} substitutions at femaleexpressed genes and K_{m} at male genes and all incompatibilities involve pairs of loci, the expected numbers of hybrid female vs. male incompatibilities are
Third, the fact that autosomal regions cause male sterility more often than female sterility when introgressed into a foreign species does not necessarily reflect faster evolution of autosomal maleexpressed than femaleexpressed genes. The reason is that introgressions confront different genetic backgrounds in males and females. Autosomal introgressions confront a “foreign” Y chromosome in males but not females. To the extent that the Y plays an important role in hybrid male sterility—and our review of the literature strongly suggests it does—one would expect more male than female sterility in introgression experiments even if male and female autosomal genes evolve at the same rate.
Last, while proponents of fastermale evolution often cite the fact that male reproductive tract proteins evolve faster than nonreproductive ones, recent studies indicate that both male and female reproductive tissues evolve at high rates. Indeed, Civetta and Singh (1995) could not reject the null hypothesis of no difference between the rates of divergence of testis and ovary proteins between Drosophila species (male:female = 1.07), although both sets of proteins evolve significantly faster than those in nonreproductive tissues.
We are not suggesting that fastermale evolution for hybrid sterility does not occur. It almost certainly does [it is hard to see how else one could explain the fact that taxa lacking a hemizygous X obey Haldane's rule for sterility (Presgraves and Orr 1998)]. We merely suggest that its extent may have been overestimated.
Our extensions to our basic model—incorporating Y and maternal effects—also have important bearings on understanding Haldane's rule. The consequences of Ylinked incompatibilities are simple: Y effects always promote Haldane's rule in both male and femaleheterogametic species. Moreover, Y effects may have disproportionately greater effects in taxa, like birds and Lepidoptera, that have relatively small X chromosomes. Our review of genetic analyses of postzygotic isolation in Drosophila leaves no doubt that Y effects are very common, at least for male sterility. The consequences of maternal effects are more subtle: maternalzygotic incompatibilities contribute to Haldane's rule in femaleheterogametic species (as the XY sex gets its cytoplasm from one species and its X from another) but work against Haldane's rule in maleheterogametic species (as the XX sex gets its cytoplasm from one species and one X from another). This asymmetry suggests that maternal effects might explain both the prevalence of exceptions (for viability) in Drosophila (Sawamura 1996) and the near absence of exceptions to Haldane's rule in birds and Lepidoptera (Laurie 1997).
The “faster X” hypothesis of Charlesworth et al. (1987) is not included as a separate factor in Table 3, as it represents, in effect, an increase in the size of the X relative to the autosomes and cannot by itself explain Haldane's rule (Orr 1997). An increase in the “effective size” of the X merely places slightly more stringent bounds on the dominance coefficient needed to obtain Haldane's rule for inviability (see Equations 5 and 8b). In contrast, a larger X promotes Haldane's rule for sterility via an increase in the prevalence of XY interactions (21). Finally, a larger X boosts the role of maternaleffectX interactions in producing exceptions to Haldane's rule for viability in male heterogametic species (25), but facilitates Haldane's rule for viability in femaleheterogametic species (26). FasterX evolution may also, of course, supplement dominance as a force contributing to the large X effect.
Large X effect: Our analysis shows that large X effects are not an inevitable consequence of backcross analysis. Substitution of a hemizygous X does not invariably lower hybrid fitness more than twice as much as substitution of a similarlysized heterozygous autosome. Instead, large X effects arise if the genes causing postzygotic isolation are fairly recessive. Here “recessive” refers to two comparisons. Because backcross hybrid males suffer from H_{0}, H_{1}, and H_{2} incompatibilities, the ratios h_{0}/h_{1} and h_{1}/h_{2} are both relevant. This highlights the shortcomings of previous attempts, including ours, to understand the implications of dominance in DobzhanskyMuller incompatibilities. Because such interactions involve both dominance and epistasis, no single dominance parameter fully captures the behavior of hybrid lethals/steriles. While this did not affect our earlier analyses of Haldane's rule—as F_{1} hybrids cannot suffer H_{2} incompatibilities—it plays a key role in backcross hybrids and thus in discussions of the large X effect.
As noted previously, we have qualitative information about h_{1}/h_{2} from the introgression experiments of Hollocher and Wu (1996) and True et al. (1996). Similarly, Orr (1992) showed that flies that are otherwise pure D. melanogaster and that are heterozygous for the dot fourth chromosome from D. simulans are essentially perfectly fertile (an H_{1} incompatibility), while those homozygous for the D. simulans fourth are completely male sterile (H_{2}). Similarly, work in the haplodiploid wasp Nasonia vitripennis shows that backcross females (who are diploid and suffer only H_{0} and H_{1} incompatibilities) are much more fit than backcross males (who are haploid and, hence, suffer only H_{2} incompatibilities between their hemizygous loci; Breeuwer and Werren 1995). Our analysis also shows that some information about dominance can be extracted from traditional backcross analyses. In particular, our reexamination of backcross data from D. mojavensis vs. D. arizonae shows that they are at least qualitatively consistent with h_{1} < h_{2}. The data from D. buzzatii vs. D. koepferae suggest that h_{1}/h_{2} may be quite small.
Evidence bearing on the dominance theory may also emerge from quantitative trait loci (QTL) studies performed for quite different reasons. Interspecific QTL analyses often uncover distorted segregation ratios at marker loci (e.g., Patersonet al. 1991; Bernacchi and Tanksley 1997). Though often referred to as “segregation distortion,” these biases likely reflect the inviability of hybrid genotypes, not meiotic drive: certain combinations of chromosome regions from two species cause partial inviability, distorting marker ratios from Mendelian expectations. Obviously, such data can be used to detect and map hybrid lethals. Less obviously, QTL data can provide information on the dominance of such factors. Bernacchi and Tanksley (1997), for example, have noted a pattern characterizing QTL studies of plants: “segregation distortion” appears more often in F_{2} analyses (where H_{2} incompatibilities arise) than in backcross analyses (where H_{2} incompatibilities do not arise in most plants as they typically lack sex chromosomes). Indeed Bernacchi and Tanksley conclude that “[t]his difference may result from increased manifestation of deleterious and subdeleterious allelic combinations in the F_{2} populations, possibly associated with recessive epistatic factors.” These QTL data also suggest that hybrid lethals are fairly recessive in plants, where we lack much direct data. (It is, however, already clear that plants often suffer DobzhanskyMuller incompatibilities, e.g., Christie and Macnair 1984.) Additional support comes from the common observation that F_{2} hybrids often exhibit lower average fitness than F_{1} hybrids, as expected from the formation of H_{2} incompatibilities.
The weight of the evidence suggests, then, that both h_{0}/h_{1} and h_{1}/h_{2} are fairly small. If so, the dominance theory provides a powerful explanation of Haldane's rule and the large X effect for both inviability and sterility in all species having heteromorphic sex chromosomes.
In sum, we believe our analysis sheds considerable light on the genetics of speciation by postzygotic isolation. Our model's one obvious merit is that it is firmly grounded on the known genetic mechanism underlying postzygotic isolation—DobzhanskyMuller incompatibilities. Such a model would seem, therefore, to provide a more solid basis for future work than the abstract theories and verbal speculations that all too often characterized discussions of speciation.
Acknowledgments
We thank N. H. Barton, A. Betancourt, J. A. Coyne, C. Jones, R. Haygood, P. D. Keightley, X. R. Maside, D. Presgraves, E. Zouros, and two anonymous reviewers for helpful comments and discussion. M.T. thanks the University of Edinburgh for providing an excellent sabbatical research environment. This research was supported in part by National Science Foundation grant DEB 9527808 (M.T.) and by National Institutes of Health grant GM51932 and the David and Lucile Packard Foundation (H.A.O.).
APPENDIX: THREELOCUS INCOMPATIBILITIES
The text focuses on the simple case of twolocus epistatic incompatibilities; but hybrid dysfunction may be caused by more complex interactions involving three or more loci. To explore the robustness of our conclusions, we consider threelocus interactions here. We denote the interacting loci A, B, and C and use subscripts to indicate species identity.
Three interacting loci produce five distinct types of incompatible genotypes, all of which are assumed to consist of two alleles from one taxon that are incompatible with a third allele from the second taxon. We again assume incompatibility effects are asymmetric, in that if allele A_{1} contributes to an incompatibility when introgressed into taxon 2, the reciprocal introgression of allele A_{2} into taxon 1 has no deleterious effect. The five types of incompatibilities are labeled according to the number of loci that are homozygous for alleles involved in the incompatibility, with the exception that there are two distinct types of incompatibilities involving two homozygous loci; these are distinguished by using H_{2} vs. H_{1,1}. The types of incompatibilities are
H_{0}, all three loci heterozygous: A_{1}A_{2}B_{1}B_{2}C_{1}C_{2};
H_{1}, one locus homozygous: e.g., A_{1}A_{2}B_{1}B_{2}C_{1}C_{1};
H_{2}, two loci homozygous for alleles from one taxon: e.g., A_{1}A_{2}B_{1}B_{1}C_{1}C_{1};
H_{1,1}, two loci homozygous for alleles from different taxa: e.g., A_{1}A_{1}B_{1}B_{2}C_{2}C_{2}; and
H_{3}, all three loci homozygous: e.g., A_{2}A_{2}B_{1}B_{1}C_{1}C_{1}.
Let h_{i} denote the average effect of H_{i} incompatibilities for i = 0, 1, 2, 3; and let h_{1,1} denote the average effect of H_{1,1} incompatibilities.
Again let n denote the number of threelocus incompatibilities that occur in a reference genotype containing one X chromosome and one complete set of autosomes from each of the hybridizing taxa. When calculating the average number of these incompatibilities that will afflict any specific hybrid genotype, it is important to note that there are two types of threelocus incompatibilities: those in which two alleles from taxon 1 interact negatively with an allele from taxon 2 and those in which an allele from taxon 1 interacts negatively with two alleles from taxon 2. More complex interactions (involving reciprocal effects of both alleles at a locus) are expected to occur infrequently and will be ignored. Among the n incompatibilities, on average half should fall into each of the two types.
To find the expected breakdown score for any particular genotype of F_{1}, backcross, or F_{2} hybrid, we need to know the proportion of the genome that is homozygous (or hemizygous) from species 1 (p_{1}), the proportion homozygous from species 2 (p_{2}), and the proportion heterozygous for material from the two species (p_{H}, where p_{H} = 1 − p_{1} − p_{2}). The expected breakdown score is
Haldane's rule for inviability: For F_{1} females, all loci are heterozygous (p_{H} = 1), and thus
We can obtain conditions for Haldane's rule for sterility and the large X effect by straightforward extensions of the arguments presented in the text.
Footnotes

This article is dedicated to our pal King Coyne on the occasion of his fiftieth birthday.

Communicating editor: P. D. Keightley
 Received September 1, 1999.
 Accepted December 21, 1999.
 Copyright © 2000 by the Genetics Society of America