Asymmetric postmating isolation, where reciprocal interspecific crosses produce different levels of fertilization success or hybrid sterility/inviability, is very common. Darwin emphasized its pervasiveness in plants, but it occurs in all taxa assayed. This asymmetry often results from Dobzhansky–Muller incompatibilities (DMIs) involving uniparentally inherited genetic factors (e.g., gametophyte–sporophyte interactions in plants or cytoplasmic–nuclear interactions). Typically, unidirectional (U) DMIs act simultaneously with bidirectional (B) DMIs between autosomal loci that affect reciprocal crosses equally. We model both classes of two-locus DMIs to make quantitative and qualitative predictions concerning patterns of isolation asymmetry in parental species crosses and in the hybrid F₁ generation. First, we find conditions that produce expected differences. Second, we present a stochastic analysis of DMI accumulation to predict probable levels of asymmetry as divergence time increases. We find that systematic interspecific differences in relative rates of evolution for autosomal vs. nonautosomal loci can lead to different expected F₁ fitnesses from reciprocal crosses, but asymmetries are more simply explained by stochastic differences in the accumulation of U DMIs. The magnitude of asymmetry depends primarily on the cumulative effects of U vs. B DMIs (which depend on heterozygous effects of DMIs), the average number of DMIs required to produce complete reproductive isolation (more asymmetry occurs when fewer DMIs are required), and the shape of the function describing how fitness declines as DMIs accumulate. Comparing our predictions to data from diverse taxa indicates that unidirectional DMIs, specifically involving sex chromosomes, cytoplasmic elements, and maternal effects, are likely to play an important role in postmating isolation.

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The degree of sterility does not strictly follow systematic affinity, but is governed by several curious and complex laws. It is generally different, and sometimes wildly different, in reciprocal crosses between the same two species.

DARWIN (1859)

ISOLATION asymmetry occurs when the strength of reproductive isolation between taxa differs significantly between reciprocal crosses. While interest in asymmetric reproductive isolation has often focused on behavioral (sexual) isolation between animal species (e.g., KANESHIRO 1980; KAWANISHI and WATANABE 1981; ARNOLD et al. 1996), postmating isolation asymmetry, expressed as reciprocal-cross differences in F₁ viability or fertility or in postmating, prezygotic isolation, is also common. It was originally reported in plants by J. G. Kölreuter in 1761, 1763, 1764, and 1766 (cited and partially translated in MAYR 1986), the first researcher to systematically create interspecific hybrids (his key work is reviewed in ROBERTS 1929, Chap. II; in OLBY 1966a, Chap. 1, 1966b; and, especially, in MAYR 1986). Isolation asymmetry was emphasized by DARWIN (1859, Chap. 8, esp. pp. 258–261), who noted, "... hybrids raised from reciprocal crosses ... generally differ in sterility in a small, and occasionally in a high degree," citing Kölreuter and Gärtner, the same plant hybridizers whose hundreds of intra- and interspecific crosses inspired Mendel in 1865 (translated in BATESON 1901). Asymmetry was subsequently found in essentially all systems subject to systematic hybridization experiments, including many invertebrates (e.g., MULLER 1942; OLIVER 1978; HARRISON 1983; COYNE and ORR 1989a; GALLANT and FAIRBAIRN 1997; PRESGRAVES and ORR 1998; NAVAJAS et al. 2000; WILLETT and BURTON 2001; PRESGRAVES 2002), vertebrates (e.g., THORNTON 1955; RAKOCINSKI 1984; BOLNICK and NEAR 2005), and fungi (e.g., DETTMAN et al. 2003). A recent analysis of reciprocal species crosses within 14 diverse angiosperm genera found significant isolation asymmetry in 35–45% of all species pairings, evaluated at three different postmating stages of reproductive isolation (TIFFIN et al. 2001). Similarly, MULLER (1942, p. 101) noted that the viability and fertility of Drosophila F₁ males derived from reciprocal crosses "... are so often very different ..." and TURELLI and ORR (1995) estimated that 15% of the cases of Haldane's rule in Drosophila show qualitative asymmetry, with males being sterile or inviable in one cross but not in the reciprocal cross. Despite its ubiquity, however, reciprocal isolation asymmetry, especially asymmetry that occurs after the interspecific transfer of gametes, has received very little theoretical attention.

Because DARWIN (1859, Chap. 8) first drew attention to both the generality and the evolutionary significance of asymmetric postmating isolation, we propose "Darwin's corollary" as a name for this phenomenon. It joins HALDANE's (1922) rule, concerning the preferential inviability/sterility of the heterogametic sex of interspecific F₁ hybrids, and "Coyne's rule" (also known as the "large X effect"; COYNE and ORR 1989b), the disproportionate contribution of the X chromosome to heterogametic F₁ inviability/sterility, as a third widespread pattern concerning intrinsic postmating isolation. We describe this reciprocal-cross asymmetry as a "corollary," both because it is less common than the two "rules" (for reasons that are elucidated by our theoretical analysis) and because it is often produced by the same genetic mechanism that explains the other two, namely interspecific epistatic incompatibilities (DOBZHANSKY 1936, 1937, p. 256; MULLER 1940, 1942; TURELLI and ORR 2000). Although our quote from DARWIN (1859) suggests that he was concerned only with hybrid sterility, he emphasizes in the opening pages of Chapter 8 that he is discussing two different classes of "sterility": "sterility of the species when first crossed," meaning an absence of progeny, which can arise from barriers to fertilization or F₁ inviability, and "sterility of the hybrids produced from them," namely F₁ sterility. Indeed, the asymmetry Darwin describes includes both pre- and postmating isolation. We emphasize the latter because of its genetic implications and close connection to previous analyses of the genetics of intrinsic postzygotic isolation.

As summarized by COYNE and ORR (2004, Chaps. 8 and 9), inviability and sterility of species hybrids can often be explained by between-locus "Dobzhansky–Muller incompatibilities" (DMIs)—inappropriate epistatic interactions between alleles that characterize independently evolving lineages. Many DMIs involve interactions between autosomal loci and affect both reciprocal crosses identically. In contrast, DMIs between autosomal loci and uniparentally inherited factors, including mitochondria (mtDNA), chloroplasts, maternal transcripts, and sex chromosomes in heterogametic hybrids, are specific to a particular direction of hybridization and can therefore contribute to asymmetric reproductive isolation. [Note that genetic imprinting has been implicated as a possible source of asymmetric postmating isolation in mammals (VRANA et al. 2000) and angiosperms (BUSHELL et al. 2003). Given that it effectively corresponds to uniparental inheritance, it can be also be included in our theoretical framework; see DISCUSSION.] ORR (1993) and TURELLI and ORR (1995, 2000) described between-sex asymmetries associated with X–autosome interactions, cytoplasmic–nuclear interactions, and maternal effects. Here we generalize those analyses by elaborating the dynamic model of ORR and TURELLI (2001) to quantitatively analyze how same-sex (including hermaphrodite) asymmetry between reciprocal crosses is expected to vary with divergence time. We provide an idealized quantitative treatment that contrasts symmetrically acting incompatibilities with asymmetrically acting ones.

In addition, because isolation asymmetry appears to be particularly common and taxonomically widespread in angiosperms (DARWIN 1859, Chap. 8; TIFFIN et al. 2001), we consider in some detail asymmetric genetic interactions that are common in angiosperms: nuclear–cytoplasmic interactions, gametophyte–sporophyte interactions, and triploid endosperm interactions (see Table 1 and Figure 1). Each involves asymmetric interactions, and only the first has been treated previously (TURELLI and ORR 2000).

View this table: In this window In a new window   TABLE 1 Summary of phenomena analyzed that contribute to asymmetric postzygotic isolation

 

View larger version (17K): In this window In a new window Download PPT slide   FIGURE 1.— Generalized angiosperm gametogenesis and double fertilization. (A) During pollination pollen is transferred to the stigma of the recipient flower. (B) During fertilization the pollen tube germinates and travels through the female stigmatic tissue into the ovary. The mature male gamete (pollen or "microgametophyte") comprises two genetically identical haploid sperm cells that result from the mitotic division of a single meiotic product. The mature female gametophyte comprises eight genetically identical haploid nuclei resulting from mitotic division of a single meiotic product. The "central cell" differs from the haploid ovule (1N) and other cells in that it is binucleate (2N). (C) Double fertilization: One haploid sperm cell fertilizes the ovule, while the other sperm cell fuses with the diploid central cell to form a triploid endosperm. (D) Postfertilization development: The triploid endosperm functions as a primary storage and nutritive tissue for the developing embryo.

 

Nuclear–cytoplasmic interactions:

Nuclear–cytoplasmic ("cytonuclear") interactions occur in all organisms where the function of haploid cytoplasmic organelles (including mitochondria, chloroplasts, and plastids) depends on coordinated expression with the diploid nuclear genome. In angiosperms, hybrid male sterility is thought to result frequently from negative epistatic interactions between cytoplasmic (most probably mitochondrial) and nuclear genes in interspecific hybrids (FRANK 1989; SCHNABLE and WISE 1998). Negative cytonuclear interactions are also known to contribute to reproductive isolation between animal species and even populations, where they have been identified as the genetic basis of both hybrid inviability and infertility [e.g., Tigriopus (WILLETT and BURTON 2001; HARRISON and BURTON 2006) and Drosophila (RAND et al. 2001; SACKTON et al. 2003)]. Assuming uniparental inheritance of the relevant organelle, cytonuclear interactions involve interactions between a cytoplasmic genome from one parent (usually maternal, GRUN 1976) and the genes in the hybrid nuclear genome derived from the second parent. For specificity, we assume maternal inheritance of cytoplasmic genomes.

X–autosome interactions:

In species with sex chromosomes, heterogametic hybrids often experience asymmetric incompatibilities between sex-chromosome alleles from one parent and autosomal alleles from the other parent. In the F₁ generation, these incompatibilities are analogous to cytonuclear interactions (as no recombination of the X chromosome has occurred). For specificity, we consider male hybrids in male-heterogametic species. Obviously, asymmetries will also arise from Y–autosome and/or X–Y interactions.

Genetic maternal effects:

In all metazoans, embryonic development begins under the control of maternal mRNAs and proteins. Early in development, control shifts from these maternal factors to zygotic transcripts (often referred to as the "maternal–zygotic transition"; WANG and DEY 2006). Incompatibilities can occur between paternally inherited alleles and the maternal factors that initiate development. Indeed, genetic analyses reveal that these incompatibilities underlie most exceptions to Haldane's rule in Drosophila (SAWAMURA 1996; TURELLI and ORR 2000). Unlike the previous two classes of incompatibilities, in which asymmetric DMIs always act simultaneously with symmetric DMIs (e.g., autosome–autosome incompatibilities that are identical in reciprocal crosses), all maternal–zygotic incompatibilities are expected to be asymmetric. However, given the gradual turnover of control from maternal factors to zygotic, these DMIs can also be expressed simultaneously with the symmetric DMIs in the hybrid nuclear genome.

Asymmetric incompatibilities in plants:

Gametophytic–sporophytic (GS) (including pollen–style) and triploid endosperm (TRE) interactions are restricted to flowering plants. Like DMIs involving early maternal effects, all GS and TRE DMIs are expected to be asymmetric. Modeling them requires a basic understanding of postpollination and early fertilization processes in angiosperms (see Figure 1). Following transfer of pollen to the stigma of a flower, each pollen grain germinates to produce a tube—containing two genetically identical haploid sperm cells—that grows down the maternal style and into the ovary (MASCARENHAS 1989; Figure 1, A and B). Importantly, angiosperm pollen is known to express many genes during this haploid phase (estimated in some cases as >70% of the total haploid genome; OTTAVIANO and MULCAHY 1989; FROVA and PE 1997), rather than solely expressing paternal gene products. GS interactions therefore occur when genes expressed by the haploid pollen (male gametophyte) interact with genes expressed in the stigma and style of the diploid (sporophyte) maternal parent (Figure 1B). For simplicity, we generally refer to these as pollen–style interactions although they can also operate between pollen and stigmatic and ovary tissues (WILLEMSE and VAN LAMMEREN 2002). In interspecific crosses, dysfunctional GS interactions frequently cause postpollination prezygotic cross failure, including failure of pollen to germinate, retardation or rupture of pollen tubes in foreign styles, and failure of pollen tubes to penetrate ovules (DE NETTANCOURT 2001 and references therein).

In contrast to GS interactions, TRE interactions predominate after fertilization, during formation and development of the new zygote. During angiosperm "double fertilization" within a single ovule, one of the pollen haploid sperm cells fertilizes a haploid egg cell to produce a diploid embryo while the remaining sperm cell fuses with a binucleate (doubled haploid) maternal "central cell" to produce a 3N endosperm (CRESTI and TIEZZI 1997) (Figure 1C). This triploid endosperm develops rapidly and sequesters maternal resources to act as the primary nutritive body for the developing embryo (Figure 1D). In crosses among angiosperms, hybrid seed failure frequently results from abnormal development of hybrid endosperm (rather than the embryo itself), likely due to dysfunctional interactions between the haploid male and doubled haploid female genetic components within the triploid endosperm (e.g., LIN 1984; KATSIOTIS et al. 1995; GUTIERREZ-MARCOS et al. 2003). Experiments can determine whether a defect in hybrid seed development is caused by TRE DMIs or symmetric DMIs acting within the diploid embryo (e.g., by assessing viability of the embryo when cultured independently of the endosperm; BRIDGEN 1994). However, when experiments simply score overall seed development, inviability can be caused by a combination of the asymmetric TRE interactions and symmetric embryonic DMIs. Hence, it is important to consider TRE interactions both in isolation and in conjunction with symmetric DMIs.

We present a theoretical analysis that encompasses all of the asymmetric DMIs discussed above. First, we generate expected fitnesses of reciprocal F₁ hybrid genotypes, on the basis of the expected number and relative effects of different classes of DMIs. Second, extending the analytical approximations for the time-dependent distribution of the cumulative effects of DMIs developed in ORR and TURELLI (2001), we examine the transient dynamics of asymmetric reproductive isolation expected during allopatric speciation. We use this first to make quantitative predictions about the magnitude and divergence-time dependence of fitness differences between reciprocal crosses, in particular to identify biological factors likely to contribute most to asymmetric isolation. Then we consider the probability that complete reproductive isolation will be seen in one cross, but incomplete isolation in the reciprocal cross (e.g., asymmetric Haldane's rule). Finally, we review data relevant to estimating the parameter values critical to our predictions and then examine plant and animal data on asymmetry to assess its likely causes. We encourage readers who are primarily interested in our predictions to read the Introduction describing our model and its parameters, look at the figures that present our numerical results, and then skip to the DISCUSSION of data.

We present a model of two-locus DMIs that distinguishes different classes of interactions in hybrids, with respect to both the magnitude and the symmetry of their expected effects. Because the bulk of available data that demonstrate asymmetry comes from initial hybridizations rather than backcrosses or the F₂ generation, our treatment focuses on isolation expressed during parental hybridization and in the resulting F₁. Our model incorporates symmetric DMIs (e.g., between autosomal loci or between sex-linked and autosomal loci in the homogametic sex) that affect reciprocal crosses equally ("bidirectional," B) and asymmetric DMIs (as described above) that differ between reciprocal crosses ("unidirectional," U). We use this framework to discuss all of the sources of asymmetry identified above. Note that, throughout our treatment, reproductive isolation due to DMIs falls into two cases: those involving only U DMIs, and those involving both U and B DMIs. For instance, GS interactions are exclusively unidirectional and occur before other interactions that affect zygote viability can act. Hence, only the U DMIs they produce need be considered to understand the resulting asymmetric failure of hybridization. Similarly, TRE effects can be experimentally isolated from embryonic B DMIs that may simultaneously affect seed development. Incompatibilities involving GS and TRE interactions act sequentially, so when we consider only one of them, we implicitly assume that viability is assayed from the beginning to the end of the relevant stage of fertilization/development. In contrast, cytonuclear DMIs (U) act simultaneously with nuclear genome DMIs (B); so both must be considered simultaneously. Similarly, in heterogametic males, X–autosome interactions (U) act simultaneously with autosomal–autosomal interactions (B). These cases are clear cut, but in others there may be no clear expectation about the relative importance or frequency of U vs. B DMIs. For instance, although maternal-effect DMIs act early in embryogenesis, some zygotic transcripts appear extremely early (e.g., TADROS and LIPSHITZ 2005; WANG and DEY 2006). As soon as zygotic transcripts are active, U maternal-effect DMIs act simultaneously with B nuclear DMIs to determine viability. Similarly, asymmetric TRE interactions will act simultaneously with symmetric embryonic incompatibilities to determine seed viability. Our idealized analyses below include parameters to weight the relative importance of U vs. B DMIs. In some cases, like X–autosome vs. autosome–autosome DMIs, we have simple a priori predictions; but in cases like maternal effects vs. zygotic effects, the biology is much less well understood and the relevant weighting will depend on the timing of expression of the loci involved. Many of these details are irrelevant to our predictions, which depend only on composite parameters that describe the cumulative consequences of U vs. B DMIs.

In general, F₁ hybrids experience only one form of B DMIs, namely those involving heterozygous loci on biparentally inherited chromosomes. However, F₁ individuals can be afflicted by several types of U DMIs simultaneously. For instance, heterogametic males may experience at least four types: those involving cytonuclear interactions, maternal effects, X-linked DMIs, and Y-linked DMIs. It is usually difficult to know the relative contributions of these to a phenotype like F₁ inviability without detailed developmental and genetic analyses. We simplify our mathematical treatment by considering only one type of U DMI. Our analysis can be easily generalized to consider both sequential and simultaneous effects of different DMIs, but we concentrate on the simplest cases to illustrate some central principles.

Basic model:

Following TURELLI and ORR (1995, 2000) and ORR and TURELLI (2001), we assume that individual DMIs contribute additively to a hybrid breakdown score, S, which maps onto fitness in a simple way. To have a single framework, we assume that U and B DMIs act simultaneously, so that situations in which only U DMIs occur (like GS and TRE) appear as special cases. Let the random variable e_U denote the effect of a specific U incompatibility in a parental cross or resulting F₁, and let e_B denote the effect of a specific B incompatibility. In general, the hybrid breakdown score after a divergence time of t depends on both the number and the kind of DMIs that have accumulated. We denote the number of B DMIs by , the number of U DMIs by , and the total number by I_t (we analyze their accumulation below). Table 2 provides a summary of repeatedly used notation. We assume that

(1)

where by definition is identical for the reciprocal F₁. The e_i are all assumed to be independent, and the () are assumed to be identically distributed (the latter assumption can be easily relaxed to allow for different types of U DMIs acting simultaneously, e.g., X–autosome and X–Y). We assume that hybrid fitness is a decreasing function of the breakdown score, v(S), that gives a relative fitness of 1 when S = 0 and declines to 0 when S reaches a threshold value C for complete sterility/inviability; i.e.,

(2)

These general conditions suffice to analyze qualitative asymmetry (see Equations 27–30); but to predict quantitative asymmetry, a particular function v(S) must be chosen (see Equation 10). Let S_ij denote the breakdown score produced with a mother from taxon i and a father from taxon j. When considering X–autosome incompatibilities in heterogametic individuals, we assume for definiteness that males are heterogametic. To understand the forces that lead to systematic differences between reciprocal crosses, we first seek conditions under which E(S₁₂) E(S₂₁).

View this table: In this window In a new window   TABLE 2 Glossary of repeatedly used notation

 

Expected differences between reciprocal crosses:

From (1), the breakdown scores from reciprocal crosses are

(3)

Thus, E(S₁₂) E(S₂₁) if and only if . Because each U DMI is assumed to follow the same distribution of effects, (1) and (3) imply that E(S₁₂) E(S₂₁) if and only if , where and denote the number of U DMIs afflicting the cross or the resulting F₁ from the reciprocal combinations. A detailed derivation of the stochastic accumulation of DMIs is presented in APPENDIX A. Here we describe only the assumptions and parameters necessary to explain our biological conclusions. Following the model of ORR (1995) as elaborated by ORR and TURELLI (2001), we assume that each pairwise interlocus allelic difference between the diverging lineages can potentially produce a DMI. Each such pair is viewed as an independent "Bernoulli trial." For pairs that may produce B DMIs, the probability of "success," i.e., the probability that the difference yields a DMI, is denoted p, as in ORR and TURELLI (2001). For pairs of loci that may produce U DMIs, the probability that a pairwise allelic difference yields a DMI is denoted p_U. From (3), only p_U enters the conditions for E(S₁₂) E(S₂₁).

To calculate the expected number of U DMIs, we must consider substitutions differentiating the diverging lineages at two sets of loci, characterized by whether their DMI effects involve maternally or paternally inherited alleles. Let K_U denote the number of substitutions in either lineage at loci that can produce maternally inherited alleles involved in the U DMIs being considered. For X–autosome incompatibilities expressed in males, K_U would be the number of X-linked substitutions. For cytonuclear incompatibilities, K_U would denote the number of substitutions in organelle genomes; whereas for pollen–style interactions, K_U would denote the number of substitutions affecting style function. We assume that K_U1 of these substitutions occurred in lineage 1 and K_U2 in lineage 2. We assume that the relevant changes are sufficiently rare that all such substitutions have occurred in only one of the two lineages (i.e., only one lineage contains a derived allele at each locus), so that the total number of substitutions differentiating the taxa at these loci is K_U = K_U1 + K_U2. Similarly, we let () denote the number of substitutions in either lineage at loci whose paternally inherited alleles can participate in the F₁ U DMIs being considered. For X–autosome incompatibilities expressed in males, would be the number of autosomal substitutions. For cytonuclear incompatibilities, would denote the number of substitutions in the nuclear genomes; whereas for pollen–style interactions, would denote the number of substitutions affecting pollen function. For brevity, we refer to the loci that contribute to K_U () as "female-acting" (male-acting) loci.

The expected number of U DMIs experienced by reciprocal crosses can differ because of differences in the rates of molecular evolution of the relevant loci. Although the female-acting and male-acting loci may overlap (for instance, some nuclear loci may contribute to both pollen and style function), we simplify our analyses by assuming that , , , and are independent Poisson processes (ORR and TURELLI 2001). (Overlap in these sets of loci can be handled more formally by ignoring products of the small parameters p and p_U, as discussed below, but the conclusions are unchanged.) The conditions for asymmetry depend on the fraction of the expected number of substitutions of each type in each lineage. Let

(4a)

and

(4b)

so that ₁ () is the fraction of the expected female-acting (male-acting) substitutions that occur in lineage 1 and ₁ measures the difference in the relative rates of evolution of these two sets of loci in the taxa being hybridized. In a parental cross with a taxon 1 mother, the expected number of U DMIs conditional on the number of substitutions of each type in each lineage [i.e., conditional on ] is

(5)

where the first, second, and third terms in parentheses describe derived–derived, derived–ancestral, and ancestral–derived U interactions, respectively (see APPENDIX A). Using our assumption that the components of K are independent, we have

(6)

For the reciprocal cross,

(7)

Equation 5 shows that reciprocal crosses can differ only because of U DMIs between derived alleles in the two lineages. In general, the parameter p_U in (6) and (7) as well as the expected effect of each U DMI will differ among alternative types of U incompatibilities. For instance, cytonuclear DMIs involving mismatches between mitochondrial and nuclear loci that disrupt ATP production may have systematically larger effects than typical X–autosome DMIs. The implications of such systematic differences are considered below.

From (6) and (7), we see that the expected number of DMIs differs between reciprocal crosses [i.e., ] only when

(8)

That is, we expect systematic asymmetries when two diverging lineages show different relative rates of evolution for the female-acting and male-acting loci . Note that it is relative rates, not absolute rates, that matter. If taxon 1 evolves uniformly twice as fast as taxon 2 at both sets of loci, we have and equal expected breakdown scores from the reciprocal crosses. In contrast, if taxon 1 exhibits a faster relative rate of evolution for female-acting loci than for male-acting loci (i.e., ₁ > ), crosses using taxon 1 females are expected to produce systematically less-fit F₁ (or lower probability of fertilization success) than the reciprocal cross, even if the overall substitution rate for taxon 1 is lower than that for taxon 2 (e.g., and ).

Stochastic dynamics of asymmetric sterility/inviability—quantitative asymmetry:

Even without lineage-specific differences in rates of accumulation of the female- vs. male-acting loci [i.e., ₁ = 0 so that E(S₁₂) = E(S₂₁)], reciprocal crosses can produce different hybrid fitnesses, i.e., v(S₁₂) v(S₂₁), because of chance differences in the numbers and effects of the separate DMIs that contribute to and (see Equation 3). To quantify the fitness asymmetry expected under allopatric divergence, we develop a time-dependent probabilistic description of intrinsic postmating isolation by extending the treatment of ORR and TURELLI (2001) to U DMIs.

Without any calculations, it is apparent that divergence time, t, must affect asymmetry. For any model of accumulating DMIs, there will be a divergence time, denoted , at which E(S_ij) = C, the value that produces complete postmating isolation. As noted above, if ₁ 0, . As shown below, a mathematically convenient reference timescale for postmating asymmetry is the geometric mean

(9)

Divergence time, t, affects asymmetry, because early in divergence (i.e., t << T_C) few DMIs have accumulated and we expect v(S₁₂) v(S₂₁) 1; whereas after extensive divergence (i.e., t >> T_C), we expect v(S₁₂) v(S₂₁) 0. Thus, asymmetry must be maximal for intermediate values of t (0 < t < T_C).

Some recent studies on isolation asymmetry report quantitative differences between the fitnesses of reciprocal F₁ (e.g., TIFFIN et al. 2001; BOLNICK and NEAR 2005). Moreover, some of those data also show how asymmetry changes with divergence time (e.g., Figure 4 of BOLNICK and NEAR 2005). To make quantitative asymmetry predictions, we need an explicit fitness function, v(S), and a model from which we can derive the time-dependent bivariate distribution of the hybrid breakdown scores (S₁₂, S₂₁). As demonstrated below, the shape of v(S) significantly affects expected levels of asymmetry. To illustrate this, we consider a family of fitness functions that satisfy (2),

(10)

with > 0. This function is displayed in Figure 2 for = 0.5, 0.75, 1, and 1.5. Because S is expected to increase roughly quadratically (ORR 1995; ORR and TURELLI 2001) as two-locus DMIs accumulate (and even faster for multilocus DMIs), = 0.5 would produce a roughly linear decline of hybrid fitness with divergence time, whereas = 1 and = 1.5 would produce a roughly quadratic or cubic decline, respectively. Given that meta-analyses reveal at most a slightly faster than linear decline of hybrid fitness with divergence (e.g., LIJTMAER et al. 2003; BOLNICK and NEAR 2005), we focus our numerical examples on an intermediate case, = 0.75.

View larger version (21K): In this window In a new window Download PPT slide   FIGURE 4.— Time-dependent medians (solid curves) and 95th percentiles (dashed curves) of asymmetry values A [i.e., P(A a) = 0.5 vs. 0.95] when only U DMIs contribute to reproductive isolation between lineages. (A) The effects of varying C with = 0.75, 1 = 0, and CV = 0.5. The curves are C = 5 (black), 10 (red), 20 (green), 100 (blue), and 1000 (orange). (B) The effects of varying with C = 20, CV = 0.5, and 1 = 0. The curves are = 0.5 (black), 0.75 (red), 1 (green), and 1.5 (blue). (C) The effects of varying 1 (which controls expected differences between reciprocal breakdown scores) with C = 20, = 0.75, and CV = 0.5. The curves are 1 = 0 [E(S12) = E(S21), black], 1 = 0.166667 (1 = 22, red), 1 = 0.3 (1 = 42, green), and 1 = 0.4 (1 = 92, blue). In addition to the median and the 95th percentiles, the 5th percentiles are shown as dotted curves. (D) The effects of varying CV with C = 20, = 0.75, and 1 = 0. The curves are CV = 0 (black), 0.25 (red), 0.5 (green), and 1.0 (blue).

 

View larger version (7K): In this window In a new window Download PPT slide   FIGURE 2.— The fitness function v(S) described by (10) with C = 100 and = 0.5 (dotted curve), 0.75 (short-dashed curve), 1.0 (solid curve), and 1.5 (long-dashed curve).

 
To quantify asymmetry, we define

(11)

where S_max = max(S₁₂, S₂₁) and S_min = min(S₁₂, S₂₁). The index A ranges from 0 (no asymmetry) to 1 (complete isolation observed in one cross and none in the reciprocal cross). Its distribution depends on divergence time, which we measure in units of T_C, defined in (9). Hence = t/T_C = 1 corresponds to the time (averaged over the two reciprocal crosses) at which their expected breakdown scores reach C, the value that produces complete postmating isolation.

To determine probable values of A, we need an approximation for the bivariate distribution of (S₁₂, S₂₁). Once that is specified, we can obtain the quantiles of A from the identity

(12)

where f(S_max, S_min) denotes the bivariate distribution of the order statistics (S_max, S_min) and v^–1(x) = C(1 – x)^1/ for 0 < x < 1, C for x 0, and 0 for x 1. Let g(S₁₂, S₂₁) denote the joint distribution of the reciprocal incompatibility scores (S₁₂, S₂₁). Then for S_max S_min, f(S_max, S_min) = g(S_max, S_min) + g(S_min, S_max). To apply (12), we must approximate g(S₁₂, S₂₁), the time-dependent bivariate distribution of (S₁₂, S₂₁).

Assuming that at least a moderate number of DMIs (on the order of 10) contribute to the incompatibility score, S, ORR and TURELLI (2001, Appendix 1) gave a heuristic analytical argument for approximate normality of S. They supported this approximation with numerical simulations of the underlying stochastic processes. We extend this Gaussian approximation to the bivariate distribution of (S₁₂, S₂₁) but must also condition the distribution so that the breakdown scores remain nonnegative. This additional approximation is inconsequential when the means of the breakdown scores are several standard deviations from 0. The adequacy of this approximation for f(S_max, S_min), which involves both truncation and applying a Gaussian approximation even when S is small, is supported by numerical results in APPENDIX B. There we show reasonable agreement between the percentiles of A obtained from (12) and the percentiles obtained from simulating an explicit stochastic model of accumulating DMIs with random effects. Using the (conditional) Gaussian approximation for (S₁₂, S₂₁), we can apply (12) once we have approximated the means, variances, and covariances of S₁₂ and S₂₁. For simplicity, we first assume that all DMIs are asymmetric.

All asymmetric (U) DMIs:

In this case, (1) implies that

(13)

We can assume without loss of generality that E() = 1, which is equivalent to measuring C in units of the expected number of DMIs required to produce complete postmating isolation. For consistency with the more general calculations below, we assume that Var() = CV². Even though the that contribute to S₁₂ and S₂₁ are assumed to be independent, S₁₂ and S₂₁ covary because of their shared dependence on K = (, , , ). However, as shown in APPENDIX C, this covariance is proportional to , which is negligible in comparison to the dominant terms in the means and variances, which are proportional to p_U. (ORR and TURELLI (2001) and PRESGRAVES (2003) estimate p to be <10^–5; and even if p_U is much larger, it is unlikely to exceed 10^–2.) Using expressions (6) and (7) for E() and E(), our Gaussian approximation for (S₁₂, S₂₁) is completely specified by the moments

(14a)

(14b)

(14c)

(14d)

and

(14e)

The variances and covariance are approximate because they ignore terms proportional to .

Our analyses require two additional parameters that describe the overall rate of molecular evolution and how it is apportioned between and . Let

(15)

denote the total number of substitutions. For each of the four independent Poisson processes, , _, , and , we denote the corresponding rate parameter by k_X, with X = U₁,U₂, etc. So, for instance, after a divergence time of t,

(16)

where . In terms of these parameters, we have ₁ = and . We introduce the new parameter

(17)

which is the fraction of substitutions relevant to U DMIs that occur at female-acting loci. For example, for nuclear–cytoplasmic (or X–autosome) interactions, g_U describes the fraction of cytoplasmic (or X chromosome) substitutions that contribute to U DMIs. Substituting into (14a), we obtain

(18a)

which implies that

(18b)

by the definition of . The other moments can be expressed similarly as explicit functions of the parameters and divergence time.

Equation 18b implies that if we measure time in units of by setting , at time , E(S₁₂) = ²C. This scaling leads to a major simplification of the expressions for the first- and second-order moments of (S₁₂, S₂₁), which demonstrates that the levels of asymmetry expected in the F₁ generation depend on C, , CV, and ₁ but do not depend on the values of , g_U, and p_U. This is easiest to see when ₁ = 0, so that E(S₁₂) = E(S₂₁) and . In this case, when t/T_C = , (14) and (18) imply that E(S₁₂) = E(S₂₁) = ²C, Var(S₁₂) = Var(S₂₁) , and Cov(S₁₂, S₂₁) 0, irrespective of p_U, g_U, and . In contrast, the cumulative distribution defined by (12) clearly depends on C, which is proportional to the means and variances, CV, which inflates the variances, and the shape of v(S). If ₁ 0, it also affects the results. When t/T_C = , (14) and (18) imply

(19a)

(19b)

and

(19c)

where

(20)

depends only on ₁. Hence, we see that only C (the number of DMIs required to produce complete postmating isolation), (the shape of the fitness function, Equation 10), CV (the coefficient of variation of DMI effects), and ₁ [the parameter that determines whether E(S₁₂) = E(S₂₁)] can affect asymmetry when only U DMIs act.

Numerical results:

To understand the levels of asymmetry expected as the parameters vary, we used (12) to approximate the quantiles of A by numerically solving the equation P(A a) = P for a at various values of P, such as 0.05, 0.5, and 0.95. This was done for a range of times, resulting in plots that display expected levels of asymmetry as a function of t/T_C. The numerical analysis was performed in Mathematica 5.2 (WOLFRAM 2003). As noted above, the quantiles of A depend only on C, , CV, and ₁. Figure 3 illustrates the time-dependent quantiles with C = 20, = 0.75, CV = 0.5, and ₁ = 0. Note that for these parameters, maximal asymmetry is observed near 0.8T_C, and the distribution of A tends to be quite broad (at each time, the dotted curves define the 90% confidence interval for A). Because the 5th percentile is generally very near zero, it is uninformative and is not displayed in most of the figures below, except when ₁ 0 produces nonnegligible values. When the lower quantile is not shown, asymmetry values indistinguishable from zero even in large experiments (e.g., A 0.05) would generally be statistically consistent with the parameter values considered.

View larger version (6K): In this window In a new window Download PPT slide   FIGURE 3.— Time-dependent quantiles of A, our measure of quantitative asymmetry defined in Equation 11 [i.e., P(A a) = P] for C = 20, = 0.75, 1 = 0, and CV = 0.5, with P = 0.05 (dotted curve), 0.5 (solid curve), and 0.95 (dashed curve).

 
Figure 4, A–D, shows how the percentiles of A change as C, , ₁, and CV vary around base values of C = 20, = 0.75, ₁ = 0, and CV = 0.5. Figure 4A shows that C has a major effect on the quantiles, with lower asymmetry expected as C increases. This supports MULLER's (1942) intuition that greater asymmetry is expected when fewer DMIs are required to produce complete postmating isolation. However, even when C = 100, moderate levels of asymmetry are produced, with the 95th percentile of A near 0.2 for t/T_C between 0.65 and 0.95. Likely levels of asymmetry are roughly doubled when C is reduced to 20 and roughly tripled (relative to C = 100) when C = 10. Figure 4B addresses the robustness of this pattern to different shapes of the fitness function v(S), with C = 20. As decreases from 0.75 to 0.5 (which produces a roughly linear decrease of hybrid fitness with divergence time), maximal asymmetry is reduced but significant asymmetry is seen over a larger range of divergence times. The intuitive explanation is that if fitness declines more quickly initially, stochastic differences in breakdown scores lead to significant asymmetry more quickly. Conversely, as increases from 0.75 to 1 (which produces a roughly quadratic decline of hybrid fitness with time) or 1.5 (roughly cubic decline), maximal asymmetry increases sharply and is markedly peaked for t/T_C near 1. The areas under these curves (corresponding to the average asymmetry) are far more consistent for different values of than the maxima. Comparing the most extreme cases, the area under the 95th percentile curve is 0.33 for = 0.5 vs. 0.39 for = 1.5.

Figure 4C examines the effect of unequal relative rates of evolution that produce E(S₁₂) E(S₂₁). Holding C = 20, = 0.75, CV = 0.5, and = 0.5, we increased ₁ from 0.5 (₁ = 0) until the rate of evolution of the female-acting loci in taxon 1 is nine times that in taxon 2 (₁ = 0.4). The qualitative result is that even when taxon 1 evolves twice as fast as taxon 2 at the female-acting loci, so that ₁ = 0.166667, probable levels of asymmetry rise only slightly—and primarily for t/T_C near 1. However, extreme relative rate differences, producing ₁ 0.3, have an appreciable effect on asymmetry. This is most clearly expressed by the fact that the 5th percentile of A rises to near 0.1 for t/T_C between 0.8 and 0.9. These analyses suggest that unless relative rate differences are extreme (e.g., female-acting loci evolve at least four times as fast in one lineage as in the other, while male-acting loci evolve at similar rates in both lineages), stochastic effects are more likely to explain observed levels of postmating asymmetry than systematic interspecific differences in the relative rates of molecular evolution.

Figure 4D considers the influence of varying the fitness effects of individual DMIs. It shows that, as expected, asymmetry increases as CV increases, corresponding to increasing variance among the effects of individual DMIs. This qualitative effect is easy to understand because as CV increases, the variances of the breakdown scores increase while their means remain fixed. However, values of CV up to 0.5 have relatively little effect on probable asymmetry values.

Both asymmetric (U) and symmetric (B) DMIs:

More generally, hybrid fitness is determined by both B and U DMIs (see Table 1), and we must consider their relative contributions to the hybrid breakdown score. In TURELLI and ORR's (2000) analysis of two-locus X–X, X–autosome, and autosome–autosome DMIs, they described three categories of two-locus DMIs. In increasing order of expected severity, they are: H₀ DMIs that involve heterozygous incompatible alleles at both loci, H₁ DMIs that involve heterozygous alleles at one locus and either a homozygous or a hemizygous incompatible allele at a second locus, and H₂ incompatibilities in which the incompatible alleles at both loci are either hemizygous or homozygous. In general, hemizygosity and homozygosity may lead to different distributions of effects; but we restrict our F₁ analyses to a single class of U DMIs and hence a single parameter will suffice to describe the relative effects of the U DMIs (which are either H₁ or H₂) and the B DMIs (which are all H₀). The three classes of DMIs are assumed to contribute on average h₀, h₁, and h₂, respectively, to the hybrid breakdown score, S, with h₀ < h₁ < h₂.

All B DMIs in the F₁ involve interactions between heterozygous nuclear loci. Hence, all are H₀ incompatibilities, each of which contributes h₀ on average to both S₁₂ and S₂₁ [see (1) and (3)]. In contrast, the U DMIs we consider may be either type H₁ (e.g., X–autosome in the heterogametic sex, cytonuclear, or maternal effects) or type H₂ (e.g., triploid endosperm). To unify our notation and facilitate comparison with the results in the previous section, we assume that the U DMIs have average effect 1, whether they are type H₁ or H₂, whereas we denote the average effect of the B DMIs by h₀ < 1. With this simplification, we can model the U DMI contributions to S₁₂ and S₂₁ by using the approximations described in the previous section. Similarly, we can model the B DMI contributions using the framework developed in ORR and TURELLI (2001). Overall, we have

(21)

where (18) provides an explicit time-dependent expression for E(). A comparable expression for E(I_B) is given by Equation 6 of ORR and TURELLI (2001). If we let p denote the probability that an allelic difference at two B loci leads to a DMI, the expected number of B DMIs in an F₁ is

(22)

where E(K_B) denotes the number of substitutions at nuclear loci (summed across both lineages) contributing to B DMIs. As in (16), we have . Unlike (6) and (7) for U DMIs, this result is independent of the expected fraction of these substitutions that occur in each lineage (ORR 1995; APPENDIX A).

To apply our bivariate Gaussian approximation and identity (12) for determining the quantiles of the asymmetry index A, we must first clarify the relationship between K_B and the stochastic process K = (, , , ) that entered our analysis of U DMIs and then approximate the variances and covariance of S₁₂ and S₂₁. From Table 1, we see that for the cases we consider, the loci contributing to K_B are either all nuclear loci or, in the case of X–autosome U DMIs, all autosomal loci. In these cases, the loci that potentially contribute to K_B are identical with the loci that potentially contribute to (namely the male-acting loci in U DMIs). Hence, the parameter g_U = introduced in (17) also can be used to differentiate the loci contributing to U vs. B DMIs (with g_B = 1 – g_U). In particular, for X–autosome DMIs in heterogametic males, g_U is the parameter g_X of TURELLI and ORR (2000), the fraction of nuclear substitutions that are X-linked. For cytonuclear interactions, g_U describes the fraction of substitutions that occur in the relevant cytoplasmic organelles. When TRE incompatibilities interact with zygotic incompatibilities to determine seed development, all nuclear DMIs in the F₁ (which are B) can act effectively simultaneously with TRE incompatibilities (which are U). In contrast, the interaction between maternal effects and the zygotic genome is much less clear cut, because the relevant zygotic loci are those expressed earliest in development. These could well be only a very small subset of the nuclear loci. However, in our idealized model of DMI origins, the zygotic incompatibilities that manifest as development moves from maternal to zygotic control can potentially involve the entire nuclear genome. Hence, in the case of simultaneous action of maternal-effect DMIs and zygotic DMIs, it seems unlikely that a single parameter g_U can capture both the relative role of male-acting loci involved in maternal–zygotic U DMIs and the relative role of zygotic B DMIs affecting early embryos (because the latter could involve a much smaller subset of loci). As discussed below, this complication can be accommodated by suitable interpretation of composite parameters that emerge in our analysis.

Given normality, the distribution of (S₁₂, S₂₁) depends only on their means, variances, and covariance. These are computed in APPENDIX D, using the simplifying assumption that all of the random variables describing DMI effects have equal coefficients of variation (CV); i.e., Var(e_U)/[E(e_U)]² = Var(e_B)/[E(e_B)]² = CV², where E(e_U) = 1 and E(e_B) = h₀. To understand how asymmetry depends on the parameters, it is useful to express the moments given in (D2), (D5), and (D7) as functions of the scaled time, = t/T_C, with T_C defined by (9). After some simplification, we obtain

(23a)

(23b)

(23c)

and

(23d)

where

(24)

Thus, asymmetry is independent of but does depend on C, CV, ₁, v(S) [i.e., in (10)], and h₀; and it depends on p, p_U, and g_U only through the ratio ß defined in (24). Multiplying the numerator and denominator of ß by (1 – g_U), we see that ß is the ratio of the expected number of B DMIs to the expected number of U DMIs, providing a straightforward biological interpretation of this parameter. ß can also be expressed as pk_B/(p_Uk_U).

When the term ß makes a negligible contribution to the variances and covariance, the implications of analyzing both B and U DMIs become much clearer. Setting ß = 0 in (23), we obtain

(25a)

(25b)

(25c)

and

(25d)

with

(26)

Hence when ß is negligible (the relevant range of parameters is discussed below), four parameters that are likely to be very difficult to estimate individually, namely p, p_U, h₀ and g_U, enter only as a single composite parameter defined by (26). Like ß, has a simple biological interpretation. It is the ratio of the expected contribution to the hybrid breakdown scores from B DMIs (which have average effect h₀) to the expected contribution from U DMIs (which have average effect 1). This interpretation of simplifies our analysis of maternal effects for which we have little data to guide our choice of g_U.

Comparing (25) with the expressions that arise with only U DMIs (19), we see that for fixed values of C, CV, and ₁, the reciprocal breakdown scores remain approximately uncorrelated, their means are brought closer to one another by the appearance of the positive term in the numerator and denominator of the ratio in , and their variances are reduced by the appearance of in the denominators of (25b) and (25c). Hence, as expected, incorporating both B DMIs and U DMIs systematically reduces asymmetry. The same result emerges from the more general approximations (23), which also have a positive covariance contributing to reduced asymmetry. A simple biological interpretation is that when both U and B DMIs act, the U DMIs, which are solely responsible for reproductive asymmetry, account for a smaller fraction of the total isolation. Hence, less asymmetry is expected.

Numerical results:

As in the case with only U DMIs, we analyze the levels of asymmetry expected as the parameters vary by using (12) to approximate the quantiles of A. We first consider the range of parameters under which the simpler moment approximations (25), which depend only on rather than on ß and h₀ separately (as in 23), are adequate for describing probable values of asymmetry. Figure 5A considers the effect of varying h₀ from 0.05 to 0.4 when is held fixed at either 1 (corresponding to equal average contributions of B and U DMIs to postmating isolation) or 10 (corresponding to 10-fold greater contribution from B DMIs than from U DMIs). As expected, asymmetry is much greater when is smaller. Less obvious is how little influence h₀ has once is known. As Figure 5A shows, for = 10, h₀ has no appreciable effect on asymmetry until h₀ = 0.4, which we consider implausibly high (as discussed below). When = 1, the value of h₀ is essentially irrelevant, even up to h₀ = 0.4. Our qualitative conclusion is that although h₀ can have a major effect on asymmetry through its role in determining , we can understand this effect by varying rather than varying ß and h₀ separately. Hence, Figure 5B explores the role of varying while holding h₀ fixed at 0.1. [Note that in Figure 5B and all subsequent figures, we use the full approximation (23) for the moments, but hold h₀ = 0.1 as we vary .] It shows that for these parameter values, on the order of 0.1 produces levels of asymmetry comparable to those seen with only U DMIs ( = 0). Conversely, once is as large as 10, relatively little asymmetry is expected (but it is likely to still be detectable in moderate-sized experiments when t/T_C is near 1).

View larger version (11K): In this window In a new window Download PPT slide   FIGURE 5.— Effects of combining symmetric (B) and asymmetric (U) DMIs on the time-dependent asymmetry values, A. The solid lines are medians, the dashed lines are 95th percentiles. (A) The effect of varying the dominance parameter h0, while holding fixed the parameter , defined in (26), which quantifies the relative contribution of B vs. U DMIs to hybrid dysfunction. Two sets of results are provided: = 1 (top) and = 10 (bottom). The curves are h0 = 0.05 (black), 0.1 (red), 0.2 (green), 0.3 (blue), and 0.4 (orange). The other parameters are C = 10, = 0.75, 1 = 0, and CV = 0.5. (B) The effect of varying the mix of symmetric (B) vs. asymmetric (U) DMIs, as measured by . The curves are = 0 (all U, black), = 0.1 (red), = 1 (green), and = 10 (blue). The other parameters are h0 = 0.1, C = 10, = 0.75, 1 = 0, and CV = 0.5.

 
Figure 6, A–D, shows how the percentiles of A change as C, , ₁, and CV vary around base values of C = 10, = 0.75, ₁ = 0, and CV = 0.5, while holding = 1 and h₀ = 0.1. Figure 6A, like Figure 4A, shows that C has a major effect on the quantiles, with lower asymmetry expected as C increases. However, even when C = 40, detectable levels of asymmetry are produced. High levels of asymmetry are probable when C is on the order of 5 or 10. It is important to realize that C has a very different interpretation in Figure 6A than in Figure 4A. In Figure 4A, C is simply the average number of U DMIs needed to produce complete postmating isolation. When = 1, we expect U and B DMIs to make equal average contributions to hybrid dysfunction, with each U DMI having average effect 1 and each B DMI having average effect h₀. Thus, when C = 10, = 1, and h₀ = 0.1, we expect that complete postmating isolation will be produced by 55 DMIs on average, with 5 U DMIs and 50 B DMIs. These numbers are halved when C = 5. Thus, even when C = 5, "simple" genetics would not underlie complete postmating isolation; yet significant asymmetry would arise by chance without systematic differences producing E(S₁₂) E(S₂₁). On the other hand, a small number of DMIs (namely the U DMIs) would account for an appreciable fraction of postmating isolation. This can be viewed as a generalization of the large X effect for heterogametic hybrids, in which X-linked DMIs (which would be U DMIs) contribute disproportionately to hybrid inviability/sterility.

View larger version (21K): In this window In a new window Download PPT slide   FIGURE 6.— Time-dependent medians (solid curves) and 95th percentiles (dashed curves) for the asymmetry index A [i.e., P(A a) = 0.5 vs. 0.95] when both U and B DMIs contribute to reproductive isolation between lineages. (A) The effects of varying C with = 1, h0 = 0.1, = 0.75, 1 = 0, and CV = 0.5. The curves are C = 5 (black), 10 (red), 20 (green), 40 (blue), and 100 (orange). (B) The effects of varying , the shape of the fitness function, with C =10, = 1, h0 = 0.1, 1 = 0, and CV = 0.5. The curves are = 0.5 (black), 0.75 (red), 1 (green), and 1.5 (blue). (C) The effects of varying 1 (which controls expected differences between reciprocal breakdown scores) with C =10, = 1, h0 = 0.1, = 0.75, and CV = 0.5. The curves are 1 = 0 [E(S12) = E(S21), black], 1 = 0.166667 (1 = 22, red), 1 = 0.3 (1 = 42, green), and 1 = 0.4 (1 = 92, blue). In addition to the median and the 95th percentiles, the 5th percentiles are shown as dotted curves. (D) The effects of varying CV with C = 20, = 0.75, and 1 = 0. The curves are CV = 0 (black), 0.25 (red), 0.5 (green), and 1.0 (blue).

 
Figure 6B looks at the effects of varying , with C = 10, = 1, h₀ = 0.1, ₁ = 0, and CV = 0.5. The results are very similar to those in Figure 4B. As increases, more extreme asymmetry is produced for t/T_C near 1, but lower asymmetry is expected at lower (earlier) divergence times. Like Figure 4C, Figure 6C examines the effect of unequal relative rates of evolution that produce E(S₁₂) E(S₂₁). Holding C = 10, = 1, h₀ = 0.1, = 0.75, CV = 0.5, and