Download Hybrid Dysfunction: Population Genetic and Quantitative Genetic

vol. 171, no. 4
the american naturalist
april 2008
Hybrid Dysfunction: Population Genetic and
Quantitative Genetic Perspectives
Benjamin M. Fitzpatrick*
Department of Ecology and Evolutionary Biology, University of
Tennessee, Knoxville, Tennessee 37996
Submitted May 22, 2007; Accepted October 5, 2007;
Electronically published February 15, 2008
abstract: In the wake of seminal work by Dobzhansky and Muller,
hybrid dysfunction is usually attributed to incompatible mutations
in different genes arising in different populations. This DobzhanskyMuller (D-M) model is among the most important contributions of
theoretical population genetics. Here I make formal connections between the D-M model and the quantitative genetic interpretation of
hybrid dysfunction as a combination of additive, dominance, and
epistatic effects. Concerns over conceptual differences between the
two approaches are unwarranted; the D-M model can be expressed
as a special case of the statistical model developed for line-cross
analysis in quantitative genetics. This unified theoretical framework
encourages application of quantitative genetic methods to the study
of speciation.
Keywords: Dobzhansky-Muller, speciation, epistasis, dominance theory, Haldane’s rule.
The genetics of hybrid fitness is a key area of research in
speciation biology. Hybrid dysfunction was once seen as
an evolutionary paradox and a key challenge for evolution
as a general explanation of biological diversity (Darwin
1859, chap. 8; Huxley 1908, p. 463; Orr 1996; Coyne and
Orr 2004, chap. 7). It was not immediately obvious how
Darwin’s theory of gradual evolution by natural selection
among subtle variants could account for the evolution of
inviability, sterility, or other dysfunction. The problem was
solved by Dobzhansky (1936, 1937) and Muller (1940),
who applied population genetic reasoning to allopatric
populations (see also Poulton 1908; Bateson 1909; and
reviews in Orr 1996; Coyne and Orr 2004, pp. 269–272).
* E-mail: [email protected].
Am. Nat. 2008. Vol. 171, pp. 491–498. 䉷 2008 by The University of Chicago.
0003-0147/2008/17104-42617$15.00. All rights reserved.
DOI: 10.1086/528991
The key insight is that independent genetic changes in
isolated populations must be compatible with their respective genetic backgrounds but need not be compatible
with each other.
A major prediction arising from the Dobzhansky-Muller
(D-M) model is that hybrid dysfunction is caused by interactions between different loci rather than by heterozygote disadvantage (Nei et al. 1983; Lynch 1991; Orr 1995;
Turelli and Orr 2000; Coyne and Orr 2004; Gavrilets 2004).
Lynch (1991) formalized this idea in a statistical model
derived from line-cross theory (Hill 1982) and involving
the quantitative genetic concepts of additivity, dominance,
and epistasis. Turelli and Orr (2000) derived an expected
hybrid breakdown score from a more mechanistic perspective, starting with an explicit model of two-locus interactions. Demuth and Wade (2005) contrasted Turelli
and Orr’s (2000) approach with a quantitative genetics
model based on the statistical analysis of line crosses. They
concluded that certain important effects were neglected or
undefined in the D-M model, making it difficult to reconcile with their quantitative genetic perspective. Here I
show that this conclusion was based on an incorrect interpretation of the basic D-M model and of Turelli and
Orr’s (2000) multilocus generalization. In fact, the TurelliOrr model can be expressed as a special case of the more
general statistical models of quantitative genetics. The Turelli-Orr formulation is based on a specific mechanistic
model of gene interactions, and formal comparison with
the quantitative genetics framework shows how interactions between loci can affect the coefficients representing
dominance and epistasis in standard quantitative genetic
analyses. My analysis clarifies conceptual connections between population genetic theory and the statistical models
of quantitative genetics.
The Dobzhansky-Muller Model
The D-M model is explained clearly by Dobzhansky (1937,
p. 282), Orr (1995), Gavrilets (2003), and Demuth and
Wade (2005, their fig. 1), among others (some refer to this
model as the Bateson-Dobzhansky-Muller, or BDM, model).
492 The American Naturalist
The problem is that natural selection should not allow the
evolution of unfit offspring. More specifically, a single
change immediately causing reproductive isolation between the mutant and its contemporaries leaves the mutant unable to produce viable or fertile outcrossed offspring. The solution begins with an ancestral species fixed
for an AA genotype at one locus and BB at another. The
species becomes split into two isolated populations. In one
isolate, an a allele becomes fixed (by either drift or selection; the point is that the genotypes AaBB and aaBB are
not less fit than the ancestral AABB). In the other population, b becomes fixed; AABb and AAbb are not less fit
than AABB (fig. 1). Hybrid genotypes combining the two
derived alleles a and b (e.g., the F1 AaBb) may suffer low
fitness, even sterility or inviability, because of interactions
between these alleles that evolved without deleterious effects within their native genetic backgrounds (fig. 1). That
is, hybrid dysfunction results from “complementary” epistasis between loci A and B (Orr 1995).
In the D-M model, the terms “dominant” and “recessive” are applied to describe how the effects of the incompatibility between alleles a and b are modified by the compatible alleles, A and B (Muller 1940; Turelli and Orr
1995). The incompatibility is said to be dominant if any
genotype including at least one a and one b allele expresses
the same level of dysfunction (fig. 1A). The incompatibility
is said to be recessive if the presence of at least one A or
B allele is adequate to mask the incompatibility; that is,
only aabb suffers hybrid dysfunction. Partially recessive
(or partially dominant) incompatibilities are those in
which the severity of the dysfunction is partially reduced
by A and B, so that the double heterozygote AaBb has
higher fitness than aabb, and genotypes that are homozygous-heterozygous (Aabb and aaBb) are intermediate
(fig. 1B). Some authors prefer to reserve the terms “dominant” and “recessive” for single-locus effects (Demuth and
Wade 2005); however, the terms are established in the
speciation genetics literature as descriptors of D-M incompatibilities (e.g., Turelli and Orr 1995, 2000; Presgraves
2003; Coyne and Orr 2004; Gavrilets 2004; Brideau et al.
2006; Sweigart et al. 2006; Willet 2006; Price 2007).
The D-M model as illustrated in figure 1 is a simple
case of a two-locus, two-allele model with interactions
between the two loci. Table 1 shows a more general twolocus, two-allele model where phenotypic effects are partitioned, in the tradition of quantitative genetics, into additive, dominance, and interaction effects. This model is
not experimentally tractable because the epistatic effects
are unconstrained, leading to a description of nine phenotypes using 13 parameters. However, the relationship
between genotype and phenotype described in table 1 provides a common foundation for comparing Turelli and
Orr’s (2000) multilocus D-M model with the multilocus
quantitative genetic models of Lynch (1991) and Demuth
and Wade (2005). Suppose there are n pairs of loci affecting
a phenotype Z, and the effect of the ith pair is described
in table 1. The expected phenotype of an individual is
obtained by adding effects across pairs of loci. For example,
if two pairs of loci affect the trait and an individual has
genotype [(AABB)1, (AaBB)2], then its phenotype should
be [⫺a 1 ⫺ b1 ⫹ I(AABB)1 ⫺ b2 ⫹ DA2 ⫹ I(AaBB)2 ]. See table A1 in the appendix for parameter definitions.
We can generalize the calculation for n independent
pairwise incompatibilities between unlinked loci (Ai and
Bi, where i p 1, …, n). Consider an individual hybrid with
a random fraction p1 of its genome homozygous for species
1 alleles (BB and aa), a random fraction p2 homozygous
for species 2 alleles (bb and AA), and the remaining fraction pH heterozygous. On average, np1p2 pairs of loci will
have genotype AABB, np1pH will have AaBB, and so on,
yielding an expected phenotype,
E[Z] p n{p1 p2[⫺a¯ ⫺ b¯ ⫹ I(AABB)] ⫹ p1 p H[⫺b¯ ⫹ DA ⫹ I(AaBB)]
⫹ p12[a¯ ⫺ b¯ ⫹ I(aaBB)] ⫹ p H p2[⫺a¯ ⫹ DB ⫹ I(AABb)]
⫹ p H2 [DA ⫹ DB ⫹ I(AaBb)] ⫹ p1 p H[a¯ ⫹ DB ⫹ I(aaBb)]
⫹ p22[⫺a¯ ⫹ b¯ ⫹ I(AAbb)] ⫹ p2 p H[b¯ ⫹ DA ⫹ I(Aabb)]
⫹ p1 p2[a¯ ⫹ b¯ ⫹ I(aabb)]}
¯ ⫹ p (b¯ ⫺ a)
¯ ⫹ p H( DA ⫹ DB ) ⫹ p12I(aaBB)
p n{p1(a¯ ⫺ b)
2
⫹ p22I(AAbb) ⫹ p H2 I(AaBb) ⫹ p1 p2[I(aabb) ⫹ I(AABB)]
⫹ p1 p H[I(aaBb) ⫹ I(AaBB)] ⫹ p2 p H[I(AABb) ⫹ I(Aabb)]},
(1)
where ā, b¯ , DA, DB, I(AABB), and so forth, are averages
across the n pairs of loci. This general expression for the
expected phenotype of an individual, given the genomic
proportions p1, p2, and pH, can be used to derive and
interpret the traditional quantitative genetic models used
by Lynch (1991; Lynch and Walsh 1998) and Demuth and
Wade (2005) as well as the Turelli-Orr formulation of the
D-M model (Turelli and Orr 2000). Throughout this article, I assume that the phenotype Z is related to fitness
by any monotone decreasing function; that is, Z is some
measure of hybrid dysfunction.
Turelli and Orr
Turelli and Orr (2000; also see Orr 1995; Orr and Turelli
2001) derived a model of hybrid dysfunction caused by
many pairs of interacting genes. They also considered
“complex” incompatibilities caused by interactions among
three or more loci, but I will focus on the pairwise case
Perspectives on Hybrid Dysfunction 493
effect of incompatibilities when both loci are homozygous
for incompatible alleles, h 1 p I (aaBb) p I (Aabb) is the
average effect when one locus is heterozygous and the
other homozygous, and h 0 p I (AaBb) is the average effect
when both loci are heterozygous.
Demuth and Wade
Figure 1: Adaptive landscapes implied by the simple diploid DobzhanskyMuller model of hybrid dysfunction (Gavrilets 2003, 2004). Bar heights
give relative fitnesses of two-locus genotypes. A, When incompatibilities
are dominant, dysfunction is fully expressed in all genotypes that include
alleles a and b. B, When incompatibilities are at least partially recessive,
the genotype that is homozygous for both incompatible alleles suffers
the greatest dysfunction.
for ease of comparison. Their model is a special case of
equation (1) where (following the asymmetrical D-M
model illustrated in fig. 1) only I (aabb), I (aaBb),
I (Aabb), and I (AaBb) are nonzero:
E[Z] p n[ p1 p2I (aabb) ⫹ p1 p HI (aaBb)
⫹ p2 p HI (Aabb) ⫹ p H2 I (AaBb)]
(2)
(see also table 2). Turelli and Orr (2000) define the phenotype (Z in the present notation) as a hybrid breakdown
score that is related to fitness by any monotone decreasing
function. In their parameters, h 2 p I (aabb) is the average
Demuth and Wade (2005) sought to compare the TurelliOrr model of hybrid dysfunction with a traditional quantitative genetics model with additive, dominance, and epistatic effects that can be estimated from line-cross data.
They concluded that the two approaches can be understood in common terms when their respective assumptions
are made clear, and the resulting unified theory leads to
novel opportunities for empirically testing hypotheses on
the genetics of reproductive isolation. However, their approach led them to two important misinterpretations.
First, they mistakenly equate the Turelli-Orr model (which
is based on the asymmetrical D-M model illustrated in fig.
1) with a symmetrical model and incorrectly infer the types
of quantitative genetic epistasis affecting hybrid dysfunction. Second, they fail to reconcile the conditions for Haldane’s rule identified using their quantitative genetic approach with those found by Turelli and Orr (2000), leading
them to suggest that quantitative genetics and the D-M
model make alternative predictions about the genetic basis
of Haldane’s rule.
Demuth and Wade (2005) equate the Turelli-Orr model
to a quantitative genetic model with symmetrical epistatic
effects illustrated in their table 1. Their model is a special
case of my table 1; the relationships between their parameters and mine are shown in table 2. Under the parameterization of Demuth and Wade (2005), and with singlelocus additive and dominance effects set to 0 (an
assumption of the simple D-M model used by Turelli and
Orr), the genotype that Demuth and Wade define as ancestral (A 0 A 0 B 0 B 0 in their table 1, AABB in my tables 1
and 2) has the same fitness (I) as the hybrid genotype that
is homozygous for both derived (and incompatible) alleles.
The symmetry of their epistatic effect I conflicts with the
basic premise of the D-M model that the ancestral genotype does not suffer dysfunction (Poulton 1908; Bateson
1909; Dobzhansky 1937; Muller 1940; Orr 1995; Coyne
Table 1: Phenotypic effects of a pair of interacting diallelic loci in terms of additive
(ai, bi), dominance (DAi, DBi), and interaction (I ) components
Genotype
(BB)i
(Bb)i
(bb)i
(AA)i
(Aa)i
(aa)i
⫺ai ⫺ bi ⫹ I(AABB)i
⫺ai ⫹ DBi ⫹ I(AABb)i
⫺ai ⫹ bi ⫹ I(AAbb)i
⫺bi ⫹ DAi ⫹ I(AaBB)i
DAi ⫹ DBi ⫹ I(AaBb)i
bi ⫹ DAi ⫹ I(Aabb)i
ai ⫺ bi ⫹ I(aaBB)i
ai ⫹ DBi ⫹ I(aaBb)i
ai ⫹ bi ⫹ I(aabb)i
Note: See table A1 for parameter definitions.
494 The American Naturalist
Table 2: Relationships among the parameters of the
general (saturated) model (table 1, eq. [1]), the
symmetrical epistatic model of Demuth and Wade,
and the asymmetrical Dobzhansky-Muller model of
Turelli and Orr
General model
ai
bi
DAi
DBi
I(AABB)i
I(AaBB)i
I(aaBB)i
I(AABb)i
I(AaBb)i
I(aaBb)i
I(AAbb)i
I(Aabb)i
I(aabb)i
Demuth and
Wade 2005
Turelli and
Orr 2000
a
b
DA
DB
⫹I
⫺K
⫺I
⫺L
⫹J
⫹L
⫺I
⫹K
⫹I
0
0
0
0
0
0
0
0
h0
h1
0
h1
h2
Nevertheless, the remainder of Demuth and Wade’s
(2005) analysis appears sound. Here I use a general formulation of their multilocus model to find explicit connections to the Turelli-Orr model and clarify the relationships between the composite parameters of quantitative
genetics and the interactions contributing to hybrid breakdown. To further explore a quantitative genetic perspective
on hybrid fitness, they used a statistical model similar to
that of Lynch (1991), except for a difference in scaling due
to use of pH (proportion of loci heterozygous for species
1 and species 2 alleles) instead of Lynch’s hybridity index
vH p 2p H ⫺ 1. Demuth and Wade (following Mather and
Jinks 1982; Lair et al. 1997) relied on tables of coefficients
for specific line crosses, but the general formula analogous
to that of Lynch’s equation (1) is
E(Z) p m ⫹ avS ⫹ dp H ⫹ I aa vS2
⫹ K ad vS p H ⫹ Jdd p H2 … ,
(3)
Note: See table A1 for parameter definitions.
and Orr 1998, 2004; Turelli and Orr 2000; Gavrilets 2003,
2004). To see this another way, consider the basic D-M
model illustrated in figure 1. When the fitnesses of species
1, species 2, and the ancestor all equal 1, the parameters
of Demuth and Wade’s model solve to a p b p I p ⫺1.
However, if the hybrid that is homozygous for both derived
alleles has fitness 0, then a ⫹ b ⫹ I p 0. Thus, equating
Demuth and Wade’s symmetrical table 1 to the asymmetrical D-M model (fig. 1) generates a contradiction.
In attempting to relate their table 1 to the Turelli-Orr
model, Demuth and Wade (2005) also misinterpret Turelli
and Orr’s parameters p1 and p2 as “the proportion of the
hybrid genome homozygous for derived alleles from species 1 or species 2” (pp. 526–527, my italics). In fact, those
parameters were defined as the proportions of an individual genome homozygous for species 1 alleles and species
2 alleles regardless of whether those alleles are derived
(Turelli and Orr 2000, p. 1665). The average fitness (expected breakdown score) of F2 offspring is an average
across all nine two-locus genotypes, not only the four genotypes included in Demuth and Wade’s (2005) equation
(2). The ancestral genotype is not undefined in the D-M
model (cf. Demuth and Wade 2005, p. 527); in the basic
model, the ancestor has fitness equal to that of both daughter species genotypes (fig. 1; Dobzhansky 1937; Coyne and
Orr 1998, 2004; Gavrilets 2003, 2004). Given these disparities between Demuth and Wade’s (2005) quantitative
genetic model and the D-M model, setting their equation
(2) equal to Turelli and Orr’s (2000) equation (2) is not
a valid way to link the Turelli-Orr model formally with
quantitative genetics.
where Z is a phenotype (measure of hybrid dysfunction),
m is the mean dysfunction after an effectively infinite
number of intercross generations (i.e., the expected dysfunction of recombinant inbred lines), a is the composite
additive genetic difference between species 1 and species
2, d is the composite dominance effect, Iaa is the composite additive # additive effect, and Kad and Jdd are the
composite additive # dominance and dominance #
dominance effects from Demuth and Wade’s (2005) table
2. The parameter vS is equal to Lynch’s (1991) source
index,
(
)
1
vS p 2 p1 ⫹ p H ⫺ 1.
2
(4)
The ellipsis in equation (3) is a reminder that interactions
involving more than two loci are possible (Lynch 1991).
The coefficients of equation (3) can be estimated via likelihood or least squares analysis of phenotypic means and
variances in an adequate set of line crosses (parentals, F1,
F2, and backcrosses in both directions; see Lynch 1991;
Lynch and Walsh 1998; Rundle and Whitlock 2001; Demuth and Wade 2005). Use of this general equation (3)
instead of a table of coefficients allows a more complete
analysis of the connection between quantitative genetics
and the Turelli-Orr formulation of the D-M model. Substituting equation (4) for vS in equation (3) and collecting
terms reveals the relationships between Demuth and
Wade’s composite coefficients and the additive, dominance, and epistatic effects of equation (1):
Perspectives on Hybrid Dysfunction 495
1
m p n[I(AABB) ⫹ I(aabb) ⫹ I(AAbb) ⫹ I(aaBB)],
4
{
{
a p n a¯ ⫺ b¯ ⫹
(5a)
}
1
[I(aaBB) ⫺ I(AAbb)] ,
2
(5b)
1
[I(AABb) ⫹ I(Aabb) ⫹ I(AaBB)
2
d p n DA ⫹ DB ⫹
⫹ I(aaBb) ⫺ I(AABB) ⫺ I(aabb)
}
⫺ I(AAbb) ⫺ I(aaBB)] ,
(5c)
1
I aa p n[I(AAbb) ⫹ I(aaBB) ⫺ I(AABB) ⫺ I(aabb)],
4
{
Jdd p n I(AaBb) ⫹
(5d)
1
[I(AABB) ⫹ I(aabb) ⫹ I(AAbb)
4
⫹ I(aaBB)]
⫺
}
1
[I(AABb) ⫹ I(Aabb) ⫹ I(AaBB) ⫹ I(aaBb)] ,
2
(5e)
1
K ad p n[I(AAbb) ⫹ I(AaBB) ⫹ I(aaBb) ⫺ I(Aabb)
2
⫺ I(AABb) ⫺ I(aaBB)].
(5f)
These relationships illustrate that even the coefficients attributed to additive (a) and dominance (d) effects in standard quantitative genetic analyses can be influenced by
interactions unless the epistatic effects are symmetrical
such that they cancel out in equations (5b) and (5c) (see
Demuth and Wade’s [2005] table 1 and Hill 1982 for alternative symmetrical schema). Under the assumptions of
the Turelli-Orr model (table 2), equations (5) become
1
1
m p nI (aabb) p nh 2 ,
4
4
(6a)
a p 0,
(6b)
1
d p n[I (Aabb) ⫹I (aaBb) ⫺I (aabb)]
2
(
)
1
p n h1 ⫺ h2 ,
2
1
1
I aa p n[⫺I(aabb)] p ⫺ nh 2 ,
4
4
{
(
1
K ad p n[I(aaBb) ⫺ I(Aabb)] p 0.
2
The quantitative genetic model of equation (3) can be seen
as a more general statistical model, while the Turelli-Orr
model is a special case with specific interpretations of the
model parameters. If the Turelli-Orr model is correct for
a given case, the estimated additive # additive effect is
caused entirely by interactions between loci that are homozygous for incompatible alleles, and the composite
dominance and dominance # dominance effects are related to the average dominance of incompatibilities, not
single-locus dominance (eqq. [6c], [6e]). All three kinds
of pairwise interactions identified by Turelli and Orr
(2000) influence the expected F2 phenotype, regardless of
which framework is used. Assuming a large number n of
pairwise incompatibilities, expected hybrid breakdown for
F2 hybrids is
(6c)
(6d)
in quantitative genetic terms, and in Turelli and Orr’s
(2000) terms,
}
)
Epistasis in F2 Hybrids
1
1
E(ZFF2 ) p m ⫹ d ⫹ Jdd ,
2
4
1
1
Jdd p n I(AaBb) ⫹ I(aabb) ⫺ [I(Aabb) ⫹ I(aaBb)]
4
2
1
p n h0 ⫺ h1 ⫹ h2 ,
4
Here, the “composite dominance effect” (d) is determined
entirely by interactions between loci, and d is 0 only if
D-M incompatibilities are “additive” in the sense of
h 0 p (1/2)h 1 p (1/4)h 2. Similar relationships can be found
for the composite effect coefficients of Lynch’s (1991)
quantitative genetics model (Fitzpatrick 2008). The differences between equation (3) and Lynch’s (1991) model
are Lynch’s use of a rescaled “hybridity index” (vH p
2p H ⫺ 1) in place of pH and Lynch’s use of the F2 as a
reference population rather than the mean of many recombinant inbred lines (m is the reference population
mean, and the other coefficients represent deviations from
that mean owing to the various genetic effects). These
equations (6) provide a basis from which to revisit two
important conclusions of Demuth and Wade (2005). First,
they conclude that their quantitative genetic description
of mean fitness in F2 hybrids implies a different role for
epistasis than does the expected F2 breakdown score of
Turelli and Orr (2000). Second, they conclude that the two
approaches make different predictions for the genetic basis
of Haldane’s rule.
(
E(ZFF2 ) p n
(7)
)
1
1
1
h2 ⫹ h1 ⫹ h0 .
16
4
4
(8)
(6e)
(6f)
Equation (8) can be obtained from equation (7) using the
substitutions described in equations (6). Thus, the quantitative genetic and D-M models do not make alternative
496 The American Naturalist
predictions; the quantitative genetic description of hybrid
fitness includes the Turelli-Orr model as a special case
derived from specific biological assumptions. Demuth and
Wade’s statement (2005, p. 534) that additive # additive
epistasis (represented by Iaa) does not contribute to F2
hybrid breakdown in the quantitative genetic model is
misleading (it may increase, decrease, or have no effect on
hybrid breakdown). The absence of Iaa from equation (7)
is due to the particular reference population used by the
joint scaling framework. When hybrid dysfunction is described by the Turelli-Orr model with large n, the mean
hybrid breakdown score of the reference population is
m p (1/4)nh 2 p ⫺I aa (eqq. [6a], [6d]), so the mean F2
breakdown is E(ZFF2 ) p ⫺I aa ⫹ (1/2)d ⫹ (1/4)Jdd. This
makes sense when we recall that Demuth and Wade’s m
is the expected hybrid breakdown of recombinant inbred
lines; that is, individuals whose genomes are mosaics of
loci homozygous for species 1 alleles and loci homozygous
for species 2 alleles. These lines are affected only by
homozygous-homozygous interactions. Equation (5d)
shows that additive # additive epistasis contributes to
average F2 hybrid breakdown (Iaa is negative) whenever
I(AABB) ⫹ I(aabb) 1 I(AAbb) ⫹ I(aaBB), that is, when the
average interaction between heterospecific alleles is more
deleterious than the average interaction between homospecific alleles.
Conditions for Haldane’s Rule
An important result from the Turelli-Orr model relates to
the dominance theory of Haldane’s rule. Haldane’s rule is
the observation that hybrid offspring with heterogametic
sex chromosomes (i.e., males in mammals and Drosophila,
females in birds and Lepidoptera) often suffer greater dysfunction (sterility and inviability) than hybrids of the
homogametic sex (Haldane 1922; Coyne and Orr 1989;
Orr 1997). The dominance theory (Muller 1940) explains
Haldane’s rule by proposing that most epistatic incompatibilities are recessive, so that they are not expressed in
F1 hybrids (fig. 1B) unless they involve the sex chromosome, which is effectively homozygous (hemizygous) in
the heterogametic sex (Orr 1997; Turelli and Orr 2000).
When interactions between loci on the X chromosome are
ignored, Turelli and Orr (1995) showed that Haldane’s
rule (E (ZFF1 male) 1 E (ZFF1 female), assuming that male
is the heterogametic sex) results when pairwise incompatibilities are partially recessive (fig. 1B):
h0 1
! .
h1 2
(9)
Based on equation (2), Turelli and Orr (2000) showed that
conditions for Haldane’s rule are slightly more restrictive
when X-X interactions are included:
h0 1 ⫺ gX
1
!
p
.
h1 2 ⫺ gX
2 ⫹ g X (1 ⫺ g X )
(10)
Here, gX is the fraction of the genome (or the fraction of
pairwise incompatibilities) represented by the X chromosome (thus, the right side of eq. [10] is between 1/2
and 1/2.25).
With the same approach (defining a fraction gX of incompatibilities involving the X chromosome), the condition for Haldane’s rule in Demuth and Wade’s (2005)
parameters is E(ZFF1 male) 1 E(ZFF1 female):
m ⫹ (1 ⫺ g X)d ⫹ (1 ⫺ g X)2Jdd ⫹ g X2I aa
Ⳳ [g X a ⫹ g X(1 ⫺ g X)K ad ] 1 m ⫹ d ⫹ Jdd ,
(11)
which reduces to
Ⳳ[a ⫹ (1 ⫺ g X)K ad ] 1 d ⫹ Jdd(2 ⫺ g X) ⫺ g XI aa .
(12)
The term in square brackets is positive when F1 males bear
X chromosomes from species 1 and negative when they
bear X chromosomes from species 2. Averaging over both
reciprocal F1 males makes the left side of equation (12)
equal to 0; therefore, a general condition for Haldane’s
rule is
d ⫹ Jdd(2 ⫺ g X) ⫺ g XI aa ! 0.
(13)
Substituting from equations (6) into equation (12) or (13)
yields equation (10), the condition found by Turelli and
Orr (2000). The condition in equation (13) is more general
than the Turelli-Orr condition because it can be satisfied
through interactions other than those considered in the
Turelli-Orr model. For example, X-linked heterozygote advantage or dominance # dominance epistasis causing high
fitness of double heterozygotes would make F1 females
tend to have higher fitness than F1 males.
Demuth and Wade’s (2005) result (their eq. [13]) differs
from the above for three reasons. First, they consider only
F1 males with X chromosomes from species 1 rather than
both reciprocal F1’s (however, this does not matter for the
Turelli-Orr model, where there is no additive difference
between species). Second, instead of the gX notation used
by Turelli and Orr (2000), they define a separate set of
coefficients to describe the composite genetic effects involving loci on the X chromosome. Finally, they ignore
interactions between loci on the X chromosome, which
are included in the Turelli and Orr (2000) model and
equations (11)–(13). If we assume (as in eq. [11]) that the
Perspectives on Hybrid Dysfunction 497
only difference between X-linked and autosomal incompatibilities is their number n, equations (6) suggest that
under the Turelli-Orr model, the X-linked additive and Xautosomal additive # additive composite effects (x and
Kxd in Demuth and Wade’s eq. [13]) are 0, the X-linked
dominance effect is d X p n X[h 1 ⫺ (1/2)h 2 ], and the Xautosomal dominance # dominance effect is JddX p
2n X[h 0 ⫺ h 1 ⫹ (1/4)h 2 ]. In the latter, the factor of 2 comes
from counting both species 1 X by species 2 autosome
and species 2 X by species 1 autosome interactions (Turelli
and Orr 1995; Demuth and Wade 2005). Substituting these
equations into Demuth and Wade’s (2005) condition for
Haldane’s rule (their eq. [13]) recovers the original condition of Turelli and Orr (1995), where they, too, ignored
X-X interactions: h 0 /h 1 ! 1/2. The same can be obtained
from Lynch’s (1991) earlier formulation of the quantitative
genetic model (Fitzpatrick 2008).
Conclusions
The multilocus D-M model of hybrid dysfunction, as described by Turelli and Orr’s (2000) equation (2), is entirely
compatible with both Lynch’s (1991) and Demuth and
Wade’s (2005) quantitative genetic descriptions of hybrid
breakdown caused by many pairwise incompatibilities (eq.
[3]). The unified theory sought by Demuth and Wade
(2005) is clarified in equations (5) and (6). Their difficulty
in reconciling the quantitative genetic perspective with the
Turelli-Orr model resulted from misinterpreting the basic
two-locus model (cf. my fig. 1 with their table 1), misinterpreting Turelli and Orr’s (2000) parameters (pi’s), and
failing to write a general equation for the joint scaling
model (eq. [3]).
The Turelli-Orr model is derived from specific assumptions about the genetics of hybrid dysfunction, that is, that
fitness is a function of the sum of many asymmetrical
pairwise interactions. My analysis shows that it is included
as a special case of the general statistical model of quantitative genetics, and its parameters can be estimated from
an adequate set of line-cross analyses (Lynch 1991; Lynch
and Walsh 1998; Demuth and Wade 2005). The TurelliOrr model lends specific meaning to the “composite effects” of the quantitative genetic model (eqq. [6]). Notably,
the composite dominance effect need have nothing to do
with single-locus dominance (eq. [6c]). My results support
Lynch’s (1991) and Demuth and Wade’s (2005) conclusions that the well-developed statistical theory of quantitative genetics can be fruitfully applied to the genetics of
hybrid dysfunction and that unified analyses of withinand between-species variation may yield important insights into the roles of dominance and epistasis in evolution.
Finally, the theory reviewed above relies on certain simplifying assumptions that merit further investigation. Perhaps of greatest concern are the assumptions that hybrid
fitness is affected by a large number of pairwise incompatibilities and that the loci involved are not physically
linked. Violation of these assumptions makes quantities
like p1p2 and vS2 unreliable indicators of the probabilities
of pairwise interactions. Therefore, additional theoretical
work is needed, and traditional line-cross analyses should
be accompanied by quantitative trait locus analyses designed to evaluate the number of incompatibilities, the
distribution of their effects, and the possibility of more
complex interactions (i.e., those involving more than two
loci; see Turelli and Orr’s [2000] appendix).
Acknowledgments
I thank N. Barton, J. Demuth, S. Gavrilets, D. Simberloff,
M. Turelli, M. Wade, and two anonymous reviewers for
their helpful comments. My financial support comes from
the University of Tennessee and the National Science
Foundation (grant DEB-0516475).
APPENDIX
Definitions of Parameters
Table A1: Definitions of parameters used in tables 1 and 2 and equation (1)
Parameter(s)
Ai, Bi
A, a
B, b
(AABB)i
ai, bi
DAi, DBi
I(AABB)i
Explanation
The ith pair of interacting loci
The alleles of an A locus
The alleles of a B locus
One of the nine genotypes for the ith pair of interacting loci
The additive phenotypic effects associated with loci Ai and Bi
The dominance effects associated with loci Ai and Bi
The interaction effect of genotypes AA and BB at loci Ai and Bi
498 The American Naturalist
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Associate Editor: Armando Caballero
Editor: Michael C. Whitlock