Copyright © 2005 by the Genetics Society of America
DOI: 10.1534/genetics.104.037754
Models of General Frequency-Dependent Selection and Mating-Interaction
Effects and the Analysis of Selection Patterns in
Drosophila Inversion Polymorphisms
José M. Álvarez-Castro*,†,1 and Gonzalo Alvarez*
*Department of Genetics, University of Santiago de Compostela, E-15782 Santiago de Compostela, Spain and
†
Section of Evolutionary Biology, Department of Biology II, Ludwig-Maximilians
University of Munich, D-82152 Planegg-Martinsried, Germany
Manuscript received October 20, 2004
Accepted for publication April 11, 2005
ABSTRACT
We investigate mechanisms of balancing selection by extending two deterministic models of selection in a
one-locus two-allele genetic system to allow for frequency-dependent fitnesses. Specifically we extend models
of constant selection to allow for general frequency-dependent fitness functions for sex-dependent viabilities
and multiplicative fertilities, while non-multiplicative mating-dependent components remain constant. We
compute protected polymorphism conditions that take the form of harmonic means involving both the
frequency- and the mating-dependent parameters. This allows for a direct comparison of the equilibrium
properties of the frequency-dependent models with those of the constant models and for an analysis of
equilibrium of the general model of constant fertility. We then apply the theory to analyze the maintenance
of inversion polymorphisms in Drosophila subobscura and D. pseudoobscura, for which data on empirical fitness
component estimates are available in the literature. Regression on fitness estimates obtained at different
starting frequencies enables us to implement explicit fitness functions in the models and therefore to perform
complete studies of equilibrium and stability for particular sets of data. The results point to frequency
dependence of fitness components as the main mechanism responsible for the maintenance of the inversion
polymorphisms considered, particularly in relation to heterosis, although we also discuss the contribution
of other selection mechanisms.
B
ECAUSE selection is known to maintain genetic polymorphisms in natural populations, extensive theoretical and experimental research on mechanisms of
balancing selection has been carried out from the first
half of the last century (see, e.g., Dobzhansky 1970). Nevertheless the relative importance of these mechanisms in
maintaining genetic variation remains an open question
but, interestingly, theoretical approaches reveal that the
extent to which balancing selection can be responsible
for standing genetic variation may indeed depend on the
specific balancing selection mechanism considered (see
Turelli and Barton 2004 and references therein). Finding out which mechanisms of selection are actually affecting genetic polymorphisms in natural populations is therefore a central topic of population genetics. In this article
we address this subject by developing and studying models
involving, at the same time, several mechanisms susceptible to leading to balancing selection, such as stage-,
sex-, frequency-, and mating-dependent fitnesses, hence
enabling a subsequent analysis of the different selection
1
Corresponding author: Section of Evolutionary Biology, Department
of Biology II, Ludwig-Maximilians University of Munich, Grosshaderner Str. 2, D-82152 Planegg-Martinsried, Germany.
E-mail: [email protected]
Genetics 170: 1167–1179 ( July 2005)
mechanisms in maintaining genetic polymorphisms for
which data on fitness component estimates are available.
Fertility differences as properties of the mating pairs
can yield complex modes of selection, thus contributing
to the maintenance of genetic polymorphisms. Equilibrium properties of models of one-locus, two-allele fertility selection were analytically investigated by assuming
several particular cases. Penrose (1949) proved that
additive fertility is analogous to the classical model of
sex-independent viability. Bodmer (1965) obtained sufficient conditions for the maintenance of the polymorphism for the multiplicative case and found that multiplicative fertility is formally analogous to sex-dependent
viability, also referred to as two-sex viability selection, which
allowed him to apply previous results for sex-dependent
viability to the multiplicative fertility model. Sex-dependent viability, first considered by Haldane (1924, 1926),
was the first genetic system under selection found to have
three simultaneous internal equilibria (Owen 1953). Further results concerning equilibrium and stability of these
equivalent models were achieved by separately analyzing
all possible patterns of selection for the two sexes (Mérat
1969; Mandel 1971; Kidwell et al. 1977).
The study of the model of one-locus fertility can be
reduced without loss of generality to the case in which
sex symmetry is assumed (Hadeler and Liberman 1975;
1168
J. M. Álvarez-Castro and G. Alvarez
Feldman et al. 1983). Hadeler and Liberman (1975)
restricted their study to a particular case of the twoallele genetic system, with more symmetries in the fitness
matrix, and they prove that up to five internal equilibria
can exist. Feldman et al. (1983) were able to perform a
complete study of equilibrium and stability by assuming
further symmetries, considering in particular that the
fitness of each mating pair depends only on the number
of heterozygotes involved. For this case, the number of
possible internal equilibria reduces to three, with up to
two of them stable, and the parameter space divides
into six regions with different equilibrium properties.
The authors analyze the six cases in relation to analogous
situations of the multiplicative-fertility model, although no
equivalent multiplicative formulation is found for most
of the cases. Several particular instances of fertility with
dominance were also considered (Nagylaki 1992, pp.
112–116, and references therein).
Frequency-dependent selection is known to be an important balancing mechanism (see, e.g., Ayala and
Campbell 1974; Clarke and Partridge 1988). Some
equilibrium studies followed a phenomenological approach by focusing on situations involving explicit or
more or less constrained types of fitness functions depending on gene or genotype frequencies (e.g., Wright
1955; Clarke and O’Donald 1964; Sacks 1967; Anderson 1969; Raveh and Ritte 1976; Anxolabéhèrre and
Périquet 1981; Curtsinger 1984; Nagylaki 1992, pp.
19 and 20). In other cases variable selection appears as a
consequence of situations in which constant parameters
for specific mechanisms are initially considered. Perhaps the most general and best known of these situations
is the pairwise-interaction model of intraspecific competition first considered by Schutz et al. (1968). Cockerham et al. (1972) performed a study of equilibrium
of the general formulation of this model and inspected
some particular cases. Mueller (1988) obtained conditions of protected polymorphism for a model with specific viability and fecundity fitness functions derived for
competition for food. Asmussen and Basnayake (1990)
highlight the potential of the frequency-dependent
model of competition in the maintenance of variation.
Variable selection caused by competition remains a focus of current research (see Bürger 2005 and references therein). Fertility selection is known to be equivalent to linear variable selection (Prout 1965, 1969,
1971; Feldman et al. 1969). Also constant parameters of
multiple-niche selection (Levene 1953) and supergene
selection (Alvarez and Zapata 1997) produce frequencydependent total fitness values. Equilibrium studies for
models including both density- and frequency-dependent selection have also been carried out (Nagylaki 1979;
Slatkin 1979; Asmussen 1983a,b; Mueller 1988).
The importance of frequency-dependent selection as
a balancing selection mechanism becomes more evident
in models considering drift (Robertson 1962) and also
when contextualized in the multiallele framework. Su-
periority of the heterozygote is neither necessary (Kimura
1956) nor sufficient (Mandel 1959) for the maintenance of more than two alleles in a population, and
constant viability selection itself does not seem to be very
likely to explain multiple-allele polymorphisms commonly
found in natural populations (Lewontin et al. 1978; Mandel 1980; Karlin 1981). Although fertility selection is
more efficient than viability selection in maintaining
two-allele genetic polymorphisms, numerical analyses
show that this result does not extend to the multiallele
framework (Clark and Feldman 1986). Rare-genotype
frequency-dependent selection on the other hand can
lead to maintenance of multiallele polymorphisms (see,
e.g., Anderson 1969). Our present study is, however, restricted to the one-locus, two-allele framework.
After their discovery, Drosophila inversion polymorphisms attracted scientific attention as likely neutral markers (see, e.g., Powell 1997, p. 73), but their frequencies
were found to be subjected to strong selective forces (Dobzhansky 1943, 1948; Dubinin and Tiniakov 1946; Dobzhansky et al. 1966; Rodrı́guez-Trelles et al. 1996), and,
moreover, epistasis among the loci involved was found
to be an essential factor for new inversions to rise in
frequency in natural populations (Charlesworth and
Charlesworth 1973). Coadaptation of inversions within
subpopulations first pointed to superiority of the heterokaryotype as a simple and reasonable explanation of the
maintenance of polymorphism (see Dobzhansky 1970).
However, several chromosomal arrangements coexist in
Drosophila natural populations (see Sperlich and Pfriem
1986; Krimbas and Powell 1992; Powell 1997), whereas
superiority of the heterozygote (or equivalently heterokaryotype) is not likely to explain the maintenance of
multiple-allele genetic systems (see above). In addition,
the absence of excess of heterokaryotypes at equilibrium, expected under heterokaryotype superiority, suggested that important selection forces act at different
stages of the life cycle (Sperlich and Pfriem 1986).
Different researchers carried out experimental work on
other selective mechanisms such as sex-dependent viability (Druger 1966; Druger and Nickerson 1972), frequency-dependent viability (Kojima and Tobari 1969;
Anderson et al. 1986), and frequency-dependent fertility
(Anderson and Watanabe 1974) and more particularly
rare-male advantage (e.g., Petit and Ehrman 1969;
Anderson and Brown 1984; Santos et al. 1986; Salceda
and Anderson 1988), multiple-niche selection (e.g.,
Fernández Iriarte and Hasson 2000), and supergene
selection or the effect of recombination in homokaryotypes (Charlesworth and Charlesworth 1975; Wasserman and Koepfer 1975). The explanation of the
maintenance of inversion polymorphisms in Drosophila
was also attempted by using a demographic approach
(Anderson and Watanabe 1997). Frequencies of Drosophila inversion polymorphisms have been carefully
measured in nature for decades (Anderson et al. 1991;
Sole et al. 2002) and molecular data have recently been
Mechanisms of Balancing Selection
1169
TABLE 1
Variables and fitnesses of the frequency-dependent multiplicative-fertility model for the three genotypes
Adult frequencies at generation t
Female fertilities
Male fertilities
Zygote frequencies at generation t ⫹ 1
Viabilities
Adult frequencies at generation t ⫹ 1
A1 A1
A1 A 2
A2A2
X
f 1(X, Z )
m 1(X, Z )
X z⬘y
v 1(X z⬘y , Z z⬘y)
X⬘
Y
f 2(X, Z )
m 2(X, Z )
Y z⬘y
v 2(X z⬘y , Z z⬘y)
Y⬘
Z
f 3(X, Z )
m 3(X, Z )
Z z⬘y
v 3(X z⬘y , Z z⬘y)
Z⬘
used to study their role in adaptation (Hoffmann et al.
2004 and references therein) and speciation (e.g., Noor
et al. 2001).
In the first part of this article we develop and study
two deterministic models of general variable selection
by following a phenomenological approach that can
involve different fitness functions for the early and late
stages, without a priori restricting the recurrence equations to particular types of functions. These models are
extensions of the selection models of Bodmer (1965)
and Prout (1969) to allow for variable selection. In the
next section, after some brief technical comments, we
apply the first model to study mechanisms of balancing
selection in two Drosophila inversion polymorphisms
for which data on fitness component estimates are available in the literature. Finally we discuss the implications
of both the theoretical results and the analysis of the
inversion polymorphisms. The theoretical framework
considered in this article, however, is not at all restricted
to the inversion polymorphism scenario.
MODELS AND THEORETICAL RESULTS
Model of early and late variable selection: Here we
consider a model with frequency-dependent viabilities
and sex- and frequency-dependent multiplicative fertilities ruling the frequencies, X, Y, and Z, of the three
genotypes, A 1 A 1, A 1A 2 , and A 2 A 2 , of a one-locus twoallele genetic system. Mating is at random and generations do not overlap. The notation for this model is set
forth in Table 1. The female and male fertility fitness
functions for the three possible genotypes, fi (X, Z ) and
m i (X, Z), i ⫽ 1, 2, 3, depend directly on independent
frequencies of the homozygous genotypes, X and Z, in
the adults, and therefore they also depend implicitly on
the heterozygous frequency Y ⫽ 1 ⫺ X ⫺ Z. The scope
of the model includes almost any kind of fertility fitness
functions (with only a minor restriction, which is mentioned below). This framework automatically encompasses variable fitness functions depending on the gene
frequencies p ⫽ X ⫹ 1⁄2Y and q ⫽ Z ⫹ 1⁄2Y since they
depend indirectly on the genotype frequencies. The
same logic holds for the sex-independent viability fitness
functions, vi (X zy , Z zy ), i ⫽ 1, 2, 3, except for the fact
that they depend in a natural way on zygote frequencies
(denoted by the subscript zy) instead of those of the
adults. To take this fact into account when building the
recurrence equations of the model, we need to consider
the expressions providing the zygote frequencies of the
three genotypes at one generation, X⬘zy , Y⬘zy , and Z ⬘zy ,
from the adult frequencies at the previous generation,
X, Y, and Z (where the primes denote “one generation
after”). These expressions are
X ⬘zy ⫽
1
1
1
[( f1(X, Z)X ⫹ f 2(X, Z)Y )(m 1(X, Z)X ⫹ m 2(X, Z)Y)],
w zy
2
2
Y ⬘zy ⫽
1
1
1
[( f 1(X, Z)X ⫹ f 2(X, Z)Y)( m 2(X, Z )Y ⫹ m 3(X, Z)Z)
w zy
2
2
1
1
⫹ ( f 2(X, Z)Y ⫹ f 3(X, Z)Z)(m 1(X, Z)X ⫹ m 2(X, Z)Y)] ,
2
2
Z ⬘zy ⫽
1 1
1
[( f 2(X, Z)Y ⫹ f 3(X, Z)Z)( m 2(X, Z)Y ⫹ m 3(X, Z)Z)] ,
w zy 2
2
(1)
where w zy ⫽ ( f1(X, Z)X ⫹ f 2(X, Z)Y ⫹ f3(X, Z)Z)(m1(X,
Z)X ⫹ m 2(X, Z)Y ⫹ m3(X, Z)Z), such that the frequencies X⬘zy , Y⬘zy , and Z⬘zy add to unity. Now we are in a
position to express the recurrence equations of the
model as
X⬘ ⫽
v 1(X z⬘y , Z z⬘y)
1
1
[( f1(X , Z)X ⫹ f 2(X , Z)Y )(m 1(X , Z)X ⫹ m 2(X , Z )Y)] ,
w
2
2
Y⬘ ⫽
v 2(X z⬘y , Z z⬘y)
1
1
[( f1(X , Z)X ⫹ f 2(X , Z )Y)( m 2(X , Z)Y ⫹ m 3(X , Z)Z)
w
2
2
1
1
⫹ ( f 2(X , Z)Y ⫹ f 3(X , Z)Z)(m 1(X , Z)X ⫹ m 2(X , Z)Y)] ,
2
2
Z⬘ ⫽
v 3(X z⬘y , Z z⬘y) 1
1
[( f 2(X , Z)Y ⫹ f 3(X , Z)Z)( m 2(X , Z)Y ⫹ m 3(X , Z)Z)] ,
w
2
2
(2)
where the zygote frequencies X ⬘zy and Z⬘zy are given by
(1), and w is such that X⬘ ⫹ Y⬘ ⫹ Z⬘ ⫽ 1. Note that in
these recurrences the frequency-dependent viability
functions v1(X⬘zy , Z⬘zy), v 2(X⬘zy , Z⬘zy ), and v 3(X⬘zy , Z ⬘zy ) depend themselves on the frequency-dependent multiplicative fertilities by means of (1).
Since Y ⫽ 1 ⫺ X ⫺ Z, (2) may be expressed in terms
of the frequencies X and Z alone, and we can refer
to them as X⬘(X, Z), Y⬘(X, Z), and Z⬘(X, Z). These
1170
J. M. Álvarez-Castro and G. Alvarez
expressions entail an extension of the constant-multiplicative fertility selection model analyzed by Bodmer (1965)
to the general variable-fitness framework. Even for the
constant-fitness multiplicative-fertility model it is not possible to obtain a general expression for the equilibrium
points (Bodmer 1965; Mérat 1969; Mandel 1971; Kidwell et al. 1977). Therefore, to gain some insight into
the equilibrium properties of our model of variable
selection, we focused on the analysis of the stability of
the trivial equilibria, (X, Z) ⫽ (1, 0) and (X, Z) ⫽ (0,
1). To this end we must assume that the fitness functions
are differentiable at the fixation points. In regard to
generality this is the only restriction we impose on the
frequency-dependent fitness functions to be considered. The capacity of some computer packages to deal
with abstract expressions (e.g., Mathematica, Wolfram
1996) allows us to build the Jacobian matrix for recurrence equations (2)—the matrix of the partial derivatives of the independent functions X⬘(X, Z) and Z⬘(X,
Z). Since these recurrence equations involve nonspecified viability functions depending on nonspecified fertility functions, the Jacobian matrix is cumbersome, but
it becomes much more treatable when evaluated at the
trivial equilibria. The nonzero eigenvalues of the two
resulting matrices at the two trivial equilibria are
␭1 ⫽
␭2 ⫽
1 v 2(1, 0)( f 1(1, 0)m 2(1, 0) ⫹ f 2(1, 0)m 1(1, 0))
,
2
v 1(1, 0)f 1(1, 0)m 1(1, 0)
1 v 2(0, 1)(f 3(0, 1)m 2(0, 1) ⫹ f 2(0, 1)m 3(0, 1))
,
2
v 3(0, 1)f 3(0, 1)m 3(0, 1)
(3)
for the fixation points of A1 and A 2 , respectively.
The protected polymorphism conditions are now given
by the inequalities that the absolute values of these eigenvalues are larger than one (Prout 1968). For fitness values
fulfilling these conditions, the maintenance of the polymorphism in the genetic system is guaranteed. Assuming,
without loss of generality—other than that the heterozygotes are not lethal at any frequency—that every heterozygous fitness component equals one in (3), the protected polymorphism conditions can be expressed in
terms of harmonic means of products of fitnesses affecting females and males as
1
⬍ 1,
⁄2(1/(v 1(1, 0)m 1(1, 0)) ⫹ 1/(v 1(1, 0)f 1(1, 0)))
1
1
⬍ 1.
⁄2(1/(v 3(0, 1)m 3(0, 1)) ⫹ 1/(v 3(0, 1)f 3(0, 1)))
(4)
1
These expressions are influenced by the fitness functions only by means of the values of these functions at
the fixation point of the genotype they affect, which
allows for direct biological interpretation. Indeed, these
values play the same role as the constant fitnesses do under
protected polymorphism conditions of the constant
multiplicative fertility model (Bodmer 1965; Kidwell
et al. 1977). Hence for every possible fitness function,
we can consider the value it takes at the fixation point
of the genotype it affects. This leads to a parameter
space for the variable model in which the parameter
values are equivalent to the fitnesses of the constant
case, and thus the regions of protected polymorphism
are the same in both situations. In the regions of no
protected polymorphism the variable selection can nevertheless lead to internal equilibria in more cases than
the constant models because of the flexibility of the fitness
functions to allow for complex situations.
From inequalities (4) it is straightforward to derive
protected polymorphism conditions for particular models in which specific fitness functions are involved. One
of the cases with real fitness estimates analyzed in the
next section of this article involves constant viabilities,
constant female fecundities, and gene-frequency-dependent linear and quadratic fitness functions for male sexual
selection, m 1(X, Z) ⫽ m 1(p) ⫽ a ⫺ bp for the genotype
A1 A1 and m3(X, Z) ⫽ m3(p ) ⫽ d ⫹ eq ⫹ gq 2 for A 2 A 2. By
just substituting these assumptions in (4), the protected
polymorphism conditions for this particular case are
1
⬍ 1,
⁄2(1/(v 1(a ⫺ b)) ⫹ 1/v 1 f1)
1
1
⬍ 1.
⁄2(1/(v 3(d ⫹ e ⫹ g)) ⫹ 1/v 3 f 3)
1
(5)
Since multiplicative fertility and two-sex viability are
formally analogous selective forces (Bodmer 1965; Kidwell et al. 1977), recurrences and inequalities given in
(2) and (4) can be applied to a frequency-dependent
two-sex viability framework. In this different context X,
Y, and Z are zygote frequencies, f i (X, Z ) and m i (X, Z ),
i ⫽ 1, 2, 3 are female and male viability fitness functions,
and the former sex-independent viabilities vi (X zy , Z zy),
i ⫽ 1, 2, 3, must be just ignored. We nonetheless directly
consider sex- and frequency-dependent viabilities in a
more general model below. The protected polymorphism conditions for the simplest case of sex-independent viability variable selection can be obtained as v1(1,
0) ⬍ 1, v3(0, 1) ⬍ 1, by just ignoring the sex-dependent
components f i (X, Z) and m i(X, Z ), i ⫽ 1, 3, in (4).
This case was already addressed by Asmussen and Basnayake (1990), and we resume it in the next section
of this article.
Model of variable selection and mating-interaction
effects: Now we analyze a more general model with
sex-, frequency-, and mating-dependent selection. We
consider both viabilities and multiplicative fertilities to
be sex and frequency dependent and also include the
contribution of constant mating-dependent effects—
nonmultiplicative fertility and nonrandom mating. This
model can be considered as the extension of the general
model of selection described by Prout (1969) to a variable-fitness framework insofar as the mating-indepen-
Mechanisms of Balancing Selection
1171
TABLE 2
Variables and fitnesses of the frequency-dependent and mating-interaction model for the three genotypes
Zygote frequencies at generation t
Female viabilities
Male viabilities
Adult female frequencies at generation t
Adult male frequencies at generation t
Female multiplicative fertilities
Male multiplicative fertilities
A1A1
A1A2
A2A2
X zy ⫽ G zy1
v f1(G zy1, G zy3)
v m1(G zy1, G zy3)
X f ⫽ G f1
X m ⫽ G m1
f 1(G f1, G f3)
m 1(G m1, G m3)
Y zy ⫽ G zy2
v f2(G zy1, G zy3)
v m2(G zy1, G zy3)
Y f ⫽ G f2
Y m ⫽ G m2
f 2(G f1, G f3)
m 2(G m1, G m3)
Z zy ⫽ G zy3
v f3(G zy1, G zy3)
v m3(G zy1, G zy3)
Z f ⫽ G f3
Z m ⫽ G m3
f 3(G f1, G f3)
m 3(G m1, G m3)
Mating-interaction parameters
Males
Females
A 1A 1
A1A2
A2A2
Zygote frequencies at generation t ⫹ 1
dent parameters are concerned. The notation for these
parameters is very similar to the one set forth in Table
1, although setting new genotype labels (see Table 2)
allows simplification of the formulas. The mating-interaction matrix A ⫽ (a ij), on the other hand, gathers
together the (nonvariable) parameters accounting for
nonmultiplicative fertility of the mating pairs and nonrandom mating. Each parameter in this matrix is the
product of these two effects on one particular mating
pair. Assuming nonoverlapping generations the three
recurrence equations describing the changes in zygote
frequencies over generations are, for l ⫽ 1, 2, 3,
G⬘zyl ⫽
冢兺G
3
zy i
Gzy j v fi(Gzy1 , G zy3)v mj(G zy1 , G zy3)
i,j ⫽1
冣
⫻ fi (G f 1 , G f 3)m j (G m1, G m3)a ij k ijl /w,
(6)
where G zy1 ⫽ X zy , G zy2 ⫽ Y zy , and G zy3 ⫽ Z zy are the
zygote frequencies at generation t. The primes denote
generation t ⫹ 1, and the subscripts f and m denote
female and male adults (in contrast to zy, zygote). G fl ⫽
Gzyl v fl(Gzy1, Gzy3)/w f , l ⫽ 1, 2, 3, with w f such that
兺 l3⫽1G fl ⫽ 1, provides the female adult frequencies at
generation t (for the multiplicative fertilities to depend
on) and analogous expressions hold for males. K ⫽ (kijl)
is the so-called Mendelian operator, given by
⎛ 1 1 ⁄2 0 ⎞
⎛0
⎟
⎜1 1
⎜1
K 1 ⫽ (k i j 1) ⫽ ⎜ ⁄2 ⁄4 0⎟ , K 2 ⫽ (k i j 2) ⫽ ⎜ ⁄2
⎝ 0 0 0⎠
⎝1
⁄2 1 ⎞
⎛0 0 0 ⎞
⁄2 1⁄2⎟ , K 3 ⫽ (k i j 3) ⫽ ⎜0 1⁄4 1⁄2⎟ ,
⎟
⎟
⎜
1
⎝ 0 1 ⁄2 1 ⎠
⁄2 0 ⎠
1
1
and w is such that 兺 l3⫽1G⬘zyl ⫽ 1.
We assume, like we did in the previous model, that
the fitness functions are differentiable at the fixation
points. This enables us to perform a study of stability
of the fixation points for recurrence equations (6) by
A 1A 1
A 1A 2
A 2A 2
a 11
a 21
a 31
a 12
a 22
a 32
a 13
a 23
a 33
X z⬘y ⫽ G z⬘y1
Y z⬘y ⫽ G z⬘y2
Z z⬘y ⫽ G z⬘y3
means of the Jacobian matrix and therefore to obtain
sufficient conditions for the maintenance of the genetic
polymorphism in the genetic system. Letting v f2(Gzy1,
Gzy3) ⫽ vm2(Gzy1, Gzy3) ⫽ f2(G f1, G f3) ⫽ m 2(G m1, Gm3) ⫽ 1,
these protected polymorphism conditions can be drawn
in terms of harmonic means as
1
⬍ 1,
⁄2(1/v m1(1, 0)m 1(1, 0)(a 11/a 12) ⫹ 1/v f1(1, 0)f 1(1, 0)(a 11/a 21))
1
1
⬍ 1.
⁄2(1/v m3(0, 1)m 3(0, 1)(a 33/a 32) ⫹ 1/v f3(0, 1)f 3(0, 1)(a 33/a 23))
1
(7)
The protected polymorphism conditions of the model
analyzed in the previous section [inequalities (4)] may
be considered as a particular case of these expressions,
in spite of the fact that the two models focus on different
stages of the life cycle. As in the previous model, the fitness functions now affect the protected polymorphism
conditions (7) only by means of the values they take at
the fixation point of the genotype they affect. Likewise,
not every mating pair’s fertility parameter plays a role
in these inequalities. The fertility factor affecting one sex
at one fixation point is in particular the fertility parameter
of the only mating pair that is present at this fixation
point, relative to the fertility of the mating pair that
differs from the previous one in one allele substitution
made in the sex in question.
Within inequalities (7) we also provide the protected
polymorphism conditions for the constant fitness case considered by Prout (1969)—in the context of the estimation
of fitness component values—in which constant fitness
values substitute the fitness functions. More particularly,
the general model of constant fertility is another interesting case we can analyze from our model of variable
1172
J. M. Álvarez-Castro and G. Alvarez
TABLE 3
Fitness components of karyotypes of Drosophila subobscura and D. pseudoobscura for different starting
frequencies, p, of the “standard” arrangement (OST and ST, respectively)
D. subobscura
Fitness component
Viability
OST/OST
p
0.125
0.25
0.375
0.50
0.625
0.75
0.875
1.10 ⫾ 0.03
Female fecundity
0.50
0.96 ⫾ 0.08
Male sexual selection
0.05
0.30
0.50
0.70
0.95
1.27 ⫾ 0.18
1.02 ⫾ 0.05
0.74 ⫾ 0.13
OST/O3⫹4⫹7
D. pseudoobscura
O3⫹4⫹7/O3⫹4⫹7
ST/ST
0.95 ⫾ 0.03 a
1
0.90 ⫾ 0.06 a
1
1
1
0.92 ⫾ 0.02
0.86 ⫾ 0.08
1.11 ⫾ 0.04
b
CH/CH
0.09 c
0.09
0.13
0.10
0.15
0.16
0.10
0.11
1.11 ⫾ 0.06
1
0.98 ⫾ 0.06 d
⫾
⫾
⫾
⫾
1
1
1
1
1
0.73
0.21
0.26
0.88
1.58
1.14
0.83
0.78
3.69
1.57
0.51
2.57
⫾
⫾
⫾
⫾
0.55
0.20
0.07
0.93
0.71
0.81
1.01
0.96
⫾
⫾
⫾
⫾
1
1
1
1
1
1
1
1.33 ⫾ 0.30
1
ST/CH
0.75 ⫾ 0.17
⫾
⫾
⫾
⫾
0.12 e
0.05
0.06
0.19
a
Tarrı́o (1993).
Santos et al. (1986).
c
Anderson et al. (1986).
d
Moos (1955).
e
Anderson and Brown (1984).
b
selection. This case considers the effect of the matrix
A alone and, if random mating is assumed, the parameters aij account for only fertility. From inequalities (7),
the protected polymorphism conditions for the general
model of constant fertility turn out to be simply
⁄2(a 12 ⫹ a 21) ⬎ a 11,
1
⁄2(a 23 ⫹ a 32) ⬎ a 33 .
1
(8)
It is known that only the means of the reciprocal matings, like the ones in the left-hand sides of these expressions, are important to describe the dynamics of the
model, and therefore the study of equilibrium reduces
to assuming that the matrix A is symmetric (Hadeler
and Liberman 1975; Feldman et al. 1983). Thus, assuming sex symmetry, as this case is biologically referred to,
the number of variables of the parameter space reduces
from nine to six and inequalities (8) may be written as
a 12 ⬎ a 11, a 23 ⬎ a 33 .
(9)
These expressions are satisfied by one-quarter of the
parameter space. Hadeler and Liberman (1975) obtained the condition for the instability of the fixation
points for the four parameters in the particular case in
which a 11 ⫽ a 33 and a 12 ⫽ a 23 . Given their assumptions
this condition can be obtained from inequalities (9) as
only the first of them, since both become equivalent. Thus
the protected polymorphism now covers one-half of the
parameter space. Feldman et al. (1983) were able to perform a complete study of equilibrium by further assuming that a 11 ⫽ a 13 . In this case the parameter space
reduces to three dimensions, the protected polymor-
phism conditions are still the same as in the previous case,
and they still cover one-half of the parameter space. The
authors describe the equilibrium and stability properties
of this model by dividing the parameter space into six
equally probable situations, one of which shows one
internal stable equilibrium without fulfilling the protected polymorphism conditions. For this case, thus, there
is no protected polymorphism in one-quarter of the situations in which internal stable equilibrium points exist.
ANALYSIS OF VARIABLE SELECTION PATTERNS
The OST and O3⫹4⫹7 arrangements of Drosophila subobscura and ST and CH arrangements of the third chromosome of D. pseudoobscura were extensively studied as
models to analyze the maintenance of genetic polymorphisms by natural selection. Table 3 summarizes the
fitness estimates we obtained from the literature to analyze patterns of frequency-dependent selection in both
species. Since these data include neither sex-dependent
viabilities nor mating-interaction parameters, the early
and late variable selection model we first considered in
this article is adequate for the analysis. Its recurrence
equations (2) on adult frequencies allow us to focus
directly on frequency-dependent fertilities—also depending on adult frequencies, which are known in both
species. The notation in Table 1 holds in this analysis
except for the labels of the arrangements that now replace the A1 and A2 alleles.
To find out what patterns of selection best fit the raw
Mechanisms of Balancing Selection
1173
TABLE 4
Adjusted coefficient of determination for different regressions on the frequency-dependent fitness
estimates of Drosophila subobscura and D. pseudoobscura summarized in Table 3
Regression
Fitness component
Variables
Viability
Gene frequencies
Sexual selection
Gene frequencies
Genotype frequencies
D. subobscura
D. pseudoobscura
Function
OST/OST
O 3⫹4⫹7/O3⫹4⫹7
ST/ST
CH/CH
Linear
Quadratic
Linear
Quadratic
Linear
Hyperbolic
—
—
0.66
0.55
0.65
0.55
—
—
0.29
0.71
0.17
0.59
0.50
0.59
Negative
0.94
Negative
Negative
Negative
0.82
Negative
0.99
Negative
Negative
Coefficients of determination for the regressions we have chosen for the study of equilibrium are shown in
italics.
frequency-dependent fitness estimates in Table 3, we
performed regressions of the data on fitness functions
dependent on gene or genotype frequencies as considered in the literature (Anderson 1969; Anxolabéhère
and Périquet 1981; Nagylaki 1992, pp. 19 and 20).
Specifically we regressed the homokaryotype fitnesses
on linear, quadratic, and hyperbolic functions. All of these
functions are linear in the parameters but have different
numbers of parameters to estimate, and therefore we
used the adjusted coefficient of determination (shown
in Table 4) to compare the fit to the data (Sokal and
Rohlf 1995, p. 654). Since the existence of replicates in
D. subobscura sexual selection estimates (see Figures 1
and 2) allows us to test the fit of the functions by means
of the lack of fit test, we also performed regressions on
a nonlinear hyperbola considered by Anxolabéhère
(1980) but it did not improve the fit to the data (results
not shown).
Once we assume the action of explicit fitness functions, inequalities (4) enable us to check if the protected
polymorphism conditions of the system are fulfilled or
not. To determine the weight that the different fitness
components have in the maintenance of the polymorphism, we first apply the expressions to the separate
action of the variable components and then to the
model including every available fitness component. Because the protected polymorphism conditions are not
necessary conditions for the maintenance of polymorphisms, and the stability of one or both fixation points
does not completely impede the maintenance of the
polymorphism, we carried out further analyses to search
for internal stable equilibrium points. For the variable
viability selection component with all fitnesses relative
to the one of the heterozygote, the equilibrium points
are given by the solutions of
p̂ ⫽
1 ⫺ v 3(p̂)
.
2 ⫺ v 1(p̂) ⫺ v 3(p̂)
(10)
The stability of the equilibria can be inspected through
the computation of the eigenvalue of the system—one
equilibrium point is stable when the absolute value of
the eigenvalue is smaller than one (e.g., Roughgarden
1979, p. 576)—which for this case is
␭⫽
冢
d
p(pv 1(p) ⫹ q)
dp p(pv 1(p) ⫹ q) ⫹ q(p ⫹ qv 3(p))
冣兩
(11)
p̂
or equivalently by using the method described by Lewontin (1958). Such an analysis is presented in Asmussen and Basnayake (1990), where expressions for the
stability of the internal equilibria and the fixation points
are provided.
Since the multiplicative fertility model is formally analogous to the two-sex viability model (Bodmer 1965; Kidwell et al. 1977), and the one-sex viability selection
model is known to have the same behavior as the simple
viability model in terms of equilibrium (Cannings 1969),
expressions (10) and (11) hold for sexual selection fitness functions by just substituting m 1(p) and m 3(p) for
v 1(p) and v 3(p). When considering the joint effect of
several fitness components, the internal equilibria can
be computed numerically for particular cases by deterministic simulations that iteratively apply the recurrence
equations (2), implemented with the specific fitness
functions and constant parameters, from different starting points. To this end we have written specific programs
in Mathematica (Wolfram 1996).
Drosophila subobscura: As shown in Table 3, viability
and female fecundity were estimated in this species for
only one starting frequency of the OST and O3⫹4⫹7 chromosomal arrangements (Tarrı́o 1993), whereas male sexual selection was estimated for three starting frequencies
and was found to be frequency dependent (Santos et
al. 1986). Since no sexual selection estimates are available at frequencies close to the fixation points we analyzed two different patterns of variable selection–on
gene and genotype frequencies. Both fit the data equally
well (see coefficients of determination in Table 4) but
show a different behavior at frequencies close to fixation.
1174
J. M. Álvarez-Castro and G. Alvarez
Figure 1.—Sexual selection estimates of the Drosophila subobscura homokaryotypes relative to the heterokaryotype (Santos
et al. 1986) and regressions on gene frequencies that best fit
the data in Table 3. The fitness functions are m 1(X , Z ) ⫽
1.67 ⫺ 1.31p and m 3(X , Z ) ⫽ 2.06 ⫺ 4.33q ⫹ 3.86q 2, where
p and q are the adult gene frequencies of OST and O3⫹4⫹7. The
functions m 1 and m 3 depend indeed on X and Z , by substituting
q ⫽ 1 ⫺ p, and p ⫽ X ⫹ (1 ⫺ X ⫺ Z ).
Figure 1 shows the regressions on gene frequencies—on
the frequencies of the arrangements—that best fit both
homokaryotype fitness estimates. A linear regression fits
the estimates well for OST, whereas a quadratic regression
improves the fit for O3⫹4⫹7 by means of the adjusted
coefficient of determination (Table 4), which penalizes
the estimation of one extra parameter. Since there are
only three starting frequencies, the quadratic regression
is just the polynomial interpolation on the three means
of the replicates of the fitness estimates. These means
are the estimates given in Table 3, but it is the existence
of the replicates shown in Figures 1 and 2—on which
we actually performed the regressions—that allows us
to compute the coefficient of determination for this
case.
Figure 2 shows the regressions on genotype frequencies—the frequencies of the karyotypes—that best fit
the data. A linear regression shows a good fit for OST
(see Table 4 and Figure 2). Fitness estimates for O3⫹4⫹7
on the other hand fit best with the constraints of the
hyperbolic function. This function is an equilateral hyperbola of the type m(Z) ⫽ a ⫹ b(1/Z), fulfilling that
the axis of ordinates is one of the asymptotes, and hence
(assuming a positive value for b) it decreases for increasing frequencies and has a lower limit of a ⫹ b in [0, 1]
at frequency one, which gives some biological meaning
to the regression parameters. The smaller the value of
b the more pronounced the curvature of the graph is.
Since the function necessarily goes to infinity when the
homokaryotype frequency goes to zero, and it does not
fulfill the constriction that it is differentiable at this
fixation point, we actually have to consider a slightly different function that takes constant values in a small neighborhood of zero instead—say [0, 10⫺4].
Figure 2.—Sexual selection estimates of the Drosophila subobscura homokaryotypes relative to the heterokaryotype (Santos et al. 1986) and regressions on genotype frequencies that
best fit the data in Table 3. The fitness functions are m 1(X ,
Z ) ⫽ 1.37 ⫺ 1.30X and m 3(X , Z ) ⫽ 0.82 ⫹ (0.03/Z ).
The results of the study of equilibrium are summarized in Table 5. We focus first on the regressions on
gene frequencies (Figure 1), and therefore inequalities
(5) describe the protected polymorphism of the genetic
system. When considering frequency-dependent sexual
selection alone, O3⫹4⫹7 is a stable fixation point that
can thus be achieved by iteration of the recurrence
equations from polymorphic situations, whereas OST is
unstable. By means of (10) and (11) we have found one
stable and one unstable internal equilibrium points,
with the unstable one being relatively close to the fixation point of O3⫹4⫹7. The interval between this unstable
equilibrium and the fixation point is the set of starting
frequencies from which the polymorphism would be
lost. The same qualitative results hold when including
the constant fitness estimates in the complete model,
the only difference being slight changes in the values
for the equilibrium points. Neither superiority of the
heterokaryotype in fecundity nor directional selection favoring OST in viability is strong enough to switch the O 3⫹4⫹7
fixation point to unstable for the regression on the genefrequencies pattern. Nevertheless they reduce the percentage of starting frequencies leading to fixation to
⬍10%, by bringing the unstable internal equilibrium
closer to the fixation of O3⫹4⫹7.
We now focus on regressions on genotype frequencies
(Figure 2). Protected polymorphism conditions given in
Mechanisms of Balancing Selection
1175
TABLE 5
Equilibria and stability predicted by the different patterns of variable selection over Drosophila subobscura and D. pseudoobscura
on gene and genotype frequencies considered for the study (see Figures 1–4 and Table 4) and by the model
including all fitness component estimates from Table 3 for both species
Drosophila subobscura
Fixation points
Fitness component
Viability
Sexual selection
All
Regression variables
O ST
O 3⫹4⫹7
Gene frequencies
Gene frequencies
Genotype frequencies
Gene frequencies
Genotype frequencies
—
Unstable
Unstable
Unstable
Unstable
—
Stable
Unstable
Stable
Unstable
inequalities (4) are satisfied when considering frequencydependent sexual selection alone. The stable equilibrium for this pattern differs only in thousandths from
the one we obtained when considering gene-frequencydependent sexual selection alone (see Table 5). Superiority of the heterokaryotype in female fecundity reinforces
the maintenance of the polymorphism if we consider both
late components at the same time. When further including
directional viability favoring OST the protected polymorphism still holds, and the internal stable equilibrium is
again virtually the same as when considering the complete
model with the gene-frequency-dependent pattern (Table
5). In general the two patterns behave almost identically
at intermediate frequencies, where the regressions are
strongly conditioned by the data, and differently close to
the fixation points, where there are no available fitness
estimates. Both regression patterns explain well the maintenance of the polymorphism when considered in the
complete model, and for the genotype-frequency pattern, moreover, protected polymorphism exists, which
more strongly prevents the loss of variability.
For both regression patterns, when considering frequency-dependent sexual selection alone, the internal
stable equilibrium points display heterokaryotype superiority. (For the gene-frequencies pattern this can be
inferred by just looking at Figure 1.) However, this is
not the case at the internal stable equilibrium point of
the complete model. The female total fitnesses of OST/
OST and O3⫹4⫹7/O3⫹4⫹7 homokaryotypes, relative to the
heterokaryotype, are respectively (for all frequencies)
1.06 and 0.86, thus showing directional selection favoring OST. We have computed the fitnesses for the same
genotypes of males according to the gene-frequencies
pattern at p̂ ⫽ 0.668 (Table 5) as respectively 0.87 and
0.99. This can be considered as directional selection
with complete dominance of the favored karyotype,
O3⫹4⫹7, or more precisely heterokaryotype superiority
very biased toward the fixation of O3⫹4⫹7. According now
to the genotype-frequencies pattern, at p̂ ⫽ 0.667 (Table
5) we found directional selection favoring O3⫹4⫹7 in
males with homokaryotype fitnesses of 0.87 and 1.04.
So for both patterns there is rather strong selection
Drosophila pseudoobscura
Internal p̂
Stable Unstable
—
0.565
0.562
0.668
0.667
—
0.183
None
0.082
None
Fixation points
ST
CH
Unstable Unstable
Stable
Unstable
—
—
Unstable Unstable
—
—
Internal p̂
Stable
Unstable
0.785
0.621
—
0.799
—
None
0.798
—
None
—
opposite in sexes at the internal stable equilibrium point
of the complete model.
Drosophila pseudoobscura: The right-hand side of Table 3 shows fitness estimates for the ST and CH inversion
polymorphism of the third chromosome of D. pseudoobscura. Female fecundity was studied assuming constant
fitnesses (Moos 1955) and both viability (Anderson et
al. 1986) and male sexual selection (Anderson and
Brown 1984) are known to be frequency dependent.
To handle these data we actually exploit the fact that
our model enables us to simultaneously consider early
and late variable selection. Although the viability estimates of D. pseudoobscura were obtained in experiments
mostly performed with only two karyotypes competing,
we pooled data with the same starting frequencies together and computed standard errors by using delta
methods (Weir 1990) when necessary.
Quadratic gene-frequency regressions show good fit to
the viability estimates (Figure 3; see coefficients of determination in Table 4). Since these experiments were not
performed at Hardy-Weinberg starting frequencies, no
regressions on genotype frequencies can be performed
on these data. The best regressions on gene frequencies
for the sexual selection estimates are also quadratic on
gene frequencies for both homokaryotypes (Figure 4),
and the negative coefficients of determination show that
the other regressions fit poorly.
As shown in Table 5 together with the rest of the
results of the study of equilibrium, viability fitness functions (Figure 3) fulfill the protected polymorphism conditions given in inequalities (4). Anderson et al. (1986)
previously inferred a protected polymorphism in their
discussion of these data on variable viabilities. By means
of (10) and (11) we have found only one internal stable
equilibrium point. Regarding now sexual selection fitness functions (Figure 4), they also lead to one internal
stable equilibrium point, although they do not prevent
the fixation of ST. The location of one unstable internal
equilibrium point at (roughly) p ⫽ 0.8 reveals that sexual selection would allow the loss of CH arrangement
when its frequency is ⬍0.2. Since fecundity selection
directionally favors ST, it reinforces both the instability
1176
J. M. Álvarez-Castro and G. Alvarez
Figure 3.—Viability estimates of the Drosophila pseudoobscura
homokaryotypes relative to the heterokaryotype (Anderson
et al. 1986) and regressions that best fit the data in Table 3.
The fitness functions are v1(X zy , Z zy) ⫽ 1.02 ⫹ 2.19p zy ⫺ 2.95
p 2zy and v 3(X zy , Z zy) ⫽ 0.20 ⫹ 2.87q zy ⫺ 2.64q 2zy , where p zy and
q zy are the zygotic gene frequencies of ST and CH. The functions v 1 and v 3 may be expressed directly in terms of the
frequencies of the genotypes in an analogous way to m 1 and
m 3 in Figure 1.
of CH and the stability of ST. Considering, last, the joint
effect of all the fitness estimates in the complete model,
inequalities (4) hold and hence there is a protected
polymorphism, preventing the system from the loss of variability. By applying the recurrence equations we have
found only one internal equilibrium point, which is stable,
so that we get to the same qualitative equilibrium results
as when considering viability alone. Thus our frequencydependent model applied to the fitness component estimates explains maintenance of the ST/CH polymorphism
in D. pseudoobscura by means of protected polymorphism.
As was also the case for D. subobscura, the internal
stable equilibrium point when considering sexual selection alone displays heterokaryotype superiority. The
same result holds for frequency-dependent viability.
However, at the internal stable equilibrium point of the
complete model, p̂ ⫽ 0.799 (Table 5), the total fitnesses
of the homokaryotypes ST/ST and CH/CH relative to
the heterokaryotype are 1.22 and 1.31 for females and
0.97 and 0.90 for males. There is, thus, heterokaryotype
disadvantage with an unstable internal equilibrium
point at p̂ f ⫽ 0.585 in females and heterokaryotype superiority with a stable internal equilibrium point at p̂m ⫽
0.769 in males. At p̂ ⫽ 0.799, therefore, selection acts
in favor of ST (toward p ⫽ 1) in females and against ST
(toward p ⫽ 0.769) in males.
DISCUSSION
Theoretical results: Protected polymorphism conditions, as sufficient conditions for the maintenance of
Figure 4.—Sexual selection estimates of the Drosophila pseudoobscura homokaryotypes relative to the heterokaryotype (Anderson and Brown 1984) and regressions that best fit the data
in Table 3. The fitness functions are m 1(X , Z ) ⫽ 10.84 ⫺
30.38p ⫹ 22.76p 2 and m 3(X , Z ) ⫽ 2.65 ⫺ 7.74q ⫹ 6.03q 2,
where p and q are as in Figure 1.
polymorphisms, provide information about whether
variability can be lost in a genetic system or not. When
protected polymorphism conditions are fulfilled, there
is no starting polymorphic frequency from which deterministic loss of variability can occur. This prevents the
fixation of alleles after an occasional switch in frequency
caused by, for instance, temporary drift effects such as
bottlenecks—as long as they do not cause the fixation
of one allele themselves. When a population is at a stable
internal equilibrium but the protected polymorphism
conditions do not hold, such occasional phenomena
could eventually bring the system to a point from which
selection alone can lead to the fixation of one allele.
Protected polymorphism conditions constitute an ideal
tool to inspect the maintenance of polymorphism in genetic systems in which frequency-dependent parameters
need to be considered. The main advantage of this
approach is that these conditions involve only the values
that the variable parameters take at the fixation points.
This holds when considering general fitness functions
that are different for the two sexes and also for the early
and late stages, which causes the input of some fitness
functions to depend on the output of the others. We
highlight two main convenient consequences of this
fact. First, it allows for notable simplicity and biological
interpretation of the formulas [see inequalities (4)].
For the variable models we consider in this article, the
protected polymorphism conditions take the same form
of harmonic means of the selective parameters as in
models with constant selection, enabling a direct comparison of the parameter spaces of the variable and
constant models. Second, it demonstrates that, for each
variable fitness component, only estimates obtained for
two genotypes competing in the vicinities of the fixation
Mechanisms of Balancing Selection
points are necessary to inspect the protected polymorphism conditions of the variable genetic systems, independent of the pattern of frequency dependence of the
variable parameters. Protected polymorphism conditions can still be obtained when further including fertility as a general mating-interaction selection force and
nonrandom mating in the variable models [inequalities
(7–9)]. Similar to variable selection, not every mating
pair’s fitness is involved in these expressions. The results
for the constant fertility model are interesting in themselves because the equilibrium studies for this model
were previously performed for particular cases (see Introduction).
However, it is noteworthy that protected polymorphism conditions are limited in three aspects. First, since
they are not necessary conditions for the maintenance
of the polymorphism, internal stable equilibria can still
occur when the inequalities are not fulfilled. Second, they
do not indicate the existence of multiple internal equilibria. Finally, although protected polymorphism conditions are sufficient conditions for the maintenance of
the polymorphisms, they do not completely guarantee
the existence of internal stable equilibrium points, because awkward instances can lead to limit cycles instead
(Nagylaki 1992, p. 65).
Drosophila inversion polymorphism: Our frequencydependent models enable us to accomplish an analysis
of the maintenance of Drosophila inversion polymorphisms, for which data on several fitness component
estimates were found to vary for different starting frequencies (see Table 3). A complete study of equilibrium
is possible since regressing on the data allows us to
obtain explicit fitness functions to enter into the model
(Figures 1–4). Regarding the type of fitness functions
to be considered for the regressions, polynomials up to
second order—linear and quadratic functions—seem
to be adaptable enough to describe the data well. When
using other functions, such as hyperbolas, that can be
suitable for depicting rare genotype advantage, it is important to pay detailed attention to the constraints these
types of functions cause in the regression curves, to distinguish them from the particularities of the curves actually
caused by the data.
The importance of superiority of the heterozygote (or
heterokaryotype) and frequency-dependent selection as
selective mechanisms in the maintenance of genetic
polymorphisms in Drosophila has been extensively discussed without conclusively tipping the scales in favor
of any of them (see Tobari 1993; Powell 1997, pp. 109–
114). We have found frequency-dependent selection to
be much more important for the Drosophila inversion
polymorphisms we have analyzed. In D. subobscura, frequency-dependent male sexual selection rules the qualitative behavior of the genetic system, in terms of both
the stability of the fixation points and the number of
internal stable and unstable equilibria. Directional viability
and superiority of the heterokaryotype in female fecundity
1177
could, however, contribute to prevent the fixation of the
O3⫹4⫹7 arrangement. In D. pseudoobscura frequency-dependent viability plays a more influential role in determining the maintenance of the polymorphism. Kojima and
Tobari (1969; see also Tobari 1993; Powell 1997, p.
112) studied frequency-dependent viability in inversion
polymorphisms of D. ananassae and found heterokaryotype superiority at the internal equilibrium points. We
have obtained the same result for D. subobscura and D.
pseudoobscura inversion polymorphisms when considering one frequency-dependent selection component at
a time. For the models considering all the selection
components available, however, we have shown that the
total fitnesses of females and males at the stable equilibrium points display more complex modes of selection
involving total selection acting in considerably strong
opposite sense in the two sexes. This suggests that sexdependent selection, more than heterokaryotype superiority, contributes with frequency-dependent selection
to the maintenance of the inversion polymorphisms we
have analyzed. More generally we have shown that very
different frequency-dependent data-based patterns of
selection—beyond mathematical coincidences in the
type of functions—in which superiority of the heterozygote is not the rule at all along the frequencies (see
Figures 1–4) generate internal stable equilibria and often fulfill the protected polymorphism conditions (Table 5). Some of the cases considered also show how
the flexibility of variable fitness functions can lead to
internal stable equilibria when the protected polymorphism conditions are not fulfilled.
The equilibrium point at roughly p̂ ⫽ 0.8 we predict
for ST/CH arrangements of D. pseudoobscura (Table 5)
is in very good agreement with the equilibrium frequencies reported for the experimental populations (see
Dobzhansky and Pavlovsky 1953; Anderson and
Brown 1984). For OST/O3⫹4⫹7 arrangements of D. subobscura, however, this does not seem to be the case, since
Zapata et al. (1986) observed that experimental populations achieve equilibrium points at (roughly) p̂ ⫽ 0.9,
in contrast with our predictions from the fitness estimates, in which p̂ ⬍ 0.7 (Table 5). This difference could
likely be due to the fact that variable viability selection,
which plays a major role in the maintenance of the
ST/CH polymorphism in D. pseudoobscura, has not been
studied for D. subobscura. However, it must also be kept
in mind that two other general selection mechanisms,
frequency-dependent female fecundity (Anderson and
Watanabe 1974) and sex-dependent viability (Druger
1966; Druger and Nickerson 1972), were found in
some experiments on chromosomal arrangements of D.
pseudoobscura. As a final point, supergene selection is an
inversion-related balancing selective force to take into
account for a precise approximation of the equilibrium
frequencies of chromosomal arrangements of Drosophila to be achieved (Wasserman 1968; Alvarez and
Zapata 1997).
1178
J. M. Álvarez-Castro and G. Alvarez
The authors thank Joachim Hermisson for valuable comments on
an earlier version of the article and kind support during part of the
investigation and Pleuni Pennings, John Parsch, and one anonymous
reviewer for discussion toward a better presentation of the work. This
survey was partially supported by a fellowship from the Secretarı́a
Xeral de Investigación e Desenvolvemento da Xunta de Galicia to
J.M.A.C.
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