Download On the Instability of Polygenic Sex Determination: The Effect of Sex

On the Instability of Polygenic Sex Determination: The Effect of Sex- Specific
Selection
William R. Rice
Evolution, Vol. 40, No. 3. (May, 1986), pp. 633-639.
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Evolution, 40(3), 1986, pp. 633-639
ON THE INSTABILITY O F POLYGENIC SEX DETERMINATION:
THE EFFECT O F SEX-SPECIFIC SELECTION
WILLIAM
R. RICE
Department of Biology, University of New Mexico, Albuquerque, NM 87131
Abstract. -A combination of analytical and simulation models is used to explore the initial evolution of genic sex determination from polygenic sex determination. Prior studies have indicated
that polygenic sex determination is rare or absent in extant species but that it has likely played an
important intermediate role in the evolution of other genetic sex-determination systems. This study
explores why polygenic sex determination does not persist. Two possibilities are considered. First
it is assumed that a major sex-determining gene also pleiotropically increases fitness. Second it is
assumed that the sex-determining gene is neutral but linked to another locus segregating for a rare
selectively favored allele. The major conclusion from the models is that sex-specific natural selection
will cause polygenic sex determination to be a transient state in most populations. Polygenic sex
determination may be an important intermediate step in the evolution of genetically controlled
sexual differentiation, but it is unlikely to persist unless there is some selective advantage compared
to genic sex determination. This may in part explain the relatively small number of extant species
that have polygenic sex determination.
Received September 2 1, 1984. Accepted December 12, 1985
An important question concerning the
evolution of sex-determination systems involves how major sex-determining genes
become established within populations. A
second question involves why certain sexdetermination systems are common while
others are rare in nature. Here, I address
both questions by examining the evolutionary stability of polygenic sex determination.
In a polygenic system, many genes, each
with a small effect, are either male- or female-determining. If the expression of the
determining genes for one sex is collectively
stronger, then the zygote differentiates as
that sex.
Polygenic sex determination is rare in extant species, and some authors argue that
reported instances are in error (see for review Bull, 1983). However, polygenic sex
determination may play an important intermediate role in the evolution of other
sex-determination systems. For example,
Kirpichnikov (198 I), in an extensive review
of the ichthyological literature, concludes
that polygenic sex determination is the
primitive state in fish, ultimately being replaced by genic sex determination. The genic system may then further evolve to semichromosomal or chromosomal sex determination. The rarity of polygenic sex determination in extant species suggests that
this mode is evolutionarily unstable. That
is, if it evolves during the evolutionary his-
tory of a taxonomic group, it is ultimately
replaced by another mode of sex determination.
Here, I provide a theoretical rationale for
the instability of polygenic sex determination. I specifically examine the effect of natural selection for genes that are selectively
favored in one sex but disfavored in the
other. I conclude that natural selection for
such "sexually antagonistic" genes (Rice,
1984) ultimately leads to the displacement
of polygenic sex determination by genic sex
determination.
The General Model
Consider a large population that initially
is balanced for a polygenic sex-determination system. One half of the zygotes will
develop into females and half into males.
Next consider a Y gene recurrently introduced by mutation. The Y gene is assumed
to be dominant and epistatic to all sex-determining polygenes. Any individual carrying Y is one sex and any individual not
carrying this gene has its sex determined by
the polygenic system. Many such major sexdetermining genes have been identified in
nature (see Mittwoch, 1973; Ohno, 1979).
For simplicity I assume that Y is male-determining, but this choice is arbitrary and
all of the following conclusions will also apply if Y is female-determining.
The assumptions I make are: 1) the pop-
634
WILLIAM R. RICE
TABLE1. Mating combinations and resulting offspring. Calculations for M (fraction males in the next
generation) assume a balanced polygenic sex determination system. See text for details.
Parents
Frequency
Q
Male
Offs~rinr
. .
Female
xx xx
Male
(Xn
Male
Female (Xx) -
-1
2
-1
2
M = P[0.5 + 0.25(1
1 P(l-FC)
- - +2
4
(xx)
-
FC)]
+ Q[0.5]
ries more female-determining polygenes (on
average) than that derived from an XXmale.
To model the female bias of X gametes
derived from XY males, I start with two
observations. First, X gametes from an XX
female (XXmale) carry on average an excess
of female-determining (male-determining)
polygenes, while those from an XY male are
intermediate. Second, since Xgametes from
an XY male fuse with an X gamete from an
XX female, the resulting XX zygotes will
have a greater than average number of female-determining polygenes. I represent this
bias by the factors (1 F*) and (1 - F*)
in Table 1. The value F * adjusts for the
female bias of XX zygotes derived from an
XY male.
I consider two possible mechanisms by
which an XX-XY sex determination system
can displace a polygenic system. First, I
briefly describe the pleiotropy case in which
Yhas some secondary phenotypic effect that
is selectively favored. Second, I evaluate
what I consider to be a more feasible model,
the linkage case. Here, Y is selectively neutral but is closely linked to another gene
which is selectively favored.
Bull (198 1, 1983) has developed a quantitative model for the evolution of major
sex-determining genes assuming the primitive state to be environmental sex determination. This model was verbally extended to the case of primitive polygenic sex
determination (Bull, 1983). Any credit for
the original idea of the pleiotropy model
belongs to Bull. Below, I more fully quantify
the pleiotropy model and extend it to consider the effects of natural selection for sexually antagonistic genes.
+
ulation is large enough to ignore sampling
error; 2) the Ygene is dominant and epistatic to all sex polygenes; 3) the Xgene is allelic
to the Y gene and has no effect (or a small
effect) on the expression of polygenic sex
determination; 4) the polygenic sex-determination system initially is balanced such
that on average one half of XX zygotes become male and one half female; 5) mating
occurs at random between the sexes; and 6)
the cost of producing a son equals the cost
of producing a daughter.
When Yis present in the population there
are two types of males, XY and XX. The
frequency of these two types will be denoted
by P and Q respectively with P
Q = 1.
All females are XX. Since there are two types
of males there are two classes of matings
(Table 1). An XX by XX mating produces
a 1:1 sex ratio in the progeny, since the polygenic system is initially assumed to be balanced. A mating between an XY male and
an XXfemale produces approximately threefourths sons and one-fourth daughters (Table 1).The proportion of sons is > '/2 because
all the Y bearing gametes plus a fraction of
the X bearing gametes from XY males produce sons (Table 1). The proportion of sons
is less than three-fourths because an X gamete derived from an XY male will produce
<'/2 males even when the polygenic system
is balanced. This occurs because the Y gene
causes a zygote to differentiate into a male
irrespective of the number of male- and female-determining polygenes. As a consequence, an X gamete from an XY male car-
+
The Pleiotropy Case
Assume that Y has two phenotypic effects: 1) a major sex-determining effect and
2) some other phenotypic effect that is selectively favored. As a result of the pleiotropy of the Y gene, XY males have increased fitness over XX males. More
specifically, the fitnesses (viability) of XY
males, XX males, and XX females are 1
S, 1, and 1, respectively. Assuming discrete
generations and constant fitness values, the
frequency of XY males in the next generation (P') is given by
+
635
POLYGENIC SEX DETERMINATION
The change in frequency of X Y males is
given by
This imbalanced sex ratio will act to increase the frequency of female-determining
polygenes and decrease the frequency of
male-determining polygenes (Bulmer and
Bull, 1982). As the polygenic sex-determination system becomes imbalanced, a
smaller proportion of X X zygotes will differentiate into males. Let K be the proportion of X X zygotes that develop into males.
From Equations (1-4) and Table 1 it can be
seen that the change in frequency of X Y
males depends on P, S, K, and F :
p'
=
P(l
P(1 + S)
- F ] )
+ S + K[l
+ 2QK
'
(5)
At equilibrium P'
At equilibrium (assuming the polygenic system remains balanced),
An exact evaluatation of Equations (1-4)
requires a knowledge of the bias factor Fa.
The dynamics of Fr are complex, but in a
polygenic system F* is always < 1. This is
because there is a finite probability that a
mating between X Y male and an X X female
will produce an X X son (assuming genic sex
determination has not entirely displaced the
polygenic mode). Throughout what follows,
the exact value of F will not be specified.
All of my conclusions only require that
FC < 1, the exact value of FC being of no
consequence.
The biological interpretation of Equation
(4) is that pleiotropy by itself cannot cause
the X Y genotype to become fixed in males.
Even when the value of S is large, some
males will have genotype X X and some will
have XY. To fully convert the polygenic system into a genic sex-determination system
a second evolutionary force must operate.
This second force is Fisherian sex-ratio selection (Fisher, 1958).
When the equilibrium value of P is greater than 0, the sex ratio of the population
will be biased in favor of males so long as
the polygenic system is balanced (Table 1).
= P,
and for a fixed K
When k > 0, the frequency of female-determining polygenes increases, and this decreases the value of K; but, from Equation
(6), as K declines the v a l u ~of P increases.
This interaction between P and K produces
a chain reaction that continues until P 1
andlK 0. To demonstrate that the value
of P ultimately goes to 1, the equilibrium
value of M (fraction males) can be expressed
for fixed values of P, K, S, and Fr:
-
-
Substituting
in terms of S and K,
Equation (8) indicates that for any values
of K and P, the population sex ratio will be
male-biased until K = 0. When K 0, P
1 [Equation (6)], and consequently the population sex ratio will remain male-biased
until all males are X Y and all females are
XX. Thus, the frequency of a Y gene that
pleiotropically increases fitness will increase
when the gene is rare. This will bias the
population sex ratio and ultimately convert
the polygenic sex-determination system into
a genic system.
- -
636
WILLIAM R. RICE
TABLE
2. Fitness model for the linkage case.
Sex
Frequency
Fitness
(viab~l~ty)
Genotype
Males:
a
P
4
C
-
[rl
a'
b'
C'
-
[xl
Females:
A
B
C
-
1.o
Average male fitness (relative):
As a check on my conclusions from Equations (1-8), I have examined the pleiotropy
model numerically via a three-locus simulation model. One locus coded for the Xand
Y alleles and the other two loci coded for
and - sex-polygenes. Sex was determined by the number of
alleles in the
genotype (see bottom Table 3). Initially Y
was rare and the numbers of and - alleles
at each locus were equal. The results of many
simulations indicate that any conditions in
which Y increased when rare [Equation (3)]
caused sex-ratio selection to increase the
frequency of female-determining polygenes
(+), ultimately converting the polygenic sexdetermination system to a genic sex-determination system.
+
+
+
The Linkage Case
If the Y gene pleiotropically increases fitness, then it is relatively easy to understand
how a major sex-determining gene can invade a polygenic system. This situation, although possible, requires a fortuitous type
of pleiotropy. A more parsimonious situation would be for the Y gene to have only
a major sex-determining effect. If this were
the case, Bull (1983) has shown that such a
gene would be unlikely to increase to substantial gene frequency unless polygenic sex
determination was poorly canalized (see below). However, one way that the Y gene
could be selectively favored is if it were
tightly linked to a second gene favored by
natural selection. I will refer to this situation
as the linkage case.
Consider the same initial conditions as
previously stated for the pleiotropy case.
Next, assume that a second locus (a viability
locus with alleles A, and A,) is segregating
for a rare selectively favored allele (A,). The
viability locus is located an arbitrary R-recombinational units away from the locus
coding for the Ygene. The fitness of all possible male genotypes is given in Table 2.
Genotype frequencies are arbitrary, subject
only to the constraint that they sum to 1.
At the bottom of Table 2, I have solved for
the relative fitness of XX and XY males. If
the population is in linkage equilibrium, then
the second term in the fitness equation for
XYmales is equal to 0, and Yis not expected
to increase in frequency.
When the A, allele is selectively favored
in both sexes, then it will ultimately go to
fixation, linkage disequilibrium will go to 0,
and the XY males will lose any advantage
that transient linkage disequilibrium may
have produced. Unless the population were
quite small, any transient linkage disequilibrium between A, and Y would have a
small effect, and Ywould remain rare in the
population. If A, were selectively favored
in males but selected against in females,
however, chronic linkage disequilibrium
could be maintained between the Y and A,
genes. This can give the supergene Y-A, a
net selective advantage, and the same genetical chain reaction outlined in the pleiotropy case could convert the polygenic sexdetermination system into a genic system.
When will the supergene Y-A, increase
when rare and convert a polygenic sex-determination system into a genic system? To
answer this question, I first assume that both
Y and A, are rare in a population with a
balanced polygenic sex-determination system. The gene A, is selectively favored in
males and selected against in females (Table
2). Two forces will act on the Y-A, supergene: selection and recombination. Selection will act to increase the frequency of the
supergene since it is only found in males.
Recombination will both build up and break
637
POLYGENIC SEX DETERMINATION
down the supergene. Assuming discrete generations, the change in frequency of Y-A, is
given by
Y LOCUS CHROMOSOME
H.---- -- - - .- 1
0
J
l
&
I + hS
It
hS h S ( I -9,'~)-9/p DISTANCE-
w8 + + + +
+
+
=a
a' 0) q b)(l hS)
where
(c c')(l S). Equation (9) is exact only
when the polygenic system is balanced,
which is the presumed starting condition.
The subsequent long term dynamics of Y-A,
are evaluated later by computer simulation.
When Yand >4,are rare,
1.O, c 0.0,
and Equation (9) is approximated by
+
+
w,
A( Y-A,)
=
FIG. 1. A schematic of the Y-bearing chromosome
indicating the three regions within which sexually antagonistic alleles could potentially be introduced by
mutation (see text for details). For simplicity, the Y
locus is assumed to be at one end of the chromosome.
If it were centrally located, a mirror image of this figure
would extend to either side of the Y locus.
+
(11) i$ hS/(l
hS). More generally, the
supergene Y-A, will increase when rare
whenever,
Ap
For A(Y-A,) > 0,
The left side of (1 1) is determined algebraically from (10). The right side of (1 1) is
determined by evaluating the minimum
value of q/p that will permit the supergene
Y-A, to increase when rare. When q/p> 1
recombination builds the supergene Y-A,
faster than it breaks it down and recombination and selection work in concert to increase its frequency. When q/p = 1, recombination has no effect since it builds and
destroys Y-A, at equal rates. When q/p is
less than 1, recombination breaks down Y-A,
faster than it builds it up, and recombination acts in opposition to selection.
When A, is selectively favored in males
and disfavored in females, the ratio q/p is
expected to be < 1, and selection and recombination act in opposition. As the value
of q/p declines below 1, the effects of recombination increase. The worst possible
case is q/p = 0. Here recombination only
breaks down Y-A,. Setting q/p = 0, it can
be seen that a lower bound for the center of
The equilibrium value of q/p depends on
the equilibrium level of linkage disequilibrium between Y andp,, i.e., D. I have been
unable to solve for D analytically and have
found no solution in the literature. Simulation analysis (see below) indicated that the
values of S and T that permit the supergene
Y-A, to increase when rare always result in
q/p values near 0. Thus, inequalities (1 112) can be reasonably, but conservatively,
approximated by assuming q/p = 0.
The biological interpretation of inequalities (1 1) and (12) is shown in Figure 1. If
the sexually antagonistic locus is within
hSl(1
hS) recombinational units of the
Y-locus, then the supergene Y-A, will increase when rare. If the sexually antagonistic locus is in the dotted region, the supergene may or may not increase depending
on the equilibrium level of linkage disequilibrium. If the recombinational distance is
greater than hSl[l
hS(1 - qlp) - qlp]
recombinational units then Y-A, cannot increase when rare. When Y-A, does increase
when rare, it should initiate the same process described in the pleiotropy case and
convert the polygenic system into a genic
sex-determination system.
The prediction that close linkage between
Y and A, can initiate the same genetical
chain reaction observed in the pleiotropy
case was tested numerically via computer
simulation. Typical starting conditions are
+
+
638
WILLIAM R. RICE
TABLE
3. Description ofthe simulation model parameters with representative values.
Locus
Sex
viability1
Sex polygenes2
Genes
Y
X
A1
A2
+
-
Initial
frequency
0.0001
0.9999
0.9999
0.000 1
0.5
0.5
The viab~litylocus is sexually antagonistic and located R recombinational units from the sex locus.
Sex determined by the number of t genes in the genome. c3: male;
3: '/2 male and % female; >3: female.
shown in Table 3. Generations were discrete, the population size was infinite, and
genotype fitnesses were as described in Table 2. Initially the population was balanced
for a polygenic sex-determination system
(Table 3). It is important to note that these
simulations permit the polygenic sex-determination system to evolve in response to
changes in the frequency of the Ygene. Thus,
they extend the domain of the above analvtical model to include the simultaneous
evolution of genic and polygenic sex-determination over long periods of time.
The results of the simulations supported
the conclusion that close linkage between Y
and A, make the polygenic system unstable.
The XX-XY sex-determination system replaced the polygenic system whenever inequality (1 1) was met (using the equilibrium
value of q/p or q/p = 0).
Thus far I have assumed that A, is a sexually antagonistic allele with S < T, In this
case, neither Ynor A, can enter a population
alone, since they are only selectively favored
in combination. As a consequence, the supergene Y-A, initially is expected to be quite
rare. In a large population this poses no
problem, but in smaller populations the diminished number of Y-A, supergenes will
be susceptible to loss via sampling error
(drift). The probability of an individual Y-A,
supergene overcoming drift is approximately equal to 2hS(1 - R) (modified from
Crow and Kimura, 1970, p. 422). Thus, in
small populations where the total number
of Y-A, supergenes is very small, the increase in frequency of Y-A, may be seriously
impeded by drift. Over evolutionary time,
however, Y-A, eventually is expected to
overcome drift and displace the polygenic
sex-determination system.
The opposing effects of drift and selection
may stall the establishment of genic sexdetermination for a protracted period of time
in small populations. One way to speed the
initial increase in frequency of Y-A, in small
populations is for A, to be sexually antagonistic but also have a net selective advantage, i.e., S > T. In this case, A, will increase
when rare independently (Mandel, 197 1;
Bull, 1983; Rice, 1984). The A, allele will
have a nontrivial equilibrium frequency
whenever
(Mandel, 197 1; Bull, 1983).
I have examined this situation numerically by first introducing A, at low frequency
in the absence of the Y gene. After the A,
allele reached its equilibrium frequency, the
Y gene was introduced at low frequency.
The sex-specific selection for A, initially
produced linkage disequilibrium between Y
and A,. When the recombinational distance
between Yand A, was sufficiently small (i.e.,
R < hSl(1
hS)) the supergene Y-A, increased when rare and initiated the same
genetical chain reaction observed in the
pleiotropy case.
+
Conclusions
An important prediction from the linkage
model is that polygenic sex determination
is invasible by genic sex determination and
is consequently evolutionarily unstable. Initially the requisite linkage may be lacking
and polygenic sex determination could persist over a protracted period of time. Ultimately mutation pressure is expected to
produce a sexually antagonistic allele at a
locus proximate to the Y locus and genic
sex determination will ultimately evolve.
Bull (198 1, 1983) has shown that genic
sex determination can invade a polygenic
system when the genetic basis of sex determination is poorly canalized, i.e., when genetic sex determination is modified by environmental variation. As the environment
varies, it produces variation in the population sex ratio. Bull was able to show that
POLYGENIC SEX DETERMINATION
genic sex determination replaces polygenic
sex determination whenever environmental
variation causes the variance in population
sex ratio to be > 0. The linkage model
presented here illustrates how genic sex determination also will invade a polygenic
system even when it is little affected by normal environmental variation, i.e., when sex
determination is well-canalized.
In summary, polygenic sex determination
will be unstable whenever 1) sex determination is poorly canalized, 2) the Y gene
pleiotropically increases fitness, or 3) the Y
gene is tightly linked to a pair of sexually
antagonistic alleles. The major conclusion
from the models presented here is that sexspecific natural selection will cause polygenic sex determination to be a transient
state in most populations. Polygenic sex determination may be an important intermediate step in the evolution of genetically
controlled sexual differentiation, but it is
unlikely to persist unless there is some selective advantage compared to genic sex determination. This may in part explain the
relatively small number of extant species
that have polygenic sex determination.
ACKNOWLEDGMENTS
I thank Steve Gaines, Mark Kirkpatrick,
Kathryn Ono, and the Ecology/Population
639
Biology Seminar Groups at the University
of California, Davis and at the University
of New Mexico for many helpful comments
on the ideas presented here. Comments by
Jim Bull and Brian Charlesworth substantially improved the manuscript.
BULL,J. J. 1981. Evolution of environmental sex
determination from genotypic sex determination.
Heredity 47: 173-184.
- 1983. Evolution of Sex Determination Mechanisms. Benjamin/Cummings, Menlo Park, CA.
BULMER,
M. G., AND J. J. BULL. 1982. Models of
polygenic sex determination and sex ratio evolution. Evolution 36: 13-26.
CROW,J. F., AND M. KIMURA. 1970. Introduction to
Population Genetics Theory. Harper & Row, N.Y.
FISHER,R. A. 1958. The Genetical Theory of Natural
Selection. Dover, N.Y.
KIRPICHNIKOV,
V. S. 198 1. Genetic Bases of Fish
Selection. Springer-Verlag, N.Y.
MANDEL,
S. P. H. 197 1. Owen's model of a genetical
system with differential viability between the sexes.
Heredity 26:49-63.
MITTWOCH,
U. 1973. Genetics of Sex Differentiation.
Academic Press, N.Y.
OHNO, S. 1979. Major Sex Determining Genes.
Springer-Verlag, N.Y.
RICE,W. R. 1984. Sex chromosomes and the evolution of sexual dimorphism. Evolution 38:735742.
Corresponding Editor: R. H. Crozier