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EVOLUTION
INTERNATIONAL JOURNAL OF ORGANIC EVOLUTION
PUBLISHED BY
THE SOCIETY FOR THE STUDY OF EVOLUTION
Vol. 57
December 2003
No. 12
Evolution, 57(12), 2003, pp. 2667–2677
THE EVOLUTION OF ENVIRONMENTAL AND GENETIC SEX DETERMINATION IN
FLUCTUATING ENVIRONMENTS
TOM J. M. VAN DOOREN1,2
1 Section
AND
OLOF LEIMAR3,4
Animal Ecology, Institute of Biology, Kaiserstraat 63, 2311 GP Leiden, The Netherlands
2 E-mail: [email protected]
3 Department of Zoology, Stockholm University, SE-106 91 Stockholm, Sweden
4 E-mail: [email protected]
Abstract. Twenty years ago, Bulmer and Bull suggested that disruptive selection, produced by environmental fluctuations, can result in an evolutionary transition from environmental sex determination (ESD) to genetic sex determination (GSD). We investigated the feasibility of such a process, using mutation-limited adaptive dynamics and
individual-based computer simulations. Our model describes the evolution of a reaction norm for sex determination
in a metapopulation setting with partial migration and variation in an environmental variable both within and between
local patches. The reaction norm represents the probability of becoming a female as a function of environmental state
and was modeled as a sigmoid function with two parameters, one giving the location (i.e., the value of the environmental
variable for which an individual has equal chance of becoming either sex) and the other giving the slope of the reaction
norm for that environment. The slope can be interpreted as being set by the level of developmental noise in morph
determination, with less noise giving a steeper slope and a more switchlike reaction norm. We found convergence
stable reaction norms with intermediate to large amounts of developmental noise for conditions characterized by low
migration rates, small differential competitive advantages between the sexes over environments, and little variation
between individual environments within patches compared to variation between patches. We also considered reaction
norms with the slope parameter constrained to a high value, corresponding to little developmental noise. For these
we found evolutionary branching in the location parameter and a transition from ESD toward GSD, analogous to the
original analysis by Bulmer and Bull. Further evolutionary change, including dominance evolution, produced a polymorphism acting as a GSD system with heterogamety. Our results point to the role of developmental noise in the
evolution of sex determination.
Key words.
Adaptive dynamics, canalization, evolutionary branching, sex determination, threshold trait.
Received May 12, 2003.
Many populations are polymorphic and consist of a small
number of distinct morphs. The division into two sexes is by
far the most widespread polymorphism, but there are many
others, such as the winged and wingless forms in some insect
groups. An individual’s morph can be environmentally determined, and thus a case of phenotypic plasticity, but morph
determination can also be genetic. It is an issue of general
relevance in evolutionary biology to determine the conditions
under which either environmental or genetic morph determination is likely to evolve. Our aim here is to investigate
this question for the case of the two sexes, using models of
the evolutionary dynamics of sex determination.
For the sexual morphs, the issue of plasticity versus genetic
polymorphism corresponds to environmental sex determination (ESD), with possible polygenic variation, versus major
gene genetic sex determination (GSD). In both cases, distinct
sexual morphs are present at the phenotypic level, but the
manner in which genes and environment affect the alterna-
Accepted June 27, 2003.
tives differs. Basically, some type of switch between alternative trajectories is needed at a certain point in development.
It was noted by Fisher (1930) that GSD can be understood
as a polymorphism for a key gene that switches development
into one of two alternatives. This general idea has been incorporated into arguments about different forms of polymorphism (e.g., Mather 1955). It can also be that the effects
of possible sex genes are small, while environmental cues
have prominent effects. Charnov et al. (1976) used the idea
of a switch in their reasoning about the evolution of hermaphrodite reproduction versus separate sexes, where they
allowed for genetic polymorphism or environmental sensitivity to do the switching. An intermediate system of polyfactorial sex determination is also possible, with comparable
contributions from genetic variation and environmental cues
(Bull 1983).
In the literature on sex determination, several explanations
have been put forward for evolutionary transitions between
2667
q 2003 The Society for the Study of Evolution. All rights reserved.
2668
T. J. M. VAN DOOREN AND O. LEIMAR
TABLE 1.
p
e
E
h(e z E), se z E
g(E), sE
x(e)
xa,b (e)
f (e), m(e)
bf , b m
h̃ (e z E)
X(E)
F(E), M(E)
x9(e)
X9(E), F9(E), M9(E)
R(x9, x)
G(x9, x)
Overview of symbols and notation.
emigration probability
individual environmental variable
patch environmental variable
probability density and conditional standard deviation of e
probability density and standard deviation of E
sex-determination reaction norm, probability of becoming a female in environment e
sex-determination strategy, which depends on genetic parameters a and b, where a determines reaction
norm location, b reaction norm slope
competitive ability function ( f , females; m, males)
slopes of competitive ability functions
distribution of e among candidate parents in patch E
proportion females in patch E
average competitive abilities in patch E (F, among females; M, among males)
mutant sex-determination strategy
equivalent to X(E), F(E), and M(E) above, but for mutants x9
average reproductive success of a mutant x9 in a resident population x
geometric mean reproductive success of a mutant x9 in a resident population x
ESD and GSD. One of them is that there may be genetic
variation in traits that differentially favor males and females,
for instance, in growth rate. A sex-determining gene tending
to produce females could then increase in frequency when
closely linked to genes with beneficial effects on females or,
in the same way, sex determination could be selected for as
a pleiotropic side effect of such genes (Rice 1986). Concerning ESD, Charnov and Bull (1977) suggested that it
would appear when there is environmental variation with
sufficiently strong differential fitness effects on the sexes—
similar to the parental control of sex ratio envisaged by Trivers and Willard (1973). Otherwise, GSD should be expected
because of developmental advantages of early sex determination. Bull (1981a) used genetic models to argue that transitions from GSD to ESD could come about in the way suggested by Charnov and Bull (1977), although it has been
noted that there may be a tendency for the evolution of sex
chromosomes to be irreversible, so that a new mechanism
cannot readily evolve (Bull 1983; Bull and Charnov 1985).
For the reverse direction, Bulmer and Bull (1982) introduced the idea that a transition from ESD to GSD is possible
when ESD leads to excessive fluctuations in the population
sex ratio over space or time. When there is variation in environmental conditions differentially favoring males and females, but also larger fluctuations in the population sex ratio
than would be warranted from these differential advantages,
selection on sex determination traits becomes disruptive, so
that genes with major effects on such traits could invade
(Bulmer and Bull 1982).
Our aim in this paper is to extend the results of Bulmer
and Bull (1982) through a more thorough examination of the
consequences of stochastic fluctuations for a system of ESD.
In particular, we investigate the idea that within-individual
variation (Lynch and Walsh 1998), for instance, in the form
of developmental noise, could introduce randomness in a
switchlike developmental map from environmentally influenced liability to sexual morph, and we examine whether
evolution of the degree of randomness can be expected. An
important consequence of such randomness would be that it
dampens fluctuations in population sex ratios, so that the
evolution of appropriate randomness in a switching mechanism could be an alternative to the evolution of GSD.
Local fluctuations in patchy environments often play a role
in evolutionary explanations of ESD (e.g., Shine 1999). At
the same time, the effects of different migration rates on sex
determination have hardly been discussed (for an exception,
see Reinhold 1998). Whereas Bulmer and Bull (1982) mainly
considered the relative magnitudes of within-generation and
between-generation environmental variation, we also take
into account within-patch and between-patch variation in a
metapopulation, allowing an investigation of the effects of
different migration rates.
DESCRIPTION
OF THE
METAPOPULATION MODEL
Table 1 lists the symbols and notations of the model. Consider a metapopulation with N patches and partial migration,
with probability p of leaving the natal patch and entering a
dispersal pool. There is no survival cost to dispersal, and
dispersers migrate into a randomly chosen patch. In each
patch, arriving adults contribute to the production of zygotes
in the same way as the local residents. The gametes forming
a zygote are drawn from the adult males and females in the
patch using a weighted distribution over the parents of either
sex. An individual’s weight in the distribution is proportional
to its competitive ability. Per patch, a fixed number of K new
individuals are formed. Development of individuals into
adults occurs in the patch of conception, without any juvenile
mortality, and the adult’s sex is determined by a combination
of its individual environment and certain of its traits. These
traits correspond to the sex-determination strategy.
An individual’s external environment during development
is assumed to influence the adult competitive ability. In our
model, competitive abilities are functions, m(e) for males and
f(e) for females, of an individual environmental parameter e
(e.g., e could be food supply during development). The parameter e is randomly drawn from a distribution that depends
on a patch environmental parameter E, and we write the conditional distribution as h(e z E). For each new generation, E
is randomly drawn from the distribution g(E), independently
for each patch, corresponding to patch level fluctuations over
space and time. When male and female competitive abilities
differ in their dependence on e, sex-determination strategies
can also become dependent on that environmental variable
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EVOLUTION OF SEX DETERMINATION
(Charnov and Bull 1977). We write such a strategy as a
function x(e) and interpret it as the probability of becoming
a female in environment e. It is the evolution of such phenotypes or reaction norms x(e) that we aim to investigate.
Shaw-Mohler Fitness Function
One feature of our model that greatly simplifies the analysis
is that, for a given resident sex determination strategy, the
distribution of individuals over sexes and competitive abilities in the next generation will not depend on the state of
the population in the current generation.
We assume that both the number of patches N and the
patch population sizes K are very large and consider a point
in time after migration but before reproduction. Note first
that the distribution of the individual environmental parameter e among the adults competing for reproduction in a patch
with parameter E is given by
h̃(e z E ) 5 (1 2 p)h(e z E ) 1 p
E
h(e z V )g(V ) dV.
(1)
The reason for the expression is that a proportion (1 2 p) of
the individuals stayed in the patch and a proportion p migrated. Integration is over the range of the environmental
variable E. When all individuals use strategy x, the proportion
of females in a patch with parameter E is
E
X(E ) 5
x(e)h̃(e z E ) de,
(2)
and the average competitive ability among the females in the
patch is
F(E ) 5
1
X(E )
E
f (e)x(e)h̃(e z E ) de.
(3a)
Similarly, the average competitive ability among the males is
M(E ) 5
1
1 2 X(E )
E
m(e)[1 2 x(e)]h̃(e z E ) de.
(3b)
Consider an individual using a rare mutant strategy x9.
When it has developed in individual environment e and attempts to reproduce in a patch with parameter E, the expected
genetic contribution of that individual to the pool of next
generation zygotes will be equal to
R(x9, x z e, E ) 5
5
1
f (e) 1
x9(e)
2
F(E ) X(E )
6
m(e)
1
1 [1 2 x9(e)]
.
M(E ) 1 2 X(E )
(4)
The logic behind this expression is that the mutant individual
first has its sex determined by the strategy x9(e) and, after
either migrating or staying, eventually reproduces in a patch
with parameter E. At reproduction, the expected genetic contribution to the pool of next generation zygotes depends on
the individual’s competitive ability relative to the average
competitive ability of the other adults of the same sex in the
patch and on the patch sex ratio. The factor one-half is present
because the individual contributes only one of the two gametes that form a zygote.
To obtain the expected reproductive success of a random
mutant individual reproducing in the patch, we must integrate
equation (4) over the distribution (1) of individual environment e, which leads to
R(x9, x z E ) 5
5
E
R(x9, x z e, E )h̃(e z E ) de
5
6
1 F9(E )X9(E )
M9(E )[1 2 X9(E )]
1
.
2 F(E )X(E )
M(E )[1 2 X(E )]
(5)
In this expression, the primes denote quantities referring to
the mutant strategy, such that
F9(E )X9(E ) 5
M9(E )[1 2 X9(E )] 5
E
E
f (e)x9(e)h̃(e z E ) de
and
m(e)[1 2 x9(e)]h̃(e z E ) de,
(6)
(7)
corresponding to equations (3a, b). A rare mutant gene will
occur predominantly in heterozygous genotypes, so the mutant
strategy x9 should be the phenotype of such a genotype. By
integrating over the distribution of patch environments E, we
get the overall average mutant reproductive success:
R(x9, x) 5
E
R(x9, x z E )g(E ) dE.
(8)
The relative rate of increase of the frequency of the mutant
gene, when small, is given by the logarithm of R(x9, x). Provided that gene flow from interpatch migration keeps the
frequency of the mutant gene spatially homogeneous, we can
regard log[R(x9, x)] as the invasion fitness of the mutant (Metz
et al. 1992). However, for very low rates of migration gene
frequencies may differ between local populations, and it is
not clear that (8) can be used to form an invasion fitness. For
the special case of no migration (p 5 0), each subpopulation
evolves independently in a temporally fluctuating environment, and we should use geometric mean reproductive success to evaluate the possibility of mutant invasion (Lewontin
and Cohen 1969). Thus,
G(x9, x) 5
E
log[R(x9, x z E )]g(E ) dE
(9)
is an appropriate invasion fitness when p 5 0.
As is often the case in simple models with two sexes, our
expression (5) has the Shaw-Mohler form (Shaw and Mohler
1953; Charnov 1982; Lessard 1989; Pen and Weissing 2002),
where the contribution of a strategy to the next generation
is expressed using a sum of ratios of functions of male and
female traits. For the special case of full migration (p 5 1),
our model corresponds to several previous models aimed at
studying ESD (e.g., Bull 1981b; Charnov and Bull 1985,
1989). With zero migration, our model becomes equivalent
to a model for a single population with environmental fluctuations within and between generations.
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T. J. M. VAN DOOREN AND O. LEIMAR
Selection Gradient and Product Rule
A traditional way to analyze sex allocation equilibria
would be to use equations like (8) or (9) directly. Since R(x,
x z E) 5 1, a mutant phenotype x9 can or cannot invade a
resident x depending on whether log[R(x9, x)] or G(x9, x) is
greater or smaller than zero. We can get additional insight
into the nature of the evolutionary stability of genetically
monomorphic equilibria by focusing on gradual evolutionary
change. For this purpose, we make use of a first-order Taylor
expansion of invasion fitness for x9 close to x, an adaptive
dynamics approximation (Dieckmann and Law 1996; Metz
et al. 1996; Geritz et al. 1998). For x9 close to x, selection
will be weak and a mutant gene frequency will be spatially
homogeneous also for small values of p. Thus, in the limit
of very weak selection, equation (8) provides a valid basis
for invasion fitness for any p . 0. For the case of no migration
(p 5 0) we should use (9) as invasion fitness. However, as
we will see below, (8) and (9) give the same equilibria under
gradual evolutionary change, so there will be continuity of
equilibria as p approaches zero.
We assume that x(e) is a function that depends on the
environment e and a limited number of genotypic parameters
(but suppress these arguments as much as possible not to
clutter notation). If c9 denotes such a parameter of the mutant
reaction norm, the partial derivative of R(x9, x z E) in (5) with
respect to c9 equals
]R(x9, x z E )
1
5
]c9
2
E 5
]x9(e)
f (e)
m(e)
2
]c9 F(E )X(E )
M(E )[1 2 X(E )]
3 h̃(e z E ) de.
6
(10)
For the functions we used, continuity properties of x, f, m,
and h̃ always allowed differentiation under the integral sign.
We obtain the selection gradient sc on c by evaluating the
derivative of invasion fitness at c9 5 c, where c is the resident
parameter value. Since R(x, x z E) 5 1, both log[R(x9, x)] from
(8) and G(x9, x) from (9) give the same expression,
sc 5
E
)
]R(x9, x z E )
]c9
g(E ) dE,
(11)
c95c
for the selection gradient.
Because of its special form in models with fitness of the
Shaw-Mohler type, the selection gradient (11) can be written
as the derivative of a function on phenotype space. For our
model such a function is
P(x) 5
1
2
E
log{F(E )X(E )M(E )[1 2 X(E )]}g(E ) dE,
(12)
]P(x)
.
]c
Sex-Determination Function
We modeled sex-determination phenotypes using a family
of sigmoid functions, xa,b(e), of the individual environmental
variable e, and with two genotypic parameters, a and b. These
give the value (a) of the environmental variable where the
probability of becoming a female equals one-half, and the
slope (b) of the reaction norm at that point (see Fig. 1 for
some examples). As sigmoid functions with these properties,
we used
x a,b (e) 5 F[Ï2p b(e 2 a)],
(14)
where F(z) is the cumulative distribution of a standard normal
variate. We can interpret equation (14) as a threshold trait
with discrete developmental responses to a continuously
varying liability (Lynch and Walsh 1998). Thus, we can regard y 5 a 2 e 1 d as the realized liability to become male,
with the threshold for alternative sexes located at y 5 0. This
liability has genotypic or breeding value a in environment e
5 0, and e and d are deviations in the liability due to external
individual environment (e) and within-individual random developmental noise (d). When d is normally distributed with
standard deviation sd 5 (Ï2p b)21, we obtain the reaction
norm (14). Parameter b measures the extent of canalization,
that is, when b increases, the relative contribution of developmental noise in sex determination decreases. In line with
the interpretation that reduced variance of developmental
noise corresponds to a better buffering of a trait (Gibson and
Wagner 2000), we say that a high value of b implies more
canalization. For our reaction norm specifying probabilities
of discrete developmental alternatives, high canalization at
the liability level corresponds to a switchlike reaction norm
and thus implies high sensitivity to an external environmental
factor at the level of the sex ratio probabilities. This is somewhat different from the idea of canalization of morphological
traits with a unimodal distribution, where developmental
buffering always decreases the environmental sensitivity
(Rendel 1967).
RESULTS
since one readily verifies that
sc 5
When evolutionary equilibria can be found by maximizing
some product of male and female traits, one often refers to
this a product rule (MacArthur 1965; Charnov 1982). In our
case, P(x) is the expected value of the logarithm of a product,
which can be seen as a generalized product rule (see Leimar
2001). It is important to note that the product rule (12) only
applies to gradual evolutionary change in a genetically monomorphic population and does not decide the issue of whether
a genetic polymorphism can evolve.
(13)
It follows that gradual evolutionary change can be understood
as a hill-climbing process that increases P(x) over the trait
space of genotypic parameters. Hill climbing also occurs
when the reaction norm is studied as an infinite-dimensional
functional trait, where sex-determination probabilities x(e)
per environment evolve independently (O. Leimar, T. J. M.
Van Dooren, and P. Hammerstein, unpubl. data).
We studied the evolution of parametric reaction norms xa,b
of the form (14), in the two-dimensional trait space given by
the parameters a and b. We made use of the fitness gradient
(sa, sb) from equations (11) or (13) to determine directions
in which evolutionary change can occur. We ignored the
evolution of competitive ability traits, assuming these to be
fixed by stabilizing selection or constraints. As competitive
ability functions, we used m(e) 5 am 1 bme and f(e) 5 af
1 bfe. In all examples, females have higher competitive abilities than males for positive e and vice versa for negative e
EVOLUTION OF SEX DETERMINATION
2671
(bf . 0 and bm , 0). Negative values of m and f and values
larger than one were replaced by zero and one, respectively.
With m and f between zero and one, we can interpret them
as the probability of succeeding when an opportunity to become parent arises. For these competitive ability functions,
female-biased sex ratios can be advantageous at positive values of e, which corresponds to the form (14) of the reaction
norm.
We have used normal distributions for the patch environmental variable E and the individual external environment e.
The patch variable has expected value equal to zero and standard deviation sE and, conditional on the patch variable E,
the individual environment e has expected value E and standard deviation sezE. The unconditional variance of e then is
the sum of the two variances, s2E 1 s2ezE . As a choice of scale,
we kept the sum equal to two and considered the relative
contributions from within- and between-patch variation to
total environmental variance.
Convergence Stable Reaction Norms
We calculated the fitness gradient with respect to the parameters a and b numerically, to determine convergence stable reaction norm strategies (Eshel and Motro 1981; Eshel
1983; Christiansen 1991). These are strategies where the fitness gradient equals zero for all strategy parameters and
where gradual evolution from nearby strategies converges
toward the stable strategy. In our model, convergence stable
parameter values (a*, b*) correspond to local maxima of the
function P(xa,b) in (12), regarded as a function on the parameter space. Such a point (a*, b*) will attract gradual evolution
starting from any point (a, b) on that peak of P(xa,b), which
means that (a*, b*) will be absolutely convergence stable
(Leimar 2001). For the cases we investigated, the function
P(xa,b) was single-peaked with a unique global maximum.
Figure 1 illustrates that the convergence stable reaction
norm changes from being relatively flat for migration probability close to p 5 0 to an abrupt sex-determination switch
when the population is completely mixing (p 5 1). A completely flat reaction norm is the classical Fisherian solution,
with 50% allocation to each sex in any environment e (Fisher
1930). An abrupt switch corresponds to the results of Charnov
and Bull (1989) for continuous patchiness, which are analogous to the results of Trivers and Willard (1973) on sex
allocation, in that a single sex is produced on either side of
a threshold value. The switch is located at e 5 0 (Fig. 1),
which is the value of the environmental variable where the
competitive ability functions of the sexes cross. For low to
intermediate values of p, there is a more gradual change from
mostly males to mostly females as e increases (Fig. 1). In
this manner, a certain amount of developmental noise becomes evolutionarily favored. Reaction norms will be completely flat only when the relative within-patch variation is
negligible or when the environmental variable has no effect
on competitive abilities.
Figures 1A and 1B differ in the ratio s2ezE/sE2 of withinpatch to between-patch environmental variance. When we
increase s2ezE/sE2 , the convergence stable reaction norm becomes more switchlike also for fairly small migration probabilities (Fig. 1B). When the variance between patches be-
FIG. 1. Convergence stable reaction norms of the adaptive dynamics are shown for different migration probabilities p and for
different ratios of the variances of the environmental parameters.
(A) s2ezE/sE2 5 0.2. (b) se2zE/sE2 5 1.0. Reaction norms with slopes .
5 are not given a separate migration rate label. m(e) 5 0.5 2 0.1e
and f(e) 5 0.5 1 0.1e; s2E 1 s2ezE 5 2.0.
comes small, a switchlike reaction norm will not cause very
strong fluctuations in the sex ratios of the local populations.
Convergence stable reaction norms will have a narrower
range of environments over which intermediate sex-determination probabilities occur when population mixing is
greater and when the relative magnitude of private to patch
environmental variation is greater. An increased level of developmental noise, corresponding to decreased canalization,
evolves when there is less population mixing and when environmental variation between patches is comparable to or
larger than variation within patches.
The qualitative pattern of the results in this example is not
2672
T. J. M. VAN DOOREN AND O. LEIMAR
FIG. 2. Pattern of invasion fitness in the space of trait parameters
(a, b). Gradual evolution can either increase or decrease trait values,
depending on the resident values of a and b. Regions with different
directions of evolution are separated by lines and the main direction
within each region is indicated by an arrow. Gradual evolutionary
change in a and b leads to a convergence stable strategy (a*, b*)
5 (0.0, 0.22), indicated by a dot. In the shaded interval of the line
a 5 a* 5 0, invasion fitness shows disruptive selection on the
location parameter a. Selection is weakly disruptive at (a*, b*).
Parameter values for the competitive abilities and environmental
variables are as in Figure 1A and the migration rate p equals 0.3.
strongly dependent on the form of competitive ability functions we assumed. For example, we found similar results with
sigmoid competitive ability functions instead of linear ones,
with female competitive ability independent of e and when
the average of E does not coincide with the point where
competitive ability functions cross. It is the difference between male and female competitive abilities per environment
e that really matters, and our conclusions seem to hold whenever there is a single point where these competitive ability
functions cross.
Stabilizing or Disruptive Selection
Once a population has evolved toward a convergence stable
strategy (a*, b*), local invasibility determines what can hap-
pen from there. With stabilizing selection in all directions
through (a*, b*) in parameter space, no single mutant can
invade and evolution comes to a stop. However, there may
be neutral or disruptive selection at (a*, b*) for some trait
or trait combination. Whether selection is stabilizing or disruptive at a point (a, b) can be determined from the secondorder partial derivatives of invasion fitness with respect to
mutant a9 and b9, which can be obtained in a similar way as
the first-order partial derivative in (10).
With disruptive selection, there may be an accumulation
of genetic variation, and a polymorphism of genetically determined strategies can appear through the process of evolutionary branching (Geritz et al. 1998). Disruptive selection
favors the extremes of the strategy distribution in the population and intermediates are at a disadvantage. As a consequence, an originally unimodal distribution of traits will
become bimodal and one can interpret this as branching of
the evolutionary tree. When a trait in a phenotypic model
shows evolutionary branching, then a corresponding singlelocus genetic model will show evolutionary branching as well
(Van Dooren 2003). For our model, branching can lead from
ESD to GSD.
Evolutionary branching is a fairly slow process, and one
would expect it to occur primarily when there is disruptive
selection at a convergence stable point in trait space, because
a population will remain near such a point for a long time.
However, if a trait evolves very slowly or not at all because
of limited genetic variation, branching can occur for other
traits that are exposed to disruptive selection. We will consider the possibility that the slope b of the reaction norm
could be constrained to evolve slowly, and that there could
then be branching in the location parameter a also where b
has not reached the equilibrium value b*.
Figure 2 shows the directions of increase of invasion fitness
at different points in the (a, b) parameter space, for the example with p 5 0.3 in Figure 1A. Selection always favors
the location parameter a to become zero, and for a* 5 0
selection will act to adjust the slope to a specific value, which
is b* 5 0.2196 in this example. At the point (a*, b*), there
is stabilizing selection on the slope b, but very weak disruptive selection on the location a. Considering that b might be
constrained to evolve very slowly, we also determined the
type of selection on a, at a* 5 0, for different values of b.
As shown in Figure 2, we found that selection on a changes
TABLE 2. Comparison of evolutionary equilibria of the reaction norm location (a) and slope (b) parameters between the adaptive
dynamics approximation and individual-based simulations. The adaptive dynamics results are illustrated in Figure 1.
Individual-based 1
Adaptive dynamics
Parameters
s2e z E
s2E
p
a
b
0.333
0.333
0.333
0.333
1.000
1.000
1.667
1.667
1.667
1.667
1.000
1.000
0.0
0.1
0.3
0.5
0.0
0.1
0.000
0.000
0.000
0.000
0.000
0.000
0.022
0.060
0.220
1.360
0.117
0.195
a
20.001
0.003
0.003
0.001
20.006
0.004
6
6
6
6
6
6
b
0.021
0.005
0.007
0.005
0.012
0.006
0.023
0.060
0.215
1.330
0.117
0.192
6
6
6
6
6
6
0.002
0.002
0.004
0.007
0.003
0.003
1 For the simulations, N 5 100 subpopulations with K 5 500 individuals used, except for the p 5 0 cases, where a single subpopulation with K 5 50,000
was used. Data are given as means 6 standard deviations over a large number of generations (50,000 or more; because of genetic drift and environmental
fluctuations, population averages of a and b will continue to fluctuate slightly over time in a simulation, also when an equilibrium has been reached). The
mutation rate was 0.0002 and the standard deviation of mutational increments was 0.01 both for a and b. We assumed additive genetics in these simulations.
EVOLUTION OF SEX DETERMINATION
from stabilizing to disruptive when the slope goes above a
certain value (b 5 0.2195) located very near the convergence
stable slope b*. The strength of disruptive selection on a is
greater for larger values of b, which means that branching
can occur more readily when the reaction norm is constrained
to have a more steplike shape. If the slope can evolve relatively fast, the population may reach (a*, b*) before branching occurs. Because there is weak disruptive selection at (a*,
b*), branching could in principle occur at this point, but our
individual-based simulations (see below) instead produced a
cluster of a genotypes, centered on a* 5 0, corresponding
to a balance between mutation, selection, and genetic drift.
Thus, the very weak disruptive selection observed at (a*, b*)
in adaptive dynamics approximations in practice seems not
to split the population into several distinct groups of genotypes.
One of the main results by Bulmer and Bull (1982) was
that ESD would be favored over GSD when the differential
fitness advantages for the sexes in different local environments were great enough in comparison with the magnitude
of fluctuations in local sex ratios. We have investigated the
importance of the extent of differential fitness advantages for
ESD-GSD transitions. We considered different parameters bm
and bf of the competitive ability functions, but kept bm 5
2bf, so that a large bf corresponds to large differential fitness
advantages for the sexes in different local environments. We
found that the equilibrium location was always at a* 5 0
and that the convergence stable slope b* increased with the
magnitude of bf, as illustrated in Figure 3A. In all cases we
examined, we observed that for bf smaller than a critical value
b̂f, there was stabilizing selection on b and very weak disruptive selection on a at the convergence stable (a*, b*).
Considering also that the slope b may be constrained to some
value different from b*, the value of b where selection on
location a (at a* 5 0) changed from stabilizing to disruptive
was very close to b* (Fig. 3A). As in Figure 2, selection on
location a was disruptive for reaction norms with large b and
stabilizing for small b, provided that bf was smaller than b̂ f.
For values of bf above the critical value, selection on both
slope and location became stabilizing at the convergence stable strategy parameters, and selection on location in fact became stabilizing for all slopes b. At the same time, the convergence stable b* increased rapidly with bf, corresponding
to a steplike reaction norm.
Figure 3B illustrates that a small migration probability p,
which enhances local fluctuations in sex ratios, makes for a
larger parameter region with disruptive section, that is, a
larger value of b̂ f. Other changes in ecological parameters
tending to increase fluctuations in sex ratios, such as a smaller
value of s2ezE/sE2 , will have a similar effect of promoting disruptive selection on the location a, which could produce evolutionary branching and GSD.
To summarize, we can get evolutionary branching and an
ESD-GSD transition if reaction norms for sex determination
are constrained to act like fairly abrupt switches, with steeper
slopes than the reaction norm favored by the population ecology. For the case of a constrained slope b, our analysis is
largely in agreement with that by Bulmer and Bull (1982).
Concerning the ecological setting, a low migration rate, little
within-patch variation in individual environments relative to
2673
FIG. 3. The convergence stable reaction norm slope b* is shown in
(A) as a function of the competitive ability parameter bf, assuming
that bm 5 2bf (the convergence stable location is always a* 5 0).
Considering that the slope b could be fixed by constraints, the region
where selection on the location parameter a is disruptive at a* 5 0
is indicated by shading. For small values of bf, the boundary separating regions with stabilizing and disruptive selection on a nearly
coincides with the curve giving b*. The part of this curve where
selection on a is disruptive is drawn as a thick line. For values of bf
above a threshold value b̂f 5 0.315, selection on a is stabilizing at
(a*, b*). For bf . b̂f, selection becomes stabilizing for any value of
b. Ecological parameters in (A) are p 5 0.3, se2zE/sE2 5 0.2, and am
5 af 5 0.5. In (B) we have drawn the threshold value b̂ f for different
values of the migration parameter p. The threshold becomes smaller
when the migration probability increases.
between-patch variation, and small differential competitive
advantages for the sexes over environments will act in favor
of a transition to GSD.
Individual-Based Simulations
We checked these conclusions and the evolution of major
segregating sex-determination effects by means of individual-based simulations. In many sex-determination systems,
2674
T. J. M. VAN DOOREN AND O. LEIMAR
a specific signal activates a key gene that controls the first
step in a short regulatory cascade (Marin and Baker 1998;
Schütt and Nöthiger 2000). It is therefore warranted to study
sex determination as a single-locus problem, where all evolutionary change is assumed to occur in the key gene.
To investigate the validity of the adaptive dynamics approximation, we ran simulations corresponding to some of
the cases in Figure 1 (see Table 2). We used large populations,
low rates of mutation, and small mutational increments, to
approach a situation of gradual, mutation-limited evolution
in an infinite population. In these simulations, we found that
GSD failed to become established. Instead, the population
evolved to the neighborhood of the convergence stable trait
values a* and b* (Table 2). Even though our adaptive dynamics approximations predicted weak disruptive selection
on a, a unimodal distribution of genetic variation in a appeared, approximately centered on a*. This distribution
seemed to reach an asymptotic standard deviation, without
any tendency toward evolutionary branching. The situation
was similar for b. Our adaptive dynamics derivation assumed
a monomorphic population, and the presence of genetic variation can thus make the approximation slightly inaccurate
when disruptive selection is weak.
To further investigate the evolution of GSD, we simulated
populations of 100 patches with 100 adults produced per
patch. Mutation rates per trait parameter and generation were
fixed at 0.001 and mutational increments in allelic parameters
were drawn from a rectangular distribution with zero mean.
We controlled the speeds of evolution of location a and slope
b by setting the standard deviation of such a distribution,
denoted sa for a and sb for b. We explored a range of initial
values of a and b. In cases where we observed branching,
the slope b of the initial reaction norm was always greater
than the convergence stable b*, and the mutational effects
on slope relative to location were small, that is, sb/sa was
usually 0.1 or smaller.
Additionally, we allowed for dominance evolution in the
determination of a and b from the alleles of a diploid genotype. For this we adopted a simple specification for a fully
linked dominance modifier, as in Van Dooren (1999). In the
simulations, haplotypes indexed i were specified by three parameters: location ai, slope bi, and dominance parameter di,
which has a positive value. For a pair of haplotypes i and j,
the resulting genotypic location and slope parameters are
a5
di ai 1 dj aj
di 1 dj
b5
di bi 1 dj bj
.
di 1 dj
and
(15a)
(15b)
Thus, the contributions of haplotypes to the genotypic traits
a and b are weighted according to their dominance parameters. Just as for the haplotypic parameters ai and bi, the
mutational increments in the dominance parameter di was
drawn from a rectangular distribution with mean zero and
specified standard deviation, sd. However, we did not allow
mutation to make di zero or negative.
We found that if the initial value of b was large enough
and mutational effects in b were small enough, disruptive
selection on a could remain substantial for long periods, lead-
ing to branching. For instance, with ecological parameters as
in Figure 3A, with bf smaller than the threshold b̂ f, and initial
conditions corresponding to fairly steplike reaction norms,
major sex-determining effects evolved in our simulations.
Figure 4 gives an overview of the typical stages in the establishment of major sex-determining alleles. First, evolutionary branching occurs and alleles with major effects become apparent. The population distribution of reaction norms
splits up into three bundles, but there is at first no sign of
dominance evolution. Once the two alternate homozygote
bundles become sufficiently separated, the dominance trait
starts evolving. The bundle of heterozygotes then aligns with
one of the homozygote bundles, so that only two bundles of
reaction norms remain. For the resulting polymorphic population, an individual’s sex is largely determined by its genotype. At the extremes of the distribution of environmental
variables, genotypic males still become females and vice versa (patterns of mixed GSD and ESD have been observed;
e.g., Lagomarsino and Conover 1993).
In the last panel of Figure 4, we changed the migration
rate to a higher value, such that the convergence stable reaction norm would be more steplike (cf. Fig. 1A). With this
higher migration rate, our individual-based simulation
showed that GSD changed back to ESD. Thus, the transition
between ESD and GSD in our model is reversible.
Another interesting aspect of our simulations is that heterogamety appears in connection with dominance evolution.
Figure 5 shows the distribution of haplotypic values for the
dominance parameter di and location parameter ai during the
simulation in Figure 4. Early in the simulation, there is a
single cluster of alleles, near to a 5 0 and the initial value
d 5 1 of the dominance parameter. After the establishment
of GSD with clear dominance, we see that haplotypes in
females cluster in two groups that differ in the location and
dominance parameters. In the example in Figure 5, alleles
with a positive location value ai have become relatively dominant over alleles with a negative value. The haplotypes in
males cluster in the same two groups, but the allele group
with positive values of ai contains few individuals. Males
have become predominantly homozygous and the females
heterozygous for two very different location alleles. We have
thus obtained a system with (still incomplete) female heterogamety and a dominant allele for femaleness. Considering
a large number of simulation repeats, female and male heterogamety seem about equally likely to evolve. This is expected, based on the entirely symmetric roles of males an
females in these simulations. We also simulated cases without
an evolving dominance parameter and observed that clear
heterogamety did not evolve.
DISCUSSION
Perhaps the main result of our analysis is that there can
be two quite different types of evolutionary responses in sexdetermination traits to fluctuating environments: either the
appearance of GSD or increased developmental noise in the
ESD mapping from liability to sexual morph. Both of these
responses have the property of reducing fluctuations in population sex ratios. When there is little developmental noise
and limited genetic variation in the level of randomness, so
EVOLUTION OF SEX DETERMINATION
2675
that a norm of reaction is constrained to show a fairly abrupt
switch from only males to only females, our analysis essentially confirmed the results by Bulmer and Bull (1982) in a
metapopulation setting. A possible difference is that Bulmer
and Bull (1982) suggested that a transition from ESD to GSD
would come about through mutations with large effects,
whereas we have shown that there can also be a more gradual
evolutionary accumulation of the extent of polymorphism in
sex determination genes, through a process of evolutionary
branching (Metz et al. 1996; Geritz et al. 1998). For the shift
from GSD to ESD, our simulations revealed that a change
in the ecological circumstances, such that ESD no longer
would cause excessive fluctuations in population sex ratios,
produced a shift from GSD to ESD more or less in the manner
suggested by Bull (1981a).
If there is sufficient genetic variation in traits influencing
the within-individual random variation in liability, so that
the slope of a norm of reaction can respond to selection, we
found that the population evolved toward the convergence
stable trait value and stayed there. While our adaptive dynamics approximation indicated very weak disruptive selection, our simulations produced a unimodal distribution of
genotypes centered on the convergence stable strategy. This
indicates the limitations of such approximations.
Bull (1981b, 1983) noted that increasing environmental
fluctuations over space and time would favor a strategy that
has a more Fisherian type of sex-determination, with a probability of producing females that tends to 50% in every environment. He interpreted sex determination probabilities
that vary gradually with environmental conditions as evidence of the presence of (polygenic) genetic variation in sex
determination traits in a population, because he assumed that
environmental sex determination entailed a threshold trait
with an abrupt switch in the probabilities of becoming either
sex (Bull 1983). In our model, smooth reaction norms can
occur in genetically monomorphic populations, because we
allow the amount of developmental noise at the level of sexdetermination liability to evolve. Imprecise information
transfer between environment and gene action could produce
such developmental noise, and the level of noise could evolve
by reducing or enhancing the precision of information transfer. Little developmental noise, implying a high degree of
canalization, corresponds to Bull’s assumptions. Our results
suggest that some form of constraint on evolutionary changes
←
FIG. 4. Illustration of an individual-based simulation where evolutionary branching occurred. For a random sample of 500 indi-
viduals from the population at a given point in time, an individual’s
genotypic probability of becoming female in the local environment
e where it was born is represented. Mutation rates are 0.001 per
trait and generation, and mutational effects have standard deviations
sa 5 0.1 for location, sb 5 0.0001 for slope and sd 5 0.1 for
dominance. The standard deviation sb was set to be extremely small
in this example, such that evolution of the slope parameter hardly
occurred. The initial population was monomorphic with haplotypic
parameters ai 5 0, bi 5 0.75 and dominance di 5 1. Ecological
parameters are as in Figure 1A, with migration probability p 5 0.3,
except that the last panel of the figure was obtained by increasing
the migration probability to p 5 0.7 and then running the population
of the third panel for 5000 additional generations.
2676
T. J. M. VAN DOOREN AND O. LEIMAR
FIG. 5. The appearance of heterogamety in the simulation in Figure 4. The panels show bubble plots of location and dominance
parameters of haplotypes in males and females (a bubble is drawn, centered at the parameter values of the haplotype, with size proportional
to the number of copies of that type among males and females). The top row shows the population of the first panel in Figure 4 and the
bottom row the population of the third panel in Figure 4. This last population displays clear differences between males and females in
the distribution of haplotypic parameters. Alleles that combine a positive location value with a high dominance value occur much more
in females than in males. Most males are homozygotes for a negative location, and most females are heterozygotes with a dominant
allele for a positive location value. Because there is little variation in the slope parameter, we do not show it.
in the level of developmental noise is needed for a transition
from ESD to GSD. Without such a constraint, the Bulmer
and Bull (1982) route toward GSD becomes less plausible.
Our analysis thus implies a crucial role for noise in the
evolution of developmental switches between alternatives,
such as the two sexes. Developmental noise is usually studied
in genetic networks having the function to produce a single
phenotype (Gavrilets and Hastings 1994; Wagner et al. 1997;
Kawecki 2000), and in such situations increased canalization
is typically favored. This differs from the situation in our
analysis, where increased canalization corresponds to a more
switchlike reaction norm. We show that it can be evolutionarily favorable to introduce more randomness in development, that is, to decrease canalization. Such noise, usually
called ‘‘within-individual variation’’ (Lynch and Walsh
1998), can play the role of a strategic randomization over
alternatives that may be favored in fluctuating environments
(cf. Haccou and Iwasa 1995; Sasaki and Ellner 1995).
Sex-determination genes seem to evolve rather easily (Marin and Baker 1998; Zarkower 2001). Different sex-determination mechanisms occur in closely related taxa and sometimes even within a single species (Bull 1983; Hodgkin
2002). It is likely that transitions between ESD and GSD will
be affected by other factors, in addition to the kind of environmental fluctuations in space and time that we have studied here. Examples of such factors could be the presence of
genetic conflict between genes with paternal and zygotic expression (Werren and Beukeboom 1998; Werren et al. 2002),
sex differences in dispersal combined with spatial heterogeneity (Reinhold 1998), or pleiotropic effects of genes in
pathways of sex determination and sexual dimorphism (Kraak
and Pen 2002). Nevertheless, for models taking such factors
into account, it may still be the case that disruptive selection
and the occurrence of evolutionary branching can serve as a
paradigm for the emergence of genes with major effects on
sex determination.
EVOLUTION OF SEX DETERMINATION
ACKNOWLEDGMENTS
We thank M. Bulmer and J. Bull for helpful comments and
P. Hammerstein for genuine interest and inspiring discussion.
This research was supported by the European ModLife Research Training Network, funded through the Human Potential Programme of the EU (contract HPRN-CT-2000-00051),
by a travel grant from the European Science Foundation Theoretical Biology of Adaptation Programme to TVD, and by
grants from the Swedish Research Council to OL.
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Corresponding Editor: S. Gavrilets