The Evolution and Consequences of Sex-Specific

Document technical information

Format pdf
Size 1.1 MB
First found May 22, 2018

Document content analysis

Category Also themed
Language
English
Type
not defined
Concepts
no text concepts found

Persons

Carl Perkins
Carl Perkins

wikipedia, lookup

Organizations

Places

Transcript

INVESTIGATION
The Evolution and Consequences of Sex-Specific
Reproductive Variance
Charles Mullon,*,† Max Reuter,†,1 and Laurent Lehmann‡
*Centre of Mathematics and Physics in the Life Sciences and Experimental Biology and †Department of Genetics, Evolution and
Environment, University College London, London WC1E 6BT, United Kingdom, and ‡Department of Ecology and Evolution,
University of Lausanne, Biophore, CH-1015 Lausanne, Switzerland
ABSTRACT Natural selection favors alleles that increase the number of offspring produced by their carriers. But in a world that is
inherently uncertain within generations, selection also favors alleles that reduce the variance in the number of offspring produced. If
previous studies have established this principle, they have largely ignored fundamental aspects of sexual reproduction and therefore
how selection on sex-specific reproductive variance operates. To study the evolution and consequences of sex-specific reproductive
variance, we present a population-genetic model of phenotypic evolution in a dioecious population that incorporates previously
neglected components of reproductive variance. First, we derive the probability of fixation for mutations that affect male and/or female
reproductive phenotypes under sex-specific selection. We find that even in the simplest scenarios, the direction of selection is altered
when reproductive variance is taken into account. In particular, previously unaccounted for covariances between the reproductive
outputs of different individuals are expected to play a significant role in determining the direction of selection. Then, the probability of
fixation is used to develop a stochastic model of joint male and female phenotypic evolution. We find that sex-specific reproductive
variance can be responsible for changes in the course of long-term evolution. Finally, the model is applied to an example of parentalcare evolution. Overall, our model allows for the evolutionary analysis of social traits in finite and dioecious populations, where
interactions can occur within and between sexes under a realistic scenario of reproduction.
I
N the absence of mutation, the change in allele frequency is
the result of natural selection and genetic drift. Natural
selection favors alleles that maximize their representation
within the gene pool, and a large body of work has investigated how alleles achieve this by increasing the expected
number of offspring produced by their carriers. However,
genetic drift, which arises from randomness in reproduction,
reduces the efficacy of natural selection, thereby slowing
down or even preventing adaptation altogether.
While many studies have investigated how natural selection affects the expected number of offspring produced by
an individual, less attention has been given to the degree
to which selection acts on the variance in offspring number,
or reproductive variance. Gillespie (1974, 1975, 1977) inCopyright © 2014 by the Genetics Society of America
doi: 10.1534/genetics.113.156067
Manuscript received August 3, 2013; accepted for publication October 16, 2013;
published Early Online October 22, 2013.
Available freely online through the author-supported open access option.
Supporting information is available online at http://www.genetics.org/lookup/suppl/
doi:10.1534/genetics.113.156067/-/DC1.
1
Corresponding author: Department of Genetics, Evolution and Environment,
University College London, Darwin Bldg., Gower St., London WC1E 6BT, United
Kingdom. E-mail: [email protected]
vestigated how natural selection dampens randomness in
within-generation fertility in a haploid population. He demonstrated that between two alleles that on average produce the same number of offspring, natural selection favors
the allele that produces offspring with lesser variance when
the breeding adults of the next generation are sampled from
a finite pool of offspring.
Reproductive variance also correlates with the intensity
of genetic drift. By decreasing effective population size,
reproductive variance mitigates the effect of selection (Wright
1931). Gillespie (1974)’s haploid model also revealed that
the level of genetic drift affecting the segregation of two
alleles increases with the reproductive variance they code
for. As a consequence, fixation of the allele coding for lower
fertility variance reduces the intensity of genetic drift and
facilitates adaptive evolution in future generations.
The variance in fertility considered in Gillespie’s (1974,
1975, 1977) seminal articles had arbitrary causes and could
have resulted from randomness at any stage of an individual’s
life history, such as its development, its fertility, or the survival
of its offspring. Extensions of Gillespie’s models have since investigated the effect of selection against reproductive variance
Genetics, Vol. 196, 235–252
January 2014
235
in the context of more specific life histories. Shpak (2007)
investigated the evolution of this variance in age-structured
populations and showed that selection favors alleles that code
for lower stochasticity in age-specific survival and fertility. Selection against reproductive variance has also been demonstrated
to affect the evolution of traits as diverse as sex allocation in
hermaphrodites (Proulx 2000), dispersal in structured populations (Shpak 2005; Shpak and Proulx 2007; Lehmann
and Balloux 2007), and helping behaviors in social animals
(Lehmann and Balloux 2007; Beckerman et al. 2011).
These models have highlighted that selection against
reproductive variance may be a subtle yet significant force in
the evolution of many different traits in natural populations
(Rice 2008, for a general discussion). However, it remains
unclear how selection on reproductive variance, and its
feedback with genetic drift, affect the reproductive biology
and life history of sexual organisms. Most models so far have
omitted sex altogether. The only study that has taken into
account selection on reproductive variance in a dioecious
(two-sexes) population approximated reproduction as the
random union of gametes and assumed that gamete production of different individuals was uncorrelated (Taylor 2009).
These assumptions miss fundamental aspects of the reproductive biology of the vast majority of organisms, where it is
individuals, rather than gametes, that unite to mate. But
considering a realistic mating system would have significant
consequences for the variances and covariances in offspring
number among individuals of one sex and how these (co)
variances differ across the sexes (Bateman 1948; Wade
1979, for examples). If the effect of the mating system on
genetic drift has successfully been captured by calculations
of the effective population sizes (Nunney 1993; Nomura
2002), a more general approach is needed to make evolutionary predictions that incorporate selection acting on the
traits that generate this variance.
In this article, we present a population-genetic model of
male and female phenotypic evolution that makes it possible
to predict the evolution of sex-specific reproductive traits
under the influence of selection and genetic drift. Using an
individual-based approach, the model incorporates a description of the mating system based on the first and second
moments (means and (co)variances) of the distribution of
individual offspring number. The article is divided into two
broad sections. In the first part of the article, we present our
model. We start by deriving the probability of fixation of
a mutant allele that affects male and/or female reproduction
in a finite dioecious population. Our derivation accounts for
sex-specific levels of reproductive variance, as well as for
covariances between members of the same sex. We then
extend the analysis of short-term evolutionary change by
deriving predictions of long-term phenotypic evolution,
which in turn makes it possible to calculate the equilibrium
for sex-specific phenotypes based on the probability of
fixation and properties of the mutational input. In the
second part of the article, we provide an illustration of
how our model can be applied and study the effects of sex-
236
C. Mullon et al.
specific reproductive variance on the evolution of various
parental-care strategies. These simple examples allow us to
demonstrate that sex-specific reproductive variance can lead
to differences between the probability of fixation for mutants
affecting female traits and those affecting male traits and
between the phenotypic equilibria of these traits.
Model and Results
Probability of fixation in dioecious populations
We derive the probability of fixation of a mutant allele (A)
introduced as a single copy into a population fixed for a resident allele (a). The population is dioecious and of any, but
constant, size, with Nm adult males and Nf females. Generations are nonoverlapping. The mutant A allele alters the
expression of a continuously varying phenotypic trait, which
may affect one or more aspects of reproductive biology, such
as mating success, fecundity, or offspring survival. The trait
can have different values in males and females and we denote by zm the phenotypic value of a male homozygous for
the resident allele (genotype aa). The phenotype of a heterozygous male (Aa) is denoted by zm + hdm, where h is the
dominance coefficient of A. A male homozygous for the mutant allele (AA) has phenotype zm + dm. Similarly, the phenotypes of the three genotypes in a female are zf (aa), zf +
hdf (Aa), and zf + df (AA).
Weak selection approximation: The fixation probability of
the mutant is derived using an individual-based approach
that builds on previous works (Rousset 2003; Roze and
Rousset 2004; Lessard and Ladret 2007; Lehmann and Rousset
2009; and supporting information, File S1). Under weak
selection (small mutant deviations dm and df), and if the
mutation rate is the same in males and females, the fixation
probability of a single mutant copy arising at random in a
monomorphic population with phenotypes zm in males and
zf in females is
1
þ dm Km zm ; zf ; h Gm zm ; zf
p zm ; zf ; dm ; df ; h ¼
2N
þ df Kf zm ; zf ; h Gf zm ; zf þ O d2 ;
(1)
where N = Nm + Nf is the total number of adults, and d is
such that dm O(d), df O(d) (File S1, Equations SI.1–
SI.39). The functions Gm(zm, zf) and Gf(zm, zf) are fitness
gradients: they measure the effect of the mutant on male
and female fitness and are further explained below (Fitness
gradients section). The functions Km(zm, zf, h) and Kf(zm, zf, h)
are measures of the variance of mutant frequencies in males
and females over the segregation process, from the appearance until the eventual fixation of the mutant. They are
inversely proportional to the intensity of genetic drift and
capture the efficacy with which selection acts on the mutant
(see Efficacy of selection section). Whether a mutant is
under positive selection [p . 1/(2N)], evolves neutrally
[p = 1/(2N)], or is counterselected (p , 1/(2N)), therefore
depends on the balance between male and female mutant
effects dm and df, the fitness gradients Gm and Gf, and the
measures of genetic variances Km and Kf. In the following
sections, we lay out how these quantities depend on the
reproductive system of a population.
u, i.e., the expected number of its adult offspring of sex u,
can then be calculated in terms of i’s reproductive success,
relative to that of the total population
Fitness gradients: The fitness gradients express how the
expected number of adult offspring of a focal individual
changes in response to small alterations of its phenotype.
They are given by
!
f z ;z
@wm
;
z
;
z
@w
;
z
z
N
mi
2mi
mi
2mi
f
f
m
m
m
Gm zm ; zf ¼
þ
zmi ¼ zm
@zmi
@zmi
Nf
u
u
u
; Jv2
; . . . ; JvN
ÞT is the realized offspring prowhere Juv ¼ ðJv1
v
duction of all parents in the population. Note that the total
number of juveniles of either sex must be the same when
counted as the offspring of males or females (i.e.,
P u
P u
¼
J ). We assume nonextinction of the populak Jmk
P u k fk
tion ð k Jvk $ Nu Þ. To describe the fitness of individual i, we
need to calculate the expectation of Equation 3 over the
distribution of Juv . Following the approach of previous work
(Gillespie 1975; Proulx 2000; Shpak and Proulx 2007; Lehmann
P u
is approximated
and Balloux 2007; Rice 2008), E½Jviu = k Jvk
using the delta method
(Oehlert
1992),
so
that
the expected
fitness wuvi ¼ E½wuvi Juv becomes
0
z2mi ¼ zm
zf ¼ zf
(2a)
Gf zm ; zf
wuv
0
1
@wff zfj ; z2fj ; zm
@wm
zfj ; z2fj ; zm N
f
f
A
þ
¼@
zfj ¼ zf ;
@zfj
@zfj
Nm
z2fj ¼ zf
zm ¼ zm
(2b)
where
is the expected number of adult offspring of sex u
of a focal individual of sex v 2 {m, f}. This fitness function
depends on the phenotype zvi of the focal individual of sex v.,
P
on the average phenotype z2vi ¼ Nk6¼v i zvk =ðNv 2 1Þ among
sex v, but excluding the focal, and on the average phenotype
P
in the population of the opposite sex (zm ¼ Ni m zmi for
PNf
males and zf ¼ j zfj for females). The model can thus
easily accommodate for sex-specific interactions based on
games, like the classic battle of the sexes. The derivatives
of focal fitness wuv are evaluated at the resident phenotypes
zmi ¼ z2mi ¼ zm ¼ zm , zfj ¼ z2fi ¼ zf ¼ zf , so that Gm and Gf
measure the effects of phenotypic changes on male and female fitness with respect to the resident population.
The fitness gradients in Equation 2a indicate the direction
of phenotypic evolution in each sex that is favored by
selection. If Gm(zm, zf) and Gf(zm, zf) are positive, then selection favors an increase in the trait in males and females; if
they are negative, then selection favors a decrease. Although
the gradients are derivatives of the expected average numf
ber of adult offspring produced (wm
m and wm ), the direction
of selection depends on how the phenotype affects the average number of juvenile offspring produced as well as reproductive variance. To demonstrate why this is the case, we
derive the fitness of a focal individual i of sex v in terms of
the distribution of juveniles in the population. Fitness is the
expected number of i’s offspring that become part of the
adult breeding population of the next generation. We separate fitness gained through male and female offspring. We
write Jviu for the number of juvenile offspring of sex u born to
i, itself of sex v. In each generation, the set of reproductive
individuals is established by independently sampling Nm
males from a pool of surviving male offspring and Nf females
from a pool of surviving female offspring. The conditional
fitness of individual i of sex v gained through offspring of sex
wuvi Juv ¼ Nu
u
Jvi
P
u þ
u ;
Jvi
Jvk
(3)
k6¼i
B
Bmu mu 2 mu
muvi X u
B
n
wuvi ¼ Nu B viu 2 T u3 vi nuvi þ u3
B mT
mT
mT k6¼i vk
@
1
C
muvi X X u
muT 2 2muvi X u C
C
þ u3
rvkl 2
rvik C þ R;
C
mT k6¼i
mu3
T
k6¼i
A
(4)
l6¼i
l6¼k
where muvi is the expected number of juveniles of sex u proP
duced by individual i, muT ¼ k muvk is the expected total
number of juveniles of sex u produced in the population,
nuvi is the variance of the number of offspring of individual
i ðnuvi ¼ V½Jviu Þ, and ruvik is the covariance between the number of offspring of sex u of individuals i and k of sex v
u
Þ. The fitness of an individual therefore
ðruvik ¼ C½Jviu ; Jvk
takes into account all first and second moments of the probability distribution that describes individual reproduction (see
Figure 1 for a depiction of those moments for a focal male).
The remainder R in Equation 4 is composed of central
cross moments of Juv of order 3 and higher. These terms
may be significant in certain scenarios (Rice 2008), but we
omit them in this analysis by assuming that the distribution
of Juv is well behaved as the population size N increases.
Previous models used the central limit theorem to justify
this assumption (Shpak and Proulx 2007; Lehmann and
Balloux 2007, Equation A6). This is not strictly valid here
because the number of offspring produced by different individuals is not necessarily independent. However, the remainder terms can be ignored if we assume that offspring
numbers are close to independence and that the “total” covariance between a given set of individuals decreases as the
number of individuals in that set increases (see Appendix,
Sex-Specific Reproductive Variance
237
Figure 1 The moments of male reproduction. At generation t, a focal
m
male i sires an expected mm
mi number of male offspring with variance nmi
and covariance rm
with
the
number
of
male
offspring
sired
by
another
mik
male k. The males of the next generation t + 1 are established by sampling the juveniles and the expected number of adult males of the focal
m
male i is wmi
. Similarly, the focal male sires an expected mfmi number of
female offspring with variance nfmi and covariance rfmik with the number
of females sired by male k. Then, the expected number of adult females
f
of the focal male i is wmi
. Finally, the number of male and female offspring of male i covary by rm;f
mi .
Equation A1). In this case, the remainder terms in Equation
4 are of order (1/N2). Therefore, while the expression for
the fixation probability (Equation 1) holds for any population size, the approximation for fitness (Equation 4) takes
into account only the first-order effects of finite population
size on offspring means and variances. If condition (A1) also
holds for the first and second moments of the distribution of
reproduction, then the effects of (co)variances on individual
fitness vanish as N / N, in agreement with previous studies
(Gillespie 1974).
Equation 4 shows that individual fitness depends on four
terms. The first is the relative expected number of offspring
produced ðmuvi =muT Þ, which has a positive effect on fitness. The
remaining three terms capture the effects of reproductive
variance. Fitness decreases with the variance in offspring
number ðnuvi Þ, increases with the variance in offspring number
produced by the remaining individuals in the population
P P
P
( k6¼i nuvk and k6¼i l6¼i ruvkl ), and decreases with the covariance between the number of offspring produced and that of
P
the remaining individuals in the population ð k6¼i ruvik Þ.
The fitness effects of the variance terms stem from the
nonlinear relationship between fitness and the offspring
production of the focal (Jviu ; see Figure 2 and Equation 3)
P
u
; see Figure 2B and
and the rest of the population ( k6¼i Jvk
Equation 3). For a given number of offspring produced by
the rest of the population, the fitness returns of a focal individual diminishes with its production of more offspring as
a consequence of the increased competition between related
juveniles for access to breeding. This results in a net negative effect of variance in the focal individual’s reproductive
output on its fitness (nuvi in Equation 4 and Figure 2A). Conversely, for a given offspring production by the focal, the
advantage of competing within a population of individuals
238
C. Mullon et al.
that are on average less fecund is expected to be greater
than the disadvantage of competing in a more productive
population. This leads to a net positive effect of population
P
variance on the focal individual’s fitness ( k6¼i nuvk and
P P u
k6¼i
l6¼i r vkl in Equation 4 and Figure 2B). Finally, using
arguments similar to those presented in Figure 2, one can
see that the benefit of overperforming in a less competitive
population is on average greater than the cost of underperforming in a more competitive population. As a consequence,
the covariance between the numbers of offspring produced
by the focal individual and the rest of the population has
P
a negative impact on focal fitness ( k6¼i ruvik in Equation 4).
Selection on a phenotype (Equation 2) then reflects the
balance between the impact of the trait on the different
terms of Equation 4. Since the difference between two
phenotypes are small [of order O(d)], we can describe the
dependence between the moments and phenotypes without
explicitly characterizing the interactions between every individual in the population. Rather, we average the sums in
Equation 2 over mean population phenotypes (File S1,
Equation SI.4). Then, if the trait of interest affects all first
and second moments of individual reproduction, the fitness
function of a focal individual of sex v can be written as
mu ðzvi ;z2vi ;zoðvÞ Þ
wuv zvi ; z2vi ; zoðvÞ ¼ NNuv mv u z ;z ;z
v ð v v oðvÞ Þ
2 u 1
nuv zvi ; z2vi ; zoðvÞ
2
Nv mv ðzv ;zv ;zoðvÞ Þ
mu ðzvi ;z2vi ;zoðvÞ Þ u
nv zv ; zv ; zoðvÞ
2 mv u z ;z ;z
v ð v v oðvÞ Þ
2 u Nv 2 1 2 ruv zvi ; z2vi ; zoðvÞ
Nv mv ðzv ;zv ;zoðvÞ Þ
mu ðzvi ;z2vi ;zoðvÞ Þ u
rv zv ; zv ; zoðvÞ
2 mv u z ;z ;z
v ð v v oðvÞ Þ
þ O 1=N 2 ; d2 ;
(5)
where zoðvÞ denotes the average phenotype zoðvÞ of the sex
opposite to that of the focal, v, (e.g., zoðfÞ ¼ zm ). The function
muv ðzvi ; z2vi ; zoðvÞ Þ is the expected number of juveniles of sex u
produced by a focal of sex v, and muv ðzv ; zv ; zoðvÞÞ is the average expected number of juveniles of sex u produced by sex v
individuals in the population; similar interpretations are
given to the variance and covariance functions (nuv and
ruv ). Therefore, calculating the individual fitness functions
wuv that go into the fitness gradients (Equation 2) requires
characterizing only the individual mean, variance, and covariance functions, (muv , nuv , and ruv ), and these depend only
on the phenotype of the focal individual (zvi), the average
phenotype in the opposite sex ðzoðvÞÞ, and the average among
other individuals of the same sex (z2vi , as the average zv is
written as zv ¼ ðNv 2 1Þz2vi =Nv þ zvi =Nv ). Examples of such
calculations are given in the Example section.
Figure 2 Effects of variance on focal fitness. (A) Fitness of a focal individual
graphed against the random number of
offspring it produces and holding the
rest of the population constant. Ignoring
the sex of parent and offspring, the focal
produces on average mi offspring with
variance s2i . It then produces more or
less than mi offspring. But fitness is a relative measure of reproductive success
(Equation 3). The advantage of producing more offspring depreciates with the
number of offspring produced because sibs also compete against each other. Then, the benefits reaped when it produces more offspring than average
(gray arrow) are outweighed by the cost when it produces less (black arrow). (B) Fitness of a focal individual graphed against the random number of
offspring produced by the rest of the population and by holding the number of offspring of the focal constant. The rest of the population produces on
average m2i offspring with variance s22i . The fitness function of a focal individual is convex with respect to the reproductive output of the rest of the
population, which means that the benefits it reaps when they produce less (gray arrow) outweighs the cost paid when they produce more (black arrow;
see also Frank 2011).
The first line in Equation 5 reflects the fact that an individual who produces on average a greater number of offspring
than the average individual in the population has higher
fitness. The second and third lines reflect the fact that an
individual with a lower variance in progeny number than
the average individual has higher fitness, as originally
described by Gillespie (1974). Finally, the last two lines of
Equation 5 reflect the fact that an individual whose offspring
production covaries with that of another individual to
a lesser degree than the average individual also has higher
fitness. In addition, we see that the effect of covariance on
fitness [of order (Nv 2 1)/Nv] is potentially greater than that
of the variance (of order 1/Nv).
Efficacy of selection: In addition to the fitness gradients, the
fixation probability (Equation 1) also depends on Km(zm, zf,
h) and Km(zf, zf, h), which weigh on the fitness gradients
and measure the sex-specific efficacy of selection in males
and females, respectively. The weight Km (Kf) is the
expected covariance between between genotype and phenotype in males (females), cumulated over the neutral segregation of the mutant. Mathematically, this is
#
"
N
X
1 11
(6)
E°
C½zui ; pui t ;
Ku ¼
2N 4 du
t¼0
where C[zui, pui]t is the covariance between the phenotype
zui of an individual of sex u and the frequency pui 2 {0, 1/2,
1} of the mutant it carries at generation t, and E°[] denotes
expectation under neutral evolution, i.e., where only genetic
drift affects fluctuations of genotypic frequency pui from one
generation t to the next (File S1, Equations SI.20, SI.21, and
SI.39). The factor 1/(2N) in Equation 6 is the initial frequency of the mutant, while the factor 1/4 is the product
of the frequency of transmission of a gene by a parent to an
offspring (i.e., 1/2; File S1, Equation SI.2) and the reproductive value of that class of offspring, which is here 1/2 for
both males and females. If mutant effects are additive (h =
1/2), then C[zui, pui]t = V[pui]t and Ku reduces to the cumulative genetic variance in sex u, highlighting that the larger
genetic variance is, the more efficient selection can be. In the
simple case of asexual haploids, this variance reduces to the
familiar pt(1 2 pt), where pt is the average frequency of the
mutant at generation t. More generally, Km (Kf) depends on
dominance and captures the association between phenotypic and genetic variance in males (females) on which selection is then able to act.
Since the calculations for Km and Kf are made over the
segregation of a neutral mutant (Equation 6), they can be
expressed in terms of coalescence times of neutral genes
(File S1, Equations SI.37–SI.38), which themselves can be
expressed in terms of how genes coalesce within individuals
of different sexes. Doing so links Km and Kf back to the
mating system and hence to the evolving reproductive traits
zm and zf that are under study. The general expressions for
Km and Kf in terms of the coalescence process depend on the
probabilities that individuals share the same parent in the
absence of selection, referred to here as “probabilities of
sibship.” We find that the coalescence process can be described by 14 probabilities of sibship. Six of these describe
the probability that a pair of individuals share the same
parent; they are written as Quv , where u 2 {♂, ♀} indicates
the sex of the parent, and v 2 {m, f, c} indicates whether
a pair of individuals consist of two males, two females or
a male and a female (Figure 3). The remaining eight probabilities of sibship describe the probability of three individuals (three males, two males and a female, two females and
a male, or three females) sharing the same parent (male or
female). Providing a general characterization of the neutral
coalescence process is complicated, but the system can be
simplified by taking into account only the first-order effects
of finite population size [O(1/N)]. In this case, we find that
the eight three-way probabilities of sibship may be expressed in terms of the pairwise probabilities Quv ’s (File S1,
Equations SI.40–SI.42), which are then sufficient to describe
the entire coalescence process.
The pairwise probabilities of sibship Quv capture different
aspects of reproductive variance. The probabilities that two
males have the same father ðQ♂
m Þ, that two females have the
Þ,
and
that
a
male
and female have the same
same father ðQ♂
f
Sex-Specific Reproductive Variance
239
Table 1 The probabilities of sibship
v
m
f
c
m;f
m
m 2
f
f 2
m f
Q♂
v 1=Nm ð1 þ nm =ðmm Þ Þ 1=Nm ð1 þ nm =ðmm Þ Þ 1=Nm ð1 þ rm =mm mm Þ
m;f
m
m 2
f
f 2
m f
Q♀
1=N
ð1
þ
n
=ðm
Þ
Þ
1=N
ð1
þ
n
=ðm
Þ
Þ
1=N
ð1
þ
r
=m
f
f
f
v
f
f
f
f
f mf Þ
f
The first row gives the paternal probabilities of sibship and the second row gives the
maternal probabilities of sibship. The moments m and v terms are defined in the
m f
main text, except for rm;f
m ¼ C½Jmi ; Jmi ; which is the covariance between the number
of male and offspring juveniles fathered by a resident male (Figure 3), and
¼ C½Jf mj ; Jf f j ; which is the covariance between the number of male and female
rm;f
f
offspring a resident female gives birth to. Therefore, the probabilities of sibship
increase with reproductive variance.
Figure 3 The paternal probabilities of sibship. With probability Q♂
m , two
males sampled at generation t + 1 have the same father from generation
t. So, with probability 1 2 Q♂
m , they come from different fathers. Similarly,
a male and a female sampled at generation t + 1 have the same father
♂
with probability Q♂
c , and two females do so with probability Qf .
father ðQ♂
c Þ measure the level of reproductive variance of
adult males and depend on the moments of the distribution
of male reproduction (Table 1). In a situation where all
males contribute equally to the next generation of individuals of either sex, the paternal probabilities of sibship are all
Q♂
x ¼ 1=Nm (x 2 {m, f, c}). With skewed paternity, the variance in the number of offspring a male has increases and so
do the paternal probabilities of sibship (Table 1). The pater♂
♂
nal probabilities of sibship Q♂
m , Qf , and Qc differ from one
another if the variance in the number of sons produced
differs from the variance in the number of daughters produced, i.e., if gene transmission is more variable through
offspring of one sex than through offspring of the other sex.
♂
The difference between maternal ðQ♀
x Þ and paternal ðQx Þ
probabilities of sibship reflect the difference between the
reproductive variances of males and females. In a monogamous population, with equal number of males and females,
males and females have the same reproductive variance and
♂
Q♀
x ¼ Qx . In contrast, a polygynous population has greater
♂
reproductive variance in males than females ðQ♀
x , Qx Þ,
while a polyandrous population exhibits the opposite pat♂
tern ðQ♀
x . Qx Þ.
Using the probabilities of sibship, we can express the
weights Ku in terms of the coalescence process of a neutral
gene. In the simplest case, where a mutant is additive and
there is no difference between the probabilities of a gene
being transmitted through a son or a daughter (Q♂ ¼ Q♂
m ¼
♂
♀
♀
♀
♀
¼
Q
and
Q
¼
Q
¼
Q
¼
Q
),
K
(Equation
A2)
simQ♂
u
f
c
m
c
f
plifies to
!
1
1
1
12
Ku zm ; zf ; 1=2 ¼
:
2N
Nu
Q ♂ zm ; zf þ Q ♀ z m ; zf
(7)
This expression decreases hyperbolically with the probabilities
of sibship. Thus, as reproductive variance increases in males
and females, the transmission of the gene from one generation
to the next becomes more stochastic and genetic variance is
reduced. As a consequence, the efficacy of selection Ku
decreases, reflecting the smaller impact of selection on the
240
C. Mullon et al.
probability of fixation in the face of increased drift (Equation 1).
Similar patterns are observed when reproductive variance varies with the sex of the parent and the sex of the
offspring. Analytical results for additive mutants (Equation
A2, Figure 4, A–C) and numerical results for nonadditive
mutants show that Km and Kf both decrease hyperbolically
with all six probabilities of sibship. Numerical results also
show that Km and Kf increase linearly with dominance (Figure 4D). This stems from the fact that dominance increases
the phenotype–genotype covariance at lower allele frequencies (pui , 1/2) and that the frequency of neutral mutants
remains on average low.
Because Km and Kf weight male (Gm) and female (Gf)
fitness gradients independently in the probability of fixation
(Equation 1), reproductive variance may affect male and
female evolution differently. For example, selection on
females has a greater impact on the probability of fixation
than selection on males when Km , Kf. In this case, a female-limited mutation (dm = 0 or Gm = 0) would have
a greater chance of reaching fixation than a male-limited
mutation (df = 0 or Gf = 0), even if both improve fitness
by the same amount. In the longer term, we would then
observe a faster rate of adaptation in females than males.
The reverse patterns are predicted when Km . Kf.
Differences between Km and Kf, and hence differences
between the efficacy of selection in males and females, occur
whenever genetic variance is lower in one sex than in the
other. For additive mutants (Equation A2 and Figure 4,
A and B), Km , Kf requires that
♀
♂
♀
Q♂
m þ Qm . Qf þ Qf ;
(8)
i.e., the probability that two males have at least one parent
in common exceeds the probability that two females have at
least one parent in common. This inequality (Equation 8)
reflects that if at each generation male offspring are more
related than female offspring, then genetic variance in males
is lower than in females, and as a consequence, selection is
less efficient in males.
Calculating the probability of fixation: Based on the above
derivations, the probability of fixation can be explicitly
calculated, taking into account the fitness change caused
by the mutant, through its effect on first and second moments
Figure 4 The effect of probabilities of sibship and dominance on the weights Km and Kf. The effects of the probabilities that two males (A) and two females (B) and that
a male and female (C) are sibs are shown for an additive
mutant (Equation A2) when all the other probabilities of
sibship are fixed at 0.1. In A and B, Km is shown in blue
and Kf is shown in red. How they change with the paternal (Q♂) and maternal (Q♀) probabilities of sibship is
shown as solid and dashed lines respectively. In C, the
effects of paternal (Q♂) and maternal (Q♀) probabilities
of sibship on Km and Kf are equal and shown as a single
solid line. In D, we solved for Km (blue) and Kf (red) (File
S1, Equation SI.38) for 100 different values of dominance
h, with probabilities of sibship randomly perturbed around
0.1 (each probabilities of sibship was sampled from a normal distribution with mean 0.1 and variance 0.02).
of the distribution of offspring production and the impact of
segregation in the two sexes on the efficacy of selection. To
calculate the fixation probability, the probabilities of sibship
(Table 1) are substituted into Km and Kf (Equation A2 if the
mutant is additive and File S1, SI.38 otherwise), and the
expressions for focal fitness (Equation 4) are substituted into
the fitness gradients Gm and Gf (Equation 2). Finally, Km, Kf,
Gm, and Gf are substituted into the fixation probability
(Equation 1).
Long-term phenotypic evolution in
dioecious populations
The fixation probability of a mutant is useful for predicting
short-term evolution and to understanding how the interplay between selection and genetic drift affects the fate of
a new mutation. However, it is often desirable to predict the
long-term evolution of phenotypes as a result of selection
and drift acting on an influx of new mutations. In this
section, we use the probability of fixation (Equation 1) to
determine the phenotypes most likely to be observed in
males and females at a selection–mutation–drift balance. To
that aim, we assume that the autosomal locus mutates at
a constant rate j. This rate is sufficiently weak compared to
the rate of fixation to ensure that there are only ever two
alleles segregating, thus complying with the weak-mutation
strong-selection limit of population genetics (e.g., Gillespie
1994; Sella and Hirsh 2005) and/or the trait substitution
sequence limit of evolutionary game and inclusive fitness
theory (e.g., Metz et al. 1995; Champagnat and Lambert
2007; Lehmann 2012). The effects (dm, df, h) of a mutation
are drawn from a distribution u(dm, df, h), which is such that
the dominance (h) of a mutant is independent from its
homozygotic effects (dm, df), and mutants have on average
no phenotypic effect E[dm] = E[df] = 0. The rate at which
a population monomorphic for (zm, zf) is substituted by
a population with traits (zm + dm, zf + df) can then be
written as
1
@p
þ df
k zm ; zf ; dm ; df ; h ¼ 2Nj u dm ; df ; h
2N
@df
@p
þ O d2 ;
(9)
þ dm
@dm
where 2N is the number of gene copies in the population,
ju(dm, df, h) is the probability that a single copy produces
a mutation of type (dm, df, h), and the term within brackets
is the probability that this mutation fixes in the population,
which is given by Equation 1.
The substitution rate k determines a jump process
(Gardiner 2009), which describes the stochastic evolution
of male and female traits as jumps between monomorphic
states in phenotypic space. Ignoring terms of order O(d2) in
Equation 9, the jump process can be described in continuous
time by a diffusion process that eventually reaches a stationary distribution c(zm,zf) (Appendix, Diffusion equation for phenotypic evolution in dioecious populations). This long-run
stationary state reflects a balance between the forces of mutation, selection, and genetic drift, and the maxima of c(zm,
zf) correspond to phenotypes around which the populations
spend the greatest amount of time. These maxima are the
most likely outcomes of phenotypic evolution, and in single
phenotype models, they are the “convergence stable” states of
the system (Lehmann 2012). A phenotype is convergent stable
if populations sitting close to this phenotype are attracted toward it. Convergence stability is an important concept of attainability of equilibrium points, common to evolutionary
game and inclusive fitness theory (Rousset and Billiard
2000; Leimar 2009). Under the trait-substitution sequence
limit, a phenotype that is convergent stable is also evolutionary stable (Wakano and Lehmann 2012).
Sex-Specific Reproductive Variance
241
Phenotypes that are multidimensional convergence stable can be found by considering the attractor points of the
system of differential equations
¼ 2Nξ bmm Km zm ; zf ; h Gm zm ; zf þ bmf Kf zm ; zf ; h Gf zm ; zf
dzf
dt ¼ 2Nξ bmf Km zm ; zf ; h Gm zm ; zf þ bff Kf zm ; zf ; h Gf zm ; zf
dzm
dt
(10)
which describes the deterministic trajectory associated to
the underlying diffusion process. Here, h ¼ E½h is the average dominance of the mutation distribution and buv = C[du,
dv] is the covariance between mutation effect in sex u and v.
If none of the attractor points ðz*m ; z*Þ
f of system Equation 10
lie on the boundary of the phenotypic space, then large deviation theory shows that as the population size grows, the
stationary distribution c(zm, zf) becomes peaked around
these attractor points (use Theorem 4.3 of Freidlin and
Wentzell 2012, p. 170, and observe that if none of the
attractor points ðz*m ; z*Þ
f lies on the boundary of the phenotypic space, then a smooth domain can be drawn around all
attractor points, thereby satisfying condition A of Freidlin
and Wentzell 2012, p. 150). Therefore, when all attractor
points ðz*m ; z*Þ
f of Equation 10 lie in the interior of the phenotypic space, they correspond to the convergence stable
states and are the most likely phenotypic outcomes of evolution. Furthermore, in the infinite size limit (N / N), the
stationary distribution becomes fully concentrated around
a single of these convergence stable states (Theorem 4.2
of Freidlin and Wentzell 2012, p. 167), which corresponds
to the highest peak of the adaptive landscape and the stochastic stable state of the system (e.g., Foster and Young
1990; Van Cleve and Lehmann 2013).
For an interior point (zm, zf) of system Equation 10 to be
convergence stable, two conditions must be satisfied. First, dzm/
dt = 0 and dzf/dt = 0 must hold, but since Km ðzm ; zf ; hÞ . 0
and Kf ðzm ; zf ; hÞ . 0, this condition is equivalent to
G m zm
* ; zf* ¼ Gf zm
* ; zf* ¼ 0;
(11)
i.e., that the male and female fitness gradients vanish, which
is equivalent to the condition for establishing singular points
in deterministic evolution (Leimar 2009). Second, the real
part of all the eigenvalues of the Jacobean matrix of Equation 10 must be negative. Combined with Equation 11 and
the properties of buv (Leimar 2009), this latter condition is
equivalent to the real part of the eigenvalues of
Km @Gm [email protected]
Kf @Gf [email protected]
Km @Gm [email protected]
Kf @Gf [email protected]
(12)
being negative at ðz*m ; z*Þ.
Conditions (11) and (12) extend
f
the one-dimensional condition of convergence stability for
finite populations (Rousset and Billiard 2000; Lehmann
2012), and when Km = Kf, it is equivalent to the condition
for multidimensional convergence stability for populations
of infinite size (Leimar 2009), which depends only on the
242
C. Mullon et al.
fitness gradients Gm and Gf. When attractor points ðz*m ; z*Þ
f of
Equation 10 lie on the boundary of the phenotypic space or
outside of it, the shape of the equilibrium distribution cannot in general be assessed (to the best of our knowledge),
and in this case, characterizing convergence stable points is
not straightforward.
Example
In this section, we illustrate a possible application of our
model by analyzing the evolution of maternal and paternal
care behaviors. The emphasis is on highlighting the effects of
selection on reproductive variance in driving the evolution of
reproductive traits. We consider a dioecious species with
a simple life cycle. An equal number N of adult males and
females randomly pair up to mate. All females mate once
with a single male, but males mate with harems of exactly
H females. If H = 1, the population is monogamous and all
males mate. If H . 1, then the population is polygynous and
some males mate H times while others do not reproduce at
all. Each female gives birth to exactly f offspring with an equal
sex ratio. The offspring survive with probabilities that depend
on the phenotypes zm and zf of their parents. Female offspring
each survive independently from each other with probability
sf(zm, zf). In contrast, the survival of males is strongly correlated within broods and the males of a brood either all survive
[with probability sm(zm, zf)] or all die [with probability 1 2
sm(zm, zf)]. This is close to the assumptions of the hermaphroditic model of Proulx (2000). The difference in male and
female survival could reflect sex differences in the susceptibility to random variation in the breeding environment provided by the male’s territory, for example. As a consequence
of their correlated survival, surviving males are more related
than surviving females. The next generation of adults is randomly sampled from the population of surviving offspring as
described in the previous section.
The phenotypes that evolve are the level zm of paternal
care provided by a male and the level zf of maternal care
provided by a female. The survival probabilities of daughters,
sf(zm, zf), and sons, sm(zm, zf), are both increasing functions of
zm and zf. We analyze the fate of four different types of mutations altering the parental-care phenotypes. These types are
characterized by the sex of the parent providing the care and
the sex of the offspring receiving it. We distinguish between
mutations that affect care for sons only and those that affect
care for daughters only. For both of those, we consider sexlimited mutations that affect care only in males or only in
females. In total, the evolution of four different traits is studied, paternal care for daughters, maternal care for daughters,
paternal care for sons, and maternal care for sons.
The covariance between the survival of male offspring
depends on the sex of the parent for which care evolves. If
female care is evolving, then for each female, her entire
male brood either survives or dies, independently from other
females, even from those that have mated with the same
male. When care is provided by males, then for each male,
his entire male brood either survives or dies, independently
from other males, but his brood includes all the male
offspring he has had with different females. We assume
that the effect of mutations is additive and identical for all
traits. Thus, compared to the resident homozygote, the level
of care is increased by an amount d/2 in heterozygotes and
by d in mutant homozygotes.
Probability of fixation of new mutants
To calculate the fitness of a focal individual (Equation 5)
and evaluate the fixation probability of the different
mutants, we need to express the means and covariances of
offspring production in terms of care phenotypes. We first
consider the offspring production of a focal female j with
phenotype zfj. Because the evolution of male and female
traits are treated separately, mutations for altered maternal
care always occur in the presence of constant resident paternal care zm, and we can therefore omit paternal care from
the survival functions. The expected number of daughters
ðmff Þ and sons ðmm
f Þ the focal female produces are given by
sf zfj
f
;
(13a)
mf zfj ¼ f
2
mm
f
sm zfj
;
zfj ¼ f
2
(13b)
where the factor 1/2 stems from the equal sex ratio.
The variance terms that contribute to the fitness of a focal
female are found by writing n♀ and n♂ as the number of
daughters and sons, respectively, at birth before survival
selection. With the sex ratio being equal, n♀ Bin(f, 1/2)
(and n♂ = f 2 n♀). Given n♀, the variance in the number of
female offspring born to a female is n♀sf (zfj)(1 2 sf(zfj)),
and the mean is n♀sf(zfj). Therefore, applying the law of
total variance, the variance in the number of female offspring born to a focal female is
nff zfj ¼ En♀ n♀ sf zfj 1 2 sf zfj þ Vn♀ n♀ sf zfj
(14)
fsf ðzfj Þ
sf ðzfj Þ
12 2
:
¼ 2
In contrast, since sons do not survive independently from
each other, we have, given n♂, that the variance in the number of male offspring is n2♂ sm ðzfj Þð1 2 sm ðzfj ÞÞ and that the
mean is n♂sm (zfj). Thus, the variance in the number of sons
produced by a focal female differs from the variance in the
number of daughters and is
2 m m z
þ Vn♂ n♂ sm zfj
nm
fj
f zfj ¼ En♂ n♂ s zfj 1 2 s
¼
fsm ðzfj Þ 1 þ fsm zfj 1 2 sm zfj :
4
(15)
Finally, since females give birth to and care for their offspring
independently of one another, the covariance between the
f
number of offspring of two females is zero ðrm
f ¼ rf ¼ 0Þ.
The means and variances of the offspring numbers produced by a focal male i are obtained similarly (see Appendix,
Calculations for the evolution of parental care), but in contrast to
singly mating females, polygyny leads to a negative covariance
between the numbers of offspring sired by two males. The
f
additional variance terms (rm
m for male offspring and r m for
female offspring) must be taken into account when determining the fitness of a focal male and are calculated here. Since
maternal care is now constant, we can omit the female care
phenotype zf from the survival functions. By definition, the
covariance between the number of male offspring fathered
by the focal male i and an “average” male other than i is
rm
m ðzmi ; z2mi Þ ¼ ðE½Nmati Nmat2i 2 1Þ
f2 m
s ðzmi Þsm ðz2mi Þ;
4
(16)
where E[NmatiNmat2i] is the expected product between the
number of matings of the focal male i, Nmati, and that of
another male, Nmat2i. Then, since N/H males are chosen
at random without replacement to mate exactly H times, E
[NmatiNmat2i] = (N 2 H)/(N 2 1), and
2
N2H
f m
2
1
s ðzmi Þsm ðz2mi Þ: (17)
ðz
;
z
Þ
¼
rm
m mi 2mi
N 21
4
Similarly, the covariance number of females fathered by the
focal male i and an average male is
2
N2H
f f
f
21
s ðzmi Þsf ðz2mi Þ:
(18)
rm ðzmi ; z2mi Þ ¼
N21
4
As expected, these covariances vanish in a monogamous
population (H = 1), but become increasingly negative as
fewer males mate. For large H, they contribute significantly
to the fitness of a focal male.
Fitness gradients: Substituting Equations 13a and 14 into
Equation 5 and deriving according to Equation 2b give the
fitness gradient for alleles that code for maternal care of
daughters. Similarly, substituting Equations 13b and 15 into
Equation 5 and differentiating according to Equation 2b give
the fitness gradient for alleles that code for maternal care of
sons. The fitness gradients for alleles that code for paternal
care are obtained in the same way using Equation 2a.
To identify the different effects of sex-specific reproductive variance, we first consider a population that is strictly
monogamous (H = 1). If maternal and paternal care have
the same effect on offspring survival [i.e., @sv(zm, zf)/@zf =
@sv(zm, zf)/@zm], the fitness gradients for mutants that increase maternal (Gf) or paternal (Gm) care of daughters are
identical and equal to
@sf zm ; zf
1
1
1
12
Gu zm ; zf ¼
þ
; (19)
N
Nf
@zu
s f z m ; zf
where u 2 {m, f} and zm and zf are the resident levels of
paternal and maternal care. Likewise, there is no difference
Sex-Specific Reproductive Variance
243
between the fitness gradients for mutations that increase
maternal (Gf) or paternal (Gm) care of sons, which are both
@sm zm ; zf
1
1
1
12
þ
G u zm ; zf ¼
N
N
@zu
s m zm ; zf
m
@s zm ; zf
1
;
¼
(20)
@zu
s m zm ; zf
u 2 {m, f}. The (1 2 1/N) term of Equations 19 and 20
describe the balance between the advantage of increasing
the expected number of offspring of the focal individual and
simultaneously increasing the total expected number of offspring in the population and the level of competition between kin. This is equivalent to the first term of Equation
5. It is equal for gradients describing care for sons and
daughters, in line with the fact that the effect of care on
the expected numbers of male and female offspring is identical. The second term in the brackets of Equations 19 and
20 captures the increase in fitness due to the reduction in
the variance of offspring number (reflecting the remaining
terms of Equation 5).
The variance term is greater for mutants that alter the care
of sons (Equation 20) because the variance in the number of
surviving sons is inherently greater than that of surviving
daughters. As a consequence, reducing that variance generates proportionately greater fitness benefits. The benefit of
reducing the variance in male offspring number is so large
that it completely offsets the reduction in fitness due to kin
competition (Equation 20). The benefits of decreasing the
variance in the number of surviving daughters vanish as
brood size f becomes large (Equation 19), due to the fact that
daughters survive independently of each other. As a result of
these different effects, selection favors the fixation of mutants
that increase the care of sons (Equation 20) with greater
strength than those that increase the care of daughters (Equation 19), especially when fertility is high.
Efficacy of selection: Differences in the patterns of male and
female survival also affect the efficacy of selection on paternal
and maternal care, Km and Kf. Both coefficients are calculated
using the probabilities of sibship (Equation A2), which are
themselves expressed in terms of the moments of offspring
production in Table 1. These moments are the same as those
appearing in the calculation of focal fitness and in addition
include the covariance between the number of male and female offspring produced by a resident individual (Appendix,
Calculations for the evolution of parental care, for calculations). We find that Km and Kf both increase with male and
female survival but that their difference
!
1 2 s m zm ; zf
(21)
Km zm ; zf ¼ Kf zm ; zf 1 2
2Nsm zm ; zf
[with f O(N)] depends only male survival rate sm(zm, zf).
In the extreme case where all males survive (sm(zm, zf) = 1),
selection is as efficient for male and female traits (Km = Kf).
244
C. Mullon et al.
But as male survival rate decreases, the efficacy of selection
falls more rapidly in males than females. This is caused by the
different modes of survival for male and female offspring.
Because male offspring tend to be more related than female
offspring, genetic variation in males is lower. As a consequence,
the efficacy of selection in females is greater than that in males
(Kf . Km), and alleles that code for maternal care are under
more efficient selection than those for paternal care.
Probability of fixation: Combining the weights Km and Kf
(Equation 21) with the fitness gradients Gm and Gf (Equations 19 and 20) for the probability of fixation (Equation 1),
we find that differences between male and female survival
affect the evolution of paternal and maternal care in two ways.
First, the correlation in male survival creates a stronger selection pressure on increased care for sons than for daughters.
Second, the effect of more stochastic male survival increases
drift in males and makes selection on paternal care less efficient than selection on maternal care. As a consequence of
these effects, the most probable form of parental care to evolve
in our model is maternal care for sons and the least probable is
paternal care for daughters (Figure 5 for H = 1). Obviously,
these conclusions are conditional on the assumptions underlying our analyses, most importantly that mutations affect only
male and female care, and do so equally, and that increases in
paternal and maternal care result in identical changes in the
expected survival of sons and daughters.
Mating system: Polygyny generates a negative correlation
between the reproductive output of different males, and this
affects the evolution of parental care in two ways. First,
reproductive skew in males decreases the strength of
selection on male care, due to intensified sib competition
among the surviving offspring of a male. This effect can be
seen when inspecting the fitness gradients on paternal care
for a female offspring
@sf zm ; zf
1
1
1
H
þ 2 2
12
Gm zm ; zf ¼
Ns
Nf N
@zm
s f z m ; zf
(22)
and that on a male offspring
@sm zm ; zf
1
H
12 2 ;
G m zm ; zf ¼
N
@zm
sm zm ; zf
(23)
where both equations are here shown for a population that
is strongly polygynous [H O(N)]. Equations 22 and 23
correspond to the gradients in a monogamous population
(Equations 19 and 20), with the exception of the last negative term. This term expresses the cost of intensified sib
competition and increases with the level of polyandry H.
The gradients for the care of female offspring are unaffected
by polygyny (see Equations 19 and Equation 20).
Second, polygyny affects the efficacies of selection Km
and Kf (Appendix, Calculations for the evolution of parental
Figure 5 Approximate probabilities of fixation of parental care strategies
against harem size. The probabilities of fixation scaled to the total gene
number, 4Np, of paternal (first column) and maternal (second column)
care are shown for populations with N = 12 (first row), N = 36 (second
row), and N = 108 (third row) males and females against harem size H.
Care for sons is shown in blue and care for daughters in red. The monogamous case corresponds to H = 1. Other parameters are set at f = 10,
s = 0.3, and d = 0.01.
care, for calculations). Polygyny, and the associated increase
in male reproductive variance, generate additional genetic
drift and reduce both Km and Kf relative to monogamy. However, when male brood are brothers through their father, the
genetic variance in male offspring decreases with harem
size. As a consequence, the depreciation in Km with H is
steeper than in Kf
!
ð1 þ HÞ 1 2 sm zm ; zf
;
Km zm ; zf ¼ Kf zm ; zf 1 2
4Nsm zm ; zf
(24)
indicating that the evolution of paternal care is more
sensitive to polygyny than the evolution of maternal care.
The joint effects of reduced selection and lower efficacy of
selection on males compromise the evolution of parental
care in polygynous populations. Figure 5 shows analytical
predictions of the probability of fixation for varying levels of
polygyny. These show that mutants for parental care become
less likely to fix as the level of polygyny increases, and this
effect is exacerbated for paternal care, demonstrating the
double effect of reproductive variance on both the intensity
and the efficacy of selection on reproductive traits.
Long-term evolutionary equilibrium
The probability of fixation captures evolutionary dynamics
over a short timescale. But as shown above (Equations 9–
12), it can be used to predict long-term phenotypic outcome
under a recurrent inflow of mutations. We explore here how
reproductive variance affects the long-term evolution of parental care.
In the above parental care example, the evolutionary
dynamics governed by the selection gradients (Equations
19, 20, 22, and 23) eventually reach the trivial equilibrium
phenotypes of maximum care for sons and daughters
[sm(zm, zf) = sf(zm, zf) = 1]. We therefore introduce the
realistic assumption that a parent’s resources are limited
and that as a consequence, there is a trade-off between
the efforts allocated to sons and daughters. The care provided to male offspring by a parent of sex u is written zu and
1 2 zu is the care allocated to daughters (with 0 # zu # 1).
As, before, the survival sv(zm, zf) of an offspring of sex v is
a function of the paternal and maternal care received. Here,
a simple additive function is assumed, where the survival of
a male offspring is sm(zm, zf) = (zm + zf)/2, while that of
a female offspring is sf(zm, zf) = (1 2 zm + 1 2 zf)/2. Then,
because we consider the evolution of care in one sex while
maintaining the care of the other sex constant, Equation 10
shows that the long-term evolution of sex-limited traits can
be inferred from selection on that sex alone. In other words,
we can predict the phenotypic equilibrium of a male-limited
trait (bmf = 0 and bff = 0) from the zeroes of the fitness
gradient Gm(zm, zf) = 0 of males and that of a femalelimited trait (bmf = 0 and bmm = 0) from the zeroes of
Gf(zm, zf) = 0.
Then, convergent stable states are found in three steps.
First, using calculations similar to those used in Equations
13–18 and Appendix, Calculations for the evolution of parental care, we calculate the moments of reproduction for a focal
male and a focal female in the presence of trade-offs to find
f
the fitness components of a focal individual (wm
vi , wvi , Equation 4). Second, we add the fitness components of a parent
of sex u derived from male and female offspring to obtain
the gradient Gu(zm, zf) as in Equation 2. Finally, solving for
Gu(zm, zf) = 0, we find the convergence-stable level of investment in sons, zu* is identical for male and female parents
when the population is monogamous, and such that male
survival is
1
f 21
:
* Þ ¼ sm zf* ¼ þ
sm ðzm
2 2 þ 2f ð2N 2 1Þ
(25)
This equation show that the equilibrium investment
approaches 1/2 as population size goes to infinity (N /
N). This prediction is in line with the fact that kin competition vanishes in infinite populations and with it the selection pressures emanating from reproductive variance.
Parents in very large populations are then expected to ensure an even survival of male and female offspring. As the
population size decreases, however, reproductive variance
starts to affect fitness and it becomes beneficial to invest
more in the care of sons ðsm ðz*Þ
u . 1=2Þ to dampen the stochasticity in reproductive output that results from their mortality. The clutch size f has an additional, weaker, effect on
equilibrium care. At the extreme of single-offspring clutches,
Sex-Specific Reproductive Variance
245
f = 1, the differences between the patterns of male and
female survival are irrelevant, and equilibrium care ensures
equal survival in males and females when sex ratio is equal
ðsm ðz*Þ
u ¼ 1=2Þ. As clutch size increases, the effects of reproductive variance come into play and, for a given population
size N, larger clutch sizes result in male bias in care
ðsm ðz*Þ
u . 1=2Þ. However, this effect rapidly plateaus with
increasing f and is weaker than that of altering population
size.
Theory about sex ratio predicts that to minimize the
variance in offspring number, hermaphroditic females
should make more offspring of the sex that is less variable
in survival (Proulx 2000, 2004). In our model population,
females should then produce more daughters. Using the
same approach as above, we can calculate female fitness
when sex ratio r at birth (ratio of males to total offspring)
is maternally controlled; i.e., the number of females at birth
of a focal female with phenotype zfj now is n♀ Bin(f, 1 2 r
(zfj)). As before, limited care is provided by females, and
covariance in survival is greater between related males. Calculating the moments of reproduction as in Equations 13–15
with appropriate sex ratio (zfj) and survival independent of
the trait, we find as expected,
1 1
ðf 2 1Þð1 2 sm Þ
r zf* ¼ 2 3
;
m
2 2 1 þ s þ fsm ð2N 2 1Þ 2 f
(26)
that the evolutionary convergent sex ratio is biased toward
females ½rðz*Þ
f , 1=2.
Discussion
Capturing sex-specific reproductive variance
It has long been known that reproductive variance impedes
adaptation by increasing genetic drift (Wright 1931; Nunney
1993; Nomura 2002; Charlesworth 2009). In parallel, a body
of work has shown that reproductive variance is itself under
selection, favoring less variable offspring production (Gillespie
1974; Courteau and Lessard 1999; Proulx 2000; Shpak
2005; Shpak and Proulx 2007; Lehmann and Balloux
2007; Rice 2008; Taylor 2009; Proulx and Adler 2010).
Together, these studies have provided a solid theoretical
basis for understanding the effects of selection on offspring
distribution in a natural world that is inherently uncertain
within generations. Despite these advances, models for the
evolution of reproductive variance and its effects on adaptation have so far ignored biologically realistic cases of
sexual reproduction, where the role of the variance can
be expected to be most important. Closest to this, Taylor
(2009) studied the effect of sex-specific reproductive variance on adaptation, but by modeling mating as the random
union of gametes, key features of reproductive biology were
neglected, since in most cases it is individuals, not gametes,
that unite to mate. Finite numbers of matings and the structure of the mating system have important evolutionary
effects. Not only do they generate correlations between the
246
C. Mullon et al.
number of offspring of different individuals of the same sex,
but they also often underlie disparities between male and
female reproductive variance.
In this article, we used an individual-based approach to
provide an analytical model for the evolution of male and
female reproductive traits within a biologically realistic
context of sexual reproduction. First, we calculated the
probability of fixation of a mutant that perturbs male and
female reproductive phenotypes (Equation 1), taking into
account all first and second moments of the probability
distribution that describes individual reproduction (Figure 1
and File S1). As a consequence, the fitness gradients, Gm and
Gf (Equations 2 and 4), which express the direction and
intensity of selection on a mutant, reveal the many components of reproductive variance that contribute to fitness and
are hence under selection. These include the variance in the
reproductive output of a focal individual (nuvi ; Equations 4
and 5), which decreases fitness (Figure 2A), and the variance in the total reproductive output of the rest of the population (nuvk , Equation 4 and 5), which increases fitness
(Figure 2B). The impact of these variances on fitness has
been accounted for in previous studies (Appendix, Link with
previous work, to see how previous works connect to the
model presented here). However, our model also takes into
account the covariance between the numbers of juveniles
produced by different individuals of the same sex (ruvik ;
Equation 4 and 5), which had been ignored so far and potentially have greater consequences for fitness than the variance alone (Equation 5). This covariance would emerge as
a direct consequence to the biological constraints that the
number of matings and female fecundity are finite and
therefore cannot be ignored.
Efficacy of selection in males and females
The probability of fixation of a mutant (Equation 1) also
depends on the efficacy with which selection can act on
mutants. This is represented here by the scaling factors Km
and Kf. They measure the degree to which neutral genetic
variation results in phenotypic variation is then exposed to
selection in males and females (Equation 6). As Km and Kf
increase, the probability of fixation of a mutant increasingly
reflects the effects it has on male and female fitnesses,
respectively.
The scalars Km and Kf express effects similar to those
captured by the traditional heritability of a trait (Falconer
and Mackay 1996). However, while heritability is a snapshot
of a population in time, Km and Kf take into account the
segregation of alleles and changes in frequency until loss
or fixation of a mutant. This is illustrated by the interpretation of Km and Kf in terms of coalescence times (File S1,
Equation SI.37), which can themselves be expressed in
terms of probabilities of sibship (File S1, Equation SI.38,
and Figures 3 and 4), or how genes coalesce in different
individuals of both sexes. The probabilities of sibship depend
on the moments of offspring production (Table 1) and
thereby establish a link between the mating system of
a population and the potential for selection to act on different traits in that population. High probabilities of sibship
reflect a situation in which reproduction is monopolized by
a subset of individuals. This reproductive skew entails
a greater likelihood that a mutant is either transmitted or
lost by chance and hence reduces levels of genetic variation.
The factors Km and Kf also increase with the dominance
coefficient h of the mutant. Dominance increases the covariation between genotype and phenotype at low allele frequency, which is the frequency dominating the segregation
process of a new mutation, and therefore increases the visibility of mutants to selection.
An important feature of Km and Kf are their sex specificity,
respectively scaling on male and female fitness gradients.
This reveals that genetic drift can influence male and female
evolution with varying strength. Traditionally, population-genetic treatments of evolution in dioecious populations express
the effect of genetic drift on the segregation of two alleles
simply as the inverse of the overall effective population size or
as some mutant frequency-dependent function (Ethier and
Nagylaki 1988; Taylor 2009), but in both cases, the effect
of genetic drift on male and female selection is the same. This
simplification stems from the requirements to obtain a diffusion limit for the segregation process, which ignore some
differences between male and female reproduction.
The method we used here to calculate the probability
incorporates all second moments of male and female
individual reproduction and shows that it is possible for
genetic drift to affect selection on males and females
differently. When Kf is larger than Km, selection on females
contributes more to the probability of fixation than does
selection on males, and vice versa. A variety of factors can
lead to differences between male and female efficacy of
selection. As shown in the Example section, discrepancies
between Km and Kf can occur as the result of differences
between male and female patterns of mortality that generate a greater level of genetic variance in females than in
males. This is not only a theoretical possibility. In the house
finch, for example, mite ectoparasitism affects related males
more strongly than related females, leading to male-biased
mortality (Badyaev et al. 2006). As a consequence, we expect Km to be smaller than Kf in this species.
Long-term sex-specific evolution
To predict the joint evolution of male and female phenotypes,
we embedded our model into a trait substitution sequence
process. We obtained a stochastic model of long-term
phenotypic evolution for dioecious populations that allows
one to conveniently evaluate convergence-stable states,
which correspond to the most likely phenotypic outcomes
of evolution at mutation–selection–drift balance (Equations
11 and 12 and Appendix, Diffusion equation for phenotypic
evolution in dioecious populations). When the reproductive
variances of males and females are such that there is
no difference in the efficacy of selection between the sexes
(Km = Kf; Equations 11 and 12), then the conditions for
phenotypes to be convergence stable depend only on the
fitness gradients, in agreement with previous deterministic
models (Leimar 2009). When the efficacy of selection differs
between the sexes (Km 6¼ Kf), however, they may affect the
evolutionary trajectory and change the stability of internal
equilibria (Equation 12). Therefore, the most likely phenotypes to be observed in natural populations can be significantly affected by sex-specific reproductive variance.
Effects of sex-specific variance on parental care
To illustrate the many effects of reproductive variance on
the evolution of dioecious species, we calculated the
probability of fixation of mutants coding for maternal and
paternal care for sons and daughters in a situation where the
survival of sons is highly correlated within broods. While
very specific, this example allows us to illustrate some of the
key effects that our model can capture. First, our results
demonstrate how phenotypic evolution can be driven by
selection against reproductive variance. Thus, care for sons
evolves more readily than care for daughters, because the
former alleviates the high degree of reproductive variance
that arises as a consequence of correlated male survival
(Equations 19 and 20). Second, we showed that the pattern
of male survival reduced the efficacy of selection on male
traits by decreasing the amount of genetic variation in males
(Equation 21). This means that mutants coding for maternal
care have a greater probability of fixation than those coding
for paternal care, even if the effect of maternal and paternal
care on offspring survival is identical. Finally, male polygyny
generates a negative correlation between the reproductive
outputs of different males, which in turn generates an additional
selection pressure on the evolution of paternal care (Equations
23 and 22). These forces mitigate the strength of selection for
paternal care because as fewer males monopolize reproduction,
kin competition between the offspring of a male increases. The
selection pressures generated by (co)variances might be
minimal when populations are very large, polygamy extensive and fecundity effectively unlimited. However, in most
biologically realistic scenarios, the complicated interactions
between the different components of reproductive variance
can be expected to affect the evolutionary process through
selection and genetic drift.
The phenotypic equilibrium predicted for both parental
sexes is, like the probability of fixation, affected by selection
against reproductive variance. Thus, fathers and mothers
will invest more in the care of sons to mitigate the
detrimental effects of their stochastic survival on parental
fitness (Equation 25). Interestingly, this prediction contrasts
with other results on the evolution of sex-ratio allocation
whereby females are expected to produce more daughters
when the survival of males within a brood is highly
correlated (Equation 26) (Proulx 2000, 2004). Therefore,
selection against reproductive variance leads to the counterintuitive equilibrium whereby females produce fewer sons
for which they care more. It would be interesting to further
explore the effect of sex-specific variance on the evolution of
Sex-Specific Reproductive Variance
247
sex allocation. In particular, we expect that the sex ratio at
birth would differ according to whether it is controlled by
the male or female parent and that the difference between
maternally and paternally controlled sex ratio depends on
the mating system. For instance, with the life cycle given in
the Example section of this article, if sex ratio is male controlled and the population is polygynous, then selection on
males to minimize their reproductive variance would favor
a bias toward females that is even more pronounced than
when the sex ratio is female controlled (Equation 26).
Our analysis of the long-term evolution of male and
female care behavior showed that both sexes evolve toward
the same equilibrium level of care (Equation 25). This
contrasts with the predicted short-term dynamics, where
greater stochasticity in male survival caused a reduction in
the probability of fixation of mutants for paternal care,
compared to that seen for maternal care mutants. This
discrepancy intuitively implies a lower rate of adaptation in
the male than female trait and a longer time to reach the
evolutionary equilibrium. In general, the stochastic model of
phenotypic evolution (Appendix, Diffusion equation for phenotypic evolution in dioecious populations) suggests that the
rate of adaptation in males and females scales with the
efficacy of selection Km and Kf, respectively.
Outlook
The framework provided in this article is ideal for studying
complex social interactions between individuals of sexual
populations. Examples of such traits are those involved in
evolutionary games between the male and female of
a mating pair or strategies in games between individuals
of the same sex, for example, in male–male competition for
mating and fertilization success. In the latter case, the covariance between the numbers of offspring produced by different males is expected to have important effects. Use of
our model to study the social and sex-specific frequency-dependent aspects of reproductive evolution is straightforward
because all parameters in Equations 1 and 10 can be derived
using only the phenotype of a focal individual and the average male and female phenotypes in the population. Another class of traits for which our model is particularly well
suited are sexually antagonistic ones (Parker 1979; Lande
1980; Bonduriansky and Chenoweth 2009; Pennell and
Morrow 2013). By taking into account the positive correlation of mutational effects in males and females (bmf . 0 in
Equation 10), different selection pressures in males and
females (Gm 6¼ Gf), and different levels of reproductive variance in the sexes, the model is well adapted to investigating
the evolution of these traits under the simultaneous influences of selection and drift.
With selection on variance being inversely proportional to
the population size, selection on the variance will be mostly
relevant in small panmictic populations, where genetic drift
may therefore mitigate its effects. But if populations are
structured into local breeding groups, then selection against
reproductive variance is inversely proportional to local patch
248
C. Mullon et al.
size (Shpak and Proulx 2007; Lehmann and Balloux 2007),
while genetic drift inversely scales with the total population
size, which can be very large. All the effects of selection on
reproductive variance described in this article may then be
particularly relevant in populations that are divided in small
patches but are globally large. In fact, if density-dependent
regulation takes place before dispersal (soft selection, Roze
and Rousset 2003), then selection against reproductive variance is as described by our panmictic model, with fitness
given by Equation 5 but with the number of individuals being
those in a local patch (Shpak 2005; Lehmann and Balloux
2007).
Explicitly taking spatial structure into account in regimes
of soft and hard selection may also reveal interesting
examples of sex-specific evolution. In structured and dioecious populations, we expect that sex-specific local
competition (e.g., Perrin and Mazalov 2000), but also reproductive variance, will drive the evolution of sex-specific
dispersal. In turn, this will generate differences in genetic
variation across the sexes (i.e., between Km and Kf), thereby
influencing the evolution of sex-specific strategies. It would
therefore be particularly interesting to extend the model to
explicitly take into account spatial structure to investigate
the evolution of sex-specific dispersal strategies and how it
interacts with the evolution of other sex-specific traits.
Future development of the model should also accommodate for a greater variety of genetic architecture of traits.
Because Km and Kf depend on the covariance between genotype and phenotype, differences in the genetic determination of traits between the sexes would also translate into
differences between the efficacy of selection in males and
females. This is not unlikely as differences between male
and female heritability have been reported for phenotypic
traits in animals (e.g., Eisen and Legates 1966; Jensen et al.
2003), including humans (Weiss et al. 2006) as well as
plants (e.g., Ashman 1999). Such differences would naturally arise for sex-linked genes. For instance, in species with
an XY sex-determining system, where males are hemizygous
for the X chromosome, dominance interactions can occur
only between the two X chromosomes of females (Wayne
et al. 2007). It would therefore be interesting to extend our
model for sex-linked genes and test whether the interaction
between selection on reproductive variance and the efficacy
of selection in males and females lead to different evolutionary dynamics than on autosomes.
To conclude, using a population-genetic approach that
takes into account all the relevant moments of reproduction
in the two sexes, we have shown that the effect of sexspecific reproductive variance and covariances and selection
on it influences the evolution of dioecious species. In
particular, we have found that even if the fitness gradients
on male and female traits have the same steepness but
opposite directions, differences in male and female reproductive variance can lead to selection in one sex dominating
selection in the other, and alter the trajectory of long term
phenotypic evolution.
Acknowledgments
We thank Andrew Pomiankowski, Rob Seymour, Lorette Noiret,
and Julie Collet for helpful comments on the manuscript. The
manuscript also vastly benefitted from the suggestions of two
anonymous reviewers. C.M. was supported by a CoMPLEX
Ph.D. studentship from the U.K. Engineering and Physical
Sciences Research Council, M.R. by funding from the U.K.
Natural Environment Research Council (NE/D009189/1, NE/
G019452/1), and L.L. by the U.S. National Science Foundation
(grant PP00P3-123344).
Literature Cited
Ashman, T.-L., 1999 Quantitative genetics of floral traits in a gynodioecious wild strawberry Fragaria virginiana: implications
for the independent evolution of female and hermaphrodite
floral phenotypes. Nat. Genet. 83: 733–741.
Badyaev, A. V., T. L. Hamstra, K. P. Oh, and D. A. Acevedo Seaman,
2006 Sex-biased maternal effects reduce ectoparasite-induced
mortality in a passerine bird. Proc. Natl. Acad. Sci. USA 103
(39): 14406–14411.
Bateman, A. J., 1948 Intra-sexual selection in Drosophila. Heredity 2(3): 349–368.
Beckerman, A. P., S. P. Sharp, and B. J. Hatchwell, 2011 Predation and kin-structured populations: an empirical perspective on the evolution of cooperation. Behav. Ecol. 22(6): 1294–
1303.
Bonduriansky, R., and S. F. Chenoweth, 2009 Intralocus sexual
conflict. Trends Ecol. Evol. 24(5): 280–288.
Champagnat, N., and A. Lambert, 2007 Evolution of discrete populations and the canonical diffusion of adaptive dynamics. Ann.
Appl. Probab. 17(1): 102–155.
Charlesworth, B., 2009 Effective population size and patterns of
molecular evolution and variation. Nat. Rev. Genet. 10(3): 195–
205.
Courteau, J., and S. Lessard, 1999 Stochastic effects in LMC models. Theor. Popul. Biol. 55(2): 127–136.
Eisen, E. J., and J. E. Legates, 1966 Genotype-sex interaction and
the genetic correlation between the sexes for body weight in
Mus musculus. Genetics 54: 611–623.
Ethier, S. N., and T. Nagylaki, 1988 Diffusion approximations of
Markov chains with two time scales and applications to population genetics, II. Adv. Appl. Probab. 20(3): 525–545.
Falconer, D. S., and T. C. F. Mackay, 1996 Introduction to Quantitative Genetics, 4th ed. Longman, London.
Frank, S. A., 2011 Natural selection. I. Variable environments and
uncertain returns on investment. J. Evol. Biol. 24(11): 2299–
2309.
Foster, D., and P. Young, 1990 Stochastic evolutionary game dynamics. Theor. Popul. Biol. 38(2): 219–232.
Freidlin, M. I., and A. D. Wentzell, 2012 Random Perturbations of
Dynamical Systems, Springer-Verlag, Berlin.
Gardiner, C., 2009 Stochastic Methods: A Handbook for the Natural
and Social Sciences, Springer Series in Synergetics, 4th ed.
Springer-Verlag, Berlin.
Gillespie, J. H., 1974 Natural selection for within-generation variance in offspring number. Genetics 76: 601–606.
Gillespie, J. H., 1975 Natural selection for within-generation variance in offspring number II. Discrete Haploid models. Genetics
81: 403–413.
Gillespie, J. H., 1977 Natural selection for variances in offspring
numbers: a new evolutionary principle. Am. Nat. 111(981):
1010–1014.
Gillespie, J. H., 1994 The Causes of Molecular Evolution, Oxford
Series in Ecology and Evolution. Oxford University Press, New
York.
Jensen, H., B.-E. Sæther, T. H. Ringsby, S. C. Tufto, J. Griffith et al.,
2003 Sexual variation in heritability and genetic correlations
of morphological traits in house sparrow (Passer domesticus). J.
Evol. Biol. 16: 1296–1307.
Lande, R., 1980 Sexual dimorphism, sexual selection, and adaptation in polygenic characters. Evolution 34(2): 292–305.
Lehmann, L., 2012 The stationary distribution of a continuously
varying strategy in a class-structured population under mutation–selection–drift balance. J. Evol. Biol. 25(4): 770–787.
Lehmann, L., and F. Balloux, 2007 Natural selection on fecundity
variance in subdivided populations: kin selection meets bet
hedging. Genetics 176: 361–377.
Lehmann, L., and F. Rousset, 2009 Perturbation expansions of
multilocus fixation probabilities for frequency-dependent selection with applications to the Hill–Robertson effect and to the
joint evolution of helping and punishment. Theor. Popul. Biol.
76(1): 35–51.
Leimar, O., 2009 Multidimensional convergence stability. Evol.
Ecol. Res. 11(2): 191–208.
Lessard, S., and V. Ladret, 2007 The probability of fixation of
a single mutant in an exchangeable selection model. J. Math.
Biol. 54(5): 721–744.
Metz, J. A. J., S. A. H. Geritz, G. Meszena, F. J. A. Jacobs, and
Heerwaarden, 1995 Adaptive dynamics: a geometrical study
of the consequences of nearly faithful reproduction. Technical
Report, International Institute for Applied Systems Analysis
A-2361. Laxenburg, Austria.
Nomura, T., 2002 Effective size of populations with unequal sex
ratio and variation in mating success. J. Anim. Breed. Genet.
119(5): 297–310.
Nunney, L., 1993 The influence of mating system and overlapping
generations on effective population size. Evolution 47(5):
1329–1341.
Oehlert, G. W., 1992 A note on the delta method. Am. Stat. 46
(1): 27–29.
Parker, G. A., 1979 Sexual Selection and Reproductive Competition
in Insects. Academic Press, San Diego.
Pennell, T. M., and E. H. Morrow, 2013 Two sexes, one genome:
the evolutionary dynamics of intralocus sexual conflict. Ecol.
Evol. 3(6): 1819–1834.
Perrin, N., and V. Mazalov, 2000 Local competition, inbreeding, and
the evolution of sex biased dispersal. Am. Nat. 155(1): 116–127.
Proulx, S., 2000 The ESS under spatial variation with applications
to sex allocation. Theor. Popul. Biol. 58(1): 33–47.
Proulx, S. R., 2004 Sources of stochasticity in models of sex allocation in spatially structured populations. J. Evol. Biol. 17(4):
924–930.
Proulx, S. R., and F. R. Adler, 2010 The standard of neutrality:
Still flapping in the breeze? J. Evol. Biol. 23(7): 1339–1350.
Rice, S., 2008 A stochastic version of the Price Equation reveals
the interplay of deterministic and stochastic processes in evolution. BMC Evol. Biol. 8(1): 262.
Rousset, F., 2003 A minimal derivation of convergence stability
measures. J. Theor. Biol. 221(4): 665–668.
Rousset, F., and S. Billiard, 2000 A theoretical basis for measures
of kin selection in subdivided populations: finite populations
and localized dispersal. J. Evol. Biol. 13(5): 814–825.
Roze, D., and F. Rousset, 2003 Selection and drift in subdivided
populations: a straightforward method for deriving diffusion
approximations and applications involving dominance, selfing
and local extinctions. Genetics 16(4): 2153–66.
Roze, D., and F. Rousset, 2004 The robustness of Hamilton’s rule
with inbreeding and dominance: kin selection and fixation probabilities under partial sib mating. Am. Nat. 164(2): 214–231.
Sex-Specific Reproductive Variance
249
Sella, G., and A. E. Hirsh, 2005 The application of statistical physics to evolutionary biology. Proc. Natl. Acad. Sci. USA 102:
9541–9546.
Shpak, M., 2005 Evolution of variance in offspring number: the
effects of population size and migration. Theory Biosci. 124(1):
65–85.
Shpak, M., 2007 Selection against demographic stochasticity in
age-structured populations. Genetics 177: 2181–2194.
Shpak, M., and S. Proulx, 2007 The role of life cycle and migration in selection for variance in offspring number. Bull. Math.
Biol. 69(3): 837–860.
Taylor, J. E., 2009 The genealogical consequences of fecundity
variance polymorphism. Genetics 182: 813–837.
Van Cleve, J., and L. Lehmann, 2013 Stochastic stability and the
evolution of coordination in spatially structured populations.
Theor. Popul. Biol. 89: 75–87.
Wade, M. J., 1979 Sexual selection and variance in reproductive
success. Am. Nat. 114(5): 742–747.
Wakano, J. Y., and L. Lehmann, 2012 Evolutionary and convergence stability for continuous phenotypes in finite populations
derived from two-allele models. J. Theor. Biol. 310: 206–215.
Wayne, M. L., M. Telonis-Scott, L. M. Bono, L. Harshman, A. Kopp
et al., 2007 Simpler mode of inheritance of transcriptional
variation in male Drosophila melanogaster. Proc. Natl. Acad.
Sci. USA 104(47): 18577–18582.
Weiss, L. A., L. Pan, M. Abney, and C. Ober, 2006 The sex-specific
genetic architecture of quantitative traits in humans. Nat. Genet.
38: 218–222.
Wright, S., 1931 Evolution in Mendelian populations. Genetics
16: 97–159.
Communicating editor: J. Wakeley
Appendix
Assumption on distribution of juveniles
Given an index set of individuals i 2 I in the population and a corresponding set of powers defined by a mapping z: I / ℤ+,
it is assumed that the following
#
!
"
P
zðiÞþ12jI j
Y
u
u zðiÞ
i2I
O N
;
(A1)
Jvi 2mvi
E
i2I
holds, where |I | is the number of individuals in set I . The remainder terms that appear in R, given by the higher-order
terms of the Taylor expansion of F, are thus of order 1/N2.
Weights for additive mutants
Using Equation SI.38 from File S1, the weights on male and female fitness gradients in the probability of fixation of an
additive (h = 1/2) mutant are given by
4þQ♀f 2 Q♀m
Km ¼ 2Nm1þ2Nf 1 2 N1m
D
Kf ¼
1
2Nm þ2Nf
1 2 N1f
4þQ♂m 2 Q♂f
D
;
(A2)
♀
♂
♂
♂
♀
♀
♀
♂
♀
♀
♂
♂ ♀
♂ ♀
where D ¼ Q♂
m þ Qf þ 2Qc þ Qm þ Qf þ 2Qc þ Qc Qf 2 Qm =2 þ Qc Qm 2 Qf =2 þ Qm Qf =2 2 Qf Qm =2:
Calculations for the evolution of parental care
Here the remaining components of the probability of fixation of alleles coding for parental care are calculated. Because male
survival is different between populations in which maternal and parental care are provided (see main text), it is simpler to
consider separately the cases when care is maternal and when care is paternal.
Maternal care: To calculate the weight Kf, we need the probabilities of sibship (Table 1) in the resident population (zfj =
zf). To calculate the maternal probabilities of sibship Q♀, in addition to Equation 13–15 of the main text, the covariance
between number of male and female offspring that a female produces is also required, and it is given by
zm ; zf ¼ En♂ n♂ sm zm ; zf ðf 2 n♂ Þsf zm ; zf 2 En♀ n♀ sf zm ; zf En♂ n♂ sm zm ; zf
rm;f
f
¼ 2
fsf ðzm ;zf Þsm ðzm ;zf Þ
:
4
(A3)
The paternal probabilities of sibship Q♂ also influence Kf (Equation A2) and their components are derived below. The
expected numbers of male and female offspring of a male are given by Equation 13 evaluated at male phenotype zm. The
250
C. Mullon et al.
variance in the number of females fathered by a male is found by conditioning on the random number of matings Nmat of
a male
"
"
!#
#
sf zm ; zf
s f zm ; zf
s f zm ; zf
f
12
þ VNmat Nmat f
:
(A4)
nm zm ; zf ¼ ENmat Nmat f
2
2
2
Because each male is equally likely to mate, and if a male does, it mates exactly H times, we have E[Nmat] = 1 and V[Nmat] =
H 2 1, so that
!
f z ;z
f z ;z
s
s
m
m
f
f
1þf
ð1 þ f ðH 2 1ÞÞ :
(A5)
nfm zm ; zf ¼ f
2
2
The variance in the number of males fathered by a male, given that each male brood entirely survives or dies, reads
sm ðzm ;zf Þ sm ðzm ;zf Þ
m z ;z
N
N
z
¼
E
1
þ
f
1
2
s
þ
V
;
z
f
f
nm
m
mat
m
mat
N
N
f
f
mat
mat
m
4
2
¼f
sm ðzm ;zf Þ 1 þ f 1 þ ðH 2 2Þsm zm ; zf :
4
(A6)
To calculate the covariance between number of males and of females produced by a male, we define X as the random product
of males and females coming from the same mating. Then, since offspring survival is independent across matings, we have
m
f
2 s ðzm ;zf Þs ðzm ;zf Þ
N
z
¼
E
;
z
X
þ
N
ðN
2
1Þ
f
rm;f
m f
mat
mat mat
Nmat ; X
m
4
sm ðzm ;zf Þ
sf ðzm ;zf Þ
E
N
;
2 ENmat Nmat f
f
mat
Nmat
2
2
(A7)
and since, E½X ¼ sm ðzm ; zf Þsf ðzm ; zf ÞEn♂ ½n♂ ðf 2 n♂ Þ;
rm;f
m
zm ; zf
fsm zm ; zf sf zm ; zf
ð1 2 f ðH 2 1ÞÞ:
¼ 2
4
(A8)
Finally, substituting Equations 13–15 and Equations A3–A8 into the probabilities of sibship (Table 1), and in turn, substituting the latter into Equation A2, we find the efficacy of selection on maternal care.
Paternal care: Most of the moments required to calculate focal male fitness and the probabilities of sibship to find Km are
the same as in the preceding section (evaluated at focal male phenotype zmi for focal fitness, and at resident male zm
instead of resident female zf). However, because male survival is different in a population in which parental care is
provided by males, the variance in the number of male offspring fathered by the focal male, and the covariance between
the number of male and female offspring fathered by a resident male (required for the probabilities of sibship) are
different. Using similar arguments as above, we find that they are respectively given by
1 m
nm
zm ; zf 1 þ f H 2 f sm zm ; zf
m zm ; zf ¼ 4 fs
(A9)
sm ðzm ;zf Þsf ðzm ;zf Þ
;
z
f
ðf
H
2
1
2
f
Þ:
rm;f
z
¼
m f
m
4
Calculating the probabilities of sibship (Table 1) using the above and substituting into Equation A2, we find the efficacy of
selection on paternal care.
Diffusion equation for phenotypic evolution in dioecious populations
The substitution rate k (Equation 9) is the jump rate of a so-called jump process (Gardiner 2009), which here describes
a population “jumping” from a monomorphic phenotypic state to another. The moments of the infinitesimal jump of the
evolving phenotypes in each sex (Dzm, Dzf) characterize the distribution of phenotypic changes over an infinitesimally small
evolutionary time period, and they are found by integrating the phenotypic effects of a substitution over the p.d.f. of all
substitution rates, E½ðDzm Þi ðDzf Þj ¼ ∭dim djf kðzm ; zf ; dm ; df ; hÞddm ddf dh. To the first order of d, we obtain for the first moment
of Dzu for u 2 {m, f},
Sex-Specific Reproductive Variance
251
au zm ; zf ¼ E½Dzu ¼ 2Nj buu Ku zm ; zf ; h Gu zm ; zf þ buv Kv zm ; zf ; h Gv zm ; zf
(A10)
for v 2 {m, f}, v 6¼ u, and where b = C[du, dv] are the second moments of mutant sex-specific effects. Because dominance is
independent of dm and df and K is linear in h (File S1, Equation SI.38), Ku ðzm ; zf ; hÞ is simply evaluated at expected
dominance E½h ¼ h: Similarly, it is possible to show that the Equation for E[Dzu] still holds if mutants have sex-specific
dominance hm and hf, as long as they are independent of dm and df and that they are on average equal hm ¼ hf ¼ h. The
second moments of infinitesimal phenotypic change of the first order of d are given by E[(Dzu)2] = jbuu and E[(Dzm)(Dzf)] =
jbmf.
Assuming that the phenotypic changes are continuous in probability, the first two moments of infinitesimal change Dzu
can be used to approximate the rate of phenotypic change by a Fokker–Planck equation (Gardiner 2009). This equation
characterizes the change of the distribution of male and female phenotypes c(zm, zf; t) in evolutionary time t. The
distribution of phenotypes here is meant over many evolutionary trajectories or experiments, rather than over the population. The population remains monomorphic: fixation of loss of mutants is instantaneous. If the male and female phenotypes
(zm, zf) at evolutionary time t have p.d.f. c(zm, zf; t), then it satisfies
@cðzm ;zf ;t Þ
@t
¼ 2 @[email protected] am zm ; zf c zm ; zf ; t 2 @[email protected] f af zm ; zf c zm ; zf ; t
@ 2 cðzm ;zf ;t Þ
@ 2 cðzm ;zf ;t Þ
@ 2 cðzm ;zf ;tÞ
j
þ 2 bmm
;
þ bff
þ bmf @zm @zf
@z2
@z2
m
(A11)
f
where the functions am(zm, zf) and af(zm, zf) are the expected infinitesimal changes of male and female phenotypes given in
Equation A10 and the stationary distribution is given by c(zm, zf) = limt/Nc(zm, zf; t).
Link with previous work
Substituting Equation 4 into Equation 2, we find fitness gradients that are consistent with previous work on the evolution of
reproductive variance. For instance, Lehmann and Balloux (2007) models the evolution of a helping trait zf that disrupts the
mean mf(zf) O(N) and variance n2f ðzf Þ OðNÞ in fertility. Mating is random, each female gives birth independently of one
another, and sex ratio is equal at birth and in the population. Substituting Equation 4 into Equation 2, we have that the
fitness gradient on zf is proportional to
2
1
1 2 Cv2 d ln mf zf
1 Cv2 d lnnf zf
(A12)
2
þ O 1=N 2 ;
G zf } 1 2
2
2 Nf
dzf
Nf
dzf
where Cv2 ¼ n2f =m2f is the squared coefficient of variation in fertility in the resident female, in agreement with Equation A37 of
Lehmann and Balloux (2007).
The original fitness gradient by Gillespie (1975, Equation 11a) or equivalently, that derived for a dioecious population by
Taylor (2009, Equation 14 for an additive mutant) may be found directly from Equation A12. These analyses use the
diffusion approximation, which requires that the difference between the mean fertilities of the resident and mutant phenotypes tend to zero as the population size tends to infinity, i.e., that d ln mf(zf)/dzf O(1/N). Applying this assumption to
Equation A12, we have
2
1 d ln mf zf
1 Cv2 d ln nf zf
G zf }
(A13)
2
þ O 1=N 2 ;
2
2 Nf
dzf
dzf
where the deleterious effects of sib competition on expected fecundity fall victim to the order condition required by the
diffusion approach of Gillespie (1975) and Taylor (2009).
252
C. Mullon et al.
GENETICS
Supporting Information
http://www.genetics.org/lookup/suppl/doi:10.1534/genetics.113.156067/-/DC1
The Evolution and Consequences of Sex-Specific
Reproductive Variance
Charles Mullon, Max Reuter, and Laurent Lehmann
Copyright © 2014 by the Genetics Society of America
DOI: 10.1534/genetics.113.156067
File S1
Deriva on of the fixa on probability of a mutant
Expected change of mutant frequency. In order to derive the probability of fixa on of a mutant, we first evaluate
the expected change of mutant frequency over one genera on. The frequency of the mutant in a male indexed
i ∈ {1, . . . , Nm } is wri en as pmi ∈ {0, 1/2, 1}, and the frequency in a female j ∈ {1, . . . , Nf } is wri en
pfj ∈ {0, 1/2, 1}. The indicator variables 1♂i and 1♀i respec vely take the value one if the paternally and
maternally inherited alleles of individual i are mutant, and zero otherwise. Then, the mutant frequencies in male
i and in female j are
pmi =
We write pm,t =
∑Nm
i=1
1♂i + 1♀i
2
pmi,t /Nm and pf,t =
∑Nf
j=1
and
pfj =
1♂j + 1♀j
2
.
(SI.1)
pfj,t /Nf for the average mutant frequencies in males and
females in the popula on and denote by qt the vector collec ng the realiza on of mutant frequencies (the realized
values of 1♂i and 1♀i ) in the popula on at me t.
If the mutant changes male and female phenotypes by δm and δf and a parent transmits its maternally or paternally
inherited gene with equal probability, the expected average male and female mutant frequencies in the next
genera on is


Nf
Nm
∑
∑
1 
m
m
E[pm,t+1 |qt ] =
pmi,t wmi
(δm , δf ) +
pfj,t wfj
(δm , δf )
2Nm i=1
j=1


Nf
Nm
∑
∑
1 
f
f
(δm , δf ) +
(δm , δf ) ,
E[pf,t+1 |qt ] =
pmi,t wmj
pfj,t wfj
2Nf i=1
j=1
(SI.2)
u
where wvi
(δm , δf ) is the expected number of adult offspring of sex u of individual i (itself is of sex v) (Price 1970).
Eq. (SI.2) extends Rice (2008)'s "selec on differen al" to a two-sexes popula ons (his cov(ϕ, Ω̂) term assuming a
constant popula on size).
If selec on is weak, it is sufficient to approximate allele frequency change to the first order of phenotypic efu
u
u
(0) +
(δm , δf ) = wvi
fect in males and females δm and δf . The fitness terms wvi
are approximated as wvi
u
u
(0)/∂δf ) + O(δ 2 ), with (0) = (0, 0). There are two things to note about the fitδm (∂wvi
(0)/∂δm ) + δf (∂wvi
ness terms and their deriva ves. First, in the absence of phenotypic differences, each individual is expected to
u
contribute equally to the next genera on, and so wvi
(0) = Nu /Nv . Second, the par al deriva ves of an individu
ual's fitness with respect to phenotypic effect in the other sex is zero ∂wvi
(0)/∂δu = 0 with u ̸= v. For instance,
when all males are the same (δm = 0), changes in female phenotype have no effect on the expected number of
u
adult offspring of a focal male. So subs tu ng for wvi
(δm , δf ) in eq. (SI.2) gives


Nf
Nm
m
m
∑
∂wfj
(0)
1
1  ∑
∂wmi
(0)
 + O(δ 2 )
E[pm,t+1 |qt ] = (pm,t + pf,t ) +
pmi,t
pfj,t
δm
+ δf
2
2Nm
∂δ
∂δ
m
f
i=1
j=1


N
N
f
m
f
f
∑
∂wfj
(0)
1
1  ∑
∂wmi
(0)
 + O(δ 2 ).
E[pf,t+1 |qt ] = (pm,t + pf,t ) +
δm
pmi,t
+ δf
pfj,t
2
2Nf
∂δ
∂δ
m
f
i=1
j=1
2 SI
C. Mullon et al.
(SI.3)
Another consequence of weak selec on is that the fitness deriva ve of an individual in eq. (SI.3) can be approximated in terms of only three phenotypic values: the phenotype of an individual, the average male phenotype and the average female phenotype. To see this, consider the expected number of female adults prof
duced by male i, wmi
. This depends on his phenotype zmi , as well as the collec on of the phenotypes of all
the other males in the popula on, z−mi = {zmk ; k : 1 → Nm , k ̸= i}, as well as those of all the females
in the popula on, zf = {zfj ; j : 1 → Nf }. Expanded about male popula on average, excluding male i,
∑
∑
f
z −mi = 1/(Nm − 1) k̸=i zmk , and female popula on average z f = j zfj /Nf , wmi
reads
f
f
wmi
(zmi , z−mi , zf ) ≈ wmi
(zmi , z −mi , z f ) +
Nm
∑
k=1,k̸=i
Nf
f
f
∑
∂wmi
∂wmi
(zmk − z −mi ) +
(zfj − z f ),
∂zmk
∂zfj
j=1
(SI.4)
and the remainder is O(δ 2 ) because the difference between any two phenotypes of the same sex is of order
O(δ). The effect of changing the phenotype of any female has the same effect on the fitness of male i, so
∑Nf
∑Nf
f
f
f
/∂zfj are equal, and j=1
(∂wmi
/∂zfj )(zfj − z f ) = (∂wmi
/∂zfj ) j=1
(zfj − z f ), but by definithat all ∂wmi
∑Nf
∑Nm
f
on, j=1 (zfj − z f ) = 0. A similar argument shows that k=1,k̸=i (∂wmi
/∂zmk )(zmk − z −mi ) = 0. Hence,
f
f
the female component of fitness of male i, wmi
(zmi , z−mi , zf ), can be approximated by wmi
(zmi , z −mi , z f ); that
is, as a func on of its phenotype, zmi , the average male phenotype excluding the focal, z −mi , and the average phenotype of females in the popula on. However, for computa onal purposes it may be more convef
nient to express wmi
in terms of zmi and the average male phenotype z m . This can be done since z −mi =
f
(Nm z m − zmi )/(Nm − 1), so from now on we write the fitness of individual i as wmi
(zmi , z m , z f ), keeping in
f
f
f
mind that with this nota on ∂wmi
(zmi , z −mi , z f )/∂zmi = ∂wmi
(x, z m , z f )/∂x + (∂wmi
(zmi , z m , z f )/∂z m )/Nm .
u
u
/∂zvi )(dzvi / dδv ) +
/∂δv = (∂wvi
Using the chain rule, the deriva ves of fitness with respect to δv is ∂wvi
u
u
(∂wvi
/∂z m )(dz m / dδv ) + (∂wvi
/∂z f )(dz f / dδv ). By observing that the average male phenotype is insensi-
ve to changes in female mutant effects (dz m / dδf = 0), and that the average female phenotype is insenu
si ve to changes in male mutant effects (dz f / dδm = 0), the deriva ves of fitness collapse to ∂wvi
/∂δv =
u
u
(∂wvi
/∂zvi )(dzvi / dδv ) + (∂wvi
/∂z v )(dz v / dδv ). This may be further simplified by no ng that since the number
of adults of either sex held constant at each genera on, any fitness gain made by a focal individual due to a change
of phenotype must be compensated by a decrease in fitness by the rest of the popula on (Rousset 2004, p. 96),
u
u
u
u
i.e., ∂wmi
/∂zmi + ∂wmi
/∂z m = 0 and ∂wfj
/∂zfj + ∂wfj
/∂z f = 0. Thus, we eventually obtain for the deriva ves
of fitness
u
u
∂wvi
∂wvi
=
∂δv
∂zvi
(
dzvi
dz v
−
dδv
dδv
)
.
(SI.5)
Eq. (SI.5) is used to subs tute for the deriva ves of fitness in eq. (SI.3). To see how, consider the subs tu on for
m
∂wmi
(0)/∂δm in
(
)
Nm
Nm
1 ∑
∂wm (0)
1 ∑
∂wm (0) dzmi (0) dz m (0)
pmi,t mi
=
pmi,t mi
−
.
Nm i=1
∂δm
Nm i=1
∂zmi
dδm
dδm
(SI.6)
C. Mullon et al.
3 SI
At (δm , δf ) = 0, i.e. where all males are the same, the rate of change of fitness of a male i with respect to its
m
m
phenotype is the same for all males ∂wmi
(0)/∂zmi = ∂wmk
(0)/∂zmk . Thus, the index i denotes a representa ve
m
male (or a focal male), rather than a specific one. Then, ∂wmi
(0)/∂zmi may be taken out of the sum in eq. (SI.6)
m
and the index dropped for the func on wmi
is dropped, giving
(
)
Nm
m
m
1 ∑
(0)
zmi
(0)
∂wmi
dz m ∂wm
= pmi
− pm
,
pmi,t
Nm i=1
∂δm
dδm
dδm
∂zmi
where the overbar with index mi denotes averaging over all males xmi =
∑Nm
i=1
(SI.7)
xi /Nm . Using a similar argument
for all deriva ves of fitness in eq. (SI.3), we obtain
E[pm,t+1 |qt ] =
1
∂wm (0)
Nf
∂wm (0)
(pm,t + pf,t ) + δm Dm,t m
+ δf
Df,t f
+ O(δ 2 )
2
∂zmi
Nm
∂zfj
E[pf,t+1 |qt ] =
1
Nm
∂wf (0)
∂wf (0)
(pm,t + pf,t ) + δm
Dm,t m
+ δf Df,t f
+ O(δ 2 ),
2
Nf
∂zmi
∂zfj
where
Dm,t
1
=
2
(
)
dzmi
dz m
pmi
− pm
dδm
dδm t
and,
Df,t
1
=
2
and the overbar with index fj denotes averaging over all females xfj =
(
)
dzfj
dz f
,
pfj
− pf
dδf
dδf t
∑Nf
j=1
(SI.8)
(SI.9)
xj /Nf . We have added the sub-
script t in eq. (SI.9) to make the me dependence of Dm,t and Df,t explicit, since they depend on the popula on
genotypic realiza on at genera on t, qt .
The expecta on of mutant frequencies in males and females from genera on t to genera on t + 1 are found by
marginalizing eq. (SI.8) over qt
pm,t+1 = E[E[pm,t+1 |qt ]] =
∑
E[pm,t+1 |qt ] Pr(qt )
qt
pf,t+1 = E[E[pf,t+1 |qt ]] =
∑
(SI.10)
E[pf,t+1 |qt ] Pr(qt ),
qt
where Pr(qt ) is the distribu on of allele frequencies at me t. By inspec on of eq. (SI.8), we see that only pm,t ,
pf,t , Dm,t and Df,t depend on qt and thus have to be marginalized over qt . Doing so will define the moments of
the distribu on Pr(qt ) required to calculate the expected allele frequency change over one genera on. Since pm,t ,
pf,t , Dm,t and Df,t are all evaluated in the absence of phenotypic differences ((δm , δf ) = 0), they are marginalized
for a neutral process, and the expecta on operator is wri en E◦ [·]. We have E◦ [pm,t ] = pm and E◦ [pf,t ] = pf , and
evaluate E◦ [Dm,t ] and E◦ [Df,t ] below.
We will calculate E◦ [pmi (dzmi /dδm )] and E◦ [pfj (dzfj /dδf )] together, and then E◦ [pm (dz m /dδm )] and
E◦ [pf (dz f /dδf )], but first, we note that individual phenotype in terms of individual allele frequencies are
given by zmi = zm + δm (2hpmi + (1 − 2h)1♂i 1♀i ), and zfj = zf + δf (2hpfj + (1 − 2h)1♂j 1♀j ). So that average
∑
male and female phenotypic values are wri en as z m = i zmi /Nm = zm + δm (2hpm,t + (1 − 2h)1♂i 1♀i t )
∑
and z f = j zfj /Nf = zf + δf (2hpf,t + (1 − 2h)1♂j 1♀j t ). We then obtain the deriva ves with respect to δ of
4 SI
C. Mullon et al.
these averages and the phenotype of male i, which are needed for the popula on sta s cs, as
dzmi
= 2hpmi + (1 − 2h)1♂i 1♀i
dδm
dz m
= 2hpm,t + (1 − 2h)1♂i 1♀i t
dδm
dz f
= 2hpf,t + (1 − 2h)1♂j 1♀j .
t
dδf
(SI.11)
Using eq. (SI.1) together with eq. (SI.11), we have
]
[
]
)
1♂i + 1♀i (
dzmi
◦
E pmi
=E
h(1♂i + 1♀i ) + (1 − 2h)1♂i 1♀i
dδm t
2
[
]t
[
]
)
(
1
+
1
♀j
dzfj
♂j
E◦ pfj
= E◦
h(1♂j + 1♀j ) + (1 − 2h)1♂j 1♀j
,
dδf t
2
◦
[
(SI.12)
t
which expanded gives
[
]
[
]
dzmi
= E◦ h/2(1♂i + 21♂i 1♀i + 1♀i ) + (1 − 2h)1♂i 1♀i
E pmi
dδm t
t
[N
]
m
1 ◦ ∑
=
E
h/2(1♂i + 21♂i 1♀i + 1♀i ) + (1 − 2h)1♂i 1♀i
Nm
i=1
◦
(SI.13)
t
= E◦ [h/2(1♂i + 21♂i 1♀i + 1♀i ) + (1 − 2h)1♂i 1♀i ]t ,
where we have used that at neutrality, all males are expected to have the same genotypic composi on. More
succinctly, we write
]
dzmi
= h(pm,t + ηt ) + (1 − 2h)ηt
E pmi
dδm t
]
[
dzfj
= h(pf,t + ηt ) + (1 − 2h)ηt ,
E◦ pfj
dδf t
◦
[
(SI.14)
where η H = E◦ [1♂i 1♀i ] is the probability that both the paternal and maternal alleles of an individual are mutants.
In the absence of phenotypic differences, this probability is equal for all individuals E◦ [1♂i 1♀i ] = E◦ [1♂k 1♀k ]
for all i and k and irrespec ve of the sexes of the individuals. To see this, consider the recurrence for η over one
genera on: ηt+1 = E◦ [1♂i 1♀i ]t+1 . If individual i of genera on t + 1 has father indexed a and mother indexed c
at genera on t,
ηt+1 =
1 ◦
E [(1♂a + 1♀a )(1♂c + 1♀c )]t ,
4
(SI.15)
since the paternally inherited mutant of i is equally likely to be the paternally or the maternally inherited mutant
of its father a, and the maternally inherited mutant of i is equally likely to be the paternally or the maternally
inherited mutant of its mother c. This argument holds whatever the sex of i, so η = E◦ [1♂i 1♀i ] does not depend
on the sex of individual i. A similar argument shows that η is also equal to the probability that a paternally inherited
allele and a maternally inherited allele of two different, randomly sampled individuals are mutants, i.e. η =
E◦ [1♂i 1♀j ] = E◦ [1♂j 1♀i ] with i ̸= j.
C. Mullon et al.
5 SI
We now calculate E◦ [pm (dz m /dδm )] and E◦ [pf (dz f /dδf )]. Using eq. (SI.11) and rearranging to collect the terms
that involve the same male i, and those that involve two different males i and k, we have E◦ [pm (dz m /dδm )]t =
∑
∑
∑
∑
E◦ [2h/Nm2 ( i p2mi + i,k,i̸=k pmi pk ) + (1 − 2h)/(Nm2 )( i pmi 1♂i 1♀i + i,k,i̸=k pmi 1♂k 1♀k )]t . Le ng expecta on run through gives 2h/Nm (E◦ [p2mi ]t + (Nm − 1)E◦ [pmi pk ]t ) + (1 − 2h)/Nm (E◦ [pmi 1♂i 1♀i ]t + (Nm −
1)E◦ [pmi 1♂k 1♀k ]t ) where i ̸= k. Finally, factoring by 1/Nm yields
[
)
]
( (
)
(
)
dz m
1
E◦ pm
=
2h E◦ [p2mi ]t − E◦ [pmi pk ]t + (1 − 2h) E◦ [pmi 1♂i 1♀i ]t − E◦ [pmi 1♂k 1♀k ]t
dδm t
Nm
(SI.16)
+ 2hE [pmi pk ]t + (1 − 2h)E [pmi 1♂k 1♀k ]t .
◦
◦
Expanding the above in terms of indicator variables for paternally and maternally inherited alleles, we have
♀
E◦ [p2mi ] = E◦ [(1♂i + 1♀i + 21♂i 1♀i )/4] = (pm + η)/2, and we write E◦ [pmi pk ] = (2η + κ♂
m + κm )/4, where
◦
κ♂
m = E [1♂i 1♂k ] is the probability that two randomly sampled males i ̸= k both inherited the mutant allele
♀
from their fathers, and κm = E◦ [1♀i 1♀k ] is the probability that they inherited the mutant allele from their moth♀
◦
♂
ers. Then, E◦ [pmi 1♂i 1♀i ] = η, and finally E◦ [pmi 1♂k 1♀k ] = (ρ♂
m + ρm )/2, where ρm = E [1♂i 1♂k 1♀k ] is
the probability that randomly sampled male i has inherited the mutant from its father and that another randomly
♀
sampled male k is homozygous for the mutant, and ρm = E◦ [1♀i 1♂k 1♀k ] is the probability that randomly sampled male i has inherited the mutant from its mother and that another randomly sampled male k is homozygous
for the mutant. A er using the similar argument for E◦ [pf dz f ], we find that at genera on t
(
♀ )}
♀)
ρ♂
κ♂
t + ρt
t + κt
E
h pm,t −
+ (1 − 2h) ηt −
2
2
t
)
(
)
♀
♀
κ♂ + κt
ρ♂
t + ρt
+ h ηt + t
+ (1 − 2h)
,
2
2
{ (
(
]
[
♀)
♀ )}
1
κ♂
ρ♂
dz f
t + κt
t + ρt
◦
=
h pf,t −
+ (1 − 2h) ηt −
E pf
dδf t
Nf
2
2
(
)
(
)
♀
♀
κ♂ + κt
ρ♂
t + ρt
+ h ηt + t
+ (1 − 2h)
,
2
2
◦
[
dz m
pm
dδm
]
1
=
Nm
(
{ (
(SI.17)
♀
◦
◦
◦
♂
where for two randomly sampled females j ̸= l, κ♂
f = E [1♂j 1♂l ] , κf = E [1♀j 1♀l ], ρf = E [1♂j 1♂l 1♀l ]
♀
and ρf = E◦ [1♀j 1♂l 1♀l ].
Subs tu ng eqs. (SI.14) and (SI.17) into eq. (SI.8), we find that the uncondi onal expected allele frequencies in
the males and females of the next genera on are given by
pm,t+1 =
pf,t+1
where
Ku,t
6 SI
1
=
2
(
1
∂wm (0)
Nf
∂wm (0)
(pm,t + pf,t ) + δm Km,t m
+ δf
Kf,t f
2
∂zmi
Nm
∂zfj
Nm
∂wf (0)
∂wf (0)
1
Km,t m
+ δf Kf,t f .
= (pm,t + pf,t ) + δm
2
Nf
∂zmi
∂zfj
1
1−
Nu
(
)[ (
♀ )]
♀ )
ρ♂
κ♂
u,t + κu,t
u,t + ρu,t
h pu,t −
+ (1 − 2h) ηt −
,
2
2
C. Mullon et al.
(SI.18)
(SI.19)
for u ∈ {m, f}. The la er can be interpreted as the neutral expecta on of the covariance between genotype and
phenotype at genera on t in an individual of sex u. Indeed, from eqs. (SI.6) and (SI.10), we have that Ku is also
equal to
Ku,t
[
)]
(
Nu
1 ◦ 1 ∑
dzui (0) dz u (0)
= E
−
,
pui,t
2
Nu i
dδu
dδu
(SI.20)
and since zui = zu + δu (2hpui + (1 − 2h)1♂i 1♀i ), this may be wri en as
Ku,t
[
]
Nu
1 1 ◦ 1 ∑
=
E
pui,t (zui − z u )
2 δu
Nu i
[
]
Nu
1 1 ◦ 1 ∑
E
=
(pui,t − pui,t ) (zui − z u ) .
2 δu
Nu i
(SI.21)
Therefore, Ku,t is propor onal to the expected covariance E◦ [C[pui,t , zui ]] at genera on t between individual
genotype and phenotype in sex u, when mutant frequencies pui,t evolve neutrally.
Closing the recursion. Eq. (SI.18) gives the change of pm and pf over one genera on, which depends on higher
♀
♀
♂
moments of the distribu on of the mutant in the popula on (ηt , κ♂
u,t , κu,t , ρu,t , and ρu,t ). These la er also
change from one genera on to the next, and in order to evaluate the change of pm,t and pf,t over more than one
genera on, we need to characterize these recursions. Since they are evaluated at (δm , δf ) = 0 in eq. (SI.18), it
♀
♀
♂
is sufficient to evaluate the recursions for ηt , κ♂
u,t , κu,t , ρu,t , and ρu,t at neutrality, where they are only affected
by gene c dri . We give these recursions below using standard popula on gene c methods (Karlin 1968, for
example).
The probability that a gene sampled in an individual is mutant does not depend on the sex of the individual as it
comes with equal probability from its father or its mother
pm,t+1 = pf,t+1 =
) 1
1( ◦
E [1♂i + 1♂i ]t = (pm,t + pf,t ).
2
2
(SI.22)
The probability that the paternally and the maternally inherited allele of individual i at me t + 1 are both mutant,
ηt+1 , is given in terms of neutral moments of gene frequency at genera on t in eq. (SI.15) which, if expanded,
gives
ηt+1 =
1
♀
(2ηt + κ♂
c,t + κc,t ).
4
(SI.23)
♀
◦
◦
where for a male i and a female j, κ♂
c = E [1♂i 1♂j ], and κc = E [1♀i 1♀j ].
The probability that two paternally inherited alleles randomly sampled in two different males are both mutants at
genera on t+1, κ♂
m,t+1 , depends on whether the two males have the same father, which occurs with a probability
♂
denoted Θ♂
m or not (which occurs with probability 1 − Θm ). These probabili es are referred to as probabili es of
sibships. If the two males have the same father, which we index a, then their paternal alleles can be either both
copies of the paternal gene of a (with probability 1/4), both copies of the maternal gene of a (with probability
C. Mullon et al.
7 SI
1/4), or one is a paternal copy and one is a maternal copy (with probability 1/2). So, if two males have the same
father, their two paternally sampled genes are mutants with probability (1/4)E◦ [(1♂a + 1♀a )2 ]t . If they have
different fathers, indexed a and b, then the paternal copy of the first male may be the paternal or maternal copy
of a (each with probability 1/2), and the paternal copy of the second male may be the paternal or maternal copy
of b (also each with probability 1/2). In this case, the paternal alleles of the two individuals are both mutants with
probability (1/4)E◦ [(1♂a + 1♀a )(1♂b + 1♀b )]t . Combining these two cases, the probability that two randomly
◦
♂
sampled paternal alleles of different males at genera on t + 1 are mutants is κ♂
m,t+1 = Θm (1/4)E [(1♂a +
◦
◦
1♀a )2 ]t + (1 − Θ♂
m )(1/4)E [(1♂a + 1♀a )(1♂b + 1♀b )]t which, a er le ng expecta on E [.] run through and
♀
♂
♂
♂
using previous defini ons, gives κ♂
m,t+1 = Θm (2ηt + pm,t + pf,t )/4 + (1 − Θm )(2ηt + κm,t + κm,t )/4. In fact,
we find more generally that the probabili es that the paternal alleles of two males (x = m), or of two females
(x = f), or of a male and female (x = c) are mutants at genera on t + 1 are given by
κm
x,t+1 =
Θ♂
1 − Θ♂
♀
x
x
(2ηt + pm,t + pf,t ) +
(2ηt + κ♂
m,t + κm,t )
4
4
(SI.24)
♂
where Θ♂
f is the probability that two females have the same father and, Θc is the probability that a male and a
female have the same father.
Using a similar argument, we find that the probabili es that the maternal alleles of two males (x = m), or of two
females (x = f), or of a male and female (x = c) are mutants at genera on t + 1 are given by
♀
♀
Θx
1 − Θx
♀
♀
κx,t+1 =
(2ηt + pm,t + pf,t ) +
(2ηt + κ♂
f,t + κf,t ),
4
4
(SI.25)
♀
where Θx is the probability that two individuals, whose sexes are given by x, have the same mother.
◦
The probability ρ♂
m,t+1 = E [1♂i 1♂k 1♀k ]t+1 that two (different) paternally inherited alleles and one mater-
nally inherited allele at genera on t + 1 are mutants depends on whether the males from which the paternal
alleles are sampled (males i and k here) have the same father (indexed a) or different fathers (a and b). Using a similar argument as in the preceding sec on, and indexing by c the mother of the male who holds the
◦
◦
♂
2
♂
maternal allele, we have ρ♂
m,t+1 = Θm (1/8)E [(1♂a + 1♀a ) (1♂c + 1♀c )]t + (1 − Θm )(1/8)E [(1♂a +
1♀a )(1♂b + 1♀b )(1♂c + 1♀c )]t . Then, expanding and le ng expecta on run through, we have: ρ♂
m,t+1 =
(
)
(
)
♀
♀
♀
♀
♀
♂
♂
♂
♂
♂
Θ♂
2ηt + κ♂
m
c,t + κc,t + 2ρc,t + 2ρc,t /8 + (1 − Θm ) ς2m,t + ς2m,t + 2ρc,t + 2ρc,t + ρm,t + ρm,t /8, where
♂ = E◦ [1 1 1 ] and ς ♀ = E◦ [1 1 1 ] are the probabili es that the paternal and maternal alleς2m,t
♀a ♀b ♀c t
2m,t
♂a ♂b ♂c t
les, respec vely, of two randomly sampled (without replacement) males a and b and a female c at genera on t
are all mutants. We find in general that for x ∈ {m, f, c}
ρ♂
x,t+1 =
8 SI
(
)
Θ♂
♀
♀
x
♂
2ηt + κ♂
c,t + κc,t + 2ρc,t + 2ρc,t
8
(
)
1 − Θ♂
x
♂ + ς ♀ + 2ρ♂ + 2ρ♀ + ρ♂ + ρ♀
ς2m,t
+
c,t
m,t
c,t
m,t
2m,t
8
C. Mullon et al.
(SI.26)
Similarly, the probability that two (different) maternally inherited alleles and one paternally inherited allele from
♀
two individuals are mutants at genera on t + 1, ρx,t+1 = E◦ [1♀i 1♀j 1♂k ]t+1 , depends on whether individuals i
and j from which maternal genes are sampled have the same mother (indexed c) or different mothers (c and d),
♀
♀
♀
ρt+1 = Θx (1/8)E◦ [(1♂c + 1♀c )2 (1♂a + 1♀a )]t + (1 − Θx )(1/8)E◦ [(1♂c + 1♀c )(1♂d + 1♀d )(1♂a + 1♀a )]t ,
where a is the father of the individual whose paternal gene is sampled. Then for x ∈ {m, f, c}
)
Θ♂ (
♀
♀
♀
♂
ρx,t+1 = x 2ηt + κ♂
c,t + κc,t + 2ρc,t + 2ρc,t
8
♀
)
1 − Θx ( ♂
♀
♀
♀
♂
+
ς2m,t + ς2m,t + 2ρ♂
c,t + 2ρc,t + ρf,t + ρf,t
8
(SI.27)
♂ = E◦ [1 1 1 ] and ς ♀ = E◦ [1 1 1 ] are the probabili es that the paternal and maternal
where ς2f,t
♀a ♀c ♀d t
2f,t
♂a ♂c ♂d t
alleles, respec vely, of a male a and of two different females c and d at genera on t are all mutants.
The probability that three alleles sampled from different individuals are mutants depends on the probabili es
of sibship of three individuals. In order to consider the itera on of the probability ςx♂ , i.e. that three randomly
chosen paternally inherited genes are mutants, we need to separate the cases where all three individuals are
males (subscript x = 3m), all three are females (x = 3f), two are males and one is female (x = 2m), or two
are females and one is male (x = 2f). The probabili es that three paternal alleles are mutants then depend on
whether all three individuals have the same father, which occurs with a probability we write as Ξ3♂
x , whether
only two have a same father (with probability Ξ2♂
x ), or if none of the three have the same father (with probability
♂
1 − Ξ3♂
x − Ξ2x ). If they all have the same father (indexed a), then they are all mutants if they have inherited
the mutant gene from the maternal or paternal locus from a. And similar arguments apply for the case when only
two have the same father (indexed a, and the other father is indexed b) or if they have three different fathers
◦
2
3
♂ ◦
♂
= Ξ3♂
(indexed a, b and c) to give ςx,t+1
x E [(1♂a + 1♀a ) ]t /8 + Ξ2x E [(1♂a + 1♀a ) (1♂b + 1♀b )]t /8 +
]
♂ ◦[
(1 − Ξ3♂
x − Ξ2x )E (1♂a + 1♀a )(1♂b + 1♀b )(1♂c + 1♀c ) t /8, which, expanding and le ng expecta on
run through, results in
♂ =
ςx,t+1
Ξ3♂
Ξ2♂
♀
♀
x
x
♂
(pm,t + pf,t + 6ηt ) +
(2ηt + κ♂
m,t + κm,t + 2ρm,t + 2ρm,t )
8
8
♂
1 − Ξ3♂
x − Ξ2x
♂ + ς ♀ + 3ρ♂ + 3ρ♀ ).
+
(ς3m,t
m,t
m,t
3m,t
8
(SI.28)
♀
Similarly, the probability that three randomly chosen maternally inherited genes ςx are mutants can be expressed
in terms of the probabili es that the individuals have the same mother,
♀
♀
Ξ3x
Ξ2x
♀
♀
♀
♂
ςx,t+1 =
(pm,t + pf,t + 6ηt ) +
(2ηt + κ♂
f,t + κf,t + 2ρf,t + 2ρf,t )
8
8
♀
♀
1 − Ξ3x − Ξ2x ♂
♀
♀
+
(ς3f,t + ς3f,t + 3ρ♂
f,t + 3ρf,t )
8
(SI.29)
♀
where Ξ3x is the probability that the three holders (whose sexes are given by x ∈ {3m, 3f, 2m, 2f}) have the
C. Mullon et al.
9 SI
♀
same mother, and Ξ2x is the probability that out of the three individuals, two have the same mother. The mo♂ and ς ♀
ments ςx,t+1
x,t+1 (x ∈ {3m, 3f}) also sa sfy the recurrences given by eqs. (SI.28)(SI.29), and complete the
necessary moments to iterate eq. (SI.18).
Probability of fixa on of an autosomal mutant. We proceed to calculate the probability of fixa on of the mutant
by itera ng its expected change over many genera ons. Eqs. (SI.22) - (SI.29) define the changes in the moments of
the popula on genotypic distribu on of a neutral mutant. Since eqs. (SI.22) - (SI.29) are all linear in the relevant
moments, we may express the set of recurrences as a matrix opera on: pt+1 = A◦ pt , where pt is a 23 × 1
♀ ♂ ♀ ♂ ♀
vector which collects the necessary moments of Pr(qt ) (pm , pf , η, κ♂
x , κx , ρx , ρx , ςy ,ςy ) for x ∈ {m, c, f},
y ∈ {3m, 3f, 2m, 2f}, and A◦ is a 23 × 23 matrix defined by eqs. (SI.22) - (SI.29).
Eq. (SI.18) adds the effects of selec on to the expected mutant frequency change. Since it is also linear in pm , pf ,
♀ ♂
♀
η, κ♂
x , κx , ρx , and ρx , it may also be represented as a matrix opera on, giving
pt+1 = Apt
.
.
.
A = A◦ + δm Am + δf Af + O(δ 2 ),
with
(SI.30)
.
where the 23 × 23 matrices Am and Af describes the first order perturba on of average frequency change due to
mutant effect in males and females respec vely. Eq. (SI.30) fully characterizes the expected frequency change of
a mutant in a sexually dimorphic popula on at any genera on i.e., the model is dynamically sufficient.
Explicit expression for these large matrices are omi ed from this paper, but they can be found straigh or-
.
.
wardly from eqs. (SI.22) - (SI.29) for A◦ and from eq. (SI.18) for Am and Af . Their entries will of course depend on the order chosen for the entries of pt . We will assume here that the first 15 entries of pt are pt =
T
♂ ♂ ♀ ♀ ♀ ♂ ♂ ♂ ♀ ♀ ♀
(pm , pf , η, κ♂
m , κc , κf , κm , κc , κf , ρm , ρc , ρf , ρm , ρc , ρf , . . .) .
We derive the expression for the fixa on probability π of the mutant by es ma ng the asympto c sum of expected
allele-frequency change of the allele in males and females (Leturque and Rousset 2002; Rousset 2004; Lessard and
Ladret 2007; Lehmann and Rousset 2009). The fixa on probability of the mutant πm in males, and πf in females
is the asympto c average frequency of the mutant in each sex
πm = lim pm,t ,
t→∞
πf = lim pf,t .
t→∞
(SI.31)
Because the mutant allele eventually is either eliminated or fixated in the popula on, the fixa on probability in
males and females is the same πm = πf = π. The fixa on probabili es in males and females could be obtained
from the asympto c vector limt→∞ At p0 , but this is difficult as it requires the calcula on of A's eigenvectors. We
rely on an alterna ve scheme to obtain π. To that aim, it is convenient to express the fixa on probability of the
mutant as the average
π = απm + (1 − α)πf ,
(SI.32)
where the weight α is chosen such that the expected frequency change of a neutral mutant in any genera on t is
10 SI
C. Mullon et al.
zero: αE[∆pm,t ] + (1 − α)E[∆pf,t ] = 0. In this case, α = 1/2 for a diploid, autosomal gene c system. Together,
eqs. (SI.31) & (SI.32) imply that π is the average sum of gene frequency change in males and females, from the
appearance to the eventual fixa on or loss of the mutant
π = αpm,0 + (1 − α)pf,0 +
∞ (
∑
)
αE[∆pm,t ] + (1 − α)E[∆pf,t ] .
(SI.33)
t=0
The probability of fixa on of a mutant with ini al frequencies pm,0 in males and pf,0 females is approximated to the first order of δ : π = αpm,0 + (1 − α)pf,0 + αδm (∂π(0)/∂δm ) + (1 − α)δf (∂π(0)/∂δf ) +
O(δ 2 ). We begin by considering the first order effects of male phenotype on π. Using eq. (SI.33), it is
∑∞
∂π(0)/∂δm = (∂/∂δm ) t=0 (αE[∆pm,t ] + (1 − α)E[∆pf,t ])δm =δf =0 ,. In matrix nota on, this is ∂π(0)/∂δm =
∑∞
α· t=0 (∂/∂δm )(pt+1 −pt )δm =δf =0 where p = pm , pf , . . . and α = (α, 1−α, 0, . . . , 0) is such that when dot mulplied with pt , it collects and sums pm,t and pf,t weighted by the reproduc ve values. Then, using eqs. (SI.30), we
.
have ∂(pt+1 − pt )/∂δm = Am pt . So the male perturba on of the probability of fixa on may be wri en as
∞
∑
∂π(0)
=α·
Am pt .
∂δm
δm =δf =0
t=0
.
The sum
∑∞
.
t=0
pt |δm =δf =0 , which we write as
∑∞
t=0
(SI.34)
p◦t where p◦t+1 = A◦ p◦t , does not converge as A◦ is not regular.
This means A cannot be factored out of the sum in eq. (SI.34). To circumvent this problem, we construct an
.
itera on around a centred variable using the zero row-sum property of matrix Am (Lehmann and Rousset 2009).
∑∞
∑∞
To that aim, we define a vector q◦t and a matrix Q◦ such that (i) t=0 Am p◦t = t=0 Am (p◦t −q◦t ), (ii) p◦t+1 −q◦t+1 =
.
.
(A◦ − Q◦ )(p◦t − q◦t ), and (iii) limt→∞ (p◦t − q◦t ) = 0. The choice of q◦t with all vector elements being equal to
αpf,t + (1 − α)pm,t , which acts as a reference variable, and Q◦ = (qij ) with all elements of column 1 being equal
to α, all elements of column 2 being equal to 1 − α, and zero otherwise sa sfies all three condi ons. In effect, this
choice of the vector q◦t centers the itera on around the mutant frequency averaged across the sexes according to
their reproduc ve class (this average is the reference variable), while Q◦ provides the itera on of the reference
variable.
.
.
∑∞
∑∞
Using proper es (i)-(iii) in the preceding paragraph, we can now factorize t=0 Am pt = Am t=0 (p◦t − q◦t ) =
∑∞
Am t=0 (A◦ − Q◦ )t (p0 − q◦0 ). With all eigenvalues of (A◦ − Q◦ ) being less than 1 in absolute value (Lehmann
∑∞
−1
and Rousset 2009, p. 47), the sum d◦ = t=0 (A◦ − Q◦ )t (p0 − q◦0 ) can be evaluated as [I − A◦ + Q◦ ] , where I
.
is the iden ty matrix, so we have
.
∂π(0)
= α · Am d◦ ,
∂δm
where d◦ = [I − A◦ + Q◦ ]
−1
(p0 − q0 ).
(SI.35)
.
All the arguments used to derive eq. (SI.35) can be used for ∂π(0)/∂δf , and we find ∂π(0)/∂δf = α · Af d◦ . Hence,
C. Mullon et al.
11 SI
the fixa on probability to the first order in selec on intensity is
.
.
π = αpm,0 + (1 − α)pf,0 + δm α · Am d◦ + δf α · Af d◦ + O(δ 2 ).
(SI.36)
The entries of d◦ can be interpreted in terms of mean coalescence mes in the resident popula on. To see
this, we first note that if the expected ini al frequency of the mutant is the same in males and females, then
pm,0 = pf,0 = p0 , which is equivalent to assuming that muta on rate is the same in males and females.
Then, if the mutant arose as a single copy, p0 = 1/(2N ), where N = Nm + Nf , and we have p0 − q0 =
(0, 0, −1/(2N ), −1/(2N ), . . . , −1/(2N ))T . In this case, element d◦i for i ≥ 3 of d◦ is
d◦i = −T(i) /(2N ),
(SI.37)
where T(i) is the mean coalescent me into a single individual of a set of gene lineages ini ally residing in state
i (Lehmann and Rousset 2009, eqs. A-28 & A-29). State here refers to the configura on of the sampled gene
lineages, which are given by the entries of pt , e.g., for i = 3, if the third entry of pt corresponds to ηt , the
probability that an individual's paternal and maternal alleles are both mutant, so d◦3 = −T(3) /(2N ), where T(3) is
the expected number of genera ons taken for the paternal and maternal genes of an individual to coalesce.
.
.
Subs tu ng for α = 1/2 (for an autosomal gene) and for matrices Am and Af into eq. (SI.36), the probability of
fixa on of a single copy mutant (pm,0 = pf,0 = 1/(2N )) can be expressed as eq. (1) in the main text, where if
♂ ♂ ♀ ♀ ♀ ♂ ♂ ♂ ♀ ♀ ♀
T
pt = (pm , pf , η, κ♂
m , κc , κf , κm , κc , κf , ρm , ρc , ρf , ρm , ρc , ρf , . . .) , the sex-specific weights Km and Kf are
given by
(
)[
( ◦
)
( ◦
)]
1
1
d4 + d◦7
d10 + d◦13
1−
−h
− (1 − 2h)
− d◦3
4
Nm
2
2
(
)[
( ◦
)
( ◦
)]
◦
1
1
d6 + d9
d12 + d◦15
Kf =
1−
−h
− (1 − 2h)
− d◦3 ,
4
Nf
2
2
Km =
(SI.38)
with di as the ith entry of the vector d◦ defined in eq. (SI.35). This shows that Km and Kf may be interpreted in
terms of coalescent mes for sampled genes (eq. SI.37). Alterna vely, using eq. (SI.21), we see that Km and Kf
can be interpreted as the expected covariance between between genotype and phenotype in males and females
respec vely, cumulated over the neutral segrega on of the mutant
Ku =
∞
∞
1 1 ∑
1 1 1 ∑ ◦
Ku,t =
E [C[pui,t , zui ]]
2 2N t=0
4 2N δu t=0
(SI.39)
where the sum runs from the appearance to the eventual fixa on or loss of the mutant.
Probabili es of sibships of three individuals. Un l now, all our results hold for any arbitrary popula on size, but
this implies tracking many gene associa ons. Indeed, as eqs. (SI.22) - (SI.29) show, the itera on of eq. (SI.18) over
♀
mul ple genera ons depends on the six probabili es of sibships over two individuals, Θ♂
x and Θx (x ∈ {m, c, f}),
♀
♂ and Ξvw
and the eight probabili es of sibships over three individuals Ξvw
(v ∈ {2, 3}, w ∈ {m, f}). Therefore,
12 SI
C. Mullon et al.
Km and Kf (eq. SI.38) also depend on these fourteen probabili es. As we show below, we can significantly reduce
the number of necessary probabili es of sibships by approxima ng the probabili es of sibship of three individuals
♀
♀
♂ and Ξvw
Ξvw
as func ons of the probabili es of sibship of two individuals Θx♂ and Θx when we only consider
the first order effects of finite popula on size O(1/N ).
∑Nm (W m ) (Nm )
◦
mi
The probability that three randomly sampled adult males have the same father is Ξ3♂
3m = E [
i
3 / 3 ].
In the absence of phenotypic differences, each male has the same distribu on of reproduc ve output and Ξ3♂
3m =
m3
m2
m
m
1/((Nm − 1)(Nm − 2))E◦ [Wmi
− 3Wmi
+ 2Wmi
]. If we assume that the distribu on for Wmi
is sufficiently well-
behaved, and that the number of adult descendants of a male stays bounded as popula ons size (N ) tends to
mx
infinity (or that E◦ [Wmi
], x ≥ 0, remains bounded as N → ∞), we find that none of the terms in Ξ3♂
3m are
2
of order 1/N or more, i.e. Ξ3♂
3m = 0 + O(1/N ), so the probability that three randomly sampled adult males
have the same father can be approximated to being zero when N is large. Similarly, we find that all probabili es
♀
2
of sibship three genes in the same individual are approximately zero, Ξ3♂
x = Ξ3x = 0 + O(1/N ) for x ∈
{3m, 3f, 2m, 2f}.
Rather than calcula ng Ξ2♂
3m the probability that out of three males only two have the same father directly, it is
easier to consider the probability that out of three males, none have the same father. These two probabili es are
2
♂
♂
♂
related by 1 − Ξ3♂
3m − Ξ23m = 1 − Ξ23m (since Ξ33m = 0 + O(1/N )). The probability that out of three males,
none have the same father is given by the expected value of the ra o of the number of ways three individuals
may be sampled from the male offspring of three different adult males to the number of ways of sampling three
(Nm )
∑Nm
m
m
m
males out of the en re male popula on 1 − Ξ2♂
3m = [
i,j,k Wmi Wmj Wmk / 3 ]i̸=j̸=k̸=i , which a er taking the
m
m
m
sum and denominator outside reduces to E◦ [Wmi
Wmj
Wmk
]i̸=j̸=k̸=i . Again by assuming that the number of adult
descendants of a male stays bounded as popula ons size tends to infinity, using the delta method (Oehlert 1992),
[ m
]
◦
m
2
m
and observing that E◦ [Wmi
] = 1, we obtain 1 − Ξ2♂
3m = 1 + 3C Wmi , Wmj i̸=j + O(1/N ).
[ m
]
m
, Wmj
may be expressed in terms of Θ♂
The covariance term C◦ Wmi
m . The probability that two males do
i̸=j
(Nm )
∑ ∑
◦
m
m
not have the same father is, by defini on, 1 − Θ♂
m , but it is also given by E [
i
j̸=i Wmi Wmj / 2 ] =
[ m
]
m
m
m
m
m
E◦ [Wmi
, Wmj
]i̸=j = C◦ [Wmi
Wmj
]i̸=j + 1, so that C◦ Wmi
, Wmj
= −Θ♂
m . Hence subs tu ng back into
i̸=j
the probability that out of three males none have the same father, and solving for Ξ2♂
3m , we obtain that the
probability that out of three males only two have the same father is
♂
2
Ξ2♂
3m = 3Θm + O(1/N ).
(SI.40)
f
The remaining probabili es can be derived by using the same argument, and that E◦ [Wmi
] = Nf /Nm , produc-
ing
♂
2
Ξ2♂
3f = 3Θf + O(1/N )
2
4
1
Ξ2♂
+ Θ♂
+ Θ♂
+ O(1/N 2 )
2m =
3Nm
3 c
3 m
(
)
2
2
1
4
1 ♂
2
Ξ2♂
=
−
+ Θ♂
2f
c + Θf + O(1/N ).
3 Nm
Nf
3
3
C. Mullon et al.
(SI.41)
13 SI
By symmetry, we find that the probabili es of sibship of three maternal genes are given to the order O(1/N )
by
♀
♀
Ξ23m = 3Θm + O(1/N 2 )
♀
♀
Ξ23f = 3Θf + O(1/N 2 )
(
)
2 2
1
4
1 ♀
♀
2
Ξ22m =
−
+ Θ♀
c + Θm + O(1/N )
3 Nf
Nm
3
3
2
4
1 ♀
♀
2
Ξ22f =
+ Θ♀
c + Θf + O(1/N ).
3Nf
3
3
(SI.42)
So assuming the popula on is large, the itera on of eq. (SI.18) over many genera ons depends only on the six
♀
probabili es of sibships over two individuals, Θ♂
x and Θx (x ∈ {m, c, f}).
Solving for Km and Kf in terms of the probabili es of sibships of two individuals. Having expressed the eight
probabili es of sibships of three individuals in terms of the probabili es of sibships of two individuals Θuv , the
matrix A◦ now only depends on these la er six probabili es of sibships, and therefore, so do Km and Kf (eq. SI.38).
Despite this simplifica on, solving explicitly for Km and Kf s ll requires inver ng a 23x23 matrix, (I − A◦ + Q◦ )
−1
,
which is computa onally expensive and unlikely to yield results easy to interpret. Numerical results for Km and
Kf with arbitrary dominance are shown in fig. 4.D of the main text. However, if h = 1/2, only the first nine entries
of pt are required to generate the expected frequency change over many genera ons, and hence the probability
−1
of fixa on. Thus, A◦ reduces to a 9 × 9 matrix. In this case, (I − A◦ + Q◦ )
can be inverted analy cally, and using
(SI.38) with h = 1/2, Km and Kf are as eq. (A.2) in the main text.
Probabili es of sibship of two individuals. The probability of fixa on of a mutant depends on the probabili es
of sibship of two individuals in the resident popula on. Here, the probabili es of sibship are expressed in terms
of the first (µ's) and second (ν and ρ) moments of the distribu on of offspring produced by a resident male and a
resident female to give table 1 of the main text.
The probability that two randomly sampled adult males have the same father, Θ♂
m , is given by the expected
value of the ra o of the number of ways two individuals may be sampled from the number of adult males produced by each male, to the number of ways of sampling two males out of the en re male popula on, i.e.,
∑Nm (W m ) (Nm )
◦
m
mi
Θ♂
m = E [
i=1
2 / 2 ], where Wmi is the random variable for the number of male breeders produced
by male i. In the absence of phenotypic differences in the popula on, each male has the same distribu on for
their reproduc ve output, so the sum may be taken out in Θ♂
m , and the subscript i now denotes a randomly samm
m
m
pled male: 1/(Nm − 1) [V◦ [Wmi
] + E◦ [Wmi
](E◦ [Wmi
] − 1)]. The expected number of male adults produced by a
m
male in the absence of phenotypic differences, E◦ [Wmi
] = 1, so the probability that two randomly sampled adult
◦
m
males have the same father reduces to Θ♂
m = V [Wmi ]/(Nm − 1). Condi oning on the number of male juveniles
produced in the popula on, and using the law of total variance, this gives
Θ♂
m =
1
Nm − 1
(
Nm2 V◦
[
]
)
m
Jmi
m
m
+ E◦ [V◦ [Wmi
|Jmi
, Jm ]] .
Jm
(SI.43)
The second variance term in eq. (SI.43) depends on how culling or regula on is assumed to take place, which is
14 SI
C. Mullon et al.
m
assumed here to occur by sampling juveniles without replacement. In this case, Wmi
follows a hypergeometric
m
distribu on with Nm draws and parameters given by the realiza on of Jm
m , with ini al probability of success Jmi /Jm
m
m
m
m
and a total popula on size of Jm . Then, E◦ [V◦ [Wmi
|Jmi
, Jm ]] = E◦ [Nm Jmi
(Jm − Jmi
)(Jm − Nm )/(Jm2 (Jm − 1))].
Both variance terms in eq. (SI.43) are approximated omi ng terms of order 1/N 2 using the delta method. With
assump on eq. (A.1) in the main text, the second variance term can be approximated as
[
]
m
m
Nm Jmi
(Jm − Jmi
)(Jm − Nm )
1
E◦ [J m ]
µm
1
E◦
≈ ◦ mi = mi
=
.
2
Nm − 1
Jm (Jm − 1)
E [Jm ]
µm
N
m
T
(SI.44)
Then, using the delta method with the variance operator, the first variance term in eq. (SI.43) is
Jm
Nm2
V◦ [J m ]
νm
2
V◦ [ mi ] = Nm ◦ mi2 + O(1/N 2 ) = Nm mii
2 + O(1/N ).
Nm − 1
Jm
E [Jm ]
µm
T
(SI.45)
Finally, subs tu ng eqs. (SI.44)(SI.45) into eq. (SI.43) gives Θ♂
m in table 1 of the main text. Using the same argument, we find a similar form for the probabili es that two females have the father Θ♂
f , that two males have the
♀
♀
same mother Θm and that two females have the same mother Θf (see table 1 in the main text).
∑Nm
◦
m
f
The probability that a male and a female have the same father Θ♂
c is given by E [
i=1 Wmi Wmi /(Nm Nf )], where
f
Wmi
is the random variable for the number of female breeders produced by male i. By condi oning on the juvenile
produc on of every individual and using the assump on that male and female offspring are culled independently,
◦ m f
we have Θ♂
c = Nm E [Jmi Jmi /(Jm Jf )]. The delta method is used to approximate the la er. Then, expanding
m
f
about the means of Jmi
, Jmi
, Jm and Jf and using condi on eq. (A.1) in the main text, we have
m
f
1
C[Jmi
, Jmi
]
1
Θ♂
+ Nm
=
c =
Nm
E[Jm ]E[Jf ]
Nm
(
ρm,f
1 + mmiif
µmi µmi
)
,
(SI.46)
m
f
where ρm,f
mii = C[Jmi , Jmi ] is the covariance between the number of male and offspring juveniles fathered by a
male. Using a similar argument, the probability that a male and a female have the same mother is found as in
table 1 of the main text.
References
K
, S., 1968, Equilibrium Behavior of Popula on Gene c Models with Non-Random Ma ng: Part II: Pedigrees,
Homozygosity and Stochas c Models. Journal of Applied Probability 5(3): 487+.
L
, L. and F. R
, 2009, Perturba on expansions of mul locus fixa on probabili es for frequency-
dependent selec on with applica ons to the Hill-Robertson effect and to the joint evolu on of helping and
punishment. Theore cal popula on biology 76(1): 35--51.
L
, S. and V. L
, 2007, The probability of fixa on of a single mutant in an exchangeable selec on model.
Journal of mathema cal biology 54(5): 721--744.
C. Mullon et al.
15 SI
L
, H. and F. R
, 2002, Dispersal, kin compe
on, and the ideal free distribu on in a spa ally het-
erogeneous popula on. Theore cal popula on biology 62(2): 169--180.
O
P
R
, G. W., 1992, A Note on the Delta Method. The American Sta s cian 46(1): 27--29.
, G. R., 1970, Selec on and covariance. Nature 227(5257): 520--521.
, S., 2008, A stochas c version of the Price equa on reveals the interplay of determinis c and stochas c
processes in evolu on. BMC Evolu onary Biology 8(1): 262+.
R
16 SI
, F., 2004, Gene c Structure and Selec on in Subdivided Popula ons . Princeton University Press.
C. Mullon et al.

Similar documents

×

Report this document