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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.