ploidy and the evolution of endosperm of flowering plants · 2010. 1. 22. · endosperm (2m:1p)....

Copyright � 2010 by the Genetics Society of AmericaDOI: 10.1534/genetics.109.110833

Ploidy and the Evolution of Endosperm of Flowering Plants

Aurelie Cailleau,1 Pierre-Olivier Cheptou and Thomas Lenormand

Centre d’Ecologie Fonctionnelle et Evolutive, 34293 Montpellier, France

Manuscript received October 9, 2009Accepted for publication November 16, 2009

ABSTRACT

In angiosperms, spermatozoa go by pair in each pollen grain and fertilize, in addition to the egg cell,one of its sister cells, called the central cell. This ‘‘double fertilization’’ leads to the embryo on the onehand and to its nutritive tissue, the endosperm, on the other hand. In addition, in most flowering plants,the endosperm is triploid because of a doubled maternal genetic contribution in the central cell. Most ofthe hypotheses trying to explain these eccentricities rest on the assumption of a male/female conflict overseed resource allocation. We investigate an alternative hypothesis on the basis of the masking ofdeleterious alleles. Using analytical methods, we show that a doubled maternal contribution and doublefertilization tend to be favored in a wide range of conditions when deleterious mutations alter thefunction of the endosperm. Furthermore, we show that these conditions vary depending on whether thesetraits are under male or female control, which allows us to describe a new type of male/female conflict.

FEMALES provision their offspring with resources.These resources can be accumulated before fertili-

zation. When some of the gametes are not fertilized,this strategy is extremely wasteful. In a number ofspecies, females have evolved conditional strategiesavoiding this problem—in particular by providing mostresources after fertilization [i.e., to diploid offspringinstead of haploid gametes (Westoby and Rice 1982;Baroux et al. 2002)]. This postfertilization provision-ing occurs, for instance, in viviparous animals, such asmammals, some reptiles (De Fraipont et al. 1996),some sharks (Wourms 1993), some scorpions (Brown

and Formanowicz 1996), and velvet worms (Tutt et al.2002). In plants, resource provisioning to offspring canalso occur before or after fertilization.

The life cycle of plants typically alternates a sporo-phytic stage (2n) and a gametophytic stage (n) that areboth multicellular. A newly formed sporophyte (thediploid zygote) will typically use the resources accumu-lated in tissues derived from the female haploid game-tophyte and not in tissues derived from the diploidsporophyte. For instance, in ferns and mosses, the newembryo develops ‘‘parasitically’’ on the free gameto-phyte. In gymnosperms, the embryo acquires its resourcefrom a nonfree female gametophyte (it is encapsulatedwithin sporophytic tissues). In some of them (cycadsand ginkgoes) the resource is fully accumulated beforefertilization, while in some others (Pinaceae) the re-source accumulates both before and after fertilization(Friedman 2001b). The former situation is comparable

to what happens in many fish while the latter is com-parable to what happens in many bird species (eggformation before fertilization and incubation after fer-tilization). In angiosperms, resource provisioning to theseed really starts after fertilization (Friedman 2001b), asituation comparable to placental mammals (Figure 1).

Compared to other seed plants, the case of angio-sperms is complicated by the occurrence of a ‘‘doublefertilization’’ (Nawaschin 1898; Guignard 1899): twoidentical (mitotically derived) spermatozoa from a sin-gle pollen grain fertilize in parallel two cells of thefemale gametophyte. The fertilization of the egg cellproduces the embryo. The fertilization of the centralcell produces the endosperm, a specific organ thatserves as an interface for resource transfer between thediploid mother and its offspring (like the placenta inmammals). This tissue can either be transitory or serveas a storage tissue in the seed. In this article, we study theevolution of this nutritive tissue. More specifically, wefocus on the evolution of its genetic constitution.

The diversity of endosperms reflects directly the diver-sity of female gametophytes (Friedman et al. 2008).In some angiosperms [Austrobaileyales (Friedman et al.2003), Oenothera (Maheshwari 1963), and Nym-pheales (Williams and Friedman 2002)], the femalegametophyte is made of four haploid cells that aremitotically derived from a single spore. One of themis the central cell and contains one maternal contribu-tion (1m). This type of gametophyte is referred to as‘‘Oenothera.’’ After double fertilization (which addsone paternal contribution, 1p), the central cell becomesa diploid endosperm (1m:1p). This endosperm is genet-ically identical to the embryo. In other angiosperms, thefemale gametophyte is eight-nucleated, but seven-celled

1Corresponding author: CEFE-CNRS UMR 5175, 1919 Route de Mende,34293 Montpellier Cedex 05, France.E-mail: [email protected]

Genetics 184: 439–453 (February 2010)

because the central cell remains binucleate (2m). Thistype of gametophyte is referred to as ‘‘Polygonum.’’Double fertilization again carries one paternal contri-bution (1p). Thus, the central cell becomes a triploidendosperm (2m:1p). Other types of female gameto-phytes can be found sporadically among angiosperms.They are characterized by an increased number of ma-ternal contributions in the central cell and/or bypolyspory. These types are derived from the Polygonumtype and we do not consider them further. As alreadymentioned, in gymnosperms and ferns the embryo feedsdirectly on the haploid female gametophyte (1m:0p),which can be considered as the ancestral state of thenutritive tissue.

In this article, we focus on the evolution of doublefertilization (DF) (0p / 1p) and maternal contributiondoubling (MCD) (1m / 2m) from this 1m:0p ancestralstate. In principle, two scenarios can be considered(Figure 2). In the first scenario, DF evolves before MCD(1m:0p / 1m:1p / 2m:1p). In the second, MCD evolvesfirst and DF second (1m:0p / 2m:0p / 2m:1p). Thelatter scenario involves an additional step (secondaryloss of the second maternal contribution, 2m:1p /1m:1p) to account for the occurrence of 1m:1p endo-sperms in Nympheales (Nuphar, Nymphea, Cabomba)and the Illiciales-Trimeniaceae-Austrobaileyaceae clade(Schisandra, Illicium). It is therefore less parsimonious.For this reason, we consider only the first scenario(1m:0p / 1m:1p / 2m:1p) in this article.

Three main theories have been proposed to explainthe evolution of DF and MCD. They all rely on the factthat some resources are accumulated to the seed after

fertilization. The first theory proposes that postfertiliza-tion resource allocation triggers a genetic conflict overresource allocation among parents and/or offspring.Indeed, an antagonistic coevolution can occur betweenthe male, the female, and/or the embryos when theiroptimal seed sizes differ, to take control over resourceallocation (Trivers 1974; Haig 2000). Because the en-dosperm plays a pivotal role in the control of resourceallocation, DF and MCD are considered in this theory tobe adaptations allowing the father and the mother totake and take back the control over resource allocation,respectively (Charnov 1979; Westoby and Rice 1982;Queller 1983; Willson and Burley 1983; Law andCannings 1984; Bulmer 1986; Shaanker et al. 1988;Haig and Westoby 1989; Friedman 1995; Shaanker

and Ganeshaiah 1997; Hardling and Nilsson 1999,2001). These ‘‘allocation conflict theories’’ include thekinship theory of genomic imprinting (Haig 2000,2004; De Jong et al. 2005) and have received the mostextensive theoretical development.

A second theory (hereafter metabolic theory) pro-poses that DF and MCD evolve because of the metabolicproperties of polyploid tissues (Stebbins 1976; Donoghue

and Scheiner 1992). More specifically, the idea is thatDF and MCD increase the endosperm ploidy level,which enhances its postfertilization metabolic efficiency(increased transcription rate, increased cell size, andresource storage). The problem with this view is thatendomitosis would achieve the same effect and is verycommon at the beginning of gametophyte/endospermdevelopment in both angiosperm and gymnosperm(Friedman 2001a,b): DF and MCD seem unnecessary

Figure 1.—Timing of resource accumulationin the seed. All the resources are allocated beforefertilization in cycads and ginkgoes. Resourcesare allocated both before and after fertilizationin conifers and all the resources are allocated af-ter fertilization in angiosperms.

Figure 2.—Possible scenarios for the evolu-tion of the genetic makeup of the embryo nu-tritive tissue, modified from Friedman andRyerson (2009). In the left scenario, thereare one 1m:0p / 1m:1p transition and two1m:1p / 2m:1p transitions and thus three steps.In the right scenario, there are one 1m:0p /2m:0p transition, one 2m:0p / 2m:1p transi-tion, and two 2m:1p / 1m:1p transitions andthus four steps. In addition, the latter scenarioinvolves a 2m:0p state that has never been ob-served in any taxa.

440 A. Cailleau, P.-O. Cheptou and T. Lenormand

to increase ploidy in the endosperm in the first place.Moreover, this theory does not account for the presence ofa male nucleus specifically (Willson and Burley 1983).

A third theory proposes that DF and MCD evolvebecause of the genetic properties of polyploid tissues(Brink and Cooper 1940). More specifically, the idea isthat DF combines two different genomes (maternal andpaternal) in the nutritive tissue, which masks the effectof recessive deleterious mutations and thus enhancesthe postfertilization metabolic efficiency of the nutritivetissue. Following Friedman et al. (2008), we refer to thistheory as the ‘‘heterozygosity theory.’’ The heterozygos-ity theory explanation for endosperm’s genetic evolu-tion seems close to explanations for ploidy evolution.Several models have been worked out to understand theeffect of the masking of deleterious mutations in theevolution of ploidy levels (Perrot et al. 1991; Bengtsson

1992; Otto and Goldstein 1992). These models haveshown that masking protects diploid individuals fromthe effect of recessive deleterious mutations, but that italso reduces the elimination of these mutations fromthe population, creating genetic association favorable tohaploidy (Otto and Goldstein 1992). The problem ofthe evolution of the genetic constitution of the endo-sperm is, however, not directly comparable to theproblem of life cycles evolution. First, the phenotypeunder selection (e.g., resource in the seed) is at leastpartly determined by the genotype of the nutritive tissue(the gametophyte or the endosperm), which can differfrom the genotype of the diploid embryo. Second,double fertilization is not symmetrical with respect tosexes: it masks maternal alleles but reveals paternal allelesin the endosperm. Third, double fertilization can beunder pollen and/or female gametophyte control, whilethe timing of meiosis in life cycle evolution is controlledby the diploid stage. In fact, no formal model has beencreated to explore specifically the heterozygosity theory.The aim of this article is to develop such models.

We build two models to study the evolution of DF andMCD. More explicitly, we study the evolution of amodifier locus that changes their probability of occur-rence. We show that masking deleterious mutations canfavor both DF and MCD. However, unlike models for theevolution of ploidy, we find conditions where DF evolvesin opposite directions depending on the sex where themodifier is expressed. In different conditions, the samescheme occurs for MCD. This situation reveals thatmale/female conflict extends well beyond the problemof resource allocation to embryos: a conflict can alsooccur over the genetic constitution of embryos’ nutri-tive tissues.

MODEL

We analyze the evolution of DF and MCD in twodistinct deterministic models. Both involve two linkeddiallelic loci. The first one (the viability locus) is subject

to deleterious mutations and determines the survival ofthe embryo while the other one (the modifier locus)controls the probability of DF or MCD.

Life cycle: To build each model, we first write exactrecursions of the frequency of the different haploidgenotypes. We follow the different events in the life cycleover one generation, which alternates a gametophytichaploid with a sporophytic diploid stage (Figure 3). Westart at the gametophytic stage. The male gametophyte(pollen) contains two identical sperm cells (they aremitotically derived). The female gametophyte (embryosac) contains also a given number of genetically iden-tical cells, among which are the egg cell and the centralcell. First, fertilization occurs in a panmictic way (pair-ing one pollen grain with one embryo sac). The geneticconstitution of the endosperm is determined at thisstage, but differently in the different models (see below).Second, the nutritive tissue and the embryo develop(seed formation). Selection occurs among seeds in thepopulation, on the basis of the genotype of their nutri-tive tissue. Third, the surviving embryos become her-maphroditic sporophytes and produce haploid sporesthrough meiosis (where the two loci recombine at arate R). Finally, the spores undergo mutation and eachproduces a new gametophyte through mitosis. To de-scribe the different events in this life cycle, we introduceseveral parameters, specific to each locus. We presenteach locus in turn.

Viability locus: The first locus is expressed in theembryo’s nutritive tissue (we refer to this tissue as theendosperm) where it potentially affects any function(e.g., the accumulation or the transfer of resources tothe embryo or the protection of the embryo). Thisfunction, in turn, determines the survival of the embryo.We model this directly by assigning different viabilitiesto embryos associated to different endosperm geno-types. We consider that recurrent deleterious mutationsoccur at a rate m at the viability locus. We denote Pa

the frequency of a (the deleterious allele) and PA the

Figure 3.—Life cycle. The sporophyte is born from the fer-tilization of a male gamete and a female gamete. These game-tes are produced by the male and female gametophytes (thepollen and the embryo sac, respectively). In the sporophyte’ssexual organs, meiosis produces spores that develop in newgametophytes.

Endosperm Evolution 441

frequency of A (the wild-type allele). The embryosassociated to a nutritive tissue that carries only thewild-type allele (A, AA, AAA) have a survival of 1. Thoseassociated to a nutritive tissue that carries only thedeleterious allele (a, aa, aaa) have a survival of 1� s. Theembryos associated to a heterozygous nutritive tissuehave survival 1� hs, 1� h1s, and 1� h2s for Aa, aaA, andAAa genotypes, respectively. The parameters h, h1,and h2 are the coefficients of dominance of a on A.We consider h1 $ h2 (a alleles are more exposed in aaAendosperms than in aAA endosperms). Because weassume that Wa ¼ Waa ¼ Waaa and WA ¼ WAA ¼ WAAA,our model excludes potential effects of gene copynumber (i.e., dosage effects in ‘‘metabolic theory’’).Because we do not introduce any competition betweenembryos or endosperms produced by the same sporo-phyte (the viability of a given embryo depends only onthe genotype at the viability locus of its own endosperm),our model excludes any effect involved in the allocationconflict theories. In fact, our model can be understood asif there was a single female gametophyte produced byeach sporophyte. Thus, by construction, our modelfocuses exclusively on the heterozygosity theory.

Modifier locus: The second locus controls the geneticconstitution of the nutritive tissue. More specifically,it controls the rate of double fertilization (i.e., 1m:0pvs. 1m:1p, in the DF model) or the rate of doubling

of maternal contribution (i.e., 1m:1p vs. 2m:1p, in theMCD model). We consider two alleles M and m withfrequency PM and Pm, respectively. In each model, themodifier locus is expressed in the pollen (paternalcontrol), the embryo sac (maternal control), or both(biparental control). In the first two cases, the expres-sion of the modifier locus is haploid. In these cases, wedenote nM or nm the probability of double fertilization(or maternal contribution doubling) when the gameto-phyte controlling the trait carries the M or the m allele,respectively. In the biparental case, we assume that themodifier effect is additive (i.e., the probabilities of DFand MCD are nM, (nM 1 nm)/2, and nm when the embryois MM, Mm, and mm, respectively). To give a concreteexample, in a DF model with a paternal control of themodifier locus, the mating between AM male gameto-phytes (pollen) and am female gametophytes will pro-duce a proportion nM of embryos whose nutritive tissueis 1m:1p (Aa, with fitness 1 � hs) and a proportion 1 �nM embryos whose nutritive tissue is 1m:0p (a, withfitness 1 � s). Tables 1 and 2 summarize all these nota-tions. Without loss of generality, we discuss the fate ofmodifier alleles that increase the probabilities of DF andMCD (nm . nM).

Recursion equations: The recursion equations de-scribing the dynamics of the model are given inappendix a for the case of the evolution of DF with apaternal control. These recursions can be written forthe frequency of the four haplotypes (paM, pAm, pAM, andpam). Equivalently, they can be expressed for the allelefrequencies (pm and pa) and linkage disequilibrium(Cam¼ pAM pam� paM pAm¼ pam� pa pm). In this case, wedefine Cam such that it is positive when the modifierallele (m) is positively associated to the deleteriousallele at the viability locus (a). To fully understand themodel, we also describe the population during the dip-loid phase. In diploids, several genetic associations CU,V

analogous to Cam can be defined (as in, e.g., Barton

and Turelli 1991), where U denotes a set of alleleson the maternally inherited chromosome and V a setof alleles in the paternally inherited chromosome. Inour case, we have two biallelic loci and we have there-fore 11 associations: Ca;m ; Cm;a ; Cam;;; C;;am ; Ca;a ; Cm;m ;

TABLE 1

Genotypic control of selection

Endospermploidy level

Endospermgenotype (m:p) Embryo’s W Model

n (1m:0p) A: ; 1 DFn (1m:0p) a: ; 1 � s DF2n (1m:1p) A: A 1 DF, MCD2n (1m:1p) A: a 1 � hs DF, MCD2n (1m:1p) a: a 1 � s DF, MCD3n (2m:1p) AA: A 1 MCD3n (2m:1p) AA: a 1 � h2s MCD3n (2m:1p) aa: A 1 � h1s MCD3n (2m:1p) aa: a 1 � s MCD

We assume h2 , h and h2 , h1, 0 # h # 1, 0 # s # 1.

TABLE 2

Genotypic control of the genetic makeup of the endosperm

Probability of

Control

Modifierlocus

genotype

DF: double fertilization(¼ proportion of

1m:1p endosperms)

MCD: doubling(¼ proportion of

1m:2p endosperms)

Biparental MM nM ¼ n nM ¼ n

Mm (nM 1 nm)/2 ¼ n 1 dn/2 (nM 1 nm)/2 ¼ n 1 dn/2mm nm ¼ n 1 dn nm ¼ n 1 dn

Uniparental M nM ¼ n nM ¼ n

m nm ¼ n 1 dn nM ¼ n 1 dn


Ca;am ; Cm;am ; Cam;a ; Cam;m ; and Cam;am . For instance, theCa,m association measures the fact that the presence ofthe a allele on the maternally inherited chromosomecovaries with the presence of the m allele on the pa-ternally inherited chromosome.

ANALYSIS

We analyze DF and MCD models with two comple-mentary methods: quasi-linkage equilibrium, (QLE)and stability analysis (see, e.g., Otto and Day 2007).In these two methods, we assume that the frequency ofdeleterious mutations pa is close to its equilibrium value(noted pa*) and small (which entails that Cam is alsosmall). This assumption requires that m > s. This is theonly common assumption to the two approaches. In theQLE approach, we also assume that s is small, R is large,and the modifier has a weak effect. In the stabilityanalysis, the only additional assumption to m > s is thatthe modifier frequency is small. To easily define theorders of magnitude, we use a small parameter j as ascaling throughout.

The QLE analysis assumes that linkage disequilibriumequilibrates much faster than allele frequencies, so thatit can be considered to always be at a ‘‘quasi-equilibrium.’’This approach works best if the modifier effect is smalland the recombination rate large (that is why we con-sider nM¼ n and nm¼ n 1 dn, where dn is proportional toj; in the case of a modifier with a specifically maternalor paternal effect, we write dn,; and d;,n instead of dn,respectively). With this approach, we focus on conver-gence stability. We perform the stability analysis to assessthe robustness of this QLE analysis and to investigatewhat is going on for low recombination rate. Thismethod considers the fate of the modifier introducedat a low frequency in a population at equilibrium at theviability locus. We first start by studying the DF model.

Mutation–selection at the viability locus in the DFmodel: To start, it is important to understand what isgoing on at the viability locus alone. According tothe assumptions given above for the QLE approach,the change over one generation (noted with D) of thefrequency of the deleterious allele at the viability locus is

Dpa ¼ m� pa1

2ð1� nÞ1 hn

� �s 1 OðjÞ3; ð1Þ

and thus, at equilibrium (Dpa¼ 0), the frequency of thea allele (noted with an asterisk) depends on the rate ofdouble fertilization (n):

pa* ¼m

ð1=2Þð1� nÞ1 hnð Þs 1 OðjÞ2: ð2Þ

The same equations are obtained under the assumptionof the stability analysis, except that the orders of thetwo equations are OðjÞ2 and OðjÞ, respectively. Withcomplete double fertilization (n ¼ 1), the endospermis diploid and we retrieve the equilibrium frequency

of a deleterious mutation under diploid selection(m/hs). However, with no double fertilization (n ¼ 0),pa* ¼ 2m=s: the efficiency of selection is halved com-pared to the classic haploid selection situation (m/s).Indeed, half the alleles (the paternal ones) are notexposed to selection since they are in the embryo, butnot in the endosperm. Interestingly, this result pointsout that having a haploid endosperm is not equivalentto haploidy. Note that the equilibrium described aboveis a good approximation only if h . s when n is close to1. A more complicated approximation can be found forsmall values of h and 1 � n (see Chasnov 2000 for asimilar treatment), but, as explained below, a betterapproximation is not necessary to accurately predicthow DF and MCD evolve in our models.

QLE analysis of the DF model: We compute theoverall frequency change at the modifier locus Dpm as asum of two terms. The first, Dpdirect

m , is proportionalto pm(1 � pm) and corresponds to direct selection.The second, Dpindirect

m , is proportional to Cam andcorresponds to indirect selection. Under the QLEassumptions given above, we obtain at leading orders

Dpm ¼ Dpdirectm 1 Dpindirect

m

Dpdirectm ¼ sdn

1

2� h

� �pmð1� pmÞpað1� paÞ1 OðjÞ5

Dpindirectm ¼ �sCam

1

2ð1� nÞ1 hn

� �1 Oðj2CamÞ: ð3Þ

We study these terms separately for two reasons. Firstthey represent qualitatively different processes. Second,their relative magnitude varies in different waysthroughout the parameter range (e.g., we expect in-direct selection to become increasingly important as Rdecreases and direct selection to vanish when h becomesclose to 1

2 ), and it is therefore quantitatively useful tocombine their leading orders.

Direct selection: The sign of Dpdirectm is given by the sign

of 12 � h. When mutations are recessive (h , 1

2 ), directselection favors the evolution of DF. In fact, h can beunderstood as the ‘‘exposition’’ (to selection) of the aallele in a 1m:1p seed, while 1

2 can be understood as itsexposition in a 1m:0p seed. Indeed, as said before, onlyhalf the a alleles (the maternally inherited ones) areexposed to selection in 1m:0p endosperms. Direct selec-tion simply tends to minimize the exposition of thedeleterious alleles. This effect is quite different from the‘‘masking effect’’ described in models of ploidy evolu-tion (Otto and Goldstein 1992). In these models,exposition of the deleterious alleles is always greaterwith haploidy (Aa individuals have a better survivalthan haploid a individuals as long as h , 1), but this iscompensated by the fact that a diploid individual hastwice the chance to carry a deleterious mutation (it hastwo alleles). As a consequence, diploidy is favored bydirect selection when exposition (h) is at least halvedto compensate for this doubled risk of inheriting a


deleterious mutation (i.e., when h , 12 ). In contrast, in

our model, all individuals have the same risk to carry adeleterious mutation (all embryos are diploid), but theexposition of the deleterious alleles is greater with1m:0p only if h , 1

2 (because only half the alleles, thematernal ones, are exposed to selection without DF).

Indirect selection: The sign of Dpindirectm directly depends

on the sign of the linkage disequilibrium Cam. Toevaluate indirect selection more precisely, it is usefulto express the value of Cam in terms of the model’sparameters. Under the QLE assumptions, Cam reaches apseudoequilibrium value (CQLE

am ) that is obtained bysolving DCam ¼ 0, yielding

CQLEam ¼ dn

sðK � hÞ2R

pmð1� pmÞpað1� paÞ1 OðjÞ4; ð4Þ

where K ¼ 1 � R, R, and 12 for maternal, paternal, or

biparental expression of the modifier locus, respec-tively. Thus, the sign of CQLE

am depends entirely on thesign of K � h, that is, of 1 � R � h, R � h, and 1

2 � h, formaternal, paternal, and biparental expression of themodifier locus, respectively. This result can be inter-preted by following step by step how the linkagedisequilibrium is built through selection and meiosis.

Building of genetic associations by selection: A lowerexposition of deleterious alleles is immediately benefi-cial to an individual but leads to their accumulation.More specifically, in diploid embryos, deleterious muta-tions tend to become associated to modifier alleles thatminimize exposition. A paternally acting modifier ex-poses the deleterious allele carried on the paternalchromosome (exposition is increased from 0 to h in thecase of DF), which enhances the purge of these pater-nally inherited chromosomes. As a consequence, apaternally acting modifier tends to become associatedto less loaded paternal chromosomes (C9;;am � C;;am , 0,where the prime indicates that the association is mea-sured after selection):

C9;;am ¼ C;;am 1 d;;nð0� hÞs pmð1� pmÞpað1� paÞ1 OðjÞ4: ð5Þ

On the contrary, a paternally acting modifier alwaysreduces the exposition of the deleterious alleles carriedon the maternal chromosome (from 1 to h when thereis DF), which reduces purging on these chromosomesand generates more loaded maternal chromosomes(C9a;m � Ca;m . 0):

C9a;m ¼ Ca;m 1 d;;nð1� hÞs pmð1� pmÞpað1� paÞ1 OðjÞ4: ð6Þ

The exact opposite occurs for a maternally acting modi-fier (C9am;; � Cam;;. 0 and C9m;a � Cm;a , 0):

C9am;; ¼ Cam;;1 dn;;ð1� hÞs pmð1� pmÞpað1� paÞ1 OðjÞ4 ð7Þ

C9m;a ¼ Cm;a 1 dn;;ð0� hÞs pmð1� pmÞpað1� paÞ1 OðjÞ4: ð8Þ

All of these effects occur when the modifier has an effectin both male and female gametophytes (i.e., in thebiparental case).

Effect of meiosis: The haploid linkage disequilibriumafter meiosis (C9am) results from the diploid associationsas

C9am ¼C9am;;1 C9;;am

2ð1� RÞ1 C9a;m 1 C9m;a

2R ð9Þ

(Barton and Turelli 1991); i.e., the diploid cis-associations are carried over at a rate 1 � R while thediploid trans-associations are transformed into Cam at arate R. In particular, as the recombination rate becomeslower, cis-associations contribute more to generating Cam.Combining Equations 5–9, one easily retrieves Equation4. For example, with a paternally acting modifier, thecis-association C;,am is proportional to �h, while thetrans-association Ca,m is proportional to 1 � h. Becauserecombination breaks up cis-associations and transformstrans-associations into haploid cis-associations (Equation9), the modifier stays overall associated to purgedgenomes (and favored by indirect selection) when(1 � h)R � h(1 � R) , 0, i.e., when R , h (Equation 4,Figure 4A). In the case of a maternally acting modifier,the roles of cis- and trans-associations (and thus theeffect of recombination) are reversed. The modifierstays overall associated to purged genomes when(1� h)(1� R)� hR , 0, i.e., when 1� R , h (Equation4, Figure 4B). In the biparental case, the four associa-tions Cam;;; C;;am ; Ca;m ; Cm;a play a role and the effect ofrecombination cancels out. The modifier stays overall as-sociated to purged genomes when (1�R)(1� h)� hR 1

R(1 � h) � h(1 � R) , 0, i.e., when h . 12 (Equation 4).

Overall selection on the modifier: To obtain the selec-tion coefficient on the modifier [defined as sm [ Dpm=pmð1� pmÞ], we replace Cam by CQLE

am in Equation 3,yielding

sm ¼ sdnpað1� paÞ

31

2� h

� �� sðK � hÞ

2R

1

2ð1� nÞ1 hn

� �� 1 OðjÞ5: ð10Þ

The sign of sm depends on the term within the brackets.In Figure 4, we illustrate the conditions where sm ispositive. When R ¼ 1

2 , jCamj is low: indirect selectionis negligible and the sign of sm is given by 1

2 � h. In thiscase two equilibria (n ¼ 0 and n ¼ 1) exist, and DF isfavored when h , 1

2 . When R is lower, indirect selectioncan counterbalance direct selection. With a paternallyacting modifier, DF is favored from h ¼ 0 up tohpat*DF ¼ 1

2 1 H DFcrit, where

H DFcrit ¼

1

2

sð1� 2RÞ4R

1 Os

R

� �2: ð11Þ

On the contrary, with a maternally acting modifier, DFis favored from h ¼ 0 up to h*DF

mat ¼ 12 �H DF

crit. In otherwords, DF is favored for h , 1

2 when R ¼ 12 . Otherwise

(R , 12 ), DF is favored for a range of h values wider than


0 , h , 12 with a paternal expression of the modifier

locus and for a range of h values narrower than 0 , h , 12

with a maternal expression of the modifier locus.Note that in both the maternal and the paternalcase, an intermediate unstable equilibrium (n differentfrom zero or one) exists in a very small portion of theparameter space when h is between h* and h* 1 DH ,where

DH ¼ s2ð1� 2RÞ2

32R2 1 Os

R

� �3: ð12Þ

In this region, the evolutionary outcome depends oninitial conditions. Above h* 1 DH , DF is disfavored.DF is selected in different conditions depending onwhether the modifier locus is paternally or maternallyexpressed (hmat*DF 6¼ hpat*DF); it is a case of male/femaleconflict. With a biparentally expressed modifier, directand indirect selection has always opposite signs, butindirect selection is never strong enough to counterbal-ance direct selection. Thus, the critical value of h belowwhich DF is favored is 1

2 regardless of other parametervalues.

Stability analysis of the DF model: To check theQLE analysis for small R values, we perform a stabilityanalysis. To do so, we consider a population where Mis fixed (pAm ¼ pam ¼ 0) and at mutation–selectionequilibrium at the viability locus (paM ¼ pa*, Equation2). We then compute the frequency change of a raremodifier allele introduced in this population. Techni-cally, we linearize the exact recursions on the Am andam haplotypes, assuming they are rare. We then find thedominant eigenvalue of the transition matrix corre-sponding to these two equations, yielding

l ¼ 1 1mðnM � nmÞð4Rðh � ð1=2ÞÞ1 sK1K2Þð1� nM 1 2hnM Þð2R 1 sð1� RÞK1Þ

1 OðjÞ2;

ð13Þ

where K1 and K2 depend on whether the modifier ismaternally, paternally, or biparentally expressed. Theirvalues are given in Table 3. Assuming that mutationsare rare, the rate of DF will evolve through successivefixations in the direction predicted by the sign of l� 1.The results of the stability analysis are qualitatively com-parable to those of QLE. The critical value of h belowwhich DF is favored predicted by this analysis is closeto the critical value predicted under QLE assumptions(hpat* , hmat* ). In fact, the absolute difference between thecritical h values predicted by the two approaches issdnð1=8� 1=16RÞ1 OðsÞ2, which is negligible when R islarge, as expected. In the stability analysis, as in the QLE,an intermediate unstable equilibrium exists in a smallportion of the parameter space. In this region, bothfull and no DF are locally convergent stable states (theoutcome depends on initial conditions). The area of thisregion is very small, �s/32 (if one considers that theparameter space is h 2 ½0; 1� and R 2 ½0; 0:5�). This smallregion is illustrated in Figure 5. In a slightly larger regionof the parameter space around the latter, a single con-vergent stable state is present, but is not stable faced withmutations of large effects. In all cases, the results betweenQLE and stability analysis are very comparable andqualitatively identical. We now switch to the MCD model.

MCD model analysis: We analyze this model in thesame way as the DF model, using the same assumptions.

Mutation–selection at the viability locus: The mutation–selection equilibrium at the viability locus is

Figure 4.—Illustration of the QLEanalysis of the DF and MCD models.The direction of selection at the mod-ifier locus is indicated for differenth (x-axis) and R (y-axis) values. In Aand B (DF model), 1m:1p is favoredin the shaded area whereas 1m:0p is fa-vored in the open area. In C and D(MCD model), 2m:1p is favored inthe shaded area whereas 1m:1p is fa-vored in the open area. A, C and B,D illustrate paternal and maternal ex-pression of the modifier locus, respec-tively. In all panels, the dashed areacorresponds to a negative linkage dis-equilibrium (the modifier is associatedto the purged chromosome). Direct se-lection favors DF on the left-hand sideof the dashed line (which is h ¼ 1

2 , Aand B), whereas it favors MCD on theright-hand side of the dashed line(which is h ¼ hm ¼ 0.3, C and D).The QLE approximation becomes in-accurate when R is very low (the points

represent critical h values obtained directly from exact simulations). Parameter values are s ¼ 5 3 10�2, m ¼ 10�3, y ¼ 0.1, dy ¼0.01, h1 ¼ 0.2, and h2 ¼ 0.4.


pa* ¼m

ðhð1� nÞ1 hmnÞs 1 OðjÞ2; ð14Þ

where hm¼ (h1 1 h2)/2. This value is comparable to theequilibrium frequency in the DF model (Equation 2).Indeed, in the DF model, the exposition of the a alleleranges from h (full DF) to 1

2 (no DF). Similarly, in theMCD model, the exposition of the a allele ranges fromhm (full MCD) to h (no MCD).

Frequency change at the modifer locus: The frequencychange at the modifier locus is determined by direct andindirect selection, as in Equation 3:

Dpm ¼ Dpdirectm 1 Dpindirect

m

Dpdirectm ¼ sdnðh � hmÞpmð1� pmÞpað1� paÞ1 OðjÞ5

Dpindirectm ¼ �sCamðhð1� nÞ1 hmnÞ1 Oðj2CamÞ:

ð15Þ

The sign of direct selection is given by the sign of h� hm;i.e., direct selection favors the state where the averageexposition of deleterious alleles is maximal. This termfavors MCD when hm , h. The sign of indirect selectiondepends directly on the sign of Cam, whose QLE valuehas the same form as in Equation 4,

TABLE 3

Values of K1 and K2 in the stability analysis for each model

Maternallyacting modifier

Paternallyacting modifier

Biparentallyacting modifier

DFK1 1� ð1� hÞnm 1 hnM 1� ð1� hÞnM 1 hnm 1� ð1� hÞ�n 1 h�n

K2 h ðh � 1Þ ðh � 1=2Þð1� 2RÞ

MCDK1 � 1

2 ððh1 � h2Þ=2ÞðnM � nmÞ 12 ððh1 � h2Þ=2ÞðnM � nmÞ 0

K2 �ððh1 � h2Þ=2Þ �ððh1 � h2Þ=2Þ 0

�n ¼ ðnM 1 nmÞ=2

Figure 5.—Illustration of thestability analysis of DF and MCDmodels. The direction of selec-tion at the modifier locus is indi-cated for different dominance (h,x-axis) and recombination values(R, y-axis). In A and B (DFmodel), 1m:1p is favored in thearea with dark shading [that cor-responds to the pairwise invasibil-ity plot (PIP) a], whereas 1m:0p isfavored in the open area (thatcorresponds to PIP e). The areaswith intermediate shading corre-spond to b, c, and d, that is, tocases where (b) DF is only conver-gent stable, (c) an intermediary,unstable equilibrium exists, and(d) DF is not convergent stable.In C and D (MCD model),2m:1p is favored in the shadedarea whereas 1m:1p is favoredin the open area. We can see thatthe intermediate cases are indis-tinguishable for the MCD model.The parameters values are h1 ¼0.2, h2 ¼ 0.4, m ¼ 10�3, and s ¼10�1.


CQLEam ¼ dn

sðK 1 h � hmÞ2R

pmð1� pmÞpað1� paÞ1 OðjÞ4;

ð16Þ

where K ¼ 0, ð1� 2RÞððh2 � h1Þ=2Þ, and ð1� 2RÞððh1 �h2Þ=2Þ for biparental, paternal, and maternal expressionof the modifier locus, respectively. In the very likelysituation where h1 . h2, K is positive for a maternallyexpressed modifier and negative for a paternally ex-pressed one. Because the sign of Cam depends entirelyon the sign of K 1 h � hm, it is therefore more easilypositive with a maternal expression of the modifierlocus. The way cis- and trans-associations are built byselection is qualitatively the same as in the DF model:

C9;;am ¼ C;;am 1 d;;nðh � h1Þs papmð1� pmÞð1� paÞ1 OðjÞ4 ð17Þ

C9a;m ¼ Ca;m 1 d;;nðh � h2Þs papmð1� pmÞð1� paÞ1 OðjÞ4 ð18Þ

C9am;; ¼ Cam;;1 dn;;ðh � h1Þs papmð1� pmÞð1� paÞ1 OðjÞ4 ð19Þ

C9m;a ¼ Cm;a 1 dn;;ðh � h2Þs papmð1� pmÞð1� paÞ1 OðjÞ4: ð20Þ

An important difference between DF and MCD is thatMCD can increase or decrease the exposition of mater-nally inherited deleterious alleles (depending on thevalue of h1, the exposition of the a allele in aaAendosperms) and can increase or decrease the exposi-tion of paternally inherited deleterious alleles (depend-ing on the value of h2, the exposition of the a allele inAAa endosperms). For instance, a paternally actingmodifier increasing MCD decreases the exposition ofthe deleterious allele carried on the paternal chromo-some from h to h1, if h . h1, but increases it if h , h1. Inthe first case, the paternally acting modifier tends tobecome associated to more loaded paternal chromo-somes (C9;;am � C;;am . 0), while in the second case ittends to become associated to less loaded chromosomes(C9;;am � C;;am , 0). Combining Equations 17–20 withEquation 9, one easily retrieves the value of CQLE

am . Withuniparental expression of the modifier locus, threeconfigurations can occur. In the case where h2 , h , h1,the situation is ‘‘reversed’’ compared to what happensin the DF model, because in these conditions MCDincreases the maternal allele’s exposition and decreasesthe paternal allele’s one. For instance, for a paternallyacting modifier, the cis-association C;,am is proportionalto h � h2 (positive), while the trans-association Ca,m isproportional to h � h1 (negative). The modifier staysoverall associated to purged genomes when (h� h2)(1�R) � (h � h1)R , 0, i.e., when R , (h � h2)/(2h � h1 �h2) (Figure 4C). For a maternally acting modifier, themodifier stays overall associated to purged genomeswhen (h � h1)(1 � R) � (h � h2)R , 0, i.e., when R ,

(h � h1)/(2h � h1 � h2) (Figure 4D). In the case of

biparental expression of the modifier locus, the effect ofrecombination cancels out as in the DF model, and themodifier stays overall associated to purged genomeswhen hm . h. In the second case, where h2 , h1 , h, bothmaternal and paternal alleles have a better exposition atthe 2m:1p state. The haploid linkage disequilibrium Cam

is always negative. On the contrary, in the third case,where h , h2 , h1, both maternal and paternal alleleshave a better exposition at the 1m:1p state and Cam isalways positive.

Overall selection on the modifier: The selection coeffi-cient at the modifier locus at QLE is of the same form asEquation 10:

sm ¼ sdnpað1� paÞ

3 ðh�hmÞ�sðK 1 h � hmÞ

2Rðhð1� nÞ1 hmnÞ

� �

1 OðjÞ5: ð21Þ

In Figure 4, we illustrate the conditions where sm ispositive. As in the DF model, there is in most cases onlyone convergent stable state (either full or no MCD).However, in a region of the parameter space full and noMCD may be both convergent stable states and theirdomain of attraction is determined by an unstableintermediate equilibrium. As in the DF model, thisregion is, however, very small. When R¼ 1

2 , the sign of sm

simply depends on the sign of h� hm. When R is smaller,indirect selection can counterbalance direct selection:with a paternally acting modifier, MCD is favored fromh ¼ 1 down to hpat*MCD ¼ hm 1 H MCD

crit , where

H MCDcrit ¼ hm

sð1� 2RÞ4R

ðh1 � h2Þ1 Os

R

� �2; ð22Þ

which is narrower than from h ¼ 1 down to h ¼ hm. Onthe contrary, with a maternally acting modifier, MCD isfavored from h ¼ 1 to hmat*MCD ¼ hm �H MCD

crit , which iswider than from h ¼ 1 to h ¼ hm. Like DF, MCD may beselected or counterselected depending on whether themodifier locus is paternally or maternally expressed.When the recombination rate is low, there is a male/female conflict regarding MCD evolution. With a bi-parentally expressed modifier, indirect selection isnever strong enough to oppose direct selection: MCDis favored when hm , h.

Stability analysis: We make a stability analysis as for theDF model. The dominant eigenvalue of the matrix is

l ¼ 1 1nM � nm

ð1� nM Þh 1 nM hm

3Rðhm � hÞ1 sð1=2� RÞðhmð1� �nÞ1 h�n 1 K1Þðhm � h 1 K2Þ

R 1 sð1� RÞðhmð1� �nÞ1 h�n 1 K1Þm

1 oðjÞ; ð23Þ

where �n ¼ ðnM 1 nmÞ=2 and where K1 and K2 depend onwhether the modifier is maternally, paternally, or bi-parentally expressed. Their values are given in Table 3.


The direction of selection is illustrated in Figure 4. As inthe DF model, the QLE and the stability analysis yieldcomparable results. The absolute difference betweenthe critical h values predicted by the two approaches issdnðh1 � h2Þ2ð1=8� 1=16RÞ1 OðsÞ2, which is negligiblewhen R is large, as expected. As in the DF model, thereis also a small region of the parameter space with anintermediate unstable equilibrium, but it is so small thatit can hardly be seen in Figure 5.

Combining the evolution of DF and MCD: So far, wehave described the conditions favoring DF and MCDindependently. We now consider how these conditionscombine to give an overall picture of the evolution ofthe different types of endosperms (1m:0p, 1m:1p, and2m:1p). There are a large number of possibilitiesdepending on the relative values of h1, h2, and h. Con-sidering that h2 , h , h1 [as one would expect followingthe classical metabolic theory of dominance (Kacser

and Burns 1981)] and that hm , 12 , we can delineate

seven cases. They are illustrated in Figure 6. However, allseven cases are not equally plausible. In particular, h isvery likely to be ,1

2 since the deleterious mutationshave always been found to be recessive on average invarious diploid taxa (see, e.g., Garcıa-Dorado andCaballero 2000; Vassilieva et al. 2000; Szafraniec

et al. 2003). With this additional condition, only six casesare possible (all but case 7 in Figure 6). With a highrecombination rate (i.e., from the conditions where amale/female conflict occurs), only two evolutionary

outcomes are possible: 1m:1p if h , hm and 2m:1p if h .

hm (cases 1 and 2 in Figure 6). For lower recombinationrates, four other possibilities can also occur, includingthe 1m:0p case (see cases 3–6 in Figure 6).

Robustness of the results: These two-locus modelsare simplified ways to investigate the effect of maskingon DF and MCD evolutions. Integrating the overallselection coefficient over a full genome with manyviability loci might be achieved by summing the fre-quency change of the modifier caused independently byeach of them (see, e.g., Otto 2003). This approach is,however, complicated by three factors. First, one wouldhave to integrate over an explicit genetic map (toaccount for different R values). Second, different locimay exhibit different parameter values (h, h1, h2, s, m)and the distributions of these parameter values are notwell known (including possible correlations betweenthose parameters, for instance, s and h). Third, selectiveinteractions may occur among viability loci and ourmodels do not explore these effects. Finally, we do notincorporate departure from random mating that mightaffect the conclusions. Future work is required toaccount for these complications.

DISCUSSION

In this article we modeled the masking of deleteriousmutations in the endosperm and its influence on theevolution of DF and MCD. More specifically, we followedthe fate of alleles modifying the probability of occur-rence of DF or MCD at a modifier locus with a sex-limited expression. In our models, a second locus isexpressed in the endosperm and controls the amount ofresources available to the embryo. At this locus, delete-rious alleles diminish the ability to accumulate resour-ces, and the seeds inheriting this allele have therefore alower survival rate. In these models, there is no resourcereallocation between the different embryos producedby the same mother plant. Thus, all of our results areunrelated to theories based on competition betweenembryos or kin selection and are attributable only tomasking and its possible side effects.

Our work is the first formal demonstration that themasking of deleterious mutations is sufficient for DFand MCD to evolve. In fact, two different effectscombine in this theory: first, a direct effect of masking,which depends only on the dominance level of delete-rious mutations, and second, an indirect and side effectof masking, proportional to linkage disequilibrium, thevalue of which depends on whether the expression ofthe modifier locus is paternal or maternal. The lattereffect depends mainly on the values of dominance (h1,h2, and h) and recombination rate. When the recombi-nation rate is high, the indirect effect is always negligibleand evolution favors the state with highest masking.Contrary to what is suggested in the literature (Willson

and Burley 1983), the 1m:1p state is not necessarily the

Figure 6.—Combined evolution of DF and MCD. The dif-ferent possible scenarios for the combined evolution of DFand MCD are shown when hm , 1

2 . (h1 ¼ 0.6 and h2 ¼ 0.2in this example, yielding hm ¼ 0.4). The solid lines indicatethe critical maternal and paternal h values for DF model.The dashed lines indicate the critical maternal and paternalh values for MCD model. In area 1, DF is favored, and MCD isnot (1m:1p is thus expected to evolve). In area 2, both DF andMCD are favored (2m:1p evolves). In area 3, DF is favored andMCD is favored only if under maternal control (thus either1m:1p or 2m:1p may evolve depending on which sex predom-inantly controls MCD). In area 4, DF is favored only if underpaternal control, and MCD is not favored (thus either 1m:0por 1m:1p may evolve). In area 5 DF is favored only if underpaternal control, while MCD is favored only if under maternalcontrol, and thus 1m:0p, 1m:1p, or 2m:1p may be favored. Inarea 6 DF is favored only if under paternal control and MCD isalways favored (either 1m:1p or 2m:1p may evolve). Eventu-ally, in area 7 neither DF nor MCD is favored, and 1m:0pevolves. The figure is drawn using the results of the stabilityanalysis and with m ¼ 10�3, s ¼ 10�1, nM ¼ 0.1, and nm ¼ 0.1.


situation with the strongest masking: for instance,masking can be stronger in 2m:1p than in 1m:1p, whenhm , h. When recombination is low, the indirect effectcan become predominant and the outcome depends onwhether DF or MCD are paternally, maternally, orbiparentally controlled.

A new type of male/female conflict: Our resultsindicate that for low recombination rates, there is amale/female conflict for the genetic constitution of theendosperm. This male/female conflict results from thefact that increasing the exposition of alleles inheritedfrom a given parent diminishes the exposition of allelesinherited from the other one. In the case of paternalexpression of the modifier locus, indirect selectionincreases the range of h values where DF evolves, whileit reduces the range of hm values where MCD evolves.Exactly the opposite occurs with maternal expression ofthe modifier locus. With biparental expression of themodifier locus, the indirect effect is always negligibleeven when recombination is low. For instance, withR , 1

2 and h ¼ 0.5, a paternal but not a maternal DFallele would spread (see Figures 4 and 5). Similarly, withR , 1

2 and h ¼ hm, a maternal but not a paternal MCDallele would spread (see Figures 4 and 5).

Evolutions of DF and MCD: Both 2m:1p and 1m:1pendosperms can be found in angiosperms. How cansuch a diversity be explained? Deleterious mutationstend to be recessive in diploids, with average h rangingbetween 0.1 and 0.4 depending on the studies (see, e.g.,Garcıa-Dorado and Caballero 2000; Vassilieva et al.2000; Szafraniec et al. 2003). As a consequence, wewould expect that paternally expressed genes favoringDF should always spread or be maintained. This is atleast the case for the genes that determine the occur-rence of two mitotically derived spermatozoa, since thischaracteristic, which is a necessary condition for DFevolution, is maintained in all angiosperms (but in-triguingly also in gymnosperms). We would also expectthat maternally expressed genes favoring DF shouldspread provided that the average recombination rate isnot too low. In other words, for low recombination rates,we do not expect DF to be systematically maintained(depending on which sex controls the trait, see Figure6). Among angiosperms with postfertilization resourceallocation by the female sporophyte, few have lost DF orat least a functional endosperm, the only ones beingPodostemaceae and Trapaceae (Chopra and Sachar

1963). These species exhibit high selfing rates (Haig

1988). Including selfing effects in our models will be thepurpose of a future study. The Orchidaceae is the otherimportant group where DF occurs but no functionalendosperm is present (Chopra and Sachar 1963). Inthis group, however, there are virtually no resourcesallocated by the female sporophyte to the seed: theseedlings acquire resources through mycorrhiza. Thus,the degeneration of endosperm in orchids may beunrelated to DF or MCD evolution.

Compared to diploids, little is known about thetypical dominance of deleterious mutations in triploids(i.e., the dominance h1 of the aaA genotype and thedominance h2 of the AAa genotype). It seems, however,likely the average dominance in triploids is close to thatin diploids [i.e., (h1 1 h2)/2¼ hm� h], so that we wouldexpect to be often close to the situation where there is amale/female conflict over the evolution of MCD. Inthese conditions, we would expect to find both types ofendosperm (1m:1p and 2m:1p, see Figure 6). Consis-tent with this prediction, some angiosperms [in Nym-pheaceae, Hydatellaceae, Cabombaceae, Illiciaceae,Schisandraceae, Austrobaileyaceae, and Oenothera(Maheshwari 1963; Friedman 2001a; Williams andFriedman 2002; Friedman and Ryerson 2009)] have a1m:1p endosperm (DF, no MCD) and others a 2m:1pendosperm (other groups, both DF and MCD evolved).

DF and diploidy: It is worth comparing our modelsof DF evolution with models for the evolution ofdiploid life cycles (in particular, the model of Otto

and Goldstein 1992, which is referred to hereafter asOG92 and is reinvestigated in appendix b). Following theOG92 model, the evolution toward diploid life cycles(2N) requires deleterious mutations to be recessive andrecombination to be ‘‘sufficiently large.’’ In fact, theseconditions are close to the conditions favoring DF whenthe expression of the modifier locus is maternal. We cancompare these conditions in Figure 7: all other param-eter values being equal, the range of R and h values inwhich DF evolves is slightly wider, but comparable withthe range of R and h values favoring 2N. In the twomodels (DF and OG92), the selection coefficient actingon the modifier can be decomposed into two terms, adirect and an indirect effect that are, superficially at least,comparable. In both models, direct selection favorsdiploidy when h , 1

2 , but for different reasons. In theOG92 model, the exposition of the deleterious alleles isalways higher with 1N (full exposition) than with 2N

Figure 7.—Comparison of the conditions for the evolutionof diploidy and DF. The overlap of the conditions of h and Ris shown, where diploidy (2N) is favored by the conditions ofh and R where DF is favored. Indeed, 2N is favored on the leftof the dashed line, while DF is favored on the left of the solidlines (the left solid line corresponds to the case of a maternalexpression of the modifier locus and the right solid line cor-responds to the case of a paternal expression of the modifierlocus). The figure is drawn using the results of the stabilityanalysis and with s ¼ 10�1, m ¼ 10�3, nM ¼ 0.1, and nm ¼ 0.1.


(exposition at a level h). In the DF model, only maternalalleles are exposed. Thus, the average exposition isgreater with ‘‘haploidy’’ (1m:0p) only if h , 1

2 . As we havealready mentioned, this is compensated by the fact that,unlike in the DF model, 2N individuals have twice thechance to carry a deleterious mutation compared to 1Nindividuals. A second important difference is that, in theDF but not in the OG92 model, the direction andstrength of indirect selection differ for paternally ormaternally expressed modifiers. In the OG92 model, cis-associations between 2N modifier and deleterious muta-tions are always positive (in other words, modifiers ofhaploidy are found more often on purged chromo-somes). In contrast, because of the asymmetry inherentto the DF model, where haploidy necessarily concernsmaternally inherited alleles at the viability locus, purgingof deleterious alleles causes positive cis-association onlywith maternally expressed modifiers. In other words,maternally acting modifiers of haploidy are found moreoften on purged chromosomes, while paternally actingmodifiers of haploidy are found more often on non-purged chromosomes. Overall, 2N and OG92 models arein fact quite different, in terms of both direct and indirectselection. In terms of parameter range, DF is necessarilyfavored if 2N is favored (the reverse is not true). This isconsistent with the view that angiosperms have bothevolved DF and the shortest haploid phase.

DF and imprinting: Our model of DF evolution canalso be closely compared to some models of imprintingevolution (Spencer and Williams 1997). Indeed, theevolution toward imprinting in animals can be viewed,at the level of a locus, as a transition from a diploid(1m:1p) to a functionally haploid state (0m:1p formaternal imprinting or 1m:0p for paternal imprinting).In the absence of double fertilization, only the maternallyinherited haploid genome is expressed in the endo-sperm. This situation is therefore similar to a genome-wide paternal imprint. More specifically, Spencer andWilliams (1997) consider a two-locus model in whichthe primary locus determines embryo survival and isexposed to deleterious mutations and the second, amodifier locus, is maternally expressed (e.g., duringfemale meiosis) and determines whether the maternalcopy of the primary locus is imprinted or not in theembryo. This model would be comparable to a DFmodel with a paternal expression of the modifier locus.Indeed, in both cases the modifier is cis-acting, meaningthat the allele causing silencing is on the same chromo-some as the silenced allele (we define the trans-actingmodifier as the reverse situation, where the allelecausing silencing is not on the same chromosome asthe silenced allele). In the DF model with paternalexpression, the silenced allele is paternal (as in all DFmodels) and the control is also paternal. In the Spencer

and Williams (1997) model, the silenced allele ismaternal and the modifier is also controlled maternally.The comparison of our model with this model of

imprinting highlights an important and possibly generalfeature of models for the evolution of gene expression.More than a maternal vs. paternal effect, the crucial issueis whether the modifier is cis- or trans-acting. It is muchmore difficult to evolve a cis- than a trans-acting silencingallele because indirect selection always benefits a modi-fier associated to purged chromosomes. In fact, for anygiven locus controlling the expression of genes exposedto deleterious mutations, there is a potential for conflictbetween its alleles. This conflict can occur if the modifieralleles do not have an additive effect (such as when thereis a sex-limited expression of the modifier, but thephenomenon may be much more general).

Kin conflict vs. heterozygosity theory: A widespreadview is that DF and MCD evolved in response to akin conflict over resource allocation (Queller 1984;Dominguez 1995; Friedman 1995; Shaanker andGaneshaiah 1997; Hardling and Nilsson 1999). Thistheory relies strongly on the idea that embryos com-pete for maternal resources. In fact, this theory isappealing because it claims to explain quite generallythe evolution of traits involved in postfertilizationresource allocation in both plants (DF, MCD, andimprinting) and animals (imprinting). It is, however,mostly supported from observations related to im-printing (see, e.g., Dominguez 1995). The centralargument is that the male/female conflict theorypredicts a male–female coevolution, which can bedetected by distinctive adaptations and counteradap-tations. There is, for instance, the well-known case ofimprinting of the fetal growth factor Igf2 and itsinhibitor Igf2r. Similarly, DF and MCD are viewed assteps in this male/female conflict in plants. There aretwo problems with this argument. First, what is inter-preted here as the signature of a male/female conflictdoes not necessarily support the kin conflict theorysince any theory involving an escalation would beconsistent with this interpretation. For instance, ourmodel shows that a male/female conflict can occurover the genetic constitution of the endosperm evenin the absence of problems of resource allocationamong kin. Second, under a conflict theory, we wouldexpect to see, at least in some cases, an escalation ofadaptations and counteradaptations. This is clearlynot the case for the evolution of the ploidy of theendosperm: no plants exhibit, for instance, 2m:2p or3m:2p endosperms, while nothing seems to limit apriori such escalation (contrary to the Igf2 and Igf2rsystem where the escalation is maximal: the expressionof these genes is totally suppressed when inheritedmaternally or paternally, respectively). Our modelsshow that, depending on parameter values, DF andMCD may be stable states (advantageous for bothmales and females), which might in fact more closelyreflect the data. Thus, the available evidence does notcritically favor one theory over the other. More specifictests would be required to discriminate between them.


Combining theories: The masking effects generatedby DF and MCD could interfere with resource allocationto the seed. Willson and Burley (1983) proposedthat DF, by masking deleterious mutations in the endo-sperm, should increase resource allocation to the associ-ated seed (and favor the transmission of paternallyinherited alleles). They viewed MCD as a return to a statecloser to 1m:0p, which, by exposing deleterious muta-tions in the endosperm, should decrease resource allo-cation to the associated seed (and favor the transmissionof maternally inherited alleles). This view attempts tobridge the different theories by linking heterosis in theendosperm with the amount of resource allocation to theseed. It may provide a mechanism explaining how DF andMCD could play a role in a male/female conflict overresource allocation. The first part of this argument holds,but could be stated more precisely. Provided h , 1

2 , DFincreases transitorily the mean fitness in a 1m:0p pop-ulation. (It is, however, not true at equilibrium where theload is higher in a 1m:1p population if h , 1

2 .) Similarly,the second part of the argument can hold, but only underspecific conditions. MCD transitorily decreases the meanfitness in a 1m:1p population only if h , hm. This lattercondition may, however, not be systematically verified.Overall, combining these two theories would requireincluding them in a proper model, to investigate theconditions where they predict different and distinguish-able outcomes. In any case, it seems necessary to includethe impact of deleterious mutations because they neces-sarily occur and because they can produce similarpatterns of genetic conflict.

Alternative nutritive tissues: Accumulating resourcesdirectly from an embryonic tissue (like cotyledons) orfrom a maternal or sporophytic tissue (like the seedcoat) could be considered as an alternative to doublefertilization. These tissues become predominant in themature seed in some angiosperms, but the endospermnearly always subsists, at least transitorily or as aninterface. We may wonder why the endosperm is nearlyalways maintained in such cases. We can tentativelysuggest two hypotheses. First, the endosperm candevelop immediately after fertilization, while the de-velopment of a nutritive embryonic tissue would beconstrained by the necessary steps of embryo develop-ment. Second, a sporophytic maternal tissue (1m1m9:0p)would confer a similar direct effect of masking butwould lead to a very inefficient purge of deleteriousmutations because it dissociates the genotype of theembryo from the phenotype on which selection occurs.Indeed, a deleterious m9 allele would decrease thesurvival of embryos not carrying the mutation, while adeleterious p allele would not be purged (because it isnot expressed in a 1m1m9:0p perisperm) although itis transmitted to the next generation. Thus, one cansee that selection should be more efficient in purg-ing deleterious mutations with a 1m:1p than with a1m1m9:0p nutritive tissue.

In some angiosperms, endosperms derived from poly-sporic gametophytes are alternatives to 1m:1p or 2m:1pendosperms. In a tetrasporic or a bisporic gametophyte,the endosperm can be made of m, m9, and p contributions(Friedman et al. 2008). As with the perisperm, polysporydissociates the genotype of the embryo from the pheno-type on which selection occurs, so that purging would beless efficient with polyspory. However, compared with amonosporic endosperm (2m:1p) where the two maternalalleles are mitotically derived, a polysporic endosperm(1m1m9:1p, for instance) is unlikely to carry two delete-rious maternal alleles, and aaA endosperms would becomparatively less frequent with polyspory. For thisreason, masking of deleterious alleles would be moreefficient with polyspory. A further development of ourtheory should therefore include a model to evaluate howthese antagonistic direct (masking) and indirect (purg-ing) effects combine and to predict conditions whenpolyspory should evolve.

Conclusion: Double fertilization and maternal con-tribution doubling can both evolve because of themasking/exposition of deleterious mutations. Thistheory is consistent with the pattern seen in seed plantsas well as with the theory of kin conflict. Interestingly,our model also predicts that conflicting selectionpressures for DF and MCD may occur in male andfemale gametophytes. The effect of deleterious muta-tions cannot therefore be neglected in a comprehensivetheory of DF and MCD evolution.

We thank E. Imbert for stimulating discussions and Dan Schoen, ananonymous reviewer, and Evolutionary Genetics and Ecology Groupstudents, in particular, F. Debarre, for useful comments on themanuscript. This work was supported by a starting grant from theEuropean Research Council to T.L. A.C. benefited from a fellowshipfrom the French Ministry of Research.

LITERATURE CITED

Baroux, C., C. Spillane and U. Grossniklaus, 2002 Evolutionaryorigins of the endosperm in flowering plants. Genome Biol. 3:1026.1021–1026.1025.

Barton, N. H., and M. Turelli, 1991 Natural and sexual selectionon many loci. Genetics 127: 229–255.

Bengtsson, B. O., 1992 Deleterious mutations and the origin of themeiotic ploidy cycle. Genetics 131: 741–744.

Brink, R. A., and D. C. Cooper, 1940 Double fertilization and de-velopment of the seed in angiosperms. Bot. Gaz. 102: 1–25.

Brown, C. A., and D. R. Formanowicz, 1996 Reproductive invest-ment in two species of scorpion, Vaejovis waueri (Vaejovidae) andDiplocentrus linda (Diplocentridae), from West Texas. Ann.Entomol. Soc. Am. 89: 41–46.

Bulmer, M., 1986 Genetic models of endosperm evolution in high-er plants, pp. 743–763 in Evolutionary Process and Theory, editedby S. Karlin and E. Nevo. Academic Press, New York.

Charnov, E. L., 1979 Simultaneous hermaphroditism and sexualselection. Proc. Natl. Acad. Sci. USA 76: 2480–2484.

Chasnov, J. R., 2000 Mutation selection balance, dominance andthe maintenance of sex. Genetics 156: 1419–1425.

Chopra, R., and R. Sachar, 1963 Endosperm, pp. 135–170 in NewAdvances in the Embryology of Endosperm, edited by P. Maheshwari.International Society of Plant Morphologists, Delhi, India.

De Fraipont, M., J. Clobert and R. Barbault, 1996 The evolutionof oviparity with egg guarding and viviparity in lizards and snakes:a phylogenetic analysis. Evolution 50: 391–400.


De Jong, T. J., H. Van Dijk and P. G. L. Klinkhamer,2005 Hamilton’s rule, imprinting and parent-offspring conflictover seed mass in partially selfing plants. J. Evol. Biol. 18: 676–682.

Dominguez, C. A., 1995 Genetic conflicts-of-interest in plants.Trends Ecol. Evol. 10: 412–416.

Donoghue, M. J., and S. M. Scheiner, 1992 The evolution of endo-sperm: a phylogenetic account, pp. 356–389 in In Ecology andEvolution of Plant Reproduction: New Approaches, edited by R.Wyatt. Chapman & Hall, New York.

Friedman, W. E., 1995 Organismal duplication, inclusive fitness the-ory, and altruism—understanding the evolution of endospermand the angiosperm reproductive syndrome. Proc. Natl. Acad.Sci. USA 92: 3913–3917.

Friedman, W. E., 2001a Comparative embryology of basal angio-sperms. Curr. Opin. Plant Biol. 4: 14–20.

Friedman, W. E., 2001b Developmental and evolutionary hypothe-ses for the origin of double fertilization and endosperm. C. R.Acad. Sci. Ser. III Sci. Vie. Life Sci. 324: 559–567.

Friedman, W. E., and K. C. Ryerson, 2009 Reconstructing the an-cestral female gametophyte of angiosperms: insights from Am-borella and other ancient lineages of flowering plants. Am. J.Bot. 96: 129–143.

Friedman, W. E., W. N. Gallup and J. H. Williams, 2003 Femalegametophyte development in Kadsura: implications for Schisan-draceae, Austrobaileyales, and the early evolution of floweringplants. Int. J. Plant Sci. 164: S293–S305.

Friedman, W. E., E. N. Madrid and J. H. Williams, 2008 Origin ofthe fittest and survival of the fittest: relating female gametophytedevelopment to endosperm genetics. Int. J. Plant Sci. 169: 79–92.

Garcıa-Dorado, A., and A. Caballero, 2000 On the average coef-ficient of dominance of deleterious spontaneous mutations. Ge-netics 155: 1991–2001.

Guignard, L., 1899 Sur les antherozoıdes et la double copulationsexuelle chez les vegetaux angiospermes. C. R. Acad. Sci. Paris128: 864–871.

Haig, D., 2000 The kinship theory of genomic imprinting. Annu.Rev. Ecol. Syst. 31: 9–32.

Haig, D., 2004 Genomic imprinting and kinship: How good is theevidence? Annu. Rev. Genet. 38: 553–585.

Haig, D., and M. Westoby, 1988 Inclusive fitness, seed resourceand maternal care, pp. 60–79 in Plant Reproductive Ecology: Patternsand Strategies, edited by J. L. Doust and L. L. Doust. Oxford Uni-versity Press, Oxford.

Haig, D., and M. Westoby, 1989 Parent-specific gene-expressionand the triploid endosperm. Am. Nat. 134: 147–155.

Hardling, R., and P. Nilsson, 1999 Parent-offspring and sexualconflicts in the evolution of angiosperm seeds. Oikos 84: 27–34.

Hardling, R., and P. Nilsson, 2001 A model of triploid endospermevolution driven by parent-offspring conflict. Oikos 92: 417–423.

Kacser, H., and J. A. Burns, 1981 The molecular-basis of domi-nance. Genetics 97: 639–666.

Law, R., and C. Cannings, 1984 Genetic analysis of conflicts arisingduring development of seeds in the Angiospermophyta. Proc. R.Soc. Lond. Ser. B Biol. Sci. 221: 53–70.

Maheshwari, P., 1963 Recent Advances in the Embryology of Angio-sperms. Ed. International Society of Plant Morphologists, Univer-sity of Delhi, Delhi, India.

Nawaschin, S. G., 1898 Resultate einer Revision der Befruchtungs-vorgange bei Lilium Martagon und Fritillaria tenella. Bull. Acad.Sci. St. Petersburg 9: 377–382.

Otto, S. P., 2003 The advantages of segregation and the evolutionof sex. Genetics 164: 1099–1118.

Otto, S. P., and T. Day, 2007 A Biologist’s Guide to Mathematical Modelingin Ecology and Evolution. Princeton University Press, Princeton, NJ.

Otto, S. P., and D. B. Goldstein, 1992 Recombination and the evo-lution of diploidy. Genetics 131: 745–751.

Perrot, V., S. Richerd and M. Valero, 1991 Transition from hap-loidy to diploidy. Nature 351: 315–317.

Queller, D. C., 1983 Kin selection and conflict in seed maturation.J. Theor. Biol. 100: 153–172.

Queller, D. C., 1984 Models of kin selection on seed provisioning.Heredity 53: 151–165.

Shaanker, R. U., and K. N. Ganeshaiah, 1997 Conflict betweenparent and offspring in plants: predictions, processes and evolu-tionary consequences. Curr. Sci. 72: 932–939.

Shaanker, R. U., K. N. Ganeshaiah and K. S. Bawa, 1988 Parent-offspring conflict, sibling rivalry, and brood size patterns inplants. Annu. Rev. Ecol. Syst. 19: 177–205.

Spencer, H. G., and J. M. Williams, 1997 The evolution of genomicimprinting: two modifier-locus models. Theor. Popul. Biol. 51: 23–35.

Stebbins, G. L., 1976 Seeds, seedlings, and the origin of angio-sperms, pp. 300–311 in Origin and Early Evolution of Angiosperms,edited by C. B. Beck. Columbia University Press, New York.

Szafraniec, K., D. M. Wloch, P. Sliwa, R. H. Borts and R. Korona,2003 Small fitness effects and weak genetic interactions betweendeleterious mutations in heterozygous loci of the yeast Saccharo-myces cerevisiae. Genet. Res. 82: 19–31.

Trivers, R. L., 1974 Parent-offspring conflict. Am. Zool. 14: 249–264.Tutt, K., C. H. Daugherty and G. W. Gibbs, 2002 Differential life-

history characteristics of male and female Peripatoides novaezea-landiae (Onychophora: Peripatopsidae). J. Zool. 258: 257–267.

Vassilieva, L. L., A. M. Hook and M. Lynch, 2000 The fitnesseffects of spontaneous mutations in Caenorhabditis elegans.Evolution 54: 1234–1246.

Westoby, M., and B. Rice, 1982 Evolution of the seed plants andinclusive fitness of plant-tissues. Evolution 36: 713–724.

Williams, J. H., and W. E. Friedman, 2002 Identification of diploidendosperm in an early angiosperm lineage. Nature 415: 522–526.

Willson, M., and N. Burley, 1983 Sporogenesis and double fertiliza-tion, pp. 75–93 in Mate Choice in Plants: Tactics, Mechanisms, and Con-sequences, edited by R. May. Princeton University Press, Princeton, NJ.

Wourms, J. P., 1993 Maximization of evolutionary trends for placen-tal viviparity in the spadenose shark, Scoliodon-laticaudus. Envi-ron. Biol. Fishes 38: 269–294.

Communicating editor: D. Charlesworth

APPENDIX A: RECURSIONS

The exact recursions of gamete frequencies (respectively, AM, aM, AM9, and aM9) are for the example of the DFmodel with a paternal expression of the modifier locus,

p9AM ¼1

2T

� � 2p2AM 1 2pAM pAm 1 pAM paM ð2� ð1� ð1� 2hÞnM ÞsÞ1 pAM pamð1� RÞð2� ð1� ð1� 2hÞnM 1 hðnM � nmÞÞsÞ1 paM pAmRð2� ð1� ð1� 2hÞnM 1 ðh � 1ÞðnM � nmÞÞsÞ

0@

1Að1� mÞ ðA1Þ

p9aM ¼1

2T

� � 2p2AM m 1 pAM paM ð2� s 1 nM s � 2hnM sÞð1 1 mÞ1 2pAM pAmm

1 2p2aM ð1� sÞ1 pAM pamð2ð1ð1� 2hÞnM 1 hðnM � nmÞÞsÞðRð1� mÞ1 mÞ

1 paM pAmð2� ð1 1 hnM � ð1� hÞnmÞsÞð1� Rð1� mÞÞ1 2paM pamð1� sÞ

0@

1A ðA2Þ

p9Am ¼1

2T

� � 2pAM pAm 1 2p2Am 1 pAM pamRð2� ð1� ð1� 2hÞn 1 hðnM � nmÞÞsÞ

1 pAmpaM ð1� RÞð2� ð1 1 2hnM 1 nm � 2nM 1 hðnM � nmÞÞsÞ1 pAmpamð2� ð1� 2nM 1 nm 1 2hnmÞsÞ

0@

1Að1� mÞ ðA3Þ


p9am ¼1

2T

� � 2p2amð1� sÞ1 2p2

Amm� 2pAM pAmm 1 2paM pamð1� sÞ1 paM pAmð2� ð1 1 hnM � ð1� hÞnmÞsÞðRð1� mÞ1 mÞ1 pAmpamð2� s 1 ð1� 2hÞnmsÞð1 1 mÞ1 pAM pamð2� ð1� ð1� 2hÞnM 1 hðnM � nmÞÞsÞð1� Rð1� mÞÞ

0BB@

1CCA; ðA4Þ

where

T ¼ ðpaM 1 pamÞððpAM 1 pAmÞð1� 2hÞnM � 1Þ1 ðpAmðpaM 1 pamÞ � ðpaM pAm 1 ðpAM 1 2pAmÞpamÞhÞðnM � nmÞ

0@

1As: ðA5Þ

APPENDIX B: QLE ANALYSIS OF THE OTTO AND GOLDSTEIN (1992) MODEL

Whatever the expression of the modifier locus (uniparental or biparental expression), the selection coefficient ofdiploidy (2N), calculated with QLE assumptions, is

sm ¼ sdnpað1� paÞ h � 1

2

� �1 sðh � 1Þ

2Rð1� nÞðR � 1Þ � nh½ �

� �1 oðjÞ5: ðB1Þ

The eigenvalue is given in their original article. [All parameters are used equivalently except d11 ¼ nM, d12 ¼ (nM 1

nM)/2, and d22 ¼ nm.]


ploidy and the evolution of endosperm of flowering plants · 2010. 1. 22. · endosperm (2m:1p)....

Documents