A Model for the Evolution of Dioecy and Gynodioecy Author(S): Brian Charlesworth and Deborah Charlesworth Source: the American Naturalist, Vol

The University of Chicago

A Model for the Evolution of Dioecy and Gynodioecy Author(s): Brian Charlesworth and Deborah Charlesworth Source: The American Naturalist, Vol. 112, No. 988 (Nov. - Dec., 1978), pp. 975-997 Published by: The University of Chicago Press for The American Society of Naturalists Stable URL: http://www.jstor.org/stable/2460344 . Accessed: 08/05/2013 08:24

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp

. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].

The University of Chicago Press, The American Society of Naturalists, The University of Chicago are collaborating with JSTOR to digitize, preserve and extend access to The American Naturalist.

http://www.jstor.org

This content downloaded from 130.238.155.129 on Wed, 8 May 2013 08:24:38 AM All use subject to JSTOR Terms and Conditions THE AM ERICAN NATURALIST

Vol. 112,No. 988 The AmericanNaturalist November-December1978

A MODEL FOR THE EVOLUTION OF DIOECY AND GYNODIOECY

BRIAN CHARLESWORTH AND DEBORAH CHARLESWORTH School of Biological Sciences,University of Sussex,Brighton BN1 9QG, England

There is muchdifficulty in understandingwhy hermaphrodite plants should ever have been rendereddioecious. [CHARLES DARWIN, 1877]

Comparativeevidence suggeststhat dioecy in floweringplants has evolved in- dependentlyin a numberof different groups, from an hermaphroditeor monoecious ancestralcondition (Lewis 1942; Westergaard1958). It is obvious thatat least two genemutations are necessaryto transforman hermaphroditeor monoeciousspecies into one withseparate sexes; one mutationmust affectovule production,and the otherthe production of pollen. Studiesof sex determinationin severalspecies have given resultswhich supportthe view that two mutationswith these effectshave indeedbeen involved (Westergaard 1958). Since it is extremelyunlikely that two such mutationsshould occur and become establishedsimultaneously, it seems likelya priorithat the evolutionof dioecy fromhermaphroditism has involvedan inter- mediate type of population that contained both hermaphroditesand some male- sterileor female-sterileplants. The formercondition is known as gynodioecy,the latteras androdioecy.Androdioecy is extremelyrare in nature(Darwin 1877),and some reasonswhy this is to be expectedare givenlater in thispaper (see also Lloyd 1975). It thus seems most likelythat dioecy has usuallyevolved fromgynodioecy. This was originallysuggested by Darwin (1877) and has been discussedrecently by Carlquist (1966), Ross (1970), Lloyd (1974a, 1974b,1975), and Arroyoand Raven (1975). This idea is givenfurther support by the factthat the relativesof dioecious species are not infrequentlygynodioecious, though gynodioecy is in generala rare condition.Some examplesare givenin SectionV below. Anotherexample is Silene vulgaris,a gynodioeciousspecies related (but not closely)to thedioecious speciesS. alba and S. dioica; Rumex hastatusand R. lunariaare also gynodioecious(Smith 1967). In the presentpaper we explore a model for the evolutionof dioecy fromthe hermaphroditecondition, via gynodioecy,using both analyticalmethods and com- putercalculations of populationtrajectories. Previous theoretical treatments of the evolutionof dioecy includeLewis (1942), Lloyd (1974a, 1974b,1975), Ho and Ross

Amer.NatUr 1978 Vol. 112. pp 975 997. ? 1978 by The Universityof Chicago. 0003-0147/78/1288-0001$0207 975

This content downloaded from 130.238.155.129 on Wed, 8 May 2013 08:24:38 AM All use subject to JSTOR Terms and Conditions 976 THE AMERICAN NATURALIST

(1974), and Ross and Weir (1976). An analysis of the conditionsfor the spread of mutationsaffecting male and femalefertility has not previouslybeen attempted.This and a studyof how theseconditions relate to theselective pressures influencing the evolutionof dioecy are the main pointsof the presentpaper. We imaginethat the initialstep takenby a speciesthat is evolvingdioecy is the occurrenceof a mutation abolishingpollen production,i.e., a male-sterilityfactor. This can be eitherdominant or recessive.The conditionsfor the establishmentof such a gene in a population, whichwould thenbe gynodioecious,are discussedin SectionII, wherewe also show thata female-sterilityfactor is unlikelyto becomeestablished. In therest of the paper we discussthe fateof a mutationat a second locus,introduced into a gynodioecious populationand modifyinghermaphrodites in thedirection of increased maleness. We show how dioecycan evolveby thismechanism. One ofthe main questionswhich we explore is: Why are the differentgenes determiningsex grouped togetherin one regionof one chromosomein most species? Our resultssuggest that there may often be a "linkageconstraint" such thata modifiermutation of the sortdescribed above would be eliminatedfrom a gynodioeciouspopulation unless it occurredat a locus that is fairlyclosely linked to the gene controllingthe gynodioecy. Dioecy and gynodioecyhave oftenbeen regardedin the botanical literatureas outbreedingdevices (Lewis 1942; Baker 1959).In orderto examinethis idea we have incorporatedpartial self-fertilization and inbreedingdepression into our models.The role of thesefactors in themaintenance of gynodioecy and androdioecyhas also been investigatedby Lloyd (1975, 1976). Darwin (1877) feltthat redistributionof repro- ductiveresources as a consequenceof failure of pollen or ovule production causing, respectively,increased seed or pollen production was an importantfactor in the evolution of dioecy and gynodioecy.This idea is a featureof many previously publishedmodels of theevolution of dioecyand has recentlybeen exploredin detail by Charnov et al. (1976) forthe random matingcase. It willbe seen that,while this type of effectis almost certainlyessential for the evolution of dioecy,inbreeding effectsprobably also play a major role.

I. MODEL AND ASSUMPTIONS Formulationof theModel Throughoutthis paper we assume an infinitelylarge population which is partially self-fertilizingand whichis segregatingfor loci affectingovule and pollenproduction. We definethe followingparameters for individuals of genotypei: zi = thefrequency of i adults in a given generation;bi = the mean pollen output of an i individual; ei = themean ovule productionof an i individual.Provided that the genotype i is not male-sterile,a nonzero fractionsi of its ovules is assumed to be fertilizedby its own pollen.The same value,s, is assignedto all genotypesthat produce pollen. The pollen used in sellingis assumed to be a negligiblefraction of thepollen outputof a plant. The last parameterrequired is the inbreedingdepression, 6. We write1 - 6 forthe mean probabilityof survival of a zygoteproduced by self-fertilization,relative to that of a zygote produced by outcrossing.We assign a fixedvalue to 6 for a given population,independent of the genotypeof the zygoteand its parentsand of the overall composition of the population. This is clearlyin general only a realistic

This content downloaded from 130.238.155.129 on Wed, 8 May 2013 08:24:38 AM All use subject to JSTOR Terms and Conditions EVOLUTION OF DIOECY 977 procedurefor an equilibriumpopulation. The introductionof a gene whichaffects the levelof outcrossingwill change the value of 3; we would normallyexpect 3 to be positivelycorrelated with the mean level of outcrossingof the population.In our computerruns, where we treat 3 as a constantduring the evolutionof dioecy or gynodioecy,we are thereforeprobably erringon the conservativeside as far as estimatingthe selectionpressure in favorof outbreedingis concerned.Furthermore, this procedureignores the geneticbasis of inbreedingdepression and in particular the effectsof linkagebetween the loci we are studyingand thoseresponsible for the inbreedingdepression. On account of the complex geneticcontrol of inbreeding depression,this is probablythe best that can be done in the way of modeling.Our resultsshould providean accurateaccount of the possible equilibriain thesystems we studyand should also be reasonablyaccurate with regard to theintroduction of a raremutant gene into a populationat equilibrium.There seems no reasonto suppose thatour qualitativeconclusions will be seriouslyaffected by thissimplification in the representationof inbreedingdepression.

Definitionof Fitness In therest of thispaper a fundamentalconcept is the"fitness" of a genotype.For a partiallyself-fertilized population, we can definewi, the net fitnessof genotypei, as the fractionof the gametespresent among adults of the nextgeneration which are contributedby individualsof genotypei, dividedby the fractionof thetotal number of gametescontained in i adults in the presentgeneration. The mean fitnessof the populationon thisdefinition is always one; wi can be obtainedas follows. Among the gametes that go to formadults of the next generation,the fraction contributedthrough the femalegametophytes produced by typei individualsis:

[si(1 - 3)eizi + (1 - sj)ejzj]/W, (la) where

W= T+ U, T= E Sk(l- )ekZk U = (1-sk)ekzk. k k The female"fertility" of genotypei, wf , is this quantitydivided by zi. The male fertility,wmi, can be calculatedin a similarway, noting that the pollen contributions have to be normalized so that the total contributionof gametes throughmale gametophytesis equal to thatthrough female gametophytes. We thereforehave:

was UV + sj(1 - )ej 1W, (lb) where

V = bk Zk k Since all zygotescontain an equal contributionof geneticmaterial from male and femalegametes, we obtain the net fitnessof genotypei as

Wi = 2(Wmi + Wfi). (lc)

This content downloaded from 130.238.155.129 on Wed, 8 May 2013 08:24:38 AM All use subject to JSTOR Terms and Conditions 978 THE AMERICAN NATURALIST

In thenext part of thispaper we show how thisdefinition of fitness can be used to obtain conditionsfor the evolution of gynodioecy and androdioecy;we thengo on to discussthe evolution of dioecy. Before doing this, however, it is usefulto illustratethe conceptby consideringthe case of a singlelocus withtwo alleles,A1 and A2. Let the frequenciesof Al Al, Al A2, and A2 A2 among the adults ofa givengeneration be z1, Z2, and Z3 respectively.Using equations (la), (lb), and (1c), fitnessesw1, w2, and W3 can be assigned to these genotypes.The frequenciesof alleles Al and A2 are Pt and P2, respectively.The gene frequenciesin the next generationcan be seen to be givenby Pi = z1w1 + -Z2W2, (2a)

P2 = 2Z2W2 + Z3W3. (2b) If A2 is fullydominant to A.,,then W2 = W3 so thatequation (2b) becomes

P2 = P2W2* (3)

Hence at equilibriumin thiscase we have w1 = W2 (= W3) = 1, i.e.,both phenotypes contributeequally to the nextgeneration (see also Lloyd 1975). Equations (2a) and (2b) can be used to investigatethe initial spread of a raregene. For example,if A2 is introducedinto an A1 A1 population,the fitness of the new type can be approximatedby givingthe summationterms (T, U, and V) in equations (1) thevalues appropriatefor the original type. The conditionfor the spread of A2 is that thisfitness must exceed one.

II. THE EVOLUTION OF GYNODIOECY AND ANDRODIOECY Lewis (1942) suggestedthat most cases of gynodioecyare due to cytoplasmic inheritanceand thereforecannot be consideredas precursorsof dioecy. It is now known thatsingle-gene inheritance of male sterilitycan be foundin gynodioecious species (Baker 1966,and personalcommunication; Jain, Boussy, and Hauptli 1978; Arroyoand Raven 1975; see also Lloyd 1974a). Genes withsuch effectsare knownin manygenetically well-studied species (Jain 1959),so it cannotbe truethat they never occur in natural populations. Their role in the evolution of dioecy is further supportedby geneticalstudies of sex determinationin severalspecies (Westergaard 1958). In this part of the paper, we presentsome models for the evolution of gynodioecy(and androdioecy)based on single-geneinheritance, which are in some respectsa generalizationof theearlier models of Lloyd (1974a) and of Valdeyronet al. (1973). Lloyd (1975, 1976) has alreadypublished an analysisof the equilibrium conditionsfor gynodioecy and androdioecyin partiallyselfing populations. We will presentour own treatmentof thisproblem as a preliminaryto SectionIII. The main addition to Lloyd's work is the derivationof the conditionsfor the spread of a rare male- or female-sterilitygene.

Gynodioecy Gynodioecyis here assumed to have evolved froma completelyhermaphrodite conditionas a resultof a singlegene mutation. The hermaphroditesare homozygous A1 Al, and a mutantallele A2 arises which we assume to be eithercompletely

This content downloaded from 130.238.155.129 on Wed, 8 May 2013 08:24:38 AM All use subject to JSTOR Terms and Conditions EVOLUTION OF DIOECY 979

dominant or completelyrecessive and which causes a complete loss of pollen production.Let the male-sterile(female) individuals have a mean ovule production of 1 + k relativeto thatof the hermaphrodites. It is assumedhere that s and k remain constant as the frequencyof femalesincreases in the population,i.e., thereis a sufficientlylarge number of pollinatorvisits to each flowerthat seed setin outcrosses is not affectedby the presenceof femalesin the population.Lloyd (1974a, 1974b, 1974c,1975) has consideredthe effect of restricted pollinator visits on theequilibria in some models ofgynodioecy. Since, as willbe shownlater, the frequency of females is rarelyhigh, this factor can be neglectedfor present purposes. In anycase it willnot affectthe conditionfor the initial spread of A2. This can be determinedby applying the methodof the previoussection. We findthat A2 will spread ifand only if: 1 + k > 2(1-sb). (4) Equation (4) shows that a male-sterilitygene can invade an hermaphrodite population even when k = 0, i.e.,when femaleshave the same ovule productionas hermaphrodites.This requiressb > 0.5, i.e.,both verystrong inbreeding depression and a highrate of selling. (This conditionis, however, less severethan that derived by Valdeyron et al. [1973], who assumed that inbreedingdepression affects ovule productiononly.) It seems likelythat abolition of pollen productionmight increase ovule productionas a resultof reallocationof theresources available forreproduc- tion. Darwin (1877) suggestedthat this typeof effect,which he called "compensation",might play a role in the evolutionof gynodioecyand dioecy. If thereis no sellingor inbreedingdepression, equation (4) requiresk > 1 forthe establishmentof A2, i.e., that,as is intuitivelyobvious, ovule productionmust be morethan doubled in compensationfor the loss ofpollen production;such a degree of compensationseems improbable.In general,the higher the value of sb,the lower the value of k thatpermits gynodioecy to evolve.The mostfavorable conditions for theestablishment of gynodioecyoccur whenthere are fairlyhigh levels of selfing and inbreedingdepression and an increasedovule productionby females.For example, withs and 6 both equal to 0.5,a k value greaterthan 0.5 is required,compared with 1 in the random-matingcase. The frequencyof femalesat equilibriumin the case of completelydominant or recessiveinheritance of male sterilityis, by the methodof SectionI, Z= (k + 2s6- 1)/2(k+ so). (5) This correspondsto the formulaeof Lloyd (1974a, 1975, 1976). It can easilybe seen that2 is nonzeroonly when equation (4) is satisfiedand thatit is an increasingfunction of k and sb, i.e.,the higherthe sellingrate and inbreeding depressionand the greaterthe degreeof compensation,the higherthe frequencyof females.Some examplesare givenin table 1. It also followsdirectly from equation (5) that 2 < , so that femalescan never exceed hermaphroditesin frequencyin an equilibriumpopulation. Unless k is high,females will always be ratherrare (see table 1). These conclusionsare identicalwith those of Lloyd (1975, 1976) and accord with observationson gynodioeciousspecies (Darwin 1877; Lloyd 1976). In the case of a dominantmale-sterility gene, the only possible genotype with A2 is A1 A2, regardlessof the level of self-fertilization,so that the equilibriumgenotype frequenciesfollow directly from equation (5). In themore plausible case ofa recessive

This content downloaded from 130.238.155.129 on Wed, 8 May 2013 08:24:38 AM All use subject to JSTOR Terms and Conditions 980 THE AMERICAN NATURALIST

TABLE 1

FATE OF A MALE-STERILITY GENE INTRODUCED INTO A POPULATION OF HERMAPHRODITES WITH POLLEN OUTPUT = 1.0 AND OVULE OUTPUT = 1.0

SELFING INBREEDING OVULE EQUILIBRIUM EQUILIBRIUM RATE DEPRESSION OUTPUT OF FREQUENCY FREQUENCY (s) (6) 9 (1 + k) OF GENE OF FEMALES

A, RecessiveGene

11.0 .3137 .1071 7 8 11.1 .4095 .1667 *.*0I 1.5 .5648 .2925 2.1 .6394 .3675 1.0 0 0 .7 .7 1.1 .2259 .0678 1.5 .5061 .2424

.7 .4 J1.5 0 0 *7 11.7 .3406 .1326

0 0 J2.1 .2164 .0455 02.5 .4242 .1667

B, Dominant Gene

.7 .8 J1.O .0536 .1071 .1.1 .0833 .1667

0 0 )2.1 .0227 .0445 02.5 .0833 .1667 male-sterilitygene, it is necessaryto considerthe recurrencerelations relating the frequencyof A2 A2 (females)to thegenotype frequencies in the previousgeneration. Let Y be thefrequency of AI A2 individualsand Z thefrequency of A2 A2 individuals. Then it is easy to show that

_ [Z(1 + k) + (1 -s)Y]Y s(1 -)Y (6 wz'- + ,(6 ~~2(1-Z) 4 where W = (1 - Z)(1 - sb) + Z(1 + k). Atequilibrium, equation (5) showsthat W = (1 + k)/2.(This is to be expectedfrom the factsthat the equilibriumfitnesses of the threegenotypes are thesame and that the femalescontribute only throughtheir ovules.) Substitutinginto equation (6) and rearranging,we obtain a quadraticequation forthe equilibrium frequency of Al A2:

- [1-Z 2 ]Y-(1 + k) = 0, (7) where2 is given-byequation (5). Table 1 shows the phenotypiccomposition and gene frequenciesof some equilibriumpopulations with dominant and recessivemale-sterility. Computer calculations showedthat in all thesecases theequilibria are globallystable. In thosecases in table

This content downloaded from 130.238.155.129 on Wed, 8 May 2013 08:24:38 AM All use subject to JSTOR Terms and Conditions EVOLUTION OF DIOECY 981

1 where s and 6 were nonzero,or in the random matingcase withA2 dominant, equilibriumwas reached in a matter of a few hundred generationsafter the introductionof the male-sterilitygene at a frequencyof 0.00025. As might be expected,however, the frequencyof a recessivemale-sterility gene increasedex- tremelyslowly in a randommating population, and thegene remained rare for many thousandsof generations. Gynodioecy due to recessivemale-sterility is thus unlikely to be establishedin random matingpopulations unless finite population size effects can be invoked.

Androdioecy The case ofa dominantor recessivegene causing female sterility can be treatedin a way similarto the above. The pollen productionof the males relativeto thatof the hermaphroditesis writtenhere as 1 + K. The conditionfor the initialspread of a female-sterilegene is 1 + K > 2(1-s6)/(1-s). (8) When thereis partial self-fertilization(s > 0), it is immediatelyapparent that this conditionis morestringent than for the case ofgynodioecy (eq. [4] above). Further- more, for constant6, the largers the largerthe minimumvalue of K requiredto establish androdioecy.This is in contrastto the case of gynodioecy,where with nonzero inbreedingdepression, self-fertilization reduces the minimumincrease that is necessaryin ovule production.The reason forthis effect is simplythat the pollen producedby themales (femalesteriles) is used onlyin outcrosses;the higher the level of self-fertilization,the less this pollen can contributeto the next generation.We concludethat androdioecy is mostlikely to be establishedin outcrossingpopulations and cannot be establishedin any population unless the pollen productionof the males is at least doubled relativeto that forhermaphrodites. This conclusionwas also reachedby Lloyd (1975). Furthermore,a recessivegene forfemale sterility will encounterthe same difficultyin establishingitself in an exclusivelyoutcrossing population,as noted above fora recessivemale-sterility gene. These considerations may be among the reasons forthe rarityof androdioecyin nature. The equilibriumfrequency of males can be determinedby the methodused in treatinggynodioecy and is givenby = [(1 + K)(1 - s)- 2(1 - sb)]/2K(l - s6). (9)

It can easilybe shownfrom this formula that Z* is alwaysless thanor equal to 2, and is a decreasingfunction of s. Withhigh s values,an extremelylarge K is required forthere to be more thana fewpercent of males in theequilibrium population, e.g., with s = 6 = 0.5 and K = 2.5, Z* = 0.067; K mustbe greaterthan 2 in orderfor androdioecyto become establishedat all in thiscase, comparedwith a k value of0.5 forthe correspondingmale-sterile case. In the followingsections of this paper we considerthe evolutionof dioecy from gynodioecyby examiningthe fate of a modifierof ovule productionintroduced into gynodioeciouspopulations at equilibrium.The mutantallele at thissecond locus can have a total or partial effectin reducingovule production;it can act only in hermaphroditesor reducethe ovule productionof females as well.It can also havean

This content downloaded from 130.238.155.129 on Wed, 8 May 2013 08:24:38 AM All use subject to JSTOR Terms and Conditions 982 THE AMERICAN NATURALIST increasedpollen production,by the compensationprinciple, and can be dominant, recessive,or partiallydominant.

III. INTRODUCTION OF A MODIFIER INTO A GYNODIOECIOUS POPULATION In this section we consider the conditionswhich determinewhether or not a mutationat a second locus can becomeestablished in a gynodioeciouspopulation at equilibrium.

Conditionsfor Spread of a Modifierof ArbitraryEffect on OvuleProduction At equilibriumfor the first(male-sterile) mutation, both hermaphroditesand femalesare present.Knowing the frequencies of these forms, as givenby equation (5), one can calculate how big an increase in pollen output would be necessaryto increase the net fitnessof the hermaphrodite,given that its ovule productionis decreased.Let the modifiedhermaphrodites have pollen production1 + K relative to that of the originalhermaphrodites and let theirovule productionbe 1 -k* (1 > k* ? 0). Then equations (1) and (5) give the followingexpression for K*, the value of K thatwill exactlycompensate for the loss in ovule production: (1 - K* + 2sb) k* (10)

Clearly,an increasein pollenproduction greater than K* is a necessarycondition for the spread of the mutationat the second locus. The factthat s and 3 are both less than one, togetherwith the conditionfor the existenceof gynodioecygiven in equation (4), impliesthat K* mustbe positive.In otherwords, some increasein pollen productionis alwaysrequired for the modifier gene to spread.When thefemales have thesame ovule productionas thehermaphrodites (k = 0), equation (10) shows that K* > k* unless 1 + 2s < 4sb, whichwould require a veryhigh level of inbreedingdepression. In most instancesof this sort, therefore,we can conclude thata reductionin ovule productioncannot be selected for unless it is accompanied by a more than proportionateincrease in pollen production.Equation (10) also shows,however, that the higherk, the lowerK*, so that if the ovule productionof femalesis sufficientlyhigh compared with that of hermaphrodites,K* need not be as highas k. Some cases illustratingthese points are givenin table 2. One furtherpoint is thatif a similarcalculation is done fora geneof thissort introducedinto an hermaphroditepopulation (eq. [8]), thenwith the same values of s and 5, a largercompensatory increase in pollen productionis always needed than thatrequired by equation (10). The presenceof femalesin thepopulation thereforemakes it easier fora modifierreducing female fertility to spread. These considerationshave involvedno assumptionsabout the dominancerela- tions at the loci concernedor about theeffect of themodifier gene on the fertilityof females.In orderto obtain sufficientconditions for the spread of the modifier,it is necessaryin generalto take thesefactors, as well as thelinkage between the two loci, into account.This is done below fortwo importantcases.

This content downloaded from 130.238.155.129 on Wed, 8 May 2013 08:24:38 AM All use subject to JSTOR Terms and Conditions EVOLUTION OF DIOECY 983

TABLE 2

CALCULATIONS OF THE MINIMUM FRACTION (K*) BY WHICH THE POLLEN PRODUCTION MUST BE INCREASED TO ENABLE A FEMALE-STERILITY MUTATION TO SPREAD IN A GYNODIOECIOUS POPULATION

OVULE INBREEDING OUTPUT OF OVULE OUTPUT OF + SELFING DEPRESSION ORIGINAL RATE (S) (6) I 1.0 1.1 1.5 1.8 2.1 2.2

1.0 1.381 1.115 .630 .382 .7 .8 .9 1.092 .903 ...... 5 .409 .358 ... .160 ...

.7 .2 1.0 - 3.737 1.821

0 0 1.0 ...... 909 ...

A DominantModifier Introduced into a Populationwith Recessive Male-Sterility If male-sterilityis recessive,crossing-over between the two loci willproduce some femaleswhich carry the modifier gene, even whenthe modifier is rare.If the modifier reduces the ovule fertilityof thesefemales, they will be at a selectivedisadvantage, and a higherpollen productionof the modifiedhermaphrodites will be requiredto overcomethis disadvantage. Only ifthe modifier does not reducethe ovule production of femaleswill equation (10) give a sufficientcondition for the modifierto spread.In othercases, we mayexpect the degree of linkagebetween the loci to affect the conditions;the maximum rate of spread ofthe modifier will be expectedto result when it is introducedin couplingwith the male-fertility (A1) allele at thefirst locus, and whenthere is no crossing-overto breakdown thiscombination. If the modifier is not sufficientlyclosely linked to thefirst locus, it maybe eliminated,i.e., there may be a linkageconstraint. These conclusionshave been checkedby thesimulations described in SectionIV of this paper. In the case of a dominant modifiergene which abolishes the ovule productionof both hermaphroditesand females,an algebraicanalysis of its initial spreadcan be made. As outlinedin the Appendix,we can studythe properties of the quantityAO, which measures the asymptoticvalue of theratio between the frequencies of the modifiergene in two successivegenerations, while it is stillrare. The maximumvalue of AOis attainedwhen R, the frequencyof recombinationbetween the two loci,is zero. It can be seen thatthis maximum value is equal to thefitness of themodified hermaphrodites discussed above. As R increases,AO decreases smoothly and, ifits maximumvalue is greaterthan one, fallsbelow one whenR is greaterthan some criticalvalue, RC. For R greaterthan RC,the modifiergene will thereforebe eliminated.The value of RC, in termsof the parametersa and b definedin the Appendix,is givenby:

This content downloaded from 130.238.155.129 on Wed, 8 May 2013 08:24:38 AM All use subject to JSTOR Terms and Conditions 984 THE AMERICAN NATURALIST

When the maximumvalue of AOis not greatlyin excess of one, R, may be rather small. For example,with k = 0, K = 1.45,s = 0.7, and 3 = 0.8, we findR, = 0.0675. There is thereforea selectivesieve forclosely linked modifiers of thissort.

A RecessiveModifier Introduced into a Populationwith Dominant Male-Sterility At equilibriumfor a dominantmale-sterility gene, there are only two genotypes, homozygotesfor the originalallele (A1 A1) and heterozygousfemales (A1 A2). When a recessivemodifier reducing ovule productionis introducedat a low frequency, femaleshomozygous for it are produced only by outcrosses,and thereforewill initiallybe negligiblein frequencycompared with modifiedhermaphrodites. The conditionsof equation (10) will thereforebe sufficientfor the modifierto spread, regardlessof the linkagevalue and the effecton females.In particular,there can be no linkageconstraint. This conclusionis borne out by the simulationresults of Section IV.

IV. RESULTS OF COMPUTER SIMULATIONS The considerationsof SectionIII suggestthat a gene whichchanges the hermaphroditesinto a more male-likeform can be selectedfor provided that the gene also increasespollen output sufficiently.These conclusionswere verifiedby computer calculationsof population trajectories. The methodused was to startwith an arrayof genotypefrequencies. Genotypes with both male and femalepotential were assumed to devote a proportions of theirovule output to selling;the selfedovules had a probability1 - 3 of survivalto maturitycompared with outcrossedovules. Male- sterilegenotypes produced all theiroffspring by fertilizationwith a common"pollen pool," which was also used for the outcrossedprogeny of hermaphrodites.The compositionof the pollen pool was determinedby consideringeach of the male- fertilegenotypes and calculatingthe overall frequencies of the four gametes produced by thesein aggregate.Different genotypes were allowed to producediffering quanti- ties of ovules and pollen. The new arrayof genotypefrequencies produced by this procedurewas used to formthe next generation. The runswere started by introduc- ing the modifierallele at a low frequency(0.00025) into an equilibriumgynodioecious population. It was introducedheterozygous and in coupling with the originalallele (A1) at the firstlocus. The populationwas thenrun to equilibrium.

RecessiveMale-Sterility Gene Followedby a DominantModifier When the modifierconverts the hermaphroditesinto males, with zero ovule production,the equilibriumpopulation consists either of predominantlymales and females,at frequenciesclose to 0.5, or of males,females, and a substantialfrequency of hermaphrodites.Some resultsof this kind are shown in table 3. The looser the linkage,the more the populationdeparts from true dioecy. Some of the parameter sets shown in table 3 provideexamples of the action of a linkageconstraint: The modifierwas eliminatedwhen the recombinationfraction was high.This is thecase forall those examplesin table 3 whereonly a low recombinationfraction is shown. As expected,a linkageconstraint is most likelyto operatewhen the modifierhas a

This content downloaded from 130.238.155.129 on Wed, 8 May 2013 08:24:38 AM All use subject to JSTOR Terms and Conditions EVOLUTION OF DIOECY 985

TABLE 3

PHENOTYPE FREQUENCIES IN POPULATIONS THAT HAVE REACHED EQUILIBRIUM FOR A MALE- AND A FEMALE-STERILITY GENE

PHENOTYPE FREQUENCIES

OVULE OUTPUT OF RECOMBINATION Modified MODIFIED FEMALES FRACTION

Male-Sterile(1st) MutationRecessive, Female-Sterile (2d) Mutation Dominant

1.0o .085 .440 .450 .025 .9 g.1 .536 .218 .186 .060 .5 .656 .181 .115 .048 0 .01 .087 .461 .452 0

Male-Sterile(1st) Mutation Dominant,Female-Sterile (2d) Mutation Recessive

9 ).01 .085 .440 .450 .025 *9 1.5 .650 .179 .121 .050 0 J.O1 .089 .459 .447 .004 A.5 .819 .169 .012 .001

Male-Sterile(1st) MutationRecessive, Female-Sterile (2d) MutationRecessive

.9 .01 .477 .262 .221 .039 0 .01 .614 .239 .142 .004

Male-Sterile(1st) Mutation Dominant,Female-Sterile (2d) MutationDominant

.9 j.01 .584 .201 .159 .056 * \.5 .715 .168 .081 .036

NOTE.-Ovule outputof originalf = 1.0; pollen outputof originalf = 1.0; ovule outputof original = 1.1; pollen outputof X = 2.4; selfingrate = 0.7; inbreedingdepression = 0.8. strongeffect in the femalesas wellas thehermaphrodites, but theeffect in thefemales need not be as strongas thatin the hermaphrodites. Examplesof the perhaps more realistic case ofa dominantmodifier which partially reducesthe ovule outputof both femalesand hermaphroditeswere also studied.As in the above extremecase of completefemale sterility, the outcomedepends on the linkagebetween the two loci,though a small enougheffect on the femalescompared withthe hermaphroditesmay permitan unlinkedmodifier to become established. With either type of modifieraction, modifiedfemales are generallyrare at equilibrium,so that the population comes to consist predominantlyof females togetherwith eithermales and hermaphrodites,or of femalesand two typesof hermaphrodite.The simulationsalso show that a modifierwith no effecton the females,and whichsatisfies condition (10), will spread to fixation,giving a population consistingof femalestogether with either males or modifiedhermaphrodites, dependingon the phenotypiceffect of the gene; such a populationwould have only one segregatinggene affectingsex.

This content downloaded from 130.238.155.129 on Wed, 8 May 2013 08:24:38 AM All use subject to JSTOR Terms and Conditions 986 THE AMERICAN NATURALIST

In summary,modifiers converting the hermaphroditesinto males, or into a more male-likeform, could be selectedfor. In cases wherea looselylinked modifier was not eliminated,we have shownthat there is selectionfor tighter linkage between the two loci. This was done by simulatinga three-locussituation, two of the loci controlling the (sub-) dioecy,as explainedabove, and a thirdlocus modifyingthe recombination frequencybetween them. Even whenthis third locus was unlinkedto thefirst two, in whichcase theeffectiveness of selection on a modifierof recombination is likelyto be weak,it increasedin frequency,reducing the frequency of recombination between the selectedloci. With the above type of model thereare two situationswhere the equilibrium population consists essentiallyof just two forms,females and the new type of hermaphrodite.The firstway that thiscan occur is by a modifierwith no effectin females,as alreadynoted. The second way is by means of a non-sex-limitedmodifier that is tightlylinked to the male-sterilitygene. In thissituation, there are negligible frequenciesof two gametetypes, so thatjust threegenotypes occur; because of the dominanceof boththe genes, only two phenotypesare seen.Supposing that this state has been reached,we can thenask whetherfurther modification of thehermaphro- dite,making it yetmore like a male,is easier or moredifficult than was thefirst step. We have approached thisquestion by thesame methodthat we used in SectionIII to ask whetherthe firstmodifier gene will get into thepopulation. In thepresent case, thehermaphrodites are givena femalefertility of less thanone, as would be thecase if theyhad already become modifiedin the directionof maleness;we thenask what proportionalincrease in pollen productionwould be needed,given some decreasein ovule production,for net fitnessto be increased.This effectcan be studiedusing equation (10), since the increasedovule fertilityof femalescompared with hermaphroditesin thiscase correspondswith increased values of k. Some examplesare given in table 2. The lower the ovule productionof the hermaphrodites,the lower the compensatoryincrease in pollen productionneeded, so that it becomes easier for furtherchanges in thesame directionto be selected.It may thereforebe possiblefor dioecy to evolve by gradual modificationof the polleniferousplants, due to the successiveaccumulation of linkedmodifiers of ovule production.

DominantMale-Sterility Gene Followedby a RecessiveModifier Some examples of this type of situationare shown in table 3. The predictionof SectionIII thatthere can be no linkageconstraint is verifiedby theseruns. Just as in the case of a recessivemale-sterility gene followedby a dominantmodifier, this systemcan lead to equilibriumpopulations that are like those seen in nature,with males, females,and hermaphroditesor withtwo typesof hermaphroditeand one common femaletype, the otherbeing veryrare. Withthis model forthe evolution of dioecy we mustalso considerthe case withno selling,since a dominantmale-sterility factor can rise to an appreciablefrequency with random mating,provided its femalefertility is sufficientlyhigh. Some runs of thissort have been carried-out. The modifiercan increasein frequencybut does so extremelyslowly. For example,when such a male-sterilitygene raises the females' ovule outputto 2.2, it reachesequilibrium with a femalefrequency of 0.083. When a

This content downloaded from 130.238.155.129 on Wed, 8 May 2013 08:24:38 AM All use subject to JSTOR Terms and Conditions EVOLUTION OF DIOECY 987

female-sterilityfactor which increased pollen output to 2.5 was introducedat a frequencyof 0.00025, as in the other simulations,it rose to 0.00030 in 2,700 generations.When introducedat a frequencyof 0.1, it reacheda frequencyof 0.755 within1,000 generations, however, and the populationconsisted almost exclusively of males and females.Dioecy can thereforeevolve in this case, but the second mutationwill have a verylow chance ofestablishment in a largepopulation, as with most recessivegenes (Haldane 1927).

OtherPossible Cases We have also simulated the two models of gynodioecydescribed above-i.e., recessiveand dominantdetermination-followed by a modifierwhich is also recessive or dominant,as the case may be. For instance,a dominant modifierwas introducedinto a population segregatingfor a dominantmale-sterility gene; the parametervalues wereotherwise the same as thosefor a recessivecase thatwould go to an equilibriumconsisting of just males and females.In thiscase, however,with a recombinationfraction of 0.01, the modifier rose to only0.06, insteadof 0.72; dioecy did not result,but a low frequency(0.08) of males was presentat equilibrium, togetherwith a frequencyof 0.17 offemales, while the majority remained hermaphrodite.With an unlinkedmodifier, the equilibriumfrequency of males became higher (0.15), the frequencyof femalesbecame 0.20, and therewas a frequencyof 0.06 of femalescarrying the modifier,which would appear as somewhatsterile females. Similarresults were obtainedin the case of a recessivefollowed by a recessive. Anothercase whichwe have consideredis that of a recessivemale-sterility gene followedby a gene reducingovule productionwhich is partiallydominant rather thancompletely dominant or recessive.The conditiondescribed in SectionIII forthe initialspread of a female-sterilitygene mustbe satisfiedby the heterozygotefor this modifier,but the homozygotes,which we assume to be morefemale-sterile than the heterozygotes,need not necessarilysatisfy this condition.The resultsfrom runs of thissort of case are not verydifferent from those with full dominance; the frequencies of thedifferent types are affected,but no qualitativelynew phenomenon is seen.This type of model gives even more variabilityamong the hermaphroditesthan the modelsdiscussed earlier in thissection, since now we can have distinctheterozygotes and homozygotesfor the modifier as well as theoriginal hermaphrodites. As above, therewill be selectionfor further linked modifiers reducing the femalefertility of the modifiedindividuals.

V. DISCUSSION In thissection we reviewthe results described earlier in the paper and tryto relate themto some of the factsabout the breedingsystems of floweringplants.

Androdioecy As alreadymentioned, androdioecy is extremelyrare. One reason forthis may be that a cytoplasmicfactor causing female-sterilitycannot be transmitted,whereas male-sterilityis ofteninherited in this way. The rarityor absence of androdioecy

This content downloaded from 130.238.155.129 on Wed, 8 May 2013 08:24:38 AM All use subject to JSTOR Terms and Conditions 988 THE AMERICAN NATURALIST

controlledby nucleargenes is probablydue to the followingeffects, discussed more fullyby Lloyd (1975) and in Section II above. In a random matingpopulation, a female-sterilemutant cannot be establishedby selectionunless it more thandoubles pollen output (or survivalto maturity).An even greatereffect is required in a partiallyself-fertilizing population, where the pollen whicha plantcontributes to the pollen pool is worthless than its ovules in termsof genes transmittedto the next generation,since a largefraction of the ovules is self-fertilized.

Gynodioecy Gynodioecy,although not common,is foundin a largenumber of species (Darwin 1877).As we have discussedin SectionII, the conditionsfor the establishmentof a male-sterilemutant in a partiallyself-fertilizing population are less severethan those for a female-sterilegene if there is inbreedingdepression. The most favorable conditionsare when thereis a high level of self-fertilizationand high inbreeding depressionand themale-sterile plants have an increasedovule productiondue to the compensationeffect of Darwin. The latterfactor is not essential,but verysevere inbreedingdepression effects are requiredin its absence (see Section II fordetails). There is a considerable amount of data which suggeststhat female plants in gynodioeciouspopulations often have higherseed productionthan hermaphrodites. For example,Darwin (1877, p. 302) found that the seeds froma femaleplant of Thymusserpyllum weighed about 1.79 timesas muchas thosefrom a hermaphrodite. McClusker(1962, quoted by Carlquist 1966) showedthat female plants of Leucopo- gon melaleucoideshad a fruitset of 1.44 timesthat of hermaphrodites.On theother hand,these differences may reflectsubsequent evolution toward dioecy (as discussed below) ratherthan a directphysiological consequence of male-sterility. Neither of these figuresis large enough to permitthe evolution of genically determinedgynodioecy if the ancestral hermaphrodite population is assumedto have been randommating. They are sufficientto promotethe evolution of gynodioecy in a partiallyselfing species, with some inbreedingdepression. The viewthat gynodioecy has evolveddue to a combinationof the two factors,compensation and avoidance of inbreeding,is furthersupported by the followingconsiderations. First, there are manyexamples of gynodioecyin self-compatiblespecies; forexample, it is partic- ularlycommon in the Labiatae, a predominantlyself-compatible group (Fryxell 1957). Second, it is difficultto understandwhy gynodioecyis more common than androdioecyunless self-fertilizationand inbreedingdepression play a part. In a random matingpopulation, androdioecy would seem more likelyto evolve than gynodioecy,rather than the other way round, because a decreasein ovule production mightmore plausibly be thoughtto give a large increasein pollen outputthan vice versa. A differentexplanation for the existenceof male- or female-sterilitygenes segregatingin natural populations is heterozygoteadvantage of these genes (Jain 1961; Ho and Ross 1974; Ross and Weir 1975, 1976). This would be much more effectivein randommating than in selfingpopulations, because therewould have to be strongheterozygous advantage in orderto maintainsuch polymorphismin the faceof a highrate of selfing(Kimura and Ohta 1971). It is thereforenot surprising thatRoss and Weir (1976,p. 437),studying the evolution of dioecy with heterozygote

This content downloaded from 130.238.155.129 on Wed, 8 May 2013 08:24:38 AM All use subject to JSTOR Terms and Conditions EVOLUTION OF DIOECY 989 advantage of the male-sterilityfactor, conclude that "dioecy does not evolve in the presentmodel with moderateselection differentials." Furthermore, if heterozygote advantageis an importantfactor, there is again no obviousexplanation for the great rarityof androdioecycompared withgynodioecy; in random matingpopulations, where heterozygoteadvantage would be most effective,both systemswould seem equally likelyto evolve. Finally,it is worthemphasizing that our resultsagree withthose of Lewis (1941), Valdeyronet al. (1973), and Lloyd (1975, 1976) in suggestingthat rather stringent conditionsmust be metif gynodioecy is to evolve,so thatit is not surprisingthat it is uncommon. It seems likelythat gynodioecy is more oftendue to recessivemale-sterility genes thanto dominantones, though few cases ofsingle-gene inheritance in gynodioecyare known.Baker (1966, and personalcommunication; Jain et al. 1978) founda case of dominantinheritance of male-sterilityin gynodioeciouspopulations of Limnanthes douglasii.Arroyo and Raven (1975) reportboth recessiveand dominantinheritance in gynodioeciousFuchsias. Lloyd (1974a) quotes otherexamples of both modes of inheritancewhich suggest a preponderanceof recessive cases. It is not surprisingthat so fewcases of single-geneinheritance are knownin gynodioeciousspecies since they may readily evolve sub-dioecy,whereas this is not true when the gynodioecyis cytoplasmicallydetermined. Perhaps subdioecious species would provide better materialfor the studyof the geneticsof male-sterilityin naturalpopulations. Lloyd (1974a, p. 32) suggeststhat recessive male-sterility is morelikely to be foundbecause it confers"the ability... to regeneratesexual dimorphismwhen females are lost."A simplerexplanation is thatmale-sterility represents a loss offunction, and mostgenes of this sort are recessive (Wright 1934). If the male-sterilityin gynodioecious populationsis mostoften recessive, this provides support for the idea thatinbreeding is an importantfactor, since recessivemutations have littlechance of being establishedby selectionin a large randommating population.

Dioecy In SectionsIII and IV we have describedmodels forthe evolution of gynodioecy into fulldioecy or subdioecy.The simplestpath is whengynodioecy is followedby a mutationcausing complete female sterility. If male-sterility is recessive, this mutation mustbe dominantfor full dioecy to evolve,and thiswill result in male heterogamety. Conversely,if male-sterilityis dominant,dioecy will not evolve unless the second mutationis recessive,and once dioecyhas evolvedthere will be femaleheterogamety. Since, as discussed above, recessivemale-sterility is probablymore common than dominant,it is not surprisingthat male heterogametyis considerablycommoner than femaleheterogamety. The latterhas been found in threegroups: Fragaria (Westergaard1958), Potentillafruticosa (Grewal and Ellis 1972),and Cotula (Lloyd 1974a). Twenty-threefirmly established cases wherethe male is theheterozygous sex are listedby Westergaard(1958). Full dioecywill resulton thismodel whenthe two genesare tightlylinked (table 3). Withmale heterogamety,the "Y' chromosomewill have a dominant female-suppressor,and the "X" will carry a recessive male- suppressor.This typeof geneticcontrol of sex determinationis essentiallythat found

This content downloaded from 130.238.155.129 on Wed, 8 May 2013 08:24:38 AM All use subject to JSTOR Terms and Conditions 990 THE AMERICAN NATURALIST in Silenedioica (Melandrium),reviewed by Westergaard(1958), althoughsome other genes affectingsex are also present.Westergaard also discussesgenetical work with severalother species and shows thatthey conform to thesame basic model-in some cases (Ecballium)exactly as above, while in others(Carica papaya) thereare other genes whichaffect sex expression.Data on Vitisalso fitthis scheme (Lewis 1942). Westergaardregards the Ecballiumsystem as mostprimitive, which seems plausible sincethere are dioecious and monoeciousraces in thisgroup, suggesting that dioecy is of recent origin. If the two genes in our models are not closely linked,the equilibriumpopulations contain males, femalesand hermaphrodites,as shown in table 3. This sortof situationis knownto occurin a numberof species (Fryxell 1957; Westergaard1958) and is possiblewith either male or femaleheterogamety. It would be interestingto know moreabout thegenetics of sex in such cases, in particularthe linkage relations of the genes involved. The model of the evolution of dioecy proposed by Ross and Weir (1976), which involvesheterozygote advantage of the male-sterilityfactor, leads to equilibria with males, females,and hermaphrodites, even withcomplete linkage; this is not possible withour model. We have shown in Sections III and IV that modifierscausing relativelyslight reductionof female fertility can be establishedin gynodioeciouspopulations and that once thishas happened the selectionpressure in favorof increasedmaleness of the hermaphroditesis intensified.Full dioecy, of the type discussed above, can only evolve given genes with the rightdominance and linkage relations.Our model thereforepredicts that intermediatestages between gynodioecyand full dioecy should be fairlycommon. Populations segregating for one or moremodifiers of this sortwill show greatvariability in the femalefertility of thehermaphrodites, with the more extrememodified hermaphrodites behaving as functionalmales. The average ovule productionof hermaphroditesin such populationswill be much less thanthat of females. Darwin (1877, p. 292) was struck by this type of situation in the spindle-treeEuonymus europaeus, which he described"as showinghow gradually an hermaphroditeplant may be convertedinto a dioecious one." Later work has shown that thereis a virtualcontinuum of stagesbetween gynodioecy and dioecy, e.g.,in severalspecies of the Hawaiian flora(Carlquist 1966) and in Fuchsia (Arroyo and Raven 1975).In Pimelea,Burrows (1960) has shownthat species vary from those witha slightdifference in ovule fertilitybetween females and hermaphroditesat one extremethrough ones witha greaterdifference, to virtuallycomplete dioecy at the other.The frequenciesof femaleplants vary concomitantly, from around 15% up to 5000. In all thesecases of subdioecy,the femalesare relativelyconstant in theirsex expression,and it is the polleniferousplants (hermaphroditesand males) whichare variable. Westergaard(1958) commentson the fact that,even in cases of female heterogamety,the males are morevariable than the females.This is quite consistent withour model. Sex chromosomes.-Inthose cases witha male-determiningY whichhave been investigatedgenetically, the sex genesare all on thesame chromosomeand behaveas ifcompletely linked (Westergaard 1958). Previouspapers on theevolution of dioecy have concentratedon derivingconditions under which only males and femaleswill be present.Complete linkageis of course such a condition(Lewis 1942; Ross and Weir 1976), and thishas been regardedas an explanationof thegenetic findings. It

This content downloaded from 130.238.155.129 on Wed, 8 May 2013 08:24:38 AM All use subject to JSTOR Terms and Conditions EVOLUTION OF DIOECY 991

does not,however, constitute an explanationof the evolution of these genetic systems since it specifiesneither the path by whichthe present state has beenreached nor the causal factorsinvolved. Our model providesa possible evolutionaryexplanation, based on considerationof the conditions(including strength of linkage) for the spread of a gene reducingfemale fertility,when introducedinto a population segregatingfor a recessive male-sterilitygene. If sufficientlytightly linked to themale-sterility locus, such a gene mayincrease in frequencyeven when it wouldbe eliminatedif unlinked. Such a "linkageconstraint" often exists even whenthe effect on the femalesis much less (in termsof percentageloss of ovule production)than thaton the hermaphrodites.This effecttherefore provides one majorreason for the existence of sex chromosomes.Bodmer and Parsons (1962) firstanalyzed the conditionsfor the operation of this typeof effect,in randommating populations subjected to epistatic selection. We have proposed it as an explanation for the supergenesof Batesian mimicry(Charlesworth and Charlesworth1975). It may well be generallyimportant in the evolution of polymorphismsinvolving linkedgenes. Once a linkedmodifier of femalefertility has been established,there is selection pressurein favorof the reductionof crossing-overbetween it and themale-sterility locus. In Silenedioica, crossing-over between the X and Y chromosomesis confined to the regionoutside the sex genes,presumably as a resultof thisprocess (Wester- gaard 1958). In the case of femaleheterogamety, where the originalmale-sterility gene must be dominant,no linkage constraintcan operate,and so selectionfor reducedrecombination is the only factoroperating. One mighttherefore expect full dioecy to be difficultto establishin thiscase. If the operationof a linkageconstraint is the selectivebasis of the linkageof the sex-determininggenes, we would expectdioecious speciesto have thesame chromosome numbersas hermaphroditeor gynodioeciousrelatives, whereas if the genes have to be broughttogether from separate chromosomes,one mightwell expect centricfusions to be involved,with a consequentlowering of chromosomenumber. Althoughthe possibilityexists that chromosome changes may have occurredsince theestablishment of dioecy,it is perhapssignificant that dioecious species often have thesame chromosomenumbers as theirrelatives. For example,the dioecious species of Silene (dioica, alba, and otites)have 2n = 24, and so do 10 otherBritish species whose chromosomenumbers are given by Clapham, Tutin,and Warburg(1962). Similarlythe dioeciousRibes alpinum has 2n = 16,the same numberas thefour other Britishspecies of Ribes. The only othercomparisons of thissort that can be made, using the Britishflora, are inconclusive,because the chromosomenumbers of the groups are very variable. In Rumex,the hermaphroditespecies all have haploid numbersthat are multiplesof 10, and so does R. scutatus,which is describedas "polygamous."The dioecious R. acetosa and R. acetosella,however, have 2n = 14 or multiplesof 14, withan extra Y chromosomein R. acetosa males. This mightbe a case of reductionof chromosomenumber during the evolution of dioecy. According to Smith(1967), a similardistinction between dioecious and bisexual speciesholds good throughoutthe Acetosa subgenus.In R. hastatulus,which is also dioecious,the chromosomenumber has been reducedeven further(Smith 1969). In Texas 2n = 10, withan X Y sex chromosomepair; in NorthCarolina, n = 4 in females,and in males

This content downloaded from 130.238.155.129 on Wed, 8 May 2013 08:24:38 AM All use subject to JSTOR Terms and Conditions 992 THE AMERICAN NATURALIST thereare threepairs of autosomes and XY1 Y2. This seems to have resultedfrom a centric fusion between the major arm of the Texas third chromosomeand the acrocentricX, followedby a translocationbetween the two Y chromosomes.The intermediatestage is foundin materialfrom South Illinois-Missouri.Whether these translocationshave any importancein relationto the sex-determininggenes is, of course,not known,but it may be significantthat in thisspecies the Y chromosome carriesa male-promotingfactor (Smith 1963), whereas the other Acetosa species studied have an X/autosomebalance system.It is possible that the X/autosome mechanismhas arisensecondarily from the male-determining Y system considered in thispaper. Our model obviouslycannot explain its origindirectly. Barlow and Wiens (1976) reporta similarbut more extremecase in an African mistletoe,Viscum fischeri, in whichthe males are heterogameticfor a set of multiple sex chromosomes(5 Y/4X) thatproduce a multivalentchain of nine chromosomes at meiosis in addition to seven autosomal bivalents.Translocation heterozygosity is common in the dioecious Africanmistletoes but is unknownin the monoecious speciesand in the otherdioecious speciesof Viscumthat have been studied,in which no sex chromosomescould be identified.This suggeststhat translocations might be involvedin theevolution of dioecy, though there is no reason to attributethe whole developmentof thiscomplex translocation system to thisfactor alone. The roles of compensationand inbreedingdepression.-The models developed in this paper seem to agree in many respectswith the resultsof observationson dioecious species.However, we have had to make a numberof assumptions which do not have directobservational support. In the firstplace, we have assumed that a mutationabolishing reproduction either as a male or as a femalewill to some extent increasethe outputof the othertype of gamete,as originallysuggested by Darwin. Althoughgynodioecy can evolve withoutany such compensationprovided that selfingand inbreedingdepression are of sufficientmagnitude, a female-sterilitygene will always be eliminatedunless pollen productionis sufficientlyraised, even in a gynodioeciouspopulation (Section III). In a gynodioeciouspopulation, the presence of male-sterileindividuals means that investmentin pollen by male-fertilesgives a highergenetical return than in a completelyhermaphrodite population under similarconditions. This is because thegenetic contributions of females and hermaphroditesare equal in an equilibriumgynodioecious population (SectionII), so that hermaphroditesmust pass on more of theirgenes through pollen than they would in the absence of females.Provided that the femaleshave a sufficientlygreater ovule outputthan the hermaphrodites, a less-than-proportional increase in pollenproduction may thereforebe enoughto compensatefor a loss in ovule output(Section III). In view of the necessityof assuming that compensationwill occur, it would be interestingto knowwhat effects mutants causing male or femalesterility have on the fertilitythat remains. Ross (1977) has reviewedthe available data, but unfortunately this is mostly only qualitative,and quantitativedata are scarce. In a study of cytoplasmicmale-sterility in cotton,Rosales and Davis (1976) foundan increased seed productionby females,provided adequate pollinatorvisits were ensured. A second assumptionwhich we have made is that inbreedingdepression can be severe in a partiallyselfing species. Most experimentalestimates of inbreeding depressionhave been made using outbredspecies. It would be interestingto have

This content downloaded from 130.238.155.129 on Wed, 8 May 2013 08:24:38 AM All use subject to JSTOR Terms and Conditions EVOLUTION OF DIOECY 993 data forinbreeders since, as discussedearlier, gynodioecy seems most likely to have evolved fromsuch species. The only data known to us that bear on this point, althoughindirectly, are those of Marshall and Allard (1970). These authorsfound excessof heterozygotes at allozymeloci inA vena barbata, a largelyselfing species. This is most simplyinterpreted as a consequence of homozygotedisadvantage at loci linkedto the allozymeloci themselvesand thusprovides evidence that the products of sellingare at a disadvantagecompared with outbred individuals. Lloyd (1972b) showed that severalmonoecious species of Cotula (a genus in whichthere are also severaldioecious species) are self-fertile.He also foundthat the percentage of seeds germinatingwas about 3000 lower for selfed than for outcrossedprogeny. The natural rate of selling is, however,not known for these plants, nor for most hermaphroditeor monoecious relativesof dioecious species. Comparativedata of this sort would be valuable. At presentall thatcan be said is thatmany dioecious species have a low frequencyof hermaphrodites,and thesecan readilybe selfedin geneticalstudies (Westergaard 1958). Furthermore,self-fertile hermaphrodite strains have been obtainedby selectionin a numberof dioecious crop plants-e.g., strawber- ries (Darwin 1877),the grapevine, cannabis, and spinach(Westergaard 1958)-and in gardenplants thatare grownfor their berries (Holly, Celastrus). This showsthat theycannot have a strongself-incompatibility system. Finally,we should point out thatwe do not suggestthat the model describedin thispaper is the onlyway in whichdioecy can evolve.Dioecy has probablyevolved frommonoecy on several occasions (Lewis 1942), for example, in Mercurialis, Ecballium,and others (Westergaard 1958), Cotula (Lloyd 1972a), and Viscum (Barlow and Wiens 1976). Lloyd (personalcommunication) has pointedout thatno case of gynodioecyis known in whichthe polleniferousplants have separatemale and femaleflowers and thatthis suggests that dioecy'has evolved from monoecy via some otherintermediate stage than gynodioecy. Now thereis one obviousdifference betweenhermaphrodite and monoecious plants with respectto the fate of genes causing a slight reductionin male fertility.In a hermaphroditeflower, a small reductionin pollen productionwould be unlikelyto reducegreatly the proportion of selfedovules and so protectagainst inbreeding depression; moreover, such a mutant type would have feweroutcrossed progeny. These two factorscan make a small reductionin pollen output selectivelydisadvantageous, even in conditionswhich enable a gene causingcomplete male-sterility to be selected.In a monoeciousspecies, however,it seems much more likelythat a reductionin pollen outputby a plant might reduce the chance of self-fertilizationof its femaleflowers sufficiently to outweighthe disadvantageof a reductionin outcrossingpotential. It can be shown by an argumentsimilar to thatin SectionIII thatonce a genereducing pollen fertility has been establishedin a population,further steps in the same directionare more difficultsince the increasedfemale character of the populationmakes pollen more geneticallyvaluable than before(Charlesworth and Charlesworth1978). The new typeof population would, however, be morelikely to acquirefactors reducing female fertility.Thus a monoeciousspecies mightincorporate in alternationgenes slightly affectingmale and femalefertility, finally becoming dioecious. This accords with Lloyd's (1972a) findingsin Cotula, where monoecy seems to have evolved into subdioecyand dioecy. There are also a numberof cases in whichdioecy seems to

This content downloaded from 130.238.155.129 on Wed, 8 May 2013 08:24:38 AM All use subject to JSTOR Terms and Conditions 994 THE AMERICAN NATURALIST

have evolved fromheterostyly. Baker (1959) describesa case in Mussaenda, and Ornduff(1966) gives a possible instancein Nymphoides.Opler et al. (1975) have studied anothercase in Cordia. The selectivecauses of this type of change are scarcelyunderstood at all. Darwin (1877) also discusses several cases where this may have happened, but no furtherinformation seems to have been obtained about thesespecies.

SUMMARY A model forthe evolutionof gynodioecyfrom the hermaphrodite or monoecious conditionis described,taking into accountthe effects of partialselfing and inbreedingdepression. It is shownthat a mutantcausing male-sterility can be selectedeven if the femaleplants have the same ovule outputas the hermaphrodites,but that the conditionsfor this are very stringent:The product of the selfingrate and the inbreedingdepression must exceed one-half.If the femaleshave an increasedovule output,gynodioecy can evolve with lower values of the selfingand inbreeding depressionparameters. Expressions for the equilibriumfrequency of femalesand of the male-sterilitygene in both the dominantand the recessivecase are given. By a similartechnique, conditions for the evolution of androdioecy are derived.In a selfingpopulation, these conditionsare much less easily satisfiedthan those for gynodioecy,though in a randomlymating population the conditionsare similar:If ovule productionis abolished,pollen production must be morethan doubled, or vice versa. Since androdioecyis knownto be a veryrare condition,it seems likelythat avoidance of selfinghas played a role in the evolutionof gynodioecy. Using the equilibriaderived for gynodioecy, the conditionsfor the evolutionof subdioecyor dioecy,by means of a partial or total female-sterilitymutation, are studied.In contrastto thesituation in a hermaphroditepopulation, a female-sterility gene can be selectedin a gynodioeciouspopulation if it confersa moderateincrease in pollen output; some increase in pollen output is essential.The fateof such a female-sterilitygene also dependson its linkagewith the male-sterility gene. If thisis recessive,and the female-sterilitygene is dominantand has an effectin thefemales as well as the hermaphroditeindividuals, then the second mutationwill usually be eliminatedunless it occurs at a locus tightlylinked to the firstgene. In othercases thereis no such "linkageconstraint," though in all situationsthere may be selection fortighter linkage between the loci; this will result in an initiallysubdioecious populationbecoming more fullydioecious. These resultsagree with some of the factsknown about theevolution of dioecy in plants. First, since gynodioecyis more often controlledby a recessive than a dominantgene, male heterogametyshould be commonerthan female, as is observed. Second,subdioecy should be common,since fulldioecy requires not onlythe correct phenotypiceffects of thetwo genes but also complementarydominance relations and tightlinkage; subdioecy is indeedknown in manyspecies. The equilibriareached by our model have only one type of female in appreciable frequency,whereas the polleniferousindividuals may fallinto severalgenotypic classes; it is oftenobserved thatin subdioeciousspecies the males are more variablethan the females, regardless of which is the heterogameticsex. Finally,the equilibriagenerated by our model agree closely with the resultsof geneticalstudies of those dioecious species with

This content downloaded from 130.238.155.129 on Wed, 8 May 2013 08:24:38 AM All use subject to JSTOR Terms and Conditions EVOLUTION OF DIOECY 995 male-determiningY chromosomesthat have been investigated,in whichboth male- and female-sterilityfactors have been found,showing complementary dominance relationsand no crossing-overbetween the loci, so thatjust two gametetypes exist. Such a situationcan be explainedby the operationof the linkageconstraint, which ensures that only linkedmutations become establishedand does not requirethat unlinkedgenes have been broughttogether. This is consistentwith the fact that dioecious species often have the same chromosome numbers as their bisexual relatives.

ACKNOWLEDGMENTS We wish to thank H. G. Baker, J. J. Bull, J. L. Harper, D. G. Lloyd, and J. Maynard Smith for their helpfulcomments on this paper. We are particularly gratefulto Dr. Bull forpointing out an errorin theoriginal version of equation (A2).

APPENDIX

ANALYSISOF THECONDITIONS FOR THE SPREADOF A DOMINANTFEMALE-STERILITY MUTATION IN A GYNODIOECIOUSPOPULATION Let B1 be the originalallele at the female-sterilitylocus and B2 be a dominantallele which rendersits carriersfemale-sterile. The equilibriumfrequencies at the male-sterilitylocus are X(A1Al), f(A1A2), and Z(A2A2), where f and 2 are given by equations (7) and (5), respectively,and X = 1 - Y - Z. B2 is assumedto be introducedinto this population at a low frequency.Since B2- A2/A2 is totally sterile,the only genotypes which we need to consider are B2A1/B1Al, B2A1/B1A2, and B2A2/B1Al. Let the frequenciesof these genotypesamong the adults of a given generationbe El, E2, and ?3, respectively.Ignoring second-orderterms in the ?'s, we obtain the followingrecurrence relations:

E' = a[El + (1 - R)92 + Re3]

2 = b[E1+ (1 - R)92 + Re3] (A1)

3 = a[RE2 + (1 -R)93] wherea =(X +! )(1 - s)(1 + K)/(1 -2)(1 + k), b = [2if(l - s) + 2(1 + k)](1 + K)/(1 - 2)(1 + k). The characteristicequation forthis system is:

i{i%2-_A[2a + b-R(a + b)] + a[(a + b)-R(a + 2b)]}= . (A2) Equation (A2) has one root of zero and the othertwo givenby thezeros ofthe quadratic in braces. Since the matrix for the linear transformation(Al) is nonnegative,the largest eigenvalueis real and positiveand musttherefore be the largerzero of the quadratic.By the rule forthe differentiationof an implicitfunction, the derivative of the largesteigenvalue, AO, withrespect to R is: dAo a(a + 2b)- %o(a + b) (A3) dR 2No-[2a + b-R(a + b)] At R = 0, equation (A2) shows that AO= a + b and (A3) that dAO/dR =-b. Together with the factthat (A3) impliesthat d2I0 /dR2has thesame signas dAo/dR at each point,this proves thatAO is a concave,decreasing function of R.

This content downloaded from 130.238.155.129 on Wed, 8 May 2013 08:24:38 AM All use subject to JSTOR Terms and Conditions 996 THE AMERICAN NATURALIST

LITERATURE CITED

Arroyo,M. T. K., and P. H. Raven. 1975. The evolutionof subdioecyin morphologicallygynodioecious speciesof Fuchsia sect.Encliandra (Onagraceae). Evolution29: 500-511. Baker,H. G. 1959.Reproductive methods as factorsin speciationin floweringplants. Cold SpringHarbor Symp.Quant. Biol. 24:177-190. 1966. The evolutionof floralheteromorphism and gynodioecismin Silene maritima.Heredity 21:689-692. Barlow,B. A., and D. Wiens. 1976.Translocation heteroygosity and sex ratio in Viscumfischeri.Heredity 37:27-40. Bodmer,W. F., and P. A. Parsons. 1962.Linkage and recombinationin evolution.Adv. Genet. 11:1-100. Burrows,C. J. 1960. Studiesin Pimelea.I. The breedingsystem. Trans. Roy. Soc. New Zeal. 88:29-45. Carlquist,S. 1966.The biota oflong-distance dispersal. IV. Geneticsystems in thefloras of oceanic islands. Evolution20:433-455. Charlesworth,D., and B. Charlesworth.1975. Theoreticalgenetics of Batesian mimicry.II. Evolutionof supergenes.J. Theoret.Biol. 55: 305-324. Charlesworth,D., and B. Charlesworth.1978. Population genetics of partialmale-sterility and the evolution of monoecyand dioecy. Heredity(in press). Charnov,E. L., J. Maynard Smith,and J. J. Bull. 1976. Why be a hermaphrodite?Nature 263:125-126. Clapham, A. R., T. G. Tutin,and E. F. Warburg.1962. Flora of the BritishIsles. 2d ed. Cambridge UniversityPress, Cambridge. 1,269 pp. Darwin,C. R. 1877.The differentforms of flowerson plantsof the same species.Murray, London. 352 pp. Fryxell,P. A. 1957. Mode of reproductionof higherplants. Bot. Rev. 23:135-233. Grewal,M. S., and J. R. Ellis. 1972. Sex determinationin Potentillafruticosa.Heredity 29:359-362. Haldane,J. B. S. 1927.A mathematicaltheory of natural and artificialselection. V. Selectionand mutation. Proc. CambridgePhil. Soc. 28:838-844. Ho, T-Y., and M. D. Ross. 1974. Maintenanceof males and femalesin hermaphroditepopulations. Heredity32:113-118. Jain,S. K. 1959. Male-sterilityin floweringplants. Bibl. Genet. 18:101-166. 1961. On the possible adaptive significanceof male sterilityin predominantlyinbreeding populations.Genetics 46:1237-1240. 1968.Gynodioecy in Origanumvulgare: computer simulation of a model. Nature 217: 764-765. Jain,S. K., I. A. Boussy,and H. Hauptli. 1978. Male-sterilityin meadowfoam.J. Hered. 69:61-63. Kimura,M., and T. Ohta. 1971. Theoreticalaspects of populationgenetics. Princeton University Press, Princeton,N.J. 219 pp. Lewis,D. 1941. Male sterilityin naturalpopulations of hermaphroditeplants. New Phytol.40: 56-63. 1942.The evolutionof sex in floweringplants. Biol. Rev. 17:46-67. Lloyd, D. G. 1972a. Breedingsystems in Cotula L. (Compositae,Anthemidae). I. The array of mono- clinousand diclinoussystems. New Phytol.71:1181-1194. 1972b.Breeding systems in Cotula L. (Compositae,Anthemidae). II. Monoecious populations. New Phytol.71:1195-1202. 1974a. Theoreticalsex ratiosof dioecious and gynodioeciousangiosperms. Heredity 32:11-34. 1974b.Female-predominant sex ratios in angiosperms.Heredity 32: 35-44. 1974c.The geneticcontributions of individualmales and femalesin dioecious and gynodioecious angiosperms.Heredity 32:45-51. 1975. The maintenanceof gynodioecyand androdioecyin angiosperms.Genetica 45:325-339. 1976. The transmissionof genes via pollen and ovules in gynodioeciousangiosperms. Theoret. Pop. Biol. 9:299-316. Marshall,D. R., and R. W. Allard.1970. Maintenance of isozyme polymorphism in naturalpopulations of Avenabarbata. Genetics 66: 393-399. Opler, P. A., H. G. Baker,and G. W. Frankie. 1975. Reproductivebiology of some Costa Rica Cordia species (Boraginacae).Biotropica 7:234-247. Ornduff,R. 1966. The originof dioecism fromheterostyly in Nymphoides(Menyanthaceae). Evolution 20:309-314. Rosales, F. E., and D. D. Davis. 1976. Performanceof cytoplasmicmale-sterile cotton undernatural crossingin New Mexico. Crop Science 16:99-102. Ross, M. D. 1970. Evolutionof dioecy fromgynodioecy. Evolution 24:827-828.

This content downloaded from 130.238.155.129 on Wed, 8 May 2013 08:24:38 AM All use subject to JSTOR Terms and Conditions EVOLUTION OF DIOECY 997

- 1977. Behaviour of a sex differentialfertility gene in hermaphroditepopulations. Heredity 38:279-290. Ross, M. D., and B. S. Weir. 1975. Maintenanceof male sterilityin plantpopulations. III. Mixed selling and randommating. Heredity 35:21-29. 1976. Maintenance of males and femalesin hermaphroditepopulations and the evolutionof dioecy.Evolution 30:425-441. Smith,B. W. 1963. The mechanismof sex-determinismin Rumexhastatulus. Genetics 48:1265-1288. . 1967. Chromosomenumber and form,sexuality and classificationin Rumexsubgenus Acetosa. (Abstr.)Amer. J. Bot. 54:629. 1969. Evolution of sex-determiningmechanisms in Rumex.Pages 172-182 in C. D. Darlington, and K. R. Lewis, eds. Chromosomestoday. Vol. 2. Oliver & Boyd, Edinburgh. Valdeyron,G., B. Dommee, and A. Valdeyron.1973. Gynodioecy:another computer simulation model. Amer.Natur. 107: 454-459. Westergaard,M. 1958. The mechanismof sex determinationin dioecious floweringplants. Adv. Genet. 9:217-281. Wright,S. 1934. Physiologicaland evolutionarytheories of dominance.Amer. Natur. 68:25-53.

This content downloaded from 130.238.155.129 on Wed, 8 May 2013 08:24:38 AM All use subject to JSTOR Terms and Conditions