On the Origin of Meiotic Reproduction: a Genetic Modifier Model

On the Origin of Meiotic Reproduction: A Genetic Modifier Model

Marcy K. Uyenoyama* and Bengt 0. Bengtssont *Department of Zoology, Duke University, Durham, North Carolina 27706, and tDepartment of Genetics, Lund University, S-223 62 Lund, Sweden Manuscript received May 3 1, 1989 Accepted for publication September 2, 1989

ABSTRACT We study the conditions under which a rare allele that modifies the relative rates of meiotic reproduction and apomixis increases in a population in which meiotic reproduction entails selfing as well as random outcrossing. A distinct locus, at which mutation maintains alleles that are lethal in homozygous form, determines viability. We find that low viability of carriers of the lethal alleles, high rates of selfing, dominance of the introduced modifier allele, and lower rates of recombination promote the evolution of meiosis. Meiotic reproduction can evolve even in the absence of linkage between the modifier and the viability locus. The adaptive value of meiotic reproduction depends on the relative viabilities of offspring derived by meiosis and by apomixis, and on associations between the modifier and the viability locus. Meiotic reproduction, particularly under selfing, generates more diverse offspring, including those with very high and very low viability. Elimination of offspring with low viability generates positive associations between enhancers of meiotic reproduction and high viability. In addition, partial selfing generates positive associations in heterozygosity(identity disequilibrium) between the modifier and the viability locus, even in the absence of linkage. The two kinds of associationstogether can compensate for initial reductions in mean offspring viability under meiotic reproduction.

EX comprises a syndromeof diverse processes, not anism to repair DNA [see BERNSTEINet al. (1984, S all of which have necessarily evolved in response 1985) and references therein],combine favorablemu- tothe same selection pressures. Diploidy, meiosis, tations (FISHER 1958,Chapter VI; MULLER 1932; recombination, and outcrossing all promote genetic CROWand KIMURA1965; MAYNARD SMITH 1968; diversity among offspring, but thiscommon effect ESHELand FELDMAN 1970), preventthe accumulation does not by itself imply a common evolutionary func- of deleterious mutations(MULLER 1964; FELSENSTEIN tion. Evolutionaryprocesses maintaining anyone com- 1974), reduce genetic load(CROW 1988), or improve ponent in highly refined sexual systems operate in a average offspringviability by increasing genetic diver- fundamentally different context from those that fa- sity (WILLIAMS1975, Chapter 1). The repair hypoth- vored its origin in primitive sexual systems. MAYNARD esis, which holds that sexuality evolved to reduce the SMITH(1978a, Chapter 1) has drawn a strong distinc- rate atwhich new mutations appear among offspring, tion between theories for the origin and the mainte- lies beyond our study. Neither do the hypothesis of nance of sex (see also CROW 1988).Our deterministic FISHERand MULLERapply to ourmodel, because they two-locus study supports the view that the evolution- require linkage. disequilibrium among multiple loci ary function of meiosis in response to recurrentdele- which affect viability (FELSENSTEIN1974; 1985). We teriousmutation is topromote the production of use our analysis to test and refine the two remaining offspring of very high quality, even at theexpense of heuristic interpretations of the adaptive significance reducing average offspringviability. Because the con- of sexual reproduction. In particular,we examine the ditions most conducive to the origin of meiosis (low generation and consequencesof disequilibrium be- rates of recombination and high rates of selfing) are generally associated with asexuality, our findings sug- tween a locus controlling viability and a modifier of gest that the original adaptive significance of compo- the rate of meiotic reproduction. nents of the sexual syndrome may reverse during the Distinction between isogamy and reduction in out- course of refinement. crossing: MAYNARDSMITH (1 978a, Chapter4; 1978b) Significance of deleterious mutations to the evo- argued that isogamy eliminates the twofold cost of lution of sex: Viewed as aresponse to mutation, meiosis. This conclusion is based on the observation sexual reproduction has been characterizedas a mech- that sexual and asexual species transmitthe same number of genes, provided that individuals can gen- The publication costs of this article were partly defrayed by the payment of page charges. This articlemust therefore be hereby marked“advedsemenf” erate one asexual zygote only at the expense of two in accordance with 18 U.S.C. $1734 solely to indicate this fact. sexual gametes [see Figure 6.2 in MAYNARDSMITH

Genetics 123: 873-885 (December, 1989) 874 M. K. Uyenoyama and B. 0. Bengtsson (1978b)l.HOLSINGER, FELDMAN, and CHRISTIANSEN mostclosely to his under negligible investment in (1 984) used the term “pollen discounting” to describe sperm relative to eggs (4 approaching unity). In agree- the reduction in cross-fertilization as a consequence ment with our findings, meiosis doesnot incur a of increased selfing. Although the two processes have twofold cost in this case (HARPER 1982). often been treated as equivalent (CHARLESWORTH Origin of meiosis: Cohen and ZOHARI (1986) de- 1980; UYENOYAMA 1984),isogamy in fact represents rived the evolutionarily stable mixture of clonal and but one energetic constraint which ensures reduced sexual reproduction that maximizes the number of genetic contributions by less sexual forms. clones at equilibrium. Primarily asexual clones allocate In reflection of this identification of isogamy witha resources either to maintaining their own genotypes reduction in the ability to fertilize outcrossed eggs, or to generating novel genotypes through sexual re- most theoretical studies of the evolution of meiotic production. A balance between the generation of new reproduction from apomixis have assumed that either clones and theextinction of existing clones determines (1) two apomictic eggs are replaced by one meiotic the equilibrium number. Higher ratesof sexuality are egg and one sperm, or (2) two apomictic eggs are favored in unsaturated habitats which harbor many replaced by two meiotic eggs with no change in out- unoccupied niches. The force favoring sexuality in crossing success. KONDRASHOV’S(1 985) generalization COHENand ZOHARI’S (1986) model can be regarded of this approach assumes that the relative investment as a formof environmental unpredictability: offspring in apomictic seeds, amphimictic seeds, and pollen are that are unlike the parent survive if they happen to (1 + y)z/2, (1 - z)/2, and (1 - yz)/2, respectively. In be able to exploit a different niche. Other theories the studies of CHARLESWORTH(1980) and MARSHALL incorporating low or negative correlations between and BROWN (1981), the parameter y is set to zero, the selective regimes imposed on parental and off- signifying that apomixis has no effect on pollen pro- spring generations have been discussed by MAYNARD duction. This assumption may hold under well-devel- SMITH(1978a, Chapter 6). oped facultative apomixis (see MAYNARDSMITH We study evolutionary changes of amodifier of 1978a, Chapter 4; MARSHALLand BROWN1981). meiotic reproduction in response to deleterious mu- Because our primary interest lies in the origin of tations at a distinct locus which determines viability. meiosis rather thanits maintenance, we envision prim- Sexual reproduction involves both selfing and random itive organisms thatreproduce either asexually or outcrossing. Because meiosis does not suffer twofolda sexually by mating with another cell capable of con- disadvantage in our formulation, relatively small dif- jugation; in either case, the structural components of ferences in offspring viability can promote its origin. each cell are invested in one daughter cell. In our Further, even if segregation reduces the average via- model, a fractiona of a given genotype reproducesby bility of meiotically-derived offspring, associations be- apomixis, and the remaining fraction (1 - u) under- tween the modifier of reproduction and the viability goes meiosis. We assume that genotypes that repro- locus may permit the evolution of meiotic reproduc- duce by apomixis at a higher rate contribute less to tion. Selfing generates such associations through two effects. First, selfing generates a higher variance in the gamete pool from which offspring are derived by viability among offspring than apomixis. Elimination outcrossing. Because we assume both pollen discount- of the lessviable offspringinduces an association ing (positive y in KONDRASHOV’S(1 985) terminology) between high viability and enhancers of meiotic re- and a constant total number of meiotic and ameiotic production. Second, partial selfing generates identity eggs, our model departs from conventional formula- disequilibrium, even in the absence of linkage (BEN- tions (CHARLESWORTH 1980; MARSHALLAND BROWN NETT and BINET1956; WEIRand COCKERHAM 1973). 1981 ; KONDRASHOV1985), which we regard as more Identity disequilibrium implies a correlation in het- appropriate to the question of the maintenance of erozygosity between loci (HALDANE1949; COCKER- meiotic reproduction than to its origin. HAM and WEIR 1968), with individuals that are het- Unlike these studies, we find that meiosis does not erozygous at the modifier locus likely to be heterozy- incur a twofold cost relative to apomixis. MAYNARD gous at the viability locus as well. In our study, the SMITH(1 978a, Chapter 4) concluded that the twofold implications of identity disequilibrium for the origin cost of meiosis does not apply to the origin of meiosis of meiosis depend upon the level of dominance ex- because meiosis probably arose in isogamous orga- pressed by the new modifier allele. nisms. In our model, the absence of the cost of meiosis reflects not isogamy, but rather that conjugants nec- BALANCEBETWEEN MUTATION AND essarily mate onlywith other conjugants. HARPER VIABILITYSELECTION (1982) assumed that individuals which invest a fraction a of the resources available for reproduction in a Changes in genotypic frequencies sexual offspring invest (1 - a)4 and (1 - 4) in sexual Reproduction: Locus M determines the fractionof eggs and sperm, respectively. Our model corresponds offspring derived by apomixis. In the resident POPU- Origin of Meiosis 875 lation (prior to the introductionof the new allele that Mutation permits the existence at the time of repro- modifies the rateof meiotic reproduction), all individ- duction of all three genotypes AIAI, A1A2,and A2A2, uals carry genotype MIMl at the modifier locus and which occur in frequencies xll, x12, and x22, respec- reproduce by apomixis at the rate all (0 d all Cl). tively. Random outcrossing involves the fusion of two Apomixis replicates the parental genotype together gametes within the gamete pool, in which A1 and A2 with any mutations that have occurred in the parent occur with frequencies5 andf2, respectively. since its conception. Meiotic reproduction generates the remainingfraction (1 - all) of offspring. Of Equilibrium genotypic frequencies the offspringgenerated by meiosis, a proportion b Solution of (l), after removal of the primes, pro- (0 b 1) are derived by selfing and (1 - b) by ran- C C vides expressions forthe genotypic frequencies at dom outcrossing. equilibrium. Ignoring terms smaller than the muta- Viability selection:Two classes ofalleles that affect tion rate, we obtain viability segregate at locus A: A1 represents the normal, well-functioning form and A2 a heterogeneous class of mutations that depress viability. Mutation generates defective alleles at the rateP. Homozygotes for defective alleles (A2A2) are inviable, with hetero- 2(1 - a) zygotes (A1A2) surviving to reproduce at the rate u 2(1 - U) + ~b(1- all) (0 d u C 1) relative to normal homozygotes (AIAI). Recursions: While the viability of offspring derived Examination of (5b) indicates thatthe population by both meiotic and ameiotic reproduction depends mean fitness (T)increases with greater rates of selfing on the genotype at the time of conception, mutations (b)or lesser rates of apomixis (all):meiotic reproduc- that occur in surviving individuals before reproduction combined with selfing reduces mutational loadto tion may be transmitted to the next generation. Let the greatest extent. u1 denote thefrequency of individuals, censused after the phase of viability selection, that carried genotype AlAl at conception. At this stage,these individuals PREDICTIONS IN THE ABSENCE OF TWO- may bear genotypes A1A2 or A2A2 as a result of muta- LOCUS ASSOCIATIONS tion. Similarly, u12denotes the frequencyat reproductive age of individuals that carriedA1A2 at conception; In order toestablish a framework within which the the remaining class (A2A2) is inviable. Inthe next joint evolution of the viability locus and a modifier of generation, these frequenciesare determined from meiotic reproduction can be assessed, we first consider Tuil = U: (14 the influence of the modifier locus on its own trans- mission, ignoring associations between the loci. TuI)~= UU~ (1b) Changes in frequencies at the modifier locus: A rare allele (M2) introducedat the modifier locus in which and u f represent the frequenciesof AIAl u: changes therate of apomixis: rare heterozygotes and A1A2 individuals at conception and T is the nor- malizer that ensures that the genotypic frequencies (MJ42) reproduce ameiotically at the rate a12, and sum to unity. rare homozygotes (M2M2)at the rateaZ2. Parameter e At conception, the frequencies of genotypes AIAl (0 d e d 1) represents a measureof dominance of MZ: and AlA2 are given by U~=ailxli +(I-ail)[b(xll +X12/4)

+ (1 - b)f2] (24 Complete dominance (e = 1) implies that both rare uf=allx12+(1 -all)[bx12/2+(1-b)2f&], (2b) genotypes (M1M2and M2M2) express a rate of meiotic in which reproduction different from the resident, and complete recessivity (e = 0) that only the rarehomozygote x11 = Ull(1 - Py (34 (M2M2)expresses a different rate.We have not consid- x12 = (U112P + u12)(1 - P) (3b) ered overdominance or underdominance in expression of the modifier locus (e lying outside (0, 1)). x22 = (U11P + U12)P. (34 Let u, v, and w denote thefrequencies of individuals and bearing genotypes MIMI, MIMz, and MzMz, respectively. For rare M2, u lies close to the sum of uI1and = x11 + x12/2 (44 fi u12 obtainedfrom (5). Upto linearterms in the f2 = x12/2 + x22. (4b) frequencies of the rare genotypes (v and w), the pop- 876 M. K. Uyenoyarna and B. 0. Bengtsson dation in the next generation is described by absence of selfing (b = 0),the selection intensity is of the orderof the square of the mutation rate. Tv’ =v[alZ’$A + (1 -a12)b’#J~/2]

+ [v( 1 - a12) + w2( 1 - a22)](1 - b)4R (74 JOINT EVOLUTION OF THE MODIFIER AND THE VIABILITYLOCUS Tw’ = v( 1 - aln)bds/4 + W[~ZZ~A+ (1 - ~zzJ)~~JR],(7b) Partial selfing generates positive associations in het- in which &A, &-, and & represent the averageviabilities of offspring derived by apomixis, meiotic selfing, and erozygosity, even between unlinked loci (WEIR and random outcrossing, respectively. Expressions for COCKERHAM1973). Such associations compel consid- these averageviabilities are obtained from (3)and (4): eration of the jointevolution of the viability locus and the modifier locus. 6A=X11 + ax12 (84 Introduction of a modifierof meiotic reproduction 4s = x1 1 + ~1z(1 + 2a)/4 (8b) Change in frequencies of rare genotypes: Recom- 6R=fi(fi + 2f26). (84 bination between the viability locus (A)and the mod- These values determine the mean fitness of the pop- ifier locus (M) occurs atthe rate r(0 I r I 1/2). ulation: Introduction of the new allele at the modifier locus generates five new genotypes. Rare homozygotes at T = Ulld’A (1 - all)[b$s + (1 - b)4R] (9) the modifier locus (M2M2) that were AIAl at conception occur with frequency wll, and those that were [compare (5b)l. AlA2 at conceptionoccur with frequency w12. The Condition for local stability:For all levels of dom- variable v1 1 represents the frequencyof M1M2individ- inance of the rare modifier allele (e), the resident uals that were AIAl at conception, and v12 and vpl population excludes M2 if double heterozygotes AlM1/A2M2and A2M1/A1M2,respectively. (all - a12)[4A - bd’S - (1 - b)&R] > 0. (10) Appendix 1 presents recursionsto the first order in Condition (10) indicates that the adaptive value of the frequencies of the rare genotypes at the modifier meiotic reproduction depends on the averageviability locus. We assume that most of the random outcross of offspring derived by apomixis (4A)relative to the gamete pool is generated by the common genotype average viability of offspringderived meiotically MIMl. If,however, the residentpopulation repro- through selfing and outcrossing (4s and &). Parents duces by meiosis at a negligible rate, then the MlMl are equally closely related to their offspring derived population is reproductively isolated from the carriers by any of the three reproductive modes. Offspring of M2. In thatcase, interpopulation or group selection generated by apomixis or selfing derive all of their presumably would direct the evolution of meiotic genes from their parent. Offspring derivedby random reproduction, implying that meiotic reproduction will outcrossing receive enhancers of meiotic reproduction invade because it reduces the mutational load [see from both parents because only meiotically reproducing individuals contribute to the gametepool. Conse- (5~1. Qualitative features: High rates of selfing (b) and quently, the “cost of meiosis” does not apply to the low viability of carriers of the lethal allele (a) promote evolution of meiotic reproduction as an alternative to meiotic reproduction (see APPENDIX qualitative apomixis. z), trends evident in the absence of two-locusassociations To terms of the order of the mutation rate, (10) is proportional to [see (1 1)and (12)]. The major qualitative differences that emerge from the full two-locus analysis include (a11 - a12)pb(a - YZ) > 0. (1 1) the possibility of evolutionarily stable mixtures of apomictic and meiotic reproduction.Further, the In the absence of selfing (b = 0),(10) reduces to: minimum viability ofAlA2 individuals that permitsthe maintenance of apomixis dependson the rates of (a11 a12)p2(a Yz) 0. (1 2) - - > selfing (b), recombination (r),and dominance of the Conditions (1 1) and (12) indicate that meiotic repro- introduced modifier (e). duction is favored if the deleterious viability allele A2 To illustrate the qualitative behavior observed in exhibits partial dominance (a < Y2). In the presence the full two-locus analysis, we present the results ob- of selfing (b > 0), the intensity of selection is of the tained in the absence of both linkage (r = %) and order of the frequency of the deleterious allele, which selfing (b = 0). The resident population resists the in turn is of the order of the mutation rate. In the invasion of M2 if Origin of Meiosis 877 1 This simple example illustrates two major qualitative differences generated by two-locus associations, even in the absence of linkage and selfing: the existence of a continuously stable state involving both apomictic and meiotic reproduction (15a), and a sub- 2 stantial change in the minimum viability that permits the maintenance of apomixis [compare(15b) and (12)]. To explore the implications of two-locus associations on the quantitative results, we describe the effects of selfing and linkage under complete dominance (e = I), and theeffects of selfing and dominance of the introduced modifier under free recombination 0 a 1 (r = 1/2). We assume in the remaining analysis that the selection intensity at the modifier locus is weak, meaning that (a11 - a22) is sufficiently small. a11 FIGURE1 .”Sketch summarizing conditions for local stability in Effects of selfing and linkage under complete the absence of selfing and linkage between the modifier and the dominance of M2 viability locus [from (1 3)]. Rare MIM~individuals express apomixis at rate al2, and MlMl individuals, rate all. Regions marked “S” Under complete dominanceof the introduced mod- represent parameter combinations in which the population resists ifier allele M2 (a12 = a22), Me fails to increase when the invasion of Mz, and “U” parameter combinations in which M:! rare if increases when rare. Thevalue a (1 5a) represents an ESS, the only rate for which M2 fails to invade under all assignments for alz; a is valid under (1 5b). (all - aZZ)[Qdorn(alI) + 8(all - a22)] > 0, (16) (all- a12)p2(3a- 2 - aa12) > O (13a) in which 3(all - a24 denotes terms of the order of (all - a22) and smaller, and or, equivalently, Qdorn(a11) (a11 - a12)p2[Q(a11) + a(a1l - UI~)]> 0, (13b) in which =2ra(l -all)[a(l -b)(l -all) 1 +~]-(1 -~)[2(1-u)+ba(I -all)]. (17) Q(Ul1) = 3a - 2 - UUll. (14) - The positive root of Q(a11) is equal to The function Qdorn(all)has a root (a) in (0, 1) if both

a = (3a - 2)/a,(1 54 a > 2/(3 - b) ( 184 which lies in (0, 1) if (1 - a)[2 - 42 - b)] r> (18b) a>% 5b) (1 2442 - b) - 11 Figure 1 depicts the ranges for parameters all and a12 [compare (1 5b)l.A resident population thatexpresses for which the state represented by the resident popu- a(all = a) excludes all modifier alleles that cause lation is locally stable (S) or unstable (U) to the intro- small changes in the rate of meiotic reproduction: a duction of the new modifier allele M2 (compare Mo- is an ESS for a22 close to all. Further, a is a CSS, TRO 1981).For all equal to a, the local stability meaning that modifier alleles that bring the rate of condition (1 3b)is always satisfied, irrespective of the meiotic reproduction closer to a increase when rare if value of a12. Resident populations that practice apo- (all - a) is large relative to (all - ~22).For parameter mixis at this rate actively exclude any rare modifier values violating (18), meiotic reproduction is uniformly allele that determines any different rate: a is an ESS. favored. Further, a is a continuously stable state (CSS; ESHEL Figure 2 presents the minimum value for the via- 1983), which means that for all not equal to a, a new bility parameter (a, from (1 8)) that permits the exist- modifier allele that causes a small change in the rate ence of the CSS as a function of the recombination of ameiotic reproduction increases when rare only if fraction for three levels of selfing (b = 0.1, 0.5, 0.9). a12lies closer to a than all does. This property is Tight linkage and high rates of selfing promote inferred from the function Q(all) in (14), which is meiotic reproduction. Figure 3 presents the CSS rate positive for a1 1 less than a and negative for all greater of apomictic reproduction under three ratesof recom- than a. If the existence condition (1 5b)for the CSS a bination (r = 0.05, 0.1, 0.5) for a set equal to 0.9. fails, then higher rates of meiotic reproduction are Higher levels of apomixis evolve under loose linkage uniformly favored. and low rates of selfing. 878 M. K. Uyenoyama and B. 0. Bengtsson

1.o - k0.50

E 0.7- .-z - b=0.90 r - b=0.50 0.6- -.- b=O.lO

0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.2 0.6 0.4 0.8 1.o Recombination Fraction Selfing Rate FIGURE2.-Effect of recombination between the modifier and FIGURE3.-Continuously stable strategy (CSS) involving partial apomixis as a function of selfing rate (b) and the recombination the viability locus on the minimum viability (u) required for the fraction (r) under complete dominance (e = 1) of the introduced maintenance of partial apomixis under complete dominance (e = 1) of the introduced modifier allele [from (1 6)and (17)]. Higher rates modifier allele [from (16) and (17)]. The viability of genotype AIA~ (u) is set equal to 0.9. of selfing (b) and reduced recombination favor meiotic reproduction; however, in every case the threshold viability exceeds 0.5, the GENERATION OF DISEQUILIBRIUM value expected in the absence of two-locus associations,even in the absence of linkage (r = 0.5). Ignoring associations thatdevelop between the modifier locus and the viability locus generates the Effects of selfing and dominanceof M2 under free predictions that no state involving both meiotic and recombination ameioticreproduction is evolutionarily stable, and that pure meiotic reproduction evolves only if the In the absence of linkage (r = !A), the new allele M2 lethal allele A2 expresses dominance (Y2 > u) [see (1 1) fails to increase when rare if and (12)]. In contrast, our results from the full two- locus model indicatethat partial apomixis is main- tained only if the viability (u) of the AlA2 individuals is sufficiently high, with pure meiotic reproduction favored in all other cases. To clarify the nature of the in which aree(all)denotes the quadratic given in AP- interaction between the modifier locus and the viabil- PENDIX 2 [compare (14)]. The function 4free(all)has a ity locus, we express the conditions for local stability root (a, the CSS) in the range (0, 1) for u sufficiently in terms of measures of two-locus disequilibria. large [compare (15)]. For complete dominance(e = 1) or complete recessivity (e = 0) of expression of the Change of variables modifier allele M2, the CSS is in fact an ESS, meaning Definitions: A change of variables to incorporate that allequal to a ensures local stability for all a12 and measures of association between the modifier locus aZ2close to but different from a. If the CSSis not and theviability locus facilitates insight into the nature valid, then meiotic reproduction is uniformly favored. of the process of selection that directs the evolution Figure 4 presents the minimum value of u that of the modifier locus. The frequencies of the rare permits the maintenance of partial apomixis, plotted genotypes in terms of the original variables are rep- as a function of the level of dominance (e) of the resented by x: introduced modifier allele M2 for threerates of selfing XT = (w11, w12, v11, VI29 v21), (20) (b = 0.1, 0.5, 0.9). Condition (1l), obtained by ignor- in which xT denotes the transpose of x. We define a ing associations between the modifier locus and the new coordinate system with basis viability locus, suggests that ameiotic reproduction is yT = (G, + 72,/2, w, q), (21) favored for all u greater than %. In contrast, Figure vw vu, 4 indicates that the minimum viability that permits in which the maintenance of apomixis depends strongly on the w = w11 + w12 dominance of the introduced modifier alleles and the rate of selfing in the full two-locus model. While the v = v11 + VI2 + vq1 minimum viability generally lies above !A for dominant modifier alleles, the threshold forrecessive mod- ifiers mayfall substantially below 1/2, depending on the rate of selfing. Two-locus associations promote meiotic reproduction under high rates of selfing and ameiotic reproduction under low rates of selfing. Origin of Meiosis 879 Within-individual disequilibrium (G), gametic disequilibrium (D), and the quantity in (23) converge to zero, with the one-locus coefficient of inbreeding (F) converging to F = b/(2 - b), (25) and identity disequilibrium to 4b(l - b)[l + X2(1 - b)] tu 0: (2 - b)2[4 - b(1 + X7 ’ (26a) in which X denotes a measureof linkage: 0.0 0.2 0.4 0.6 0.8 1 .o X = (1 -(26b) 2r). Dominance BENNETTand BINET(1956) derived (26) under free FIGURE4.-Minimum viability of genotype AIA2 (u) that permits recombination (X = 0). Positive identity disequilibrium the maintenance of partial apomixis in the absence of linkage [tuin (26a) for D = G = 01 implies an excess of double between the modifier and the viabilitylocus [from (19)]. Pure meiotic reproduction is favored for values of u falling below the heterozygotes and double homozygotes and a defi- curves. Increasing dominance (e) of the introduced modifier allele ciency of single heterozygotes relative to expectation and higher rates of selfing (b)promote meiotic reproduction. based on independence between two inbred loci. We obtain expressions identical to (25)and (26) after with f(A ]A and f(A 1A2) denoting genotypic frequen- incorporating partialapomixis into this neutral model. cies. Inaddition tothe one-locus variables w (fre- Local stability analysis: Reformulation of the quency of M2Mz individuals) and q (frequency of allele analysis in terms of the new variables (21) entails a MZ), the new basis includes analogues of the two-locus change of basis of the state space. Matrix represents disequilibrium measures (G,tw, and tu)introduced by M the linearized transformation in terms of the original COCKERHAMand WEIR(1968, 1977). Within-individ- basis (see APPENDIX I), and the new basis; and ual disequilibrium (COCKERHAMand WEIR 1977) cor- N M N are related by responds in our model to the difference between the frequencies of the two double heterozygotes [see G in N = AMB (27) (22)]. Variables twand tuare related to identity disequilibrium (COCKERHAMand WEIR 1968), which de- in which A and B are determined from notes an association in heterozygosity between loci: y = Ax (284 for example, zero implies that the frequency of the double heterozygote is equal to the product of the x = By, (28b) frequencies of the single heterozygotes. for x and y vectors that represent a given point with The quantity (tw+ a/2) is related to a conditional respect to the original and new basis, respectively (see probability: APPENDIX 3). Because matrices M and N have the same eigenval- tw+ td2 = ~z + (~IZ+ vz1)/2 -f(AlAz)f(Mz) ues, the sole conditionfor local stability for small = [f(AlAP I M2) -f(A1Az)lf(Mz) (23) (all - aZ2)(see APPENDIX 3) is given by in which ~(AIAzI Mz) denotesthe frequency with Det(1 - N) > 0, (29) which A1Az occurs with M2, and f(Mz)the frequency of MZ [q in (22)]. Gametic phase disequilibrium,a inwhich Det denotesthe determinant. Over one measure of association between alleles borne on the generation, the population state changesby same chromosome, is given by (Y’ - Y) = -(I - N)y, (30) D =f(A1Ml)f(AzMz) -f(AzM~)f(AzMz) in which y and y’ represent the population in the = (u11 + u12/2)(w12 + v12)/2 present and next generation. Define 9 such that all but one of the elements of the vector (I - N)f are - u12(2W1 + ~IZ+ vi1 + vzi)/4 equal to zero. The number of such vectors corresponds to the dimension of the space. If one of the = (tw + tu/2)/2 + G/4, (24) eigenvalues of N were equal to unity, then 9 would in which f denotes the frequency of the indicated correspond to the eigenvector associated with unity, haplotype. and (30) would be equal to the zero vector. Because WEIRand COCKERHAM(1 973) providedexpressions unity is not in fact an eigenvalue of N, the one nonzero for these measures of disequilibrium under partial element of (I - N)? is equal tothe characteristic selfing in the absence of apomixis and selection. equation of N evaluated at unity. Applying (30) to f 880 M. K. Uyenoyamaand B. 0. Bengtsson and summingthe elements of the resulting vector Ignoringthe measures of disequilibrium D and produces (vu + vu/2), we recover(1 l), which predictsthat z(y' - f) = -z(I - N)f = -Det(I - N), (31) meiotic reproduction evolves only under partial dom- in which z represents a row vector with all elements inance of the deleterious viability allele (u < Y2). equal to unity. Equation 31 indicates that the condi- Clearly, the two-locus associations cannot be ignored, tion for local stability (29) is equivalent to because they are of the same order of magnitude as the other terms [compare (35) and (36)]. -z(y' - f) > 0. (32) Expressions (36a) and (36b) indicate that resident This equivalence implies that we can use the behavior modifier alleles thatenhance meiotic reproduction over one generation of the system initiated at f to (a11 < a22) tend to occur with the advantageous viabil- infer the asymptotic behavior of the system initiated ity allele A1 on the gametic level (D > 0), and with from arbitrary states close to the original equilibrium. genotype AlAl on the genotypic level vu/2) > 0; We examine the effect of the expression of the mod- [(vu + see (23)]. Tight linkage (r small) promotes meiotic ifier of the rate of meiotic reproduction on the meas- reproduction by increasing the absolute values of the ures of association at this particular population vector disequilibrium measures in (36), as Figure 2 suggests. f * Free recombination: Figure4 indicates thatthe Interpretation of the conditions for local stability minimum viability that permits the maintenance of In terms of the new variables, the necessary condi- apomixis may fall below Yz under recessivity of the tion for stability in (32) becomes introduced modifierallele, in contrast with the results obtained under complete dominance (see Figure 2). (a11- a22)([qe + 41- e)]2f2b(a- %) This reduction in the threshold viability implies that + vu( 1 - e)(a - 1 + b/2)/2) enhancers of meiotic reproduction can become associated with the deleterious viability allele, unlike the + C( 1 - 2r)( 1- a12)[b + 2( 1 - b)( 1 - a)]/4 case of complete dominance [see (36)]. In the absence of linkage (r = %) and under com- + (vw + vu/2)[(1- 41+ a22) + b(1 - a22)/2] > 0, (33) plete recessivity ofM2 (e = 0), (33) reduces upto terms in which of the first order in (all - a22) to w = q6/(2 - b) (344 (all - a~2)[w2f2b(a- %) + vu( 1 - e) "VU 441 - b)a(1 - I)( 1 + A') (34b) .(u 2qf2 - (2 - b)[4(1 - I) -ba( 1 - a1 L)(1 + A')]' - 1 + b/2)/2] + (vw+ vu/2)[(1 - a) While the expression for the frequency of genotype M2M2 (34a) corresponds to the neutral expectation .(I + all) + 41 - a11)/2] > 0, (37) [qF, for F given in (25)], the expression for vu (34b) reflects viability selection at the A locus as well as in which w and 77, are obtained from (34), and partial inbreeding [compare (26)]. (vw+ vJ2) = -(a11 - a22)qf2ba(4b(l - UUll) Complete dominance of the introduced modifier allele: Up to terms of the first order in (all - a22), - a(1 - all)[1 + (1 - b)2])/C (38a) condition(33) reduces under completedominance (e = 1) to with C now equal to (all - a22)q2f2b(a - 1/2) [2 - 41 + a11)](2 - b)

+ (1 - all)[b + 2(1 - b)(l - .)][D(l - 2r) .(1 - a[all+ b(1 - a11)/4]). (38b) + r(vw + vu/2)1 As expected from (1 l), ignoring the two-locus associations vu and (vw + vU/2)in (37) generates the predic- + (vw + vu/2)(1 - u)[b(l - all) + 2a111 > 0, (35) tion that meiotic reproduction evolves only if the in which deleterious viability allele A2 is partially dominant D = -(a1 I - ~z)qfzba[1 - gal I (364 (a < %). Unlike the case of dominant M2 [see (36)], (38a) indicates that a resident modifier allele MI that +u(l -a11)(r"/2)]/C promotes meiotic reproduction relative to the introduced modifier (all < a2~)becomes negatively associ- (vw+ vu/2) = -(a1 1 - az~)qf26~[2- U( 1 + 1) ated with genotype AlAl for sufficiently low rates of + 2ru( 1 - a1 l)]/C (36W selfing (b near 0). with Selfing promotes associations between loci: Our analysis indicates that selfing generates two kinds of c = (1 - aa11)[2 - a(1 + all)] - a(1 - all) associations between the modifier of reproduction and (1 - 2r)[2(1 - aa11) - ba(1 - all)].(36c) the viability locus. First, partial selfing generates as- Origin of Meiosis 88 1 sociations in heterozygosity betweenneutral loci low rates of selfing (small e and b), this relationship (WEIR and COCKERHAM 1973).While the genotypic can reverse. analog of identity disequilibrium [see (34b)l depends on viability selection (a) and the rateof apomixis (all) DISCUSSION in our model, the qualitative aspects are preserved: Sexual reproductionas a response to deleterious individuals that are heterozygous at themodifier locus mutation are more likely to be heterozygous at the viability locus as well. In agreement with intuition, tight link- Reduction of load: Underthe assumption that age (X large) maintains higher levels of association [see mutations at different loci depress viability in a multi- (36)]; however, associations arise even in the absence plicative fashion, the mean fitnesses of a sexually of linkage. reproducing population and a clonally reproducing Second, elimination of offspring with low viability population are identical (MAYNARDSMITH 1968; tends to generate associations between high viability ESHELand FELDMAN1970; CROW 1970). Although and enhancers of meiotic reproduction, which pro- some form of epistasis is necessary for sexuality to duces greater variance in offspring viability. Forma- confera population level advantage (CROW 1988; tion of the advantageous AIAl genotype occurs at a MAYNARDSMITH 1988), many forms of epistatic se- higher rate throughmeiotic reproduction with selfing. lection in fact promote asexuality (ESHELand FELD- The frequency with which AIAl offspring are pro- MAN 1970).In our model, mean fitness uniformly duced by apomixis is xll[see (3)]: the parent must increases under meiotic reproduction with selfing itself have been AIAl at its own conception and have [lower all with positive b; see (5b)l.However, our escaped subsequent mutation. Among the offspring study of a genetic modifier of meiotic reproduction generated by meiosis with selfing (b),AlAl offspring indicates that higher viability of heterozygous carriers occur with frequency x1 + x12/4, andby meiosis with of the homozygous lethal (larger a), lower rates of random outcrossing,fi2 [see (3) and (4)]. Meiotic re- selfing (smaller b), and looser linkage (larger r) pro- productionproduces more AlAl offspringthan mote the maintenance of apomixis: meiotic reproduc- ameiotic reproduction if tion is not uniformly favored. While sexual reproduction mayin fact have arisen by group selection, as b(xll + x12/4) + (1 - b)fi2 > x11, (39a) FISHER(1958, Chapter VI) suggested, inferences derived from between-population selection do not apply which reduces, to terms of the first order in the to within-population selection. mutation rate, to Improvement of expected offspring viability: bf2/2 > 0. (39b) While identical to the load argument with respect to reproductively isolated populations, the hypothesis Meiotic reproduction increases the rate of formation that sexual reproduction improves average offspring of AIAI offspring through selfing, which recovers both viability (WILLIAMS1975, Chapter 1; see also EMLEN the fully viable and the lethal homozygotes from het- 1973, Chapter 3; KONDRASHOV1982, 1984) can be erozygotes. This argument implies that modifier al- applied within populations and within broods as well. leles that promote meiotic reproduction with selfing Consider two symmetric, unimodal distributions rep- tend tobecome associated with AIAI, with the intensity resenting the probability thatoffspring in the two of selection imposed on themodifier locus ofthe order groups carry agiven number of deleterious mutations of the product of the rates of selfing and mutation [see Figure 1 in KONDRASHOV(1984); Figure 3.1 in [see expression for u12in (5a)l. This comparison holds EMLEN 973)].(1 Sexual reproduction increases the var- only initially: both identity disequilibrium and these iance while preserving the mean. Under a form of differences in the genotypic distribution among off- truncation selection for which only offspring having spring derived by meiotic and ameiotic production fewer than a certain number of mutants survive, a induce a dependence of the frequencies represented greater fraction of theoffspring described by the by XII and x12in (39a) on the genotypeat themodifier distribution with the largervariance survive, provided locus. that the truncation line falls below the mean (KON- While the association between enhancersof meiosis DRASHOV 1984). WILLIAMS(1975; see his Figure 1) and high viability promotes meiotic reproduction, the generalized this argument to include other forms of effect of identity disequilibrium dependson the epistatic selection. Strong sib competition may favor expression of dominance at the modifier locus. Our the production of diverse offspring even under con- results indicate that enhancers of meiotic reproduc- stant selection regimes (WILLIAMS1975 Chapter 2; tion tend to become positively associated with the MAYNARDSMITH 1976, 1978a (Chapter 6). advantageous allele at theviability locusfor high levels Applied to our model, this approach entails com- of dominance of the introduced modifier allele and paring the expectedviabilities of offspring derived by high rates of selfing [see (36)]; under recessivity and meiotic and ameiotic reproduction [see (1 O)]. We find 882 M. K. Uyenoyama and B. 0. Bengtsson that the expected viability of meiotically-derived off- mediate rate between biparentalreproduction and spring is higher only if the deleterious viability allele apomixis. We suggest that selfing may promote asso- shows partial dominance [% > u, see (1 1) and (12)]. ciations between highly fit genotypes and enhancers However, inferring evolutionary trends from a com- of meiotic reproduction in multilocus systems as well. parison between the expected viabilities of offspring Second,partial selfing generatesidentity disequili- derived by sexual and asexual reproduction is tanta- brium in multilocus systems (CHRISTIANSEN1989), a mount to ignoring associations between the modifier factor which promotes associations between the mod- of reproductionand the viability locus [see (33)]. ifier of reproduction and thelocus controlling viability Omission of the associations qualitatively alters the in our model. Consequently, selfing by multiply het- evolutionary outcome because the associations gen- erozygous individuals is expected to generate aneven erate effects of the same magnitude as the differences higher variance among offspring. between the average offspring viabilities. We conclude thatmodifiers of meiotic reproduction Offspring diversity promotes associations between may interact with multiple viability loci in a fashion loci similar to that observed in our simple model. How- ever, the specific nature of that interaction and its Modification of meiotic reproduction in response consequences for the origin of meiosis remain open to a single viability locus: In the absence of prior questions. Further, the relaxation of our assumption two-locus associations, meiotic reproduction with self- of homozygous lethality of deleteriousmutations ing increases the variance in genotypeamong off- would permit the accumulation of mutations (“Mull- spring of A1A2 parents, generating more highly fit as er’s ratchet,” FELSENSTEIN1974), a factor that is en- well as more lethal genotypes [see (39)]. Elimination tirely unexplored in our model. of less fit offspring generates positive associations be- Modification of recombination: FELDMAN,CHRIS- tween enhancers of meiotic reproduction and high TIANSEN,and BROOKS(1 980) studied themodification viability. High genotypic diversity amongoffspring of recombination between two loci at which mutation promotes sexual reproduction,not by improving maintains alleles that depress viability. Alleles intro- mean offspring viability, but rather by providing a duced at a third locus that enhance recombination mechanism through which enhancers of meiosis de- between the two viability loci increase from low fre- velop associations with alleles that improve viability. quencies if the depression of viability by deleterious Such associations can compensatefor initial reduc- alleles is greater than multiplicative, and if linkage tions in offspring number under meiotic reproduc- between the modifier andthe loci under viability tion. selection is sufficiently tight. We test our interpreta- Although outcrossing and recombination are gen- tion developed for the modification of meiotic repro- erally regarded as the essence of sexuality, selfing duction by applying it to the modification of recom- represents the centralforce favoring the origin of bination. sexual reproduction.Identity disequilibrium arises In the haploid case studied by FELDMAN,CHRIS- under partial selfing, resulting in acorrelation in TIANSEN and BROOKS(1980), the viabilities of geno- heterozygosity between the modifier of reproduction types AB, Ab, aB, and ab are designated 1, W1,W1, and the viability locus, even in the absence of linkage and W2 (1 > W1> W2), respectively, in which upper (BENNETTand BINET 1956; WEIR and COCKERHAM case letters denote wild-type alleles and lower case 1973). Most important, selfing promotes the forma- deleterious alleles. Deleterious alleles show a form of tion of very fit offspring by increasing the variance in multiplicative epistasis if offspring viability. Modification of meiotic reproduction in response w1 > K (40) to multilocus systems: We speculate that the princi- ples operating in our model apply to more complex Under (40),deleterious alleles occur more commonly systems which involve multiple loci affecting viability. in repulsion than in coupling in the initial population First, selfing is expected to increase the variance in (D< 0; FELDMAN,CHRISTIANSEN and BROOKS1980). We first comparethe expected viabilitiesof off- viability among offspring in the multiple locus case as springderived from parental and recombinant ga- well. PAMILO,NEI and LI (1 987) derived approxima- metes. Recombination affects the genotypic distributions for the variance in the number of mutations tion only in matings involving all four alleles. In carried by individuals under genetic drift and weak matings involving AB and ab, recombination gives rise purifying selection. Pure selfing maintains the highest to offspring with greater viability if the deleterious variance, followed by pure biparental reproduction, alleles show a form of additive epistasis: and finally pure apomixis. In agreement with HELLER and MAYNARDSMITH (1979), PAMILO,NEI and LI w1 > (1 + W2)/2. (41) (1987) found that deleterious mutations tend to fix under selfing as well as apomixis, though at an inter- Recombinant offspring produced by matings involv- Origin of Meiosis 883 ing Ab and aB have higherviability if the inequality in CHARLESWORTH,B., 1980 The cost of sex in relation to mating (41) is reversed. Considering both kinds of matings, system. J. Theor. Biol. 84: 655-671. CHRISTIANSEN,F.B., 1989 Linkage equilibrium inmulti-locus we find that recombination improves offspring viabil- genotypic frequencies with mixed selfing and random mating. ity if Theor. Popul. Biol. 35 307-336. COCKERHAM,C. C., and B. S. WEIR, 1968 Sib mating withtwo 2D[W1 - (1 + W*)/2] > 0. (42) linked loci. Genetics 60629-640. COCKERHAM,C. C., and B. S. WEIR, 1977 Digenic descent meas- Satisfaction of (41) implies (40), which in turn ensures ures for finite populations. Genet. Res. 30 121-147. negative gametic linkage disequilibrium between the COHEN,D., and D. ZOHARI,1986 The selection operating on the viabilityloci (D < 0); (42) is violated in this case, evolution equilibrium of the frequency of sexual reproduction in predominantly asexual populations, pp. 765-782 in Evolu- implying that recombination reduces offspring viabil- tionary Processes and Theory, edited by S. KARLINand E. NEVO. ity. In the absence of prior associations between the Academic Press, New York. modifier of recombination and the two loci under CROW,J. F., 1970 Genetic loads and the cost of natural selection, viability selection, parents that express enhanced re- pp. 128-177 in MathematicalTopics in Population Genetics, combination generate more offspring only if W1 falls edited by K. KOJIMA. Springer-Verlag, New York. between the arithmetic andgeometric means of 1 and CROW,J. F., 1988 The importance of recombination, pp. 56-73 in TheEvolution of Sex, edited by R. E. MICHODand B. R. W2[see (40) and (42)]. LEVIN.Sinauer Associates, Sunderland, Mass. Wenow examine thegeneration of associations CROW,J. F., and M. KIMURA,1965 Evolution in sexual and asexual between enhancers of recombination and thefavored populations. Am. Nat. 99: 439-450. genotype (AB). Recombination promotes the forma- EMLEN,J. M., 1973 Ecology: AnEvolutionary Approach. Addison- Wesley, Reading, Mass. tion of AB offspring in matings between Ab and aB, ESHEL,I., 1983 Evolutionary and continuous stability. J. Theor. but reduces their formation in matings between AB Biol. 103: 99- 1 1 1. and ab. Enhancers of recombination tend to become ESHEL,I., AND M. W. FELDMAN,1970 On the evolutionary effect associated with AB if the former mating occurs more of recombination. Theor. Popul. Biol. 1: 88-100. frequently [D< 0; or, equivalently, if (40) holds]. FELDMAN,M. W., F.B. CHRISTIANSENand L. D. BROOKS, In the absence of prior associations between the 1980 Evolution of recombination in a constant environment. Proc. Natl. Acad. Sci. USA 77: 4838-4841. recombination modifier and the loci under viability FELSENSTEIN,J., 1974 The evolutionary advantage of recombi- selection, enhancers of recombination reduce the ex- nation. Genetics 78: 737-756. pected viability of offspring unless W1 falls in a re- FELSENSTEIN,J., 1985 Recombination and sex: is Maynard Smith stricted range. However, recombination promotesthe necessary? pp 209-220 in Evolution, Essays in Honour of John Maynard Smith, edited by P. J. GREENWOOD,P. H. HARVEY and formation of highly fit offspring if deleterious alleles M. SLATKIN.Cambridge University Press, Cambridge. have greater than multiplicative effects on viability. FISHER,R. A., 1958 The Genetical Theory of Natural Selection, Ed. We suggest that associations between the modifier 2. Dover Publications, New York. and the viability loci generated by the latter effect of GANTMACHER,F. R., 1959 The Theory of Matrices, Vol. 2. Chelsea, recombination can compensate for the reduction in New York. offspring number if the associations are sufficiently HALDANE,J. B. S., 1949 The association of characters as a result of inbreeding and linkage. Ann. Eugenics 15: 15-23. strong. Tight linkage between the modifier and the HARPER,A. R., 1982 The selective significance of partial apo- viability loci promotes the evolution of increased re- mixis. Heredity 48 107-1 16. combination between the two viability loci by main- HELLER, R.,and J. MAYNARDSMITH, 1979 Does Muller’s ratchet taining stronger associations. work with selfing? Genet. Res. 32: 289-293. HOLSINGER,K. E., M. W. FELDMANand F. B. CHRISTIANSEN, 1984 The evolution of self-fertilization in plants: a population We thank F. B. CHRISTIANSENfor discussion and for observing genetic model. Am. Nat. 124: 446-453. that the significance of recombination and outcrossing for sexual reproduction may reverse during the course of evolution; and L. KONDRASHOV,A. S., 1982 Selection against harmful mutations in D. BROOKSand two anonymous reviewers for their suggestions. large sexual and asexual populations. Genet. Res. 40325-332. KONDRASHOV,A. S., 1984 A possible explanation of cyclical par- U.S. Public Health Service grant GM-37841 and the Swedish Nat- thenogenesis. Heredity 52: 307-308. ural Science Research Council provided supportfor this study, which was conducted at the Department of Genetics, Lund Univer- KONDRASHOV,A. S., 1985 Deleterious mutations as an evolution- sity. ary factor. 11. Facultative apomixis and selfing. Genetics 111: 635-653. MARSHALL,D. R., and A. H. D. BROWN,1981 The evolution of LITERATURE CITED apomixis. Heredity 47: 1-15. MAYNARDSMITH, J., 1968 Evolution in sexual and asexual popu- BENNETT,J. H., and F.E. BINET, 1956 Association between lations. Am. Nut. 102: 469-473. Mendelian factors with mixed selfing and random mating. MAYNARDSMITH, J., 1976 A short-term advantage for sex and Heredity 10 51-55. recombination through sib-competition. J. Theor. Biol. 63: BERNSTEIN,H., H. C. BYERLY,F. A. HOPF and R.E. MICHOD, 245-258. 1984 Origin of sex. J. Theor. Biol. 110 323-351. MAYNARDSMITH, J., 1978a The Evolution of Sex. Cambridge Uni- BERNSTEIN,H., H. C. BYERLY,F. A. HOPF and R. E. MICHOD, versity Press, Cambridge. 1985 Genetic damage, mutation andthe evolution of sex. MAYNARDSMITH, J., 1978b. The ecology of sex, pp. 159-1 79 in Science 229: 1277-1281. Behavioural Ecology, An Evolutionary Approach, edited by J. R. 884 M. K. Uyenoyamaand B. 0. Bengtsson

KREBS~~~N. B. DAVIES. Sinauer Associates, Sunderland, Mass. carrying MI)are given by (4a), and gl and g2, the frequencies MAYNARDSMITH, J., 1988 The evolution of recombination, pp. of gametes carryingM2, by 106-125 in The Evolution of Sex, edited by R. E. MICHOD and B. R. LEVIN.Sinauer Associates, Sunderland, Mass. g1=(l -a22)(z11+212/2) MOTRO, U., 1981 Optimal rates of dispersal. I. Haploid popula- +(I -a12)[y11 +~12r+~21(1-r)1/2 (A1.3) tions. Theor. Popul. Biol. 21: 394-41 1. MULLER,H. J., 1932 Some genetic aspects of sex. Am. Nat. 66: g2 = (1 - a22)(222 + 212/2) 118-138. +(1 -a12)[y22+y12(1 -r)+YzIr1/2> MULLER,H. J., 1964 The relation of recombination to mutational advance. Mutat. Res. 1: 2-9. in which PAMILO,P., M. NEI and W.-H. LI, 1987 Accumulation of mutations in sexual and asexual populations. Genet. Res. 49 135- 146. UYENOYAMA,M. K., 1984 On the evolution of parthenogenesis: a genetic representation of the "cost of meiosis." Evolution 38: 87-102. WEIR, B. S., and C. C. COCKERHAM, 1973Mixed self and random (Al.4) mating at two loci. Genet. Res. 21: 247-262. WILLIAMS,G. C.,1975 Sex and Evolution. Princeton University Press, Princeton, N.J.

Communicating editor: B. S. WEIR

APPENDIX 2 APPENDIX 1 Determination of CSS states Linearized recursions Viability: Under complete dominance of the lethal allele (u = 0), the resident population excludesM2 if Recursions in the frequencies of the rare genotypes, up to linear terms in those frequencies, aregiven by -(a11 - a22)bp > 0, (A2.1)

which implies that selectionfavors meiotic reproduction (reduced apomixis) under all levels of recombination and dominance of expression of M2. Under complete recessivity of the lethal alleleA2 (u = l), theinitial monomorphism at themodifier locus is locally stable if the inequality in (A2.1) is reversed. Selfing rate: In the absence of random outcrossing (b = l), the condition forlocal stability corresponds to(A2.1) with unity substituted for b. High rates of selfing favor meiotic reproduction. If all meiotic reproduction entails randomoutcrossing (b = 0), then the resident population resists the invasion of M2 [compare (l)], in which the primed variables denote the fre- if quencies in the next generation and the starred variables frequencies at conception. At conception, the frequencies of the (all - Ul*)Pc"(U(l- a12)(1 - 2r)[(l - a) - u(l - an)] rare genotypes are - (1 - uaI2)[2(1- u) - u(1 - aln)])> 0; (A2.2a)

or, equivalently,

(all - UI~)P~[Q(UII)+ ~~(UII - ale)] > 0, (A2.2b)

in which

Q(a1,)= -(1 - ua11)[2(1 - u) - u(1 - all)] (A2.3)

+ u(l - Ull)(l - 2r)[(l - u) - u(l - all)]

[compare (12)]. A root of Q(a11)(a) lies in (0, 1) if both

u > 2/3 (A2.4a)

(1 - u)' r> (A2.4b) u(2u - 1)

The rate a corresponds to both a CSS and an ESS, provided that aI2is restricted to values close to all. Complete meiotic In (Al.2), f, and f2 (the equilibrium frequencies of gametes reproduction is favored if (A2.4a) and (A2.4b) areviolated. Origin of Meiosis 885

Absence of linkage: The function Qf.e(a~I) in (1 9) is given 71)]. For positive M,the dominant eigenvalue is positive and by occurs with multiplicity one. In theabsence of differences amongmodifier genotypes (all = a12 = az2), this dominant 4frCe(aI~)=-(1 -a)(l -6/2)(1-e)[2(1 -4 eigenvalue corresponds to unity; for sufficiently small (all - aZ2), positivity of the determinant of (I - M) ensures local + 41- a11)(2 + 41 stability. Matrices A and B are determined from(28), using (22). For +(l-~[~II+b(l-~11)/4])((1-6)[2(1-~) (A2.5) rare M2,f(AlA1) in (22) lies close to UII andf(AIA2) to u12;under +dl -a11)/2] this substitution, A and its inverse B are given by: 000 - e[( 1 - u)(4 - 36) - u( 1 - alI)( 1 b)']]. - -u12 UII -u12/2u11/2 u11/2 0 0 -%I2 u11' u11-' ) (A3.1) APPENDIX 3 110 00 1 1 1 Y2'/s Y2 Local stability analysis Matrix M is anon-negative five-dimensional matrix,the 0 1 -% u12 dominant eigenvalue of which is non-negative. Local stability 0 0 -1 -2ull 2Ull (A3.2) requires positivity of the determinants of thesuccessive princi- Y2 0 v2 -u12 UI2 pal minors of the matrix (1") [see GANTMACHER(1959, p. -% 0 Y2 -u12 2112