(:opyt.ight 0 1992 by the Genetics Societyof America

Heterozygote Advantage and the Evolutionof a Dominant Diploid Phase

David B. Goldstein

Department of Biological Sciences, Stanford University, Stanford, Cal$ornia 94305-5020 Manuscript received March 27, 1992 Accepted for publication August 15, 1992

ABSTRACT The life cycle ofeukaryotic, sexual speciesis divided into haploid and diploid phases. In multicellular animals and seed plants, the diploid phase is dominant, and the haploid phase is reduced to one, or a very few cells, which are dependent on the diploid form. In other eukaryotic species, however, the haploid phase may dominate or the phases may be equally developed. Even though an alternation between haploid and diploid forms is fundamental to sexual reproduction in eukaryotes, relatively little is known about the evolutionary forces that influence the dominance of haploidy or diploidy. An obvious genetic factor that might result in selection for a dominant diploid phase is heterozygote advantage, since onlythe diploid phase can beheterozygous. In this paper, I analyze a model designed to determine whether heterozygote advantage could lead to the evolution of a dominant diploid phase. The main result is that heterozygote advantage can lead to an increase in the dominance of the diploid phase, but only if the diploid phase is already sufficiently dominant. Because the diploid phase is unlikely to be increased in organisms that are primarily haploid, I conclude that heterozygote advantage is not a sufficient explanation of the dominance of the diploid phase in higher plants and animals.

LL eukaryotic, sexual species pass through a hap- sibility of this theory, OTTO and GOLDSTEIN(1992) A loid phase (one copy of each chromosome) fol- showed that masking only favors diploids under cer- lowing meiosis and a diploid phase (two copies of each tain conditions. In particular, when recombination chromosome) following syngamy. The two phases may rates are low, masking is unlikely to favor diploids. It or may not be morphologically distinct, and each of is therefore important to evaluate other indirect ad- the phases may or may not undergo mitotic divisions. vantages of diploidy that might have played a role in In all metazoans and seed plants, the diploid phase is the evolution of a dominant diploid phase. the multicellular, dominantform, and the haploid An obvious possible indirect advantage to diploidy phase is highly reduced. In other eukaryotes, how- is heterozygote advantage. As suggested by CROWand ever, allpossibilities are found:either the diploid KIMURA(1 965),it seems that evolution should favor phase may dominate (e.g., some brown algae,diatoms), an extension of the diploid phase if some loci are or the haploid phase may dominate (e.g., some species subject to heterozygote advantage, since only diploids of fungi, of green algae, and of red algae) or both the can be heterozygous. Although heterozygote advan- haploid and the diploid phase may be multicellular tage has rarely been demonstrated in nature,the and complex, neither seeming to dominate the life possibility remainsthat heterozygotes often have a cycle (e.g., marine members of Ulvophyceae). Numer- small fitness advantage which is hard to detect. In this paper, I assume that variation for the degree of dom- ous theories involving both direct and indirect advan- inance of the diploid phase exists and determine the tages of diploidy have been proposed to explain the conditions under which heterozygote advantage can dominance of the diploid phase in higher plants and be expected to lead to an increase in the dominance animals. Possible direct advantages include the repair of the diploid phase. The degree of dominance of the of damaged DNA (BERNSTEIN,HOPF and MICHOD diploid phase is measured by the probability that 1988) andprotection from somatic mutation (EFROIM- natural selection is experienced during the diploid as SON 1932).Among the theories involving indirect opposed to the haploid phase. It seems clear intuitively advantages of diploidy, only the masking hypothesis, that heterosis should always promote an increase in which holds that a dominant diploid phase evolved the diploid phase; we will see, however, that as with due to its capacity to mask deleterious recessive mu- the rigorous analysis of the masking hypothesis (OTTO tations, has been rigorously investigated (PERROT, and GOLDSTEIN1992), intuition does not provide the RICHERDand VALERO199 1; KONDRASHOVand CROW whole story. 1991 ; OTTOand GOLDSTEIN1992). Despite the plau-

Genetics 134: 1195-1 198 (December, 1992) TABLE 1 Recombination and The genotypic control of ploidy level Meiosis

TABLE 2 The genotypic control of selection

I consider the same life cycle and its genetic control as was used in OTTO and GOLDSTEIN(1992) (Table 1). A single viability locus with two alleles (AI, A,) is suhject to overdominant selection in diploids and di- 1 rectional selection in haploids (see Table 2). A I homo- 0.8 zygotes and haploids have selective disadvantage $1 relativeto heterozygotes, and A, homozygotes and 0.6 haploids have selective disadvantage $2. A second locus % 0.4 (ploidy locus) with two alleles (Cl, C,) controls the timing of meiosis and, in consequence, the probability 0.2 of undergoing selection as either a haploid or a diploid (seeFigure 1). Genotype C;C, atthe ploidy locus produces probabilities d, and (1 - d,) of undergoing selection as a diploid and as a haploid, respectively. Thus when dlI = 1, selection operates only on diploids and when dlI = 0 selection operates only on haploids. Note that we are assuming that selection operates on only one phase, eitherthe haploid or the diploid. Finally, a fraction, r, of the offspring are recombinant with respect to the two loci. The recursion equations describingthe dynamics of this nlodel are given in the Appendix. Suppose that thatthe ploidylocus is initially fixed on allele C1, resulting in a fraction dl I of the population undergo- ing selection as diploids, and (1 - dl I) undergoing selection as haploids. Then, theequilibrium frequency at the viability locus is

A x*=1 - x1 (2)

where i1and 22 are the equilibriunl frequencies of chromosonles AICl and A2CI, respectively. For con- venience, label the alleles such that sl > s?. Then, the Evolution of Diploidy 1197 is low (0.03) and when it is high (0.50). Notice that 1, since the initial equilibrium is assumed stable. The when the probability is 0.03, so that on average 3% other two eigenvalues describe the initial change in of the population experiences diploid selection, only therare chromosomefrequencies X3 and X,. The a tiny sliver of the parameter space allows the equilib- conditions resulting in an eigenvalue greater than 1 rium to exist. Thus, when there is very little diploidy are foundmost easily by considering thecharacteristic in thepopulation, the fitnesses of the two haploid polynomial,fiX), of the Jacobian matrix, at X = 1. We types (1 - sl, 1 - sp) must be nearly identical if the find thatfil) has the same sign as: polymorphic equilibriumis to exist, even though there is heterozygoteadvantage. This restrictionon the (dl1 - dlZ)k(Sl + s2 - %IS2 + 2d12S1.32) relative fitnesses of the haploid types becomes much + SISZ(dl1 - d12)l (3) stronger as the amount of diploidy decreases. When the haploid fitnesses are quite different, a polymor- assuming that dl, > (sl - s2)/sl (ie., the polymorphic phic equilibrium can exist only if the amount of diplo- equilibrium exists). Since the Jacobian matrix is posi- idy is large, as shown by the line for dl1 = 0.50 in tive, the Perron-Frobenius theorem guarantees a sin- Figure 2. gle, real eigenvalue which is positive and greatest in The dependence of the polymorphic equilirium on magnitude. In addition to this, we need to use the theamount of diploidy results fromthe fact that sign of the first derivative of the characteristic poly- constant viability selection in haploids must be direc- nomial evaluated at 1 for thecase of no recombination tional. Consequently, variation at the viability locus (r = 0). This derivative, f(l)lr = o, has the same sign cannot be maintained, unless the fitnesses of the alleles as (dl, - d12). Consider first the case of a new allele are exactly the same. When the fitnesses are different, increasing the probability of diploid selection (d12> haploid selection pushes the allele frequency toward dl]) when there is complete linkage (r = 0). In this the fitter allele. Therefore, the more haploidy in the case,fil) > 0 andf(l)lr=o< 0. Because the Jacobian population, the more the frequency is pushed toward matrix is positive, these two facts guarantee the exist- fixation on the fitter allele. This directional selection ence of two eigenvalues greater than 1. When (d12 < in the haploid phase is represented in the genetic load, dl,), however, we have bothfil) > 0 andf( l)lr=0 > which is easily seen to be an increasing function of the 0, which guaranteesthat both eigenvalues are less amount of haploidy in the population (assuming the than 1. When recombination exists (r > 0), and (dl2 polymorphic equilibrium exists). > dl,), thenfil) may change from positive to negative as r increases from zero, since in Equation 3 the term INVASION OF NEW PLOIDY ALLELES involving r is positive, while the other term in the braces is negative. The linearity in r of Equation 3, To determine whether overdominant selection will however, guarantees that there will be no more than favor an increase in the dominance of the diploid one sign change. This implies that for r sufficiently phase (increase in the probability of undergoing dip- small two eigenvalues are greater than 1, and as r loid selection), I will identify the type of new ploidy increases, one and only one of the these eigenvalues allele that can increase in frequency when introduced may become smaller than 1. Therefore, whenever (dlZ at low frequency (by mutation, say). Thus, we would > dl I), at least one eigenvalue is greater than 1. When like to know whether the equilibrium with C1 fixed is the inequality is reversed, however, fil) is strictly locally stable or unstable to the introduction of new positive, regardless of the value of r, andf(l)l,= is ploidy alleles. In the case of a mutation-selection bal- positive. These two facts guarantee that, when (dll > ance at theviability locus, evolution at theploidy locus dls), both eigenvalues are less than 1. depends critically on the recombination rate between In summary, the local stability analysis shows that the ploidy and viabilityloci (OTTOand GOLDSTEIN provided the polymorphic equilibrium exists, then for 1992). We might, therefore, expect some dependence any degree of linkage (0 5 r 5 1/2), alleles increasing on therecombination rate in the modification of the probability of undergoing selection as a diploid ploidy level when the viability locus is overdominant. (dl2 > dl I) always invade at an initially geometric rate, We answer the question of what ploidy alleles can and alleles decreasing this probability never invade. invade when initially rare by performing a local linear Unlike the case of mutation-selection balance, when stability analysis about the polymorphic equilibrium the viability locus is overdominant the recombination (RI,22, 0,0). This is done by taking a Taylor Series rate does not affect the condition for initial increase. approximation pf the recursions given in the APPEN- Numerical iterations: I have also iterated the equa- DIX about (gI,X2, 0, 0) and determining whether the tions in the APPENDIX in order to see what happens resultant Jacobian matrix has an eigenvalue greater after invasion, since the localanalysis provides no than 1 (implying that the equilibrium is unstable). Of information about this. In a limited exploration of the thethree eigenvalues, one relatesto the dynamics parameter space with dl, > (sl - sp)/sl, I found that when the rare chromosomes are absent; it is less than alleles increasing diploidy invaded when rare (as 1198 B. D. Goldstein shown analytically) and continued to increase until jix- rate in populations that already have a high rate of ation. Conversely, alleles decreasing diploidy were al- diploidy, but could not increase the amount of diplo- ways eliminated, even if introduced in high frequency. idy in populations that are mostly haploid, since the The only exception I found was in the case of com- polymorphic equilibrium is not likely to exist. plete linkage (r= 0) and a perfectly dominant modifier I thank M. W. FELDMANand S. P. OTTOfor helpful discussions; increasing diploidy. Under these special conditions, F. B. CHRISTIANSEN,A. G. CLARK,M. W. FELDMAN,A. S. KON- the allele increasing diploidy does not continue until DRASHOV, s. P. OTTO, s. SHAFIR, s. D. TULJAPURKAR,P. L. WIENER fixation,but hits aneutral curve after invading. and two anonymous reviewers for comments on the manuscript; [Along this curve, the frequencyof the new modifier and A. S. KONDRASHOVfor translating an important paper from theoriginal Russian. This research was supported by National allele ranges from - sl(1- d12))/d12(s1 sy) to 1.0, Is:! + Institutes of Health grant GM 28016 to MARCUSW. FELDMAN. and the frequency of AI is {s2-s1( 1 - dl2)]/dl2(s1+ s4.1 This result does not seem of much biological interest, LITERATURE CITED however, since even very small rates of recombination BERNSTEIN,H., F. HOPFand R. MICHOD, 1988 Is meiotic recom- or slightly incomplete dominance resulted in fixation bination an adaptation for repairing DNA, producing genetic of the new allele. variaton, or both? in The Evolution of Sex, an Examination of CurrentIdeas, edited by R. MICHOD and B. LEVIN. Sinauer CONCLUSIONS Associates, Sunderland, Mass. CROW,J., and M. KIMURA,1965 Evolution in sexual and asexual I have shown that beginning from a polymorphic populations. Am Nat. 99 439-450. equilibrium, alleles increasing the diploidy rate always EFROIMSON,V. 1932 On some problems of the accumulation of lethals (in Russian). J. Biol. 1: 87-102. increase when rare. This seems to confirm the intui- KONDRASHOV,A., and J. CROW,1991 Haploidy or diploidy: which tion that overdominance always helps diploids. For is better? Nature 351: 314-315. the diploidy rate to be increased, however, the poly- OTTO,S., and D. GOLDSTEIN,1992 Recombination and the evo- morphic equilibrium must exist. Otherwise, the via- lution of diploidy. Genetics 131: 745-751. bility locus will be fixed for the superiorallele and no PERROT,V., S. RICHERDand M. VALERO,1991 Transition from haploidy to diploidy. Nature 351: 3 15-3 17. evolution will occur at the modifier locus since ho- Communicating editor: A. G. CLARK mozygote diploids and haploids have the same fitness. I have also shown, however, that the overdominant APPENDIX equilibrium is not likely to exist in a primarily haploid population. Therefore, heterozygote advantage could The recursion equations describing the dynamics of the over only drive an increase in the dominance of diploidy if dominance modeare given below. mutation first introduced variation atthe viability Tx; = x] - x]sl(l - dllX2 - dI2x4) - r{l - sl(1 - dlP)]D locus. If such variation existed simultaneously with variation at the ploidy locus, then a model similar to TX; = x2 - X2s2(1 - dllXl - d12xs)+ r(1 - sS(1 - dlP))D that described in (PERROT,RICHERD and VALERO TX; = Xg - X:,sI(l - dIvX2 - dpsX4) + r(l - sl(1 - dln)lD 1991) and OTTOand GOLDSTEIN(1 992) can be used TX; = X, - X4S2(1 - dl2XI - d22X3) - r(l - SY(~- d12))D to show that diploidy would be increased. However, since mutation to alleles capable of forming overdom- where T is the normalizing constant, equal to inant heterozygotes is probably quiterare (unlike the sum of the right hand sides, and, mutation to deleterious alleles), it seems unlikely that variation would exist simultaneously at both the via- XI= frequency of chromosomeCIAI X2 = frequency of chromosomeCIA? bility and ploidy loci in predominantly haploid popu- Xs = frequency of chromosomeCPAI lations. Consequently, it seems that heterozygote ad- X4 = frequency of chromosomeCPA~ vantage could cause a further increase in the diploidy D = XlX4- X2X3 is the linkage disequilibrium