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(:opyt.ight 0 1992 by the Genetics Societyof America Heterozygote Advantage and the Evolution of 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 of eukaryotic, 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 haploid phase (one copy of each chromosome) following meiosis and a diploid phase (two copies of each chromosome) following syngamy. The two phases may or may not be morphologically distinct, and each of the phases may or may not undergo mitotic divisions. In all metazoans and seed plants, the diploid phase is the multicellular, dominantform,andthe haploid phase is highly reduced. In other eukaryotes, however, allpossibilities are found:eitherthe diploid phase may dominate (e.g., some brown algae,diatoms), o r the haploid phase may dominate (e.g., some species of fungi, of green algae, and of red algae) or both the haploid and the diploid phase may be multicellular and complex, neither seeming to dominate the life cycle (e.g., marine members of Ulvophyceae). Numerous theories involving both direct and indirect advantages of diploidy have been proposed to explain the dominance of the diploid phase in higher plants and animals. Possible direct advantages include the repair of damaged DNA (BERNSTEIN, HOPF and MICHOD 1988) andprotection from somatic mutation (EFROIMSON 1932).Amongthetheories involving indirect advantages of diploidy, only the masking hypothesis, which holds that a dominant diploid phase evolved due to its capacity to mask deleterious recessive mutations, has been rigorously investigated (PERROT, RICHERD and VALERO199 1; KONDRASHOVand CROW 1991 ; OTTOand GOLDSTEIN 1992). Despite the plau- A Genetics 134: 1195-1 198 (December, 1992) sibility of this theory, OTTO and GOLDSTEIN (1992) showed that masking only favors diploids under certain conditions. In particular, when recombination rates are low, masking is unlikely to favor diploids. It is therefore important to evaluate other indirect advantages of diploidy that might have played a role in the evolution of a dominant diploid phase. An obvious possible indirect advantage to diploidy is heterozygote advantage.As suggested by CROWand KIMURA(1 965),it seems that evolution should favor an extension of the diploid phase if some loci are subject to heterozygote advantage, since only diploids can be heterozygous. Although heterozygote advantage has rarely been demonstrated in nature,the possibility remainsthatheterozygotesoften have a small fitness advantage which is hard to detect. In this paper, I assume that variation for the degree of dominance of the diploid phase exists and determine the conditions under which heterozygote advantage can be expected to lead to an increase in the dominance of the diploid phase. T h e degree of dominance of the diploid phase is measured by the probability that natural selection is experienced during the diploid as opposed t o the haploid phase. It seems clear intuitively that heterosis should always promote an increase in the diploid phase; we will see, however, that as with the rigorous analysis of the masking hypothesis (OTTO and GOLDSTEIN 1992), intuition does not provide the whole story. TABLE 1 Recombination and Meiosis The genotypic control of ploidy level TABLE 2 The genotypic control of selection I consider the same life cycleand its genetic control as was used i n OTTOand GOLDSTEIN(1992) (Table 1). A single viability locus with two alleles (AI, A,) is suhject to overdominant selection in diploids and directional selection in haploids (see Table 2). A I homozygotes and haploids have selective disadvantage $ 1 relativetoheterozygotes, and A, homozygotes and haploids have selective disadvantage $2. A second locus (ploidy locus) with two alleles ( C l , C,) controls the timing of meiosis and, i n consequence, the probability 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 d l I = 1 , selection operates only on diploids and when d l I = 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. T h e recursion equations describingthe dynamics of this nlodel are given in the Appendix. Suppose that thatthe ploidylocus is initially fixed on allele C 1 , resulting in a fraction dl I of the population undergoing selection as diploids, and ( 1 - dl I ) undergoing selection as haploids. Then, theequilibrium frequency at the viability locus is x * = 1 - x1 A (2) where i 1 and 2 2 are the equilibriunl frequencies of chromosonles A I C l and A 2 C I , respectively. For convenience, label the alleles such that sl > s?. Then, the 1 0.8 0.6 % 0.4 0.2 1197 Evolution of Diploidy is low (0.03) and when it is high (0.50). Notice that when the probability is 0.03, so that on average 3% of the population experiences diploid selection, only a tiny sliver of the parameter space allows the equilibrium to exist. Thus, when there is very little diploidy in thepopulation, the fitnesses of the two haploid types (1 - s l , 1 - sp) must be nearly identical if the polymorphic equilibriumis to exist, even though there is heterozygoteadvantage. This restrictionon the relative fitnesses of the haploid types becomes much stronger as the amount of diploidy decreases. When the haploid fitnesses are quite different, a polymorphic equilibrium can exist only if the amount of diploidy is large, as shown by the line for dl1 = 0.50 in Figure 2. The dependence of the polymorphic equilirium on theamount of diploidy results fromthe fact that constant viability selection in haploids must be directional. Consequently, variation at the viability locus cannot be maintained, unless the fitnesses of the alleles are exactly the same. When the fitnesses are different, haploid selection pushes the allele frequency toward the fitter allele. Therefore, the more haploidy in the population, the more the frequency is pushed toward fixation on the fitter allele. This directional selection in the haploid phase is represented in the genetic load, which is easily seen to be an increasing function of the amount of haploidy in the population (assuming the polymorphic equilibrium exists). INVASION OF NEW PLOIDY ALLELES To determine whether overdominant selection will favor an increase in the dominance of the diploid phase (increase in the probability of undergoing diploid selection), I will identify the type of new ploidy allele that can increase in frequency when introduced a t low frequency (by mutation, say). Thus, we would like to know whether the equilibrium with C1 fixed is locally stable or unstable to the introduction of new ploidy alleles. In the case of a mutation-selection balance at theviability locus, evolution at theploidy locus depends critically on the recombination rate between the ploidy and viabilityloci (OTTOand GOLDSTEIN 1992). We might, therefore, expect some dependence on therecombinationrate in the modification of ploidy level when the viability locus is overdominant. We answer the question of what ploidy alleles can invade when initially rare by performing a local linear stability analysis about the polymorphic equilibrium (RI,22, 0,0). This is done by taking a Taylor Series approximation pf the recursions given in the APPENDIX about (gI, X 2 , 0, 0) and determining whether the resultant Jacobian matrix has an eigenvalue greater than 1 (implying that the equilibrium is unstable). Of thethree eigenvalues, one relatesto the dynamics when the rare chromosomes are absent; it is less than 1, since the initial equilibrium is assumed stable. The other two eigenvalues describe the initial change in therare chromosomefrequencies X 3 and X,. The conditions resulting in an eigenvalue greater than 1 are foundmost easily by considering thecharacteristic polynomial,fiX), of the Jacobian matrix, at X = 1. We find thatfil) has the same sign as: (dl1 - dlZ)k(Sl + s2 - %IS2 + 2d12S1.32) + SISZ(dl1 - d12)l (3) assuming that d l , > (sl - s2)/sl ( i e . , the polymorphic equilibrium exists). Since the Jacobian matrix is positive, the Perron-Frobenius theorem guarantees a single, real eigenvalue which is positive and greatest in magnitude. In addition to this, we need to use the sign of the first derivative of the characteristic polynomial evaluated at 1 for thecase of no recombination ( r = 0). This derivative, f(l)lr= o, has the same sign as ( d l , - d 1 2 ) . Consider first the case of a new allele increasing the probability of diploid selection (d12 > d l ] ) when there is complete linkage ( r = 0). In this case,fil) > 0 a n d f ( l ) l r = o < 0. Because the Jacobian matrix is positive, these two facts guarantee the existence of two eigenvalues greater than 1. When (d12 < dl,), however, we have bothfil) > 0 a n d f ( l ) l r =0 > 0, which guaranteesthatboth eigenvalues are less than 1. When recombination exists ( r > 0), and (dl2 > dl,), thenfil) may change from positive to negative as r increases from zero, since in Equation 3 the term involving r is positive, while the other term in the braces is negative. The linearity in r of Equation 3, however, guarantees that there will be no more than one sign change. This implies that for r sufficiently small two eigenvalues are greater than 1, and as r increases, one and only one of the these eigenvalues may become smaller than 1. Therefore, whenever (dlZ > d l I ) , at least one eigenvalue is greater than 1. When the inequality is reversed, however, f i l ) is strictly positive, regardless of the value of r, andf(l)l,= is positive. These two facts guarantee that, when ( d l l > dls), both eigenvalues are less than 1. In summary, the local stability analysis shows that provided the polymorphic equilibrium exists, then for any degree of linkage (0 5 r 5 1/2), alleles increasing the probability of undergoing selection as a diploid ( d l 2 > d l I ) always invade at an initially geometric rate, and alleles decreasing this probability never invade. Unlike the case of mutation-selection balance, when the viability locus is overdominant the recombination rate does not affect the condition for initial increase. Numerical iterations: I have also iterated the equations in the APPENDIX in order to see what happens after invasion, since the localanalysis provides no information about this. In a limited exploration of the parameter space with d l , > (sl - sp)/sl, I found that alleles increasing diploidy invaded when rare (as 1198 B. D. Goldstein shown analytically) and continued to increase until jixation. Conversely, alleles decreasing diploidy were always eliminated, even if introduced in high frequency. T h e only exception I found was in the case of complete linkage ( r = 0) and a perfectly dominant modifier increasing diploidy. Under these special conditions, the allele increasing diploidy does not continue until fixation,but hits aneutralcurveafter invading. [Along this curve, the frequencyof the new modifier allele ranges from Is:! - sl(1- d12))/d12(s1 s y ) to 1.0, and the frequency of A I is {s2-s1( 1 - dl2)]/dl2(s1 s4.1 This result does not seem of much biological interest, however, since even very small rates of recombination o r slightly incomplete dominance resulted in fixation of the new allele. + + CONCLUSIONS I have shown that beginning from a polymorphic equilibrium, alleles increasing the diploidy rate always increase when rare. This seems to confirm the intuition that overdominance always helps diploids. For the diploidy rate to be increased, however, the polymorphic equilibrium must exist. Otherwise, the viability locus will be fixed for the superiorallele and no evolution will occur at the modifier locus since homozygote diploids and haploids have the same fitness. I havealso shown, however, that the overdominant equilibrium is not likely to exist in a primarily haploid population. Therefore, heterozygote advantage could only drive an increase in the dominance of diploidy if mutation first introduced variation atthe viability locus. If such variation existed simultaneously with variation at the ploidy locus, then a model similar to that described in (PERROT,RICHERDand VALERO 1991) and OTTOand GOLDSTEIN (1992) can be used to show that diploidy would be increased. However, since mutation to alleles capable of forming overdominant heterozygotes is probably quiterare (unlike mutation to deleterious alleles), it seems unlikely that variation would exist simultaneously at both the viability and ploidy loci in predominantly haploid populations. Consequently, it seems that heterozygote advantage could cause a further increase in the diploidy rate in populations that already have a high rate of diploidy, but could not increase the amount of diploidy in populations that are mostly haploid, since the polymorphic equilibrium is not likely to exist. I thank M. W. FELDMAN and S. P. OTTOfor helpful discussions; F. B. CHRISTIANSEN, A. G . CLARK,M. W. FELDMAN, A. S. KONDRASHOV, s. P. OTTO, s. SHAFIR, s. D. TULJAPURKAR, P. L. WIENER and two anonymous reviewers for comments on the manuscript; for translating an important paper from and A. S. KONDRASHOV theoriginalRussian.Thisresearch was supported by National Institutes of Health grant GM 28016 to MARCUSW. FELDMAN. LITERATURE CITED BERNSTEIN,H., F. HOPFand R. MICHOD, 1988 Is meiotic recombination an adaptation for repairing DNA, producing genetic variaton, or both? in T h e Evolution of Sex, a n Examination of CurrentIdeas, edited by R. MICHOD and B. LEVIN. Sinauer Associates, Sunderland, Mass. CROW,J., and M. KIMURA,1965 Evolution in sexual and asexual populations. Am Nat. 99 439-450. EFROIMSON, V. 1932 On some problems of the accumulation of lethals (in Russian). J. Biol. 1: 87-102. KONDRASHOV,A., and J. CROW,1991 Haploidy or diploidy: which is better? Nature 351: 314-315. OTTO,S., and D. GOLDSTEIN, 1992 Recombination and the evolution of diploidy. Genetics 131: 745-751. PERROT,V., S. RICHERDand M. VALERO,1991 Transition from haploidy to diploidy. Nature 351: 3 15-3 17. Communicating editor: A . G . CLARK APPENDIX T h e recursion equations describing the dynamics of the over dominance modeare given below. - d l l X 2 - dI2x4) - r { l - sl(1 - dlP)]D TX; = x2 - X2s2(1- d l l X l - d 1 2 x s )+ r ( 1 - sS(1 - dlP))D T X ; = Xg - X:,sI(l - dIvX2 - dpsX4) + r ( l - s l ( 1 - dln)lD T X ; = X , - X4S2(1 - dl2XI - d22X3) - r ( l - S Y ( ~- d12))D Tx; = x ] - x ] s l ( l where T is the normalizing constant, equal to the sum of the right hand sides, and, X I = frequency of chromosomeC I A I X 2 = frequency of chromosomeCIA? X s = frequency of chromosomeC P A I X4 = frequency of chromosomeC P A ~ D = X l X 4 - X 2 X 3 is the linkage disequilibrium