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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