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Copyright 2003 by the Genetics Society of America

The Effect of Neutral Nonadditive Gene Action on the Quantitative Index of

Population Divergence

Carlos Lo ´pez-Fanjul,*,1Almudena Ferna ´ndez†and Miguel A. Toro†

*Departamento de Gene ´tica, Facultad de Ciencias Biolo ´gicas, Universidad Complutense, 28040 Madrid, Spain and

†Departamento de Mejora Gene ´tica Animal, SGIT-INIA, 28040 Madrid, Spain

Manuscript received July 17, 2002

Accepted for publication April 10, 2003

ABSTRACT

Forneutraladditivegenes,thequantitativeindexofpopulationdivergence(QST)isequivalenttoWright’s

fixation index (FST). Thus, divergent or convergent selection is usually invoked, respectively, as a cause of

the observed increase (QST? FST) or decrease (QST? FST) of QSTfrom its neutral expectation (QST? FST).

However, neutral nonadditive gene action can mimic the additive expectations under selection. We have

studied theoretically the effect of consecutive population bottlenecks on the difference FST? QSTfor two

neutral biallelic epistatic loci, covering all types of marginal gene action. With simple dominance, QST?

FSTfor only low to moderate frequencies of the recessive alleles; otherwise, QST? FST. Additional epistasis

extends the condition QST? FSTto a broader range of frequencies. Irrespective of the type of nonadditive

action, QST? FSTgenerally implies an increase of both the within-line additive variance after bottlenecks

over its ancestral value (VA) and the between-line variance over its additive expectation (2FSTVA). Thus,

both the redistribution of the genetic variance after bottlenecks and the FST? QSTvalue are governed

largely by the marginal properties of single loci. The results indicate that the use of the FST? QSTcriterion

toinvestigatetherelativeimportanceofdriftandselectioninpopulationdifferentiationshouldberestricted

to pure additive traits.

A

tion for quantitative traits is an important issue, in both

evolutionary and conservation genetics (Wright 1978;

Endler 1986). In the absence of selection, inbreeding

affects all genes to the same average degree, and the

effect of the breeding structure on population diver-

gence can be described by Wright’s among-population

fixation index FST. In parallel, a dimensionless measure

of the quantitative genetic variance among populations

(termed QSTby Spitze 1993) can be defined as QST? Vb/

(Vb? 2Vw), where Vband Vware, respectively, the be-

tween- and the additive within-population components

of the genetic variance for the trait considered. For

neutral genes, with additive action between and within

loci, it is expected that Vb ? 2FSTVA and Vw ? (1 ?

FST)VA,whereVAistheancestraladditivegeneticvariance

(Wright1951).Inthissituation,QSTistheneutralquan-

titative analog of FST. This result holds quite generally,

regardless of the model of population structure (Whit-

lock 1999). The computation of the expected diver-

gence of population means due to drift requires the

estimation of generally unknown parameters, such as

the rate of mutation, the time since divergence, and

the effective population size (Lande 1977). However,

SSESSING the relative contributions of natural se-

lection and genetic drift to population differentia-

for a given set of populations, an FSTestimate can be

obtained from marker loci, assumed to be neutral, and

it can be used as the null expectation that can be com-

pared to the corresponding QSTestimate for a quantita-

tive trait, assumed to be additive. Thus, divergent or con-

vergentselectionmaybeinvoked,respectively,asacause

of the observed increase (QST? FST) or decrease (QST?

FST) of QSTfrom its neutral expectation (QST? FST).

Experimentally, this approach has been used in many

studies (see Merila ¨ and Crnokrak 2001 and McKay

and Latta 2002 for reviews). In most instances, the

errors in the estimation of FSTand QSTwere large, re-

sulting in nonsignificant pairwise comparisons of these

estimates. Nevertheless, a meta-analysis carried out by

Merila ¨ and Crnokrak (2001) indicated that QSTwas

generally larger than FST, and this result was interpreted

in the sense that a considerable part of the observed

population divergence for quantitative traits should be

attributed to differential selection pressures imposed by

local environmental conditions.

Notwithstanding, the correspondence between QST

and FSTdepends crucially on the assumption of pure ad-

ditive gene action. This may not be an important restric-

tion to the study of morphological traits, typically show-

ing substantial additive genetic variation and little or no

inbreeding depression, but will markedly affect that of

life-history traits, usually exhibiting larger levels of non-

additive variance and, correspondingly, higher inbreed-

ing depression (Crnokrak and Roff 1995; DeRose and

Roff 1999). In the absence of selection, it has been

1Corresponding author: Departamento de Gene ´tica, Facultad de Cien-

cias Biolo ´gicas, Universidad Complutense, 28040 Madrid, Spain.

E-mail: clfanjul@bio.ucm.es

Genetics 164: 1627–1633 (August 2003)

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1628 C. Lo ´pez-Fanjul, A. Ferna ´ndez and M. A. Toro

TABLE 1

shown theoretically that inbreeding can change the

magnitude of the contribution of dominant and/or epi-

static loci to the values of Vband Vw, relative to their

additive expectations (Robertson 1952; Goodnight

1988; Willis and Orr 1993; Cheverud and Routman

1996; Lo ´pez-Fanjul et al. 1999, 2000, 2002). In parallel,

increases of the additive variance with inbreeding have

been reported for viability in Drosophila melanogaster

(Lo ´pez-Fanjul and Villaverde 1989; Garcı ´a et al.

1994) and Tribolium castaneum (Ferna ´ndez et al. 1995)

and for morphological and behavioral traits in the

housefly (reviewed by Meffert 2000). Thus, nonaddi-

tive gene action can potentially modify the expected

additive relationship between FSTand QST.

In this article, we have investigated theoretically the

effectofsuccessivepopulationbottlenecksonthediffer-

ence FST? QSTfor two-locus neutral epistatic systems,

covering all possible types of marginal gene action at

the single-locus level (excluding overdominance). Our

approach follows that of Robertson (1952), where the

expected values of the derived within-line additive vari-

ance and the between-line variance, after consecutive

bottlenecks of size N, are obtained from the expressions

giving the corresponding ancestral values in an infinite

populationatequilibriumandthemomentsoftheallele

frequency distribution in populations of size N with

binomial sampling. Explicit equations in terms of the

genetic effects and allele frequencies derived by Lo ´pez-

Fanjul et al. (2002) have been used, allowing the speci-

fication of the necessary conditions to observe a depar-

ture of QSTfrom the pertinent FSTvalue.

Genotypic values for the neutral two-locus epistatic system

A1A1

A1A2

A2A2

B1B1

B1B2

B2B2

11 ? sh1

1 ? s(h1? h2)

1 ? s(1 ? h1)

1 ? s

1 ? s(1 ? h2)

1 ? ks

1 ? sh2

1 ? s

s/2 (s ? 0) and hi(0 ? hi? 1) are, respectively, the basic

homozygous effect at each locus and the basic coefficient of

dominance at the ith locus, and k (k ? 2) is the epistatic

factor.

?i? s[hi? (1 ? 2hi)qi? (k ? 2)qiq2

j], (1)

?i? s[(1 ? 2hi) ? (k ? 2)q2

j]/2,(2)

?i? hi/[1 ? (k ? 2)q2

j].(3)

Thus, epistasis (k ? 2) modifies the basic properties of

single loci, as ?i, ?i, and ?ibecome dependent on the

allelic frequencies at the other locus (qj); i.e., they are

contingent on the genetic background. For a given k

value, the basic (hi) and the marginal (?i) degrees of

dominance become closer to each other as qjdecreases.

On the other hand, ?iapproaches zero (complete re-

cessivity) as k and qjincrease.

In an infinitely large panmictic population, expres-

sions for the mean (ancestral mean M) and the additive

component of the genetic variance (ancestral additive

variance VA) can be obtained from Table 1, as

M ? 1 ? 2s?hiqi? s?(1 ? 2hi)q2

i ? (k ? 2)q2

1q2

2, (4)

VA? ??2

iHi, (5)

where Hiis the ancestral heterozygosity at the ith locus

(Hi?2piqi).Theseexpressionsarepolynomialfunctions

of pm

equilibrium, after t consecutive bottlenecks of N ran-

domly sampled parents each (derived mean Mt* and

additivevariance VAt*),can readilybe deducedby substi-

tuting pm

exact mth moment of the allelic frequency distribution

with binomial sampling, given by Crow and Kimura

(1970, p. 335). In parallel, the between-line variance

V(Mt) after t consecutive bottlenecks can be derived by

taking variances in Equation 4, the resulting expression

being also a function of the first four moments of the

allelic frequency distribution at each locus. As these

moments can also be written in terms of the inbreeding

coefficient after t generations (Ft), expressions for Mt*,

VAt*, and V(Mt) also apply when bottleneck sizes are not

constant from generation to generation. Those expres-

sions(givenbyLo ´pez-Fanjuletal.2002)areanalytically

unmanageable, but numerical solutions can be calcu-

lated for any combination of allele frequencies, as well

as the corresponding value of the quantitative index of

populationdivergenceQt?V(Mt)/(V(Mt)?2VAt*)(see

next section). In the following, the notations Ft, Qt,

THE MODEL

i(i ? 1, 2; m ? 1–4) and their expected values at

We consider the model developed by Lo ´pez-Fanjul

et al. (2002), where the variation is due to segregation

at two neutral independent loci (i ? 1, 2) at Hardy-

Weinberg equilibrium. At each locus there are two al-

leles, with frequencies piand qi(?1 ? pi). Both loci have

equal homozygous effect (s/2), showing any degree of

dominance in the absence of epistasis(hi? 0,1⁄2, or 1 for

complete recessive, additive, or complete dominance

action, respectively). This basic gene action can be

viewed as that shown by single loci segregating against

a fixed genetic background. Epistasis has been imposed

on that basic system, and it is represented by a factor k

affecting the genotypic value of the double homozygote

for the negative allele at each locus (k ? 2, k ? 2,

or k ? 2 for diminishing, reinforcing, or no epistasis,

respectively). A full specification of the genotypic values

isgiveninTable1.Attheithlocus,themarginalaverage

effect of gene substitution (?i), the marginal genotypic

value of the heterozygote (?i, expressed as deviation

from the midhomozygote value), and the marginal de-

gree of dominance (?i) are given by

iin Equations 4 and 5 by the corresponding

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1629 Epistasis and Population Divergence

V(Mt), and VAt* are kept for the two-locus system and

the subdivided population studied, but FST, QST, Vb, and

Vware used, respectively, with reference to the whole

set of loci affecting a metric trait and the relevant popu-

lation structure.

It can be shown (Lo ´pez-Fanjul et al. 2002) that the

change in mean after t bottlenecks is always negative

forbasicrecessivegeneaction(completeorincomplete,

0 ? hi?

epistasis (k ? 2). Nevertheless, diminishing epistasis

(k ? 2) and/or basic dominance (incomplete or com-

plete,1⁄2? hi? 1) result in an unrealistic enhancement

of the mean with inbreeding and, therefore, they are

not considered further.

For pure additive action it is expected that Qt? Ft,

as VAt* ? (1 ? Ft)VAand V(Mt) ? 2FtVA(Wright 1951).

For additive-by-additive epistasis, Qt ? Ft for Ft ? 1

(Whitlock 1999) as VAt* ? (1 ? Ft)VA? 4Ft(1 ? Ft)VAA

and V(Mt) ? 2FtVA? 4Ft2VAA, where VAAis the additive-

by-additive variance component (Goodnight 1988).

With dominance (with or without epistasis) equations

giving VAt* and V(Mt) cannot be written in terms of

summary statistics, as in the previous cases, but as com-

plex functions of the allele frequencies and effects at

each locus and the pertinent epistatic factor (Robert-

son 1952; Willis and Orr 1993; Lo ´pez-Fanjul et al.

1999, 2000, 2002). However, for nonepistatic complete

recessive action (hi? 0, k ? 2), some further insight

on the redistribution of the genetic variance induced

by bottlenecks and, consequently, on the relationship

between Qtand Ft, can be achieved as follows.

In general, from Equation 5, the expected additive

variance after bottlenecks can be given as

tion 2). Equation 7 shows that VAt* always exceeds its

additive expectation, i.e., VAt* ? (1 ? Ft)VA. Further-

more, as ? cov(?2

be given as VA? 2(1 ? Ft)VD, implying qi? (1 ? Ft)/

(2 ? Ft), i.e., qi?1⁄2. These results also apply for incom-

plete recessives (0 ? hi?1⁄2).

In parallel, the between-line variance can be written as

i, Hi) ? 0, the condition VAt* ? VAcan

V(Mt) ? ?4? FtVA? F2

tVD,

1⁄2) and reinforcing epistasis (k ? 2) or no

where ?4? s2?[E(q4

its additive expectation (2FtVA) for ?4? 3FtVA? F2

Moreover, Qt will be larger than Ft if ?4 ? 3FtVA ?

5F2

slightly more restrictive than those for V(Mt) ? 2FtVA,

unless Ftis large [e.g., V(Mt) ? 2FtVAfor ?4? 3FtVA?

F2

Summarizing, for neutral loci and nonadditive gene

action (dominant and/or epistatic), Qtwill generally

depart from Ft, except in the particular case of V(Mt) ?

2FtVAt*/(1 ? Ft).

i) ? q4

i] ? 0. Thus, V(Mt) equals only

tVD.

tVD and smaller otherwise, these conditions being

tVD].

NUMERICAL EVALUATION

Three representativecases were studied,with additive

(hi?1⁄2) or recessive (hi? 0) basic gene action at both

loci (s ? 0.1) and strong reinforcing epistasis (k ? 6)

or with recessive nonepistatic action (hi? 0, k ? 2). For

each case, surfaces were represented (Figure 1), giv-

ing the values of the following contrasts after one bottle-

neck (N ? 2, F1? 0.25) for all possible combinations

of allelefrequencies atboth loci: (1)the ratioof derived

to ancestral additive components of variance VA1*/VA,

(2) the ratio of the between-line variance to its expected

value for additive gene action V(M1)/2F1VA, and (3) the

difference F1? Q1between the inbreeding coefficient

and the quantitative index of population divergence.

Withdifferentbasicgeneactionateachlocusandepista-

sis, intermediate results were obtained (not shown).

For complete recessive nonepistatic action, Q1? F1

for only low to moderate frequencies of the recessive

allele at both loci (or for the recessive allele fixed in one

locus and segregating at low frequency in the other);

otherwise, Q1? F1. The absolute value of the difference

F1? Q1increased as the corresponding allele frequen-

cies became more extreme. With additional epistasis,

the condition Q1? F1holds for a much broader range

of allele frequencies, and Q1? F1for only high frequen-

cies of the recessive allele at both loci (or for the domi-

nant allele fixed in one locus and the recessive one

segregating at high frequency in the other). This situa-

tion is similar to that obtained with basic additive action

and epistasis but, for low frequencies of both negative

alleles,theexcessofF1overQ1was,comparatively,much

reduced. As shown by Equation 3, this can be ascribed

to the marginal degrees of dominance (?i) becoming

closer to the basic ones (hi?1⁄2) as the frequencies of

both negative alleles diminish. However, that excess was

VAt* ? ? cov(?2

i, Hi) ? ?E(Hi)E(?2

i)

? ? cov(?2

i, Hi) ? (1 ? Ft)?Hi(V(?i) ? ?it*2) (6)

(Lo ´pez-Fanjul et al. 2002), where E(Hi) ? (1 ? Ft)Hi,

and?it*andV(?i)are,respectively,thederivedmarginal

average effect of gene substitution at the ith locus after

t bottlenecks and its variance, which can be deduced by

taking expectations or variances, respectively, in Equa-

tion 1. Equation 6 shows that any excess of VAt* over its

ancestralvaluecanbeassignedtothespatialandtempo-

ral changes in ?i(represented by V(?i) and ?it*, respec-

tively), which are both induced by drift, as well as to

the covariance term, which depends on the ancestral

properties of dominant loci [? cov(?2

additive or additive-by-additive gene action]. Of course,

dominance may be basic (0 ? hi?1⁄2) or marginal (hi?

1⁄2, k ? 2). For nonepistatic complete recessives (hi? 0,

k ? 2), ?it* ? ?i, V(?i) ? Fts2Hi/2, and ? cov(?2

0. Thus,

i, Hi) ? 0 for pure

i, Hi) ?

VAt* ? ? cov(?2

i, Hi) ? (1 ? Ft)(VA? 2FtVD), (7)

where VDis the dominance component of the ancestral

genetic variance, VD? ??2

iH2

i ? ?s2H2

i/4 (from Equa-

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1630C. Lo ´pez-Fanjul, A. Ferna ´ndez and M. A. Toro

Figure 1.—Ratio of derived to ancestral additive components of variance VA1*/VA(log scale), ratio of the between-line variance

to its additive expectation V(M1)/2F1VA(log scale), and difference F1? Q1between the inbreeding coefficient and the quantitative

index of population divergence, after one bottleneck (N ? 2, F1?1⁄4), for (1) nonepistatic complete recessive action at both

loci (hi? 0, k ? 2), (2) basic complete recessive action at both loci and reinforcing epistasis (hi? 0, k ? 6), and (3) basic

additive action at both loci and reinforcing epistasis (hi?1⁄2, k ? 6). Darker zones correspond to ratios smaller than one or to

negative F1? Q1values.

preserved when the negative allele is fixed in one locus

and segregates at low frequency in the other as, in this

case, the marginal degree of dominance of this second

locus approaches zero (i.e., the locus becomes increas-

ingly recessive). These results apply to populations sub-

jected to a single bottleneck of any size, albeit the abso-

lute value of F1 ? Q1 decreased as the size of the

bottleneck increased. With basic recessive action, in-

creasing values of the epistatic factor k did not affect

much the absolute value of the contrast (not shown)

as, in this case, the basic and marginal degrees of domi-

nance are the same. However, with epistasis and basic

additive action, that absolute value was positively corre-

lated with k, as the marginal degree of dominance tends

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1631 Epistasis and Population Divergence

to zero for increasing k values. Of course, for basic

additive action without epistasis F1? Q1.

As shown in Figure 1, Q1? F1holds approximately

for the whole range of allele frequencies implying both

V(M1) ? 2F1VAand VA1* ? VA, irrespective of the type

ofbasicgeneaction,andthereversewasalsotrue.These

conditions also hold after consecutive bottlenecks, but

the absolute value of Ft? Qtinitially increases with the

number of bottlenecks until a maximum is reached for

Ftclose to 0.5 and then subsequently decreases to zero

(not shown). These changes have also been described

by Robertson (1952) for nonepistatic recessives at low

frequency. Summarizing, with epistasis, Qt? Ftfor all

combinations of allele frequencies at both loci, excep-

ting for high frequencies of the negative (recessive)

alleles. Without epistasis, however, Qt? Ftfor only low

to moderate frequencies of those alleles.

only as a modulating factor (Lo ´pez-Fanjul et al. 1999,

2000, 2002).

Sofar,thisdiscussionhasbeenlimitedtoinvestigating

the consequences of population bottlenecks on the Ft?

Qtdifference generated by two-locus nonadditive neu-

tral systems. An extension of these results to the whole

set of loci determining the additive variance of a quanti-

tative trait will, in principle, require a complete specifi-

cation of their genotypic effects and allele frequencies,

as the contribution of loci with the same type of gene

actiontothetotalFST?QSTvaluecanevenbeofdifferent

sign, depending on their respective allele frequencies.

Generalizations into multilocus systems can be made

only if individual loci show the same type of gene action

and segregate with similar frequencies. Only in this situ-

ation do our theoretical results provide a framework

within which some experimental data can be inter-

preted. The following discussion is restricted to D. mela-

nogaster and T. castaneum, where detailed genetic infor-

mation on relevant traits is available.

At one extreme of the spectrum, we have traits such

as abdominal bristle number or wing size and shape

characteristics of the wing. In natural populations of

Drosophila, very little or no inbreeding depression has

been detected for those characters and their between-

and within-line additive variances after bottlenecks very

closely approached the expectations under the pure

additive model (Lo ´pez-Fanjul et al. 1989; Whitlock

and Fowler 1999). In parallel, spontaneous mutations

affecting bristle number, wing length, and wing width

occur at a low rate and have a relatively large average

homozygous effect (Garcı ´a-Dorado et al. 1999). Fur-

thermore, those mutations with an effect smaller than

one-half phenotypic standard deviation of the pertinent

trait were predominantly additive and quasi-neutral

(Santiago et al. 1992; Lo ´pez and Lo ´pez-Fanjul 1993;

Merchante et al. 1995). Thus, a large fraction of the

corresponding genetic variance in natural populations

will be contributed by a small number of quasi-neutral

additive loci segregating at intermediate frequencies

(Robertson 1967; Gallego and Lo ´pez-Fanjul 1983).

At the other end of the spectrum, we consider viability

in natural populations of Drosophila and Tribolium.

After bottlenecks, viability showed strong inbreeding

depression and its within-line additive variance signifi-

cantly increased above the ancestral value (Lo ´pez-Fan-

jul and Villaverde 1989; Garcı ´a et al. 1994; Ferna ´n-

dezetal.1995).Therefore,muchofthegeneticvariance

of the trait should be due to partially (or totally) reces-

sive deleterious alleles segregating at low frequencies.

In Drosophila, most newly arisen mutations affecting

viability had, on the average, a relatively large homozy-

gous disadvantage and their gene action was close to

recessive (Garcı ´a-Dorado et al. 1999; Garcı ´a-Dorado

and Caballero 2000; Chavarrı ´as et al. 2001). Using

mutational information, Wang et al. (1998) have been

DISCUSSION

We have shown that the Qtvalue generated by neutral

dominantand/orepistaticloci,aftertconsecutivepopu-

lation bottlenecks, will always be larger or smaller than

its additive expectation Ft, with the trivial exception de-

termined by those particular combinations of allele fre-

quencies fixing the boundary lines between the positive

and negative regions of the Ft? Qtsurface. Therefore,

the use of the Ft? Qtdifference as a criterion to investi-

gate the relative importance of genetic drift and natural

selection in population differentiation is restricted to

pure additive traits, as nonadditive action at neutral

loci can mimic the expectations for additive loci under

divergent (Qt? Ft) or convergent selection (Qt? Ft).

Moreover, for nonneutral nonadditive loci, selection

will also affect (positively or negatively) the Ft ? Qt

value and this additional effect could even change the

expected sign of that difference under neutrality.

For nonadditive gene action, previous theoretical

work concerned with the divergence between Ftand Qt

was restricted to the neutral additive-by-additive model,

where Qt? Ftfor Ft? 1 (Whitlock 1999). However,

the effect of dominance (with or without epistasis) can

qualitatively alter that result, as Qtmay be smaller or

larger than Ft, depending on the relevant allele frequen-

cies. With simple dominance, Qt? Ftfor only low to

moderate frequencies of the recessive alleles, but addi-

tional epistasis extends that condition to higher values

of the frequencies of the negative (recessive) alleles at

one of the loci involved (but not at both). Irrespective

of the type of gene action considered, we have also

shown that Qt? Ftgenerally implies an increase of the

within-line additive variance after bottlenecks (VAt* ?

VA), as well as an excess of the between-line variance

over its additive expectation (V(Mt) ? 2FtVA). Thus,

both the redistribution of the genetic variance after

bottlenecksand thevalue ofFt? Qtare governedlargely

by the marginal properties of single loci, epistasis acting