Epistasis and the Conversion of Non-additive to Additive Genetic Variance at Population Bottlenecks

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Theoretical Population Biology 58, 49�59 (2000)

Epistasis and the ConversioNon-additive to Additive GPopulation Bottlenecks

Carlos Lo� pez-Fanjul1

Departamento de Gene� tica, Facultad de Ciencias Biolo� gicas, Univ28040 Madrid, Spain

and

Almudena Ferna� ndez and Miguel A. ToroDepartamento de Mejora Gene� tica y Biotecnolog@� a, SGIT-INIA, Ca28040 Madrid, Spain

Received August 18, 1999

The effect of population bottlenecks on the mneutral independent epistatic loci has been stuthe analysis of binary disease traits, were copanmictic population were compared withsecutive bottlenecks of equal size N (derived vbottlenecks (inversely related to N and t) wiallele at each locus are: (1) low, invariably ahigh, always accompanied by an enhancemenundesirable property, making the pertinent mease. For the epistatic models considered, itaccelerated after population bottlenecks, inadditive variance over its ancestral value. ] 20

INTRODUCTION

The evolutionary relationship between the additiveand non-additive components of the genetic variance ofquantitative traits in subdivided populations has recentlybeen the object of considerable theoretical andexperimental research. Within small demes, randomgenetic drift has been shown to convert non-additive toadditive variance, and this phenomenon has beenassumed to increase the potential for adaptation to localenvironments and, therefore, to enhance the genetic dif-ferentiation among demes. In this context, the role and

doi:10.1006�tpbi.2000.1470, available online at http:��www.idealibra

1 To whom correspondence should be addressed. E-mail: clfanjul�eucmax.sim.ucm.es.

49

ofnetic Variance at

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era de La Corun~ a km. 7,

an and the additive variance generated by twoed theoretically. Six epistatic models, used indered. Ancestral values in an infinitely largeir expectations at equilibrium, after t con-es). An increase in the additive variance afterccur only if the frequencies of the negative

ociated to strong inbreeding depression; (2)f the mean with inbreeding. The latter is an

els unsuitable for the genetic analysis of dis-unlikely that the rate of evolution may be

ite of occasional increments of the derivedcademic Press

implications of epistasis have been specially emphasized(see Wade and Goodnight, 1998, for a comprehensivereview).

However, epistasis is not a necessary condition for theconversion of non-additive to additive variance, asdominance can also result in increased additive variancefollowing population bottlenecks (Robertson, 1952;Willis and Orr, 1993). Moreover, for those models imply-ing dominance at the single-locus level, additionalepistasis does not greatly affect the value of the additivevariance after bottlenecks over the corresponding single-locus expectations, as those combinations of allelefrequencies resulting in increased additive variancealso result in small epistatic variance (Lo� pez-Fanjulet al., 1999). Experimentally, the additive variance of

om on

0040-5809�00 K35.00

Copyright ] 2000 by Academic PressAll rights of reproduction in any form reserved.

viability has been shown to increase after inbreeding,but the relative contributions of dominance and epistasiscould not be disentangled. Notwithstanding, usingempirical mutational parameters and the equilibriumgene frequency distribution in large populations undermutation�selection balance, Wang et al. (1999) were ableto show that, for Drosophila melanogaster viability, theobserved changes in mean, additive variance andbetween-line variance following bottlenecks can bemainly attributed to lethals and partial recessive muta-tions of large effect. Therefore, dominance can be con-sidered the primary cause of an increase in additivevariance after bottlenecks.

In parallel, theoretical investigation of a wide range oftwo-locus epistatic models indicated that enhancedadditive variance after bottlenecks will occur only forsimultaneous segregation at both loci of unfavourablealleles with intermediate frequencies, or of favourablerecessives at low frequencies, and will be accompanied bylarge inbreeding depression (Lo� pez-Fanjul et al., 1999).Thus, epistasis is not a sufficient condition for conversionof non-additive to additive variance either, as those allelefrequencies cannot easily be conceived in natural popula-tions undergoing selection, unless there is stronggenotype�environment interaction implying a reversal ofthe sign of the allelic effects. With this possible exception,it is unlikely that the rate of evolution may be acceleratedafter population bottlenecks, in spite of occasionalincrements of the additive variance over its ancestral value.

In this paper, we have theoretically investigated theeffect of successive population bottlenecks on the mean,the additive variance, and the between-line variancegenerated by two-loci epistatic systems, following theapproach outlined by Lo� pez-Fanjul et al. (1999). Wefocus on a set of epistatic models that have beenproposed as those most likely for a binary disease trait.For those models, the behaviour of the components ofthe genetic variance in large panmictic populations hasbeen recently investigated by Tiwari and Elston (1998).

THE MODEL

We consider the variation due to segregation at twoneutral independent loci (A and B) at Hardy�Weinbergequilibrium. At each locus there are two alleles, with fre-quencies p1 (q1=1&p1) and p2 (q2=1&p2) at locus A

50

and B, respectively.Six epistatic models were studied (Table I). In these,

only two genotypic values are possible (1 and 1&s, 0<s�1), where s can be taken as the penetrance, equal for

TABLE I

Genotypic Values for Different Two-Loci Epistatic Models (0<s�1)

Model A1A1 A1A2 A1A2 A2A2

I B1B1 1 1 1&sB1B2 1 1 1&sB2B2 1&s 1&s 1&s

II B1B1 1 1 1B1B2 1 1 1B2B2 1 1 1&s

III B1B1 1 1 1B1B2 1 1 1B2B2 1&s 1&s 1

IV B1B1 1&s 1 1B1B2 1&s 1 1B2B2 1&s 1&s 1

V B1B1 1 1 1&sB1B2 1 1&s 1&sB2B2 1&s 1&s 1&s

VI B1B1 1 1 1&sB1B2 1 1 1&sB2B2 1&s 1&s 1

all susceptible genotypes. Models I, II, and III correspondto classical epistatic models with F2 segregation ratiosequal to 9 :7 (model I, duplicate recessive genes), 15:1(model II, duplicate dominant genes), and 13:3 (modelIII, dominant and recessive interaction) (Phillips, 1998).

The mean and the additive variance in an infinitelylarge panmictic population (ancestral mean M andvariance VA) are compared to their expected valuesat equilibrium, after t consecutive bottlenecks of Nrandomly sampled parents each (derived mean Mt* andvariance V*At). For convenience, the change in thepopulation mean after bottlenecks will be expressed as2Mt=Mt*&M and that of the additive variance asV*At �VA . For any set of genotypes considered (Table I),the average effect of gene substitution at each locus (:and ;, respectively) can be obtained from the corre-sponding marginal genotypic values (Crow and Kimura,1970, p. 125), and VA is given by

VA=2:2p1q1+2;2p2q2 .

We can also compute the rate of divergence betweenlines V(Mt), all of them independently started from theancestral population and subsequently maintained with

Lo� pez-Fanjul, Ferna� ndez, and Toro

equal effective size N in each of t consecutive generations.For each genetic model in Table I, this can be accom-plished by taking variances V(M) in the correspondingexpression giving the ancestral population mean M.

In general, equations for M, VA , and V(M) are polyno-mial functions of pk

i (i=1, 2; k=1�4). Expressions forMt*, V*At , and V(Mt) can be readily obtained by sub-stituting pk

i in M, VA , and V(M) by the exact kth momentof the allelic frequency distribution with binomial sampl-ing, given by Crow and Kimura (1970, p. 335).

Six models complementary to those in Table I, withthe genotypic values 1 and 1&s interchanged, can alsobe studied (models VII to XII). Of course, equationsgiving VA , V*At , and V(Mt) are the same for each modeland its complement. However, for a given set of allele fre-quency values, 2Mt has, in both cases, the same absolutevalue but opposite sign. For all models, expressions forVA are given by Tiwari and Elston (1998). Models I andVI (and complementary models VII and XII) are par-ticular cases of the multiple dominant (recessive)genotype favoured model, with diminishing epistasis,studied by Lo� pez-Fanjul et al. (1999), where the corre-sponding behaviour of 2Mt , V(Mt), and V*At �VA hasbeen analysed.

The change of the remaining components of the totalgenetic variance after bottlenecks (dominance, additive_additive, additive_dominance, and dominance_domi-nance) can also be obtained by the same procedure usedfor the additive variance. Formulae for the ancestralvalues of those components are given in Tiwari andElston (1998) and Lo� pez-Fanjul et al. (1999).

ANALYTICAL RESULTS

To illustrate the general procedure followed, onlyresults pertaining to model III will be presented. Theaverage effects of gene substitution at each locus are

:=&sq1q22 and ;=s(1&q2

1) q2 .

Thus, the effect s becomes a scale factor.The ancestral additive variance is given by

VA=2s2p1 q31q4

2+2s2(1&q21)2 p2 q3

2 , (1)

in agreement with Tiwari and Elston (1998), after cor-recting for a typographical error.

Differentiating expression (1) with respect to q1 and q2

and setting the two equations equal to zero, maxima of

Epistasis at Population Bottlenecks

the VA surface are given by

q41 q2

2 (3&4q1)= p21 p2 (4q2&3)(1+q1)3.

It follows that there are two equal maxima when onelocus is fixed and the negative allele at the othersegregates at a frequency 3�4. There are no minima but,of course, VA=0 for q1=0 and q2=1, and also for q1=1or q2=0 as, in these cases, all genotypic values at thesegregating locus are equal.

Equation (1) can also be expressed as

VA=2s2[(q31&q4

1) q42+(q3

2&q42)(1+q4

1&2q21)].

To obtain the derived additive variance V*At after t con-secutive bottlenecks of equal size N, we substitute qk

i bythe corresponding expected values (Crow and Kimura,1970),

E(q2i )=qi&pi qi* t

2 ,

E(q3i )=qi&

32 pi qi *t

2& 12 pi qi (2qi&1) * t

3 ,

and

E(q4i )=q i&\18N&11

10N&6 + piqi* t2&piqi (2qi&1) * t

3

+piqi _piqi&\ 2N&110N&6+& * t

4 ,

where *2=1&1�2N, *3=*2 (1&2�2N), and *4=*3 (1&3�2N) are the roots of the transition matrix for the allelefrequency moments. The previous equations giving E(qk

i )can also be expressed in terms of the inbreeding coef-ficient after t generations (Ft=1&* t

2) and, therefore, canalso be applied when bottleneck sizes are not constantfrom generation to generation. When N is not too small(N>10; Crow and Kimura, 1970), * t

3 and * t4 can be

approximated by (1&Ft)3 and (1&Ft)

6, and the ratios(18N&11)�(10N-6) and (2N-1)�(10N-6) by 9�5 and 1�5,respectively. Thus, a single parameter describes theoutcome of an arbitrary bout of random drift.

As indicated, an expression for V*At can easily beobtained but the inequality V*At>VA becomes analyti-cally intractable, even in the simple case of a singlebottleneck and equal frequencies at both loci. Of course,numerical solutions for any combination of allelefrequencies can be computed from the corresponding

51

formula.In parallel, the dominance VD , additive_additive VAA ,

additive_dominance VAD , and dominance_dominanceVDD , ancestral components of variance are given by

VD=:2p21+;2p2

2 ,

VAA=4s2p1 q31 p2q3

2 ,

VAD=2s2p1q21 p2q2

2 ( p1q2+q1 p2),

and

VDD=s2p21q2

1 p22q2

2 ,

in agreement with those obtained by Tiwari and Elston(1998) using a different procedure. Thus, expressions forV*Dt , V*AAt , V*ADt , and V*DDt can also be derived by sub-stituting pk

i by its expected value, but they can only bemanaged numerically.

From the genotypic values in Table I, the ancestralpopulation mean is given by

M=1&sq22(1&q2

1). (2)

Taking expectations in (2) we obtain the change in meanafter t bottlenecks

2Mt=&sp1q2 (1&* t2)[ p2&q1 [1& p2 (1+* t

2)]], (3)

where the coefficient of the quadratic term *2t2 is equal to

the dominance_dominance standard deviation, asindicated by Crow and Kimura (1970). From Eq. (3), itcan be easily shown that 2Mt<0 only if q1< p2 �[1& p2 (1+* t

2)], implying p2<1�(2+*t2). Thus, the

maximum frequency of the positive allele B1 giving 2Mt<0is quite insensitive to both bottleneck size and number,ranging from 1�3 (t=1, N=�) to 1�2 (t=�, N=2).For complementary model IX, the positive allele is B2

and 2Mt<0 for q2>(1+*t2)�(2+* t

2), ranging fromq2<1�2 (t=�, N=2) to q2<2�3 (t=1, N=�). Atlocus A, however, the condition 2Mt<0 is compatiblewith all possible allele frequencies.

Maxima and minima of expression (3) are given by

q1 (q2&p2)[ p2&(q1&p1)]

= p2(q1&p1)[q1+(q2&p2)].

It follows that there are one maximum ( p1=q1 , q2=1)and one minimum ( p2=q2 , q1=0). Of course, q1=1 orq2=0 give 2Mt<0, as all genotypic values at thesegregating locus are equal in these cases.

Taking variances in expression (2), we have

52

V(M)=s2[V(q22)+[E(q2

1)]2 V(q22)+[E(q2

2)]2 V(q21)

+V(q21) V(q2

2)&2E(q21) V(q2

2)].

The between-line variance V(Mt) after t consecutivebottlenecks can be obtained by substituting V(q2

i )=E(q4i )

&[E(q2i )]2. Although the equation giving V(Mt) is analy-

tically unmanageable, numerical solutions for anycombination of allele frequencies can be computed fromthe formula.

NUMERICAL EVALUATION

For each epistatic model in Table I (s=1) and allpossible combinations of allele frequencies at both loci,surfaces were represented giving the correspondingvalues of the following parameters: (1) change of themean after 1 bottleneck 2M1 (N=2), (2) ancestraladditive variance VA , (3) ratio of derived to ancestraladditive variances V*A1 �VA after 1 bottleneck (N=2), and(4) between-line variance after 1 bottleneck V(M1) or 10bottlenecks V(M10) (N=2).

Change of the Mean

Surfaces giving the change of the mean after onebottleneck are shown in Fig. 1. Those corresponding tosuccessive bottlenecks (N=2) had the same shape butthe magnitude of the change increased with bottlenecknumber (not shown).

Inbreeding depression after bottlenecks, unconditionalto allele frequencies, was only observed for epistaticmodels I and II. This implies, however, that an enhance-ment of the mean after bottlenecks will always ensue forcomplementary models VII and VIII. In the remainingcases, inbreeding depression was generally restricted torelatively large frequencies of the positive allele B1

(models III�VI) or B2 (complementary models IX�XII).

Change of the Additive Variance

The ancestral additive variance is plotted against two-locus allele frequencies in Figs. 2a�4a. For models I�V,the VA surface shows no minima and two equal maxima,when one locus is fixed and the recessive (dominant)negative allele at the other locus segregates at a frequency3�4 (1�4). For model VI, there are four equal maxima,when one locus is fixed and the other segregates at a fre-quency 3�4, and a minimum at q1=q2=3�4, as shown by


Lo� pez-Fanjul et al. (1999).After a bottleneck (Figs. 2b�4b), the derived variance

exceeds the ancestral one (V*A1�VA>1) only for thosecombinations of allele frequencies resulting in lower


FIG. 1. Change of the mean after one bottleneck (N=2) plotted againcorrespond to 2M1>0 (models III�VI).

53

st two-locus allele frequencies for epistatic models I�VI. Darker zones

54

FIG. 2. Ancestral additive variance (a), ratio of derived to ancestral adafter 1 (c) or 10 (d) bottlenecks (N=2), plotted against two-locus allele freqratios greater than one.


ditive variances after 1 bottleneck (b, N=2), and between-line varianceuencies, for models I and II. Darker zones in (b) correspond to variance


FIG. 3. Ancestral additive variance (a), ratio of derived to ancestral adafter 1 (c) or 10 (d) bottlenecks (N=2), plotted against two-locus allele frequratios greater than one.

55

ditive variances after 1 bottleneck (b, N=2), and between-line varianceencies, for models III and IV. Darker zones in (b) correspond to variance

56

FIG. 4. Ancestral additive variance (a), ratio of derived to ancestral adafter 1 (c) or 10 (d) bottlenecks (N=2), plotted against two-locus allele frequratios greater than one.


ditive variances after 1 bottleneck (b, N=2), and between-line varianceencies, for models V and VI. Darker zones in (b) correspond to variance

values of VA . Milder bottlenecks (N>2, not shown) gavethe same qualitative results, but the variance ratiodecreased as the size of the bottleneck increased. Withsuccessive bottlenecks of equal size, the shape of theV*At �VA surface did not change, but the correspondingvalues decreased (not shown).

When the ancestral frequencies of the positive allele ateach locus are high (A1 , models I, II, V, VI, IX, and X;A2 , models III, IV, VII, VIII, XI, and XII; B1 , modelsI�VI; B2 , models VII�XII), an excess of the derivedadditive variance over its ancestral value was alwaysobserved; however, it was invariably accompanied byinbreeding depression. On the other hand, low ancestralfrequencies of the positive allele resulted in both anexcess of the additive variance and an enhancement ofthe mean after bottlenecks. The only exceptions to thatgeneral pattern were models I and II, where no excess ofthe additive variance was detected at low frequencies,and complementary models VII and VIII, where nodepression of the mean occurs for any combination ofallele frequencies.

Between-Line Variance

For each type of epistasis considered, the surfacesgiving the ancestral additive variance (Figs. 2a�4a) andthe between-line variance after one bottleneck (N=2,Figs. 2c�4c) have the same shape but, in all cases, the valuesof the latter were, approximately, one-half those of theformer. Thus, at low levels of inbreeding, the between-line variance with epistasis behaves similarly to that withpure additive gene action as, at that stage, the contribu-tion of the ancestral non-additive variance to thebetween-line variance is always small.

For models I�V, the shape of the between-linevariance surface did not change much after consecutivebottlenecks, but the dynamics of the redistributionprocess was much slower than expected with additivegene action (Figs. 2d�4d). Thus, after 10 bottlenecks(N=2), the between-line variance was about one-half ofthe additive expectation V(Mt)=2(1&*$2) VA , givingV(M10) = 1.89 VA . Model VI differs from the rest inhaving: (1) a two-peaked ancestral mean surface of thesaddle type (single peaked in all other models), and (2)minimum VA and maximum V*At�VA for the allele frequen-cies determining the saddle (q1=q2=3�4). In this instance,after 2�3 bottlenecks (N=2, not shown) the only impor-


tant change of the between-line variance surface was theconversion of the initial minimum to a level top. There-after, the shape of the between-line variance surface didnot change substantially (Fig. 4d).

DISCUSSION

For all models considered, an increase in the additivevariance after consecutive bottlenecks of equal size willoccur only if its ancestral value is very small. In thisinstance, the difference between the additive and non-additive ancestral components of variance is large and,therefore, the potential for conversion of the second to thefirst is also large. However, the magnitude of the excesswas inversely related both to bottleneck size and to thenumber of bottlenecks.

There are two situations of interest, characterized bythe frequency values of the positive allele at each locus. Athigh frequencies, an increase of the derived additivevariance relative to the ancestral value was generallyfound, but it was invariably accompanied by stronginbreeding depression. In practice, these cases reduce to:(1) one (model IV) or two simple dominant loci (modelsI, V, and VI), with the recessive negative allele(s)segregating at low frequencies; (2) one (model X) or twosimple dominant loci (models VII, XI, and XII), with therecessive positive allele(s) segregating at high frequencies;(3) all genotypes with appreciable frequencies havingequal values and, therefore, generating very little variance(models II, III, VIII, and IX). Thus, dominance can beconsidered the primary cause of the increase of additivevariance after bottlenecks, as previously shown byRobertson (1952) and Willis and Orr (1993) for one-locusmodels under a broad range of dominance coefficients. Inthe symmetrical case of low frequencies of the positiveallele at each locus, an excess of the derived variance overits ancestral value was always associated with an enhance-ment of the mean after bottlenecks.

Strictly, our analysis assumes neutrality, but the con-clusions can be qualitatively extended to fitness. In thissituation, it is difficult to conceive simultaneous segrega-tion of low-frequency favourable recessives at a number ofloci, except with strong genotype�environment interac-tion converting harmful alleles to beneficial ones. Inparallel, for recessive deleterious alleles at low fre-quencies, the increase of the additive variance afterbottlenecks will always be penalised by strong inbreedingdepression. Moreover, models III�VI (and complemen-tary models IX�XII) are highly unrealistic, as low-fre-quency values of the favourable alleles at both loci resultin an enhancement of the mean after bottlenecks. ModelsVII and VIII are extreme cases in which inbreedingdepression is not observed for any allele frequency com-

57

bination. This undesirable property contradicts theubiquitous observation of fitness-related traits beingsubjected to inbreeding depression (Charlesworth andCharlesworth, 1987). Thus, the above-mentioned models

are unsuitable for the genetic analysis of disorders whichdo not follow simple Mendelian single-locus inheritance(see Tiwari and Elston, 1998, for references).

For additive gene action within and between loci, thebetween-line variance after a bottleneck (N =2) is equalto VA�2. This prediction holds approximately for theepistatic models studied. The reason for that is that onlysmall fractions of the non-additive components ofvariance contribute to the between-line variance, thosecomponents being generally smaller than the additive one.Thus, at low levels of inbreeding, one can safely make thegeneralization that the behaviour of the between-linevariance will not be greatly affected by the type of geneaction of the loci involved. After several bottlenecks,however, differentiation proceeds at a much slower ratethan predicted by the additive model in all cases.

The behaviour of the additive variance afterbottlenecks has also been studied by Cheverud andRoutman (1996) for specific two-loci models in whichgenotypic values equal the epistasis values, concludingthat the derived additive variance will always exceed theancestral variance after one or several bottlenecks.However, these models are very restrictive, implying: (1)minimum ancestral additive variance with intermediatefrequencies at both loci, and (2) underdominance oroverdominance at one or both loci considered, withadditive_dominance or dominance_dominance epistasis,respectively. Those models are also unrealistic, as theyimply an enhancement of the population mean withinbreeding (additive_dominance epistasis, for q1 >1�2;dominance_dominance epistasis, for extreme frequenciesat any one locus). Moreover, Cheverud and Routmanconsidered only the special case of an ancestral populationsegregating with intermediate frequencies at both loci,i.e., the case of maximum potential for conversion ofnon-additive to additive variance. These limitations havea marked influence on the dynamics of the additivevariance in bottleneck populations. As we have shown,their conclusion of the additive variance invariablyincreasing after bottlenecks cannot be extended to otherepistatic models, where an excess of the additive variancewas only detected for specific combinations of allelefrequencies, which may be extreme (models I, IV, V, andVI, and complementary models VII, X, XI, and XII) orintermediate at one (model III and complementarymodel IX) or both loci involved (model II and com-plementary model VIII). For other epistatic models(additive_additive, multiple dominant genotype favoured,

58

and Dobzhansky-Muller type), similar conclusions havebeen obtained (Lo� pez-Fanjul et al., 1999).

The effect of linkage on the ratio V*At�VA has beenstudied for additive_additive epistasis (Goodnight,

1988; Tachida and Cockerham, 1989) and rare non-epistatic recessives (Wang et al., 1998). In both instances,linkage equilibrium in the ancestral population has beenassumed: i.e., disequilibrium is only due to sampling. Foradditive_additive epistasis, recombination does notaffect the contribution of the ancestral additive varianceto either the derived additive variance or the between-line variance. In parallel, large recombination ratesincrease the contribution of the ancestral epistaticvariance to the derived additive variance, but decrease itto the between-line variance. Nevertheless, both effectswere shown to be small. With rare recessives, linkage dis-equilibrium can lead only to a small increase in V*At�VA ,above that expected for drift alone. In this case, thebetween-line variance is not affected by linkage dis-equilibrium generated by sampling (Avery and Hill,1979). Interestingly, the excess in the additive varianceafter bottlenecks induced by additive_additive epistasisdeclines with decreasing recombination rates, but thereverse was found for rare recessives. Thus, for morecomplex systems, as those considered in this paper, thoseeffects will tend to cancel each other. Moreover, linkagedisequilibriumgeneratedby drift will be transient, so that theoutcome some time after a bottleneck depends only on thedistribution of allele frequencies. Therefore, it is unlikelythat linkage can qualitatively affect our conclusions.

An extension of our results to multilocus systemsrequires a complete specification of genotypic effects andallele frequencies, as differences in any of these factorscan even change the sign of the contribution of specificloci to the total additive variance after bottlenecks. Thus,inferences from theory to experimental data can only bemade if individual loci show the same type of gene actionand segregate with similar frequencies. A review of perti-nent Drosophila and Tribolium results can be found inLo� pez-Fanjul et al. (1999). It was concluded that, formorphological traits such as bristle number, where mostof the genetic variance in natural populations has beenshown to be contributed by quasi-neutral additive allelessegregating at intermediate frequencies, no inbreedingdepression was detected and the behaviour of the within-line additive variance after bottlenecks very closelyapproached the expectation under the pure additivemodel. Recent data on wing size and shape traits in D.melanogaster also conform with those predictions(Whitlock and Fowler, 1999). For these traits, inbreed-ing depression after a single bottleneck (N=2) was lessthan 10 of the outbred mean, and the additive variance


decreased proportionately to the inbreeding coefficient.This result is in agreement with an analysis of polygenesaffecting wing shape in chromosome 3 in D. melanogaster(Weber et al., 1999), showing that the vast majority of

the trait variation in the population considered wasexplained by additive effects, epistatic variation beingminor. It is also consistent with spontaneous mutationsaffecting wing length and width showing predominantadditive gene action (Santiago et al., 1992). Data on micepopulations passing through four consecutive bottle-necks (N=4) have been reported for 10-week bodyweight (Cheverud et al., 1999). No inbreeding depressionnor heterosis was detected, suggesting additive geneaction at most loci involved. However, the averagewithin-line full-sib component of variance did notsignificantly differ from that of an outbred control, andwas significantly larger than expected under a purelyadditive model. This result has been considered evidencefor epistatic gene action, reducing the loss of additivegenetic variance under inbreeding. Notwithstanding, thefull-sib component of variance not only contains one-halfof the additive variance, but also part of the non-additivevariance, as well as the whole component of variance dueto environmental effects common to full-sibs (includingmaternal effects). Thus, a more parsimonious interpreta-tion of the behaviour of the full-sib component ofvariance after bottlenecks, is an increase of the common-environment component of variance in the inbred lines,relative to that of the control (see review by Falconer andMackay, 1996, pp. 267�269). On the other hand, fitness-component traits as viability, where most geneticvariance in natural populations is due to partially (ortotally) recessive deleterious alleles segregating at lowfrequencies, show strong inbreeding depression and theadditive variance after bottlenecks exceeded the ancestralvalue. For both kinds of traits, the between-line varianceconformed with the additive predictions as indicated byour analysis and Lo� pez-Fanjul et al. (1999) results. Areview covering many species (Fowler and Whitlock,1999) also points out that fitness-component traits aremuch more likely to increase in phenotypic variance afterbottlenecks than morphological traits.

Summarizing, for the epistatic models considered,although occasional increases in the derived additivevariance can be observed, it is unlikely that the rate ofevolution may be accelerated after population bottle-necks, unless unrealistic parameter values are assumed.This conclusion is in agreement with the results obtainedby Lo� pez-Fanjul et al. (1999) for other two-loci epistaticsystems.


ACKNOWLEDGMENT

This study was supported by a grant from the Direccio� n General deInvestigacio� n Cient@� fica y Te� cnica (PB98-0814-C03-01).

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Epistasis and the Conversion of Non-additive to Additive Genetic Variance at Population Bottlenecks