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

Theoretical Population Biology 58, 4959 (2000) doi:10.1006tpbi.2000.1470, available online at http:www.idealibrary.com on

Epistasis and the Conversion of Non-additive to Additive Genetic Variance at Population Bottlenecks Carlos Lopez-Fanjul 1 Departamento de Genetica, Facultad de Ciencias Biologicas, Universidad Complutense, 28040 Madrid, Spain

and Almudena Fernandez and Miguel A. Toro Departamento de Mejora Genetica y Biotecnolog@ a, SGIT-INIA, Carretera de La Corun~a km. 7, 28040 Madrid, Spain Received August 18, 1999

The effect of population bottlenecks on the mean and the additive variance generated by two neutral independent epistatic loci has been studied theoretically. Six epistatic models, used in the analysis of binary disease traits, were considered. Ancestral values in an infinitely large panmictic population were compared with their expectations at equilibrium, after t consecutive bottlenecks of equal size N (derived values). An increase in the additive variance after bottlenecks (inversely related to N and t) will occur only if the frequencies of the negative allele at each locus are: (1) low, invariably associated to strong inbreeding depression; (2) high, always accompanied by an enhancement of the mean with inbreeding. The latter is an undesirable property, making the pertinent models unsuitable for the genetic analysis of disease. For the epistatic models considered, it is unlikely that the rate of evolution may be accelerated after population bottlenecks, in spite of occasional increments of the derived additive variance over its ancestral value. ] 2000 Academic Press

implications of epistasis have been specially emphasized (see Wade and Goodnight, 1998, for a comprehensive review). However, epistasis is not a necessary condition for the conversion of non-additive to additive variance, as dominance can also result in increased additive variance following population bottlenecks (Robertson, 1952; Willis and Orr, 1993). Moreover, for those models implying dominance at the single-locus level, additional epistasis does not greatly affect the value of the additive variance after bottlenecks over the corresponding singlelocus expectations, as those combinations of allele frequencies resulting in increased additive variance also result in small epistatic variance (Lopez-Fanjul et al., 1999). Experimentally, the additive variance of

INTRODUCTION The evolutionary relationship between the additive and non-additive components of the genetic variance of quantitative traits in subdivided populations has recently been the object of considerable theoretical and experimental research. Within small demes, random genetic drift has been shown to convert non-additive to additive variance, and this phenomenon has been assumed to increase the potential for adaptation to local environments and, therefore, to enhance the genetic differentiation among demes. In this context, the role and 1 To whom correspondence should be addressed. E-mail: clfanjul eucmax.sim.ucm.es.

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0040-580900 K35.00 Copyright ] 2000 by Academic Press All rights of reproduction in any form reserved.

50 viability has been shown to increase after inbreeding, but the relative contributions of dominance and epistasis could not be disentangled. Notwithstanding, using empirical mutational parameters and the equilibrium gene frequency distribution in large populations under mutationselection balance, Wang et al. (1999) were able to show that, for Drosophila melanogaster viability, the observed changes in mean, additive variance and between-line variance following bottlenecks can be mainly attributed to lethals and partial recessive mutations of large effect. Therefore, dominance can be considered the primary cause of an increase in additive variance after bottlenecks. In parallel, theoretical investigation of a wide range of two-locus epistatic models indicated that enhanced additive variance after bottlenecks will occur only for simultaneous segregation at both loci of unfavourable alleles with intermediate frequencies, or of favourable recessives at low frequencies, and will be accompanied by large inbreeding depression (Lopez-Fanjul et al., 1999). Thus, epistasis is not a sufficient condition for conversion of non-additive to additive variance either, as those allele frequencies cannot easily be conceived in natural populations undergoing selection, unless there is strong genotypeenvironment interaction implying a reversal of the sign of the allelic effects. With this possible exception, it is unlikely that the rate of evolution may be accelerated after population bottlenecks, in spite of occasional increments of the additive variance over its ancestral value. In this paper, we have theoretically investigated the effect of successive population bottlenecks on the mean, the additive variance, and the between-line variance generated by two-loci epistatic systems, following the approach outlined by Lopez-Fanjul et al. (1999). We focus on a set of epistatic models that have been proposed as those most likely for a binary disease trait. For those models, the behaviour of the components of the genetic variance in large panmictic populations has been recently investigated by Tiwari and Elston (1998).

THE MODEL

Lopez-Fanjul, Fernandez, and Toro TABLE I Genotypic Values for Different Two-Loci Epistatic Models (0
A1 A1

A1 A 2

A1 A 2

A2 A 2

I

B1 B1 B1 B2 B2 B2

1 1 1&s

1 1 1&s

1&s 1&s 1&s

II

B1 B1 B1 B2 B2 B2

1 1 1

1 1 1

1 1 1&s

III

B1 B1 B1 B2 B2 B2

1 1 1&s

1 1 1&s

1 1 1

IV

B1 B1 B1 B2 B2 B2

1&s 1&s 1&s

1 1 1&s

1 1 1

V

B1 B1 B1 B2 B2 B2

1 1 1&s

1 1&s 1&s

1&s 1&s 1&s

VI

B1 B1 B1 B2 B2 B2

1 1 1&s

1 1 1&s

1&s 1&s 1

all susceptible genotypes. Models I, II, and III correspond to classical epistatic models with F 2 segregation ratios equal to 9 : 7 (model I, duplicate recessive genes), 15 : 1 (model II, duplicate dominant genes), and 13 : 3 (model III, dominant and recessive interaction) (Phillips, 1998). The mean and the additive variance in an infinitely large panmictic population (ancestral mean M and variance VA ) are compared to their expected values at equilibrium, after t consecutive bottlenecks of N randomly sampled parents each (derived mean M t* and variance V* At ). For convenience, the change in the population mean after bottlenecks will be expressed as 2M t =M t*&M and that of the additive variance as V* 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 corresponding marginal genotypic values (Crow and Kimura, 1970, p. 125), and VA is given by VA =2: 2p 1 q 1 +2; 2p 2 q 2 .

We consider the variation due to segregation at two neutral independent loci (A and B) at HardyWeinberg equilibrium. At each locus there are two alleles, with frequencies p 1 (q 1 =1&p 1 ) and p 2 (q 2 =1&p 2 ) at locus A and B, respectively. Six epistatic models were studied (Table I). In these, only two genotypic values are possible (1 and 1&s, 0< s1), where s can be taken as the penetrance, equal for

We can also compute the rate of divergence between lines V(M t ), all of them independently started from the ancestral population and subsequently maintained with equal effective size N in each of t consecutive generations. For each genetic model in Table I, this can be accomplished by taking variances V(M) in the corresponding expression giving the ancestral population mean M.

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Epistasis at Population Bottlenecks

In general, equations for M, VA , and V(M) are polynomial functions of p ki (i=1, 2; k=14). Expressions for M t*, V* At , and V(M t ) can be readily obtained by substituting p ki in M, VA , and V(M) by the exact k th moment of the allelic frequency distribution with binomial sampling, given by Crow and Kimura (1970, p. 335). Six models complementary to those in Table I, with the genotypic values 1 and 1&s interchanged, can also be studied (models VII to XII). Of course, equations giving VA , V* At , and V(M t ) are the same for each model and its complement. However, for a given set of allele frequency values, 2M t has, in both cases, the same absolute value but opposite sign. For all models, expressions for VA are given by Tiwari and Elston (1998). Models I and VI (and complementary models VII and XII) are particular cases of the multiple dominant (recessive) genotype favoured model, with diminishing epistasis, studied by Lopez-Fanjul et al. (1999), where the corresponding behaviour of 2M t , V(M t ), and V* At VA has been analysed. The change of the remaining components of the total genetic variance after bottlenecks (dominance, additive_ additive, additive_dominance, and dominance_dominance) can also be obtained by the same procedure used for the additive variance. Formulae for the ancestral values of those components are given in Tiwari and Elston (1998) and Lopez-Fanjul et al. (1999).

It follows that there are two equal maxima when one locus is fixed and the negative allele at the other segregates at a frequency 34. There are no minima but, of course, VA =0 for q 1 =0 and q 2 =1, and also for q 1 =1 or q 2 =0 as, in these cases, all genotypic values at the segregating locus are equal. Equation (1) can also be expressed as VA =2s 2[(q 31 &q 41 ) q 42 +(q 32 &q 42 )(1+q 41 &2q 21 )]. To obtain the derived additive variance V* At after t consecutive bottlenecks of equal size N, we substitute q ki by the corresponding expected values (Crow and Kimura, 1970), E(q 2i )=q i & p i q i * t2 , E(q 3i )=q i & 32 p i q i * t2 & 12 p i q i (2q i &1) * t3 , and

i

ANALYTICAL RESULTS To illustrate the general procedure followed, only results pertaining to model III will be presented. The average effects of gene substitution at each locus are := &sq 1 q 22

and

;=s(1&q 21 ) q 2 .

Thus, the effect s becomes a scale factor. The ancestral additive variance is given by VA =2s 2 p 1 q 31 q 42 +2s 2(1&q 21 ) 2 p 2 q 32 ,

(1)

in agreement with Tiwari and Elston (1998), after correcting for a typographical error. Differentiating expression (1) with respect to q 1 and q 2 and setting the two equations equal to zero, maxima of the VA surface are given by q 41 q 22 (3&4q 1 )= p 21 p 2 (4q 2 &3)(1+q 1 ) 3.

18N&11

\ 10N&6 + p q * & p q (2q &1) * 2N&1 +p q p q & _ \10N&6+& * ,

E(q 4i )=q i &

i

i

i

i

i

t 2

i

i

i

t 3

t 4

where * 2 =1&12 N, * 3 =* 2 (1&22N), and * 4 =* 3 (1& 32N) are the roots of the transition matrix for the allele frequency moments. The previous equations giving E(q ki ) can also be expressed in terms of the inbreeding coefficient after t generations (F t =1&* t2 ) and, therefore, can also be applied when bottleneck sizes are not constant from generation to generation. When N is not too small (N>10; Crow and Kimura, 1970), * t3 and * t4 can be approximated by (1&F t ) 3 and (1&F t ) 6, and the ratios (18N&11)(10N-6) and (2N-1)(10N-6) by 95 and 15, respectively. Thus, a single parameter describes the outcome of an arbitrary bout of random drift. As indicated, an expression for V* At can easily be obtained but the inequality V* >V becomes analytiAt A cally intractable, even in the simple case of a single bottleneck and equal frequencies at both loci. Of course, numerical solutions for any combination of allele frequencies can be computed from the corresponding formula. In parallel, the dominance VD , additive_additive VAA , additive_dominance VAD , and dominance_dominance VDD , ancestral components of variance are given by

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Lopez-Fanjul, Fernandez, and Toro

VD =: 2p 21 +; 2p 22 , 2

3 1

The between-line variance V(M t ) after t consecutive bottlenecks can be obtained by substituting V(q 2i )=E(q 4i ) &[E(q 2i )] 2. Although the equation giving V(M t ) is analytically unmanageable, numerical solutions for any combination of allele frequencies can be computed from the formula.

3 2

VAA =4s p 1 q p 2 q , VAD =2s 2 p 1 q 21 p 2 q 22 ( p 1 q 2 +q 1 p 2 ), and VDD =s 2 p 21 q 21 p 22 q 22 , in agreement with those obtained by Tiwari and Elston (1998) using a different procedure. Thus, expressions for V* Dt , V* AAt , V* ADt , and V* DDt can also be derived by substituting p ki by its expected value, but they can only be managed numerically. From the genotypic values in Table I, the ancestral population mean is given by M=1&sq 22(1&q 21 ).

(2)

Taking expectations in (2) we obtain the change in mean after t bottlenecks 2M t = &sp 1 q 2 (1&* t2 )[ p 2 &q 1 [1& p 2 (1+* t2 )]],

(3)

where the coefficient of the quadratic term * 2t 2 is equal to the dominance_dominance standard deviation, as indicated by Crow and Kimura (1970). From Eq. (3), it can be easily shown that 2M t <0 only if q 1 < p 2  [1& p 2 (1+* t2 )], implying p 2 <1(2+* t2 ). Thus, the maximum frequency of the positive allele B1 giving 2Mt <0 is quite insensitive to both bottleneck size and number, ranging from 13 (t=1, N=) to 12 (t=, N=2). For complementary model IX, the positive allele is B 2 and 2M t <0 for q 2 >(1+* t2 )(2+* t2 ), ranging from q 2 <12 (t=, N=2) to q 2 <23 (t=1, N=). At locus A, however, the condition 2M t <0 is compatible with all possible allele frequencies. Maxima and minima of expression (3) are given by q 1 (q 2 &p 2 )[ p 2 &(q 1 &p 1 )] = p 2(q 1 &p 1 )[q 1 +(q 2 &p 2 )]. It follows that there are one maximum ( p 1 =q 1 , q 2 =1) and one minimum ( p 2 =q 2 , q 1 =0). Of course, q 1 =1 or q 2 =0 give 2M t <0, as all genotypic values at the segregating locus are equal in these cases. Taking variances in expression (2), we have V(M)=s 2[V(q 22 )+[E(q 21 )] 2 V(q 22 )+[E(q 22 )] 2 V(q 21 ) +V(q 21 ) V(q 22 )&2E(q 21 ) V(q 22 )].

NUMERICAL EVALUATION For each epistatic model in Table I (s=1) and all possible combinations of allele frequencies at both loci, surfaces were represented giving the corresponding values of the following parameters: (1) change of the mean after 1 bottleneck 2M 1 (N=2), (2) ancestral additive variance VA , (3) ratio of derived to ancestral additive variances V* A1 VA after 1 bottleneck (N=2), and (4) between-line variance after 1 bottleneck V(M 1 ) or 10 bottlenecks V(M 10 ) (N=2). Change of the Mean Surfaces giving the change of the mean after one bottleneck are shown in Fig. 1. Those corresponding to successive bottlenecks (N=2) had the same shape but the magnitude of the change increased with bottleneck number (not shown). Inbreeding depression after bottlenecks, unconditional to allele frequencies, was only observed for epistatic models I and II. This implies, however, that an enhancement of the mean after bottlenecks will always ensue for complementary models VII and VIII. In the remaining cases, inbreeding depression was generally restricted to relatively large frequencies of the positive allele B 1 (models IIIVI) or B 2 (complementary models IXXII). Change of the Additive Variance The ancestral additive variance is plotted against twolocus allele frequencies in Figs. 2a4a. For models IV, 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 frequency 34 (14). For model VI, there are four equal maxima, when one locus is fixed and the other segregates at a frequency 34, and a minimum at q 1 =q 2 =34, as shown by Lopez-Fanjul et al. (1999). After a bottleneck (Figs. 2b4b), the derived variance exceeds the ancestral one (V* A1 VA >1) only for those combinations of allele frequencies resulting in lower

Epistasis at Population Bottlenecks

53

FIG. 1. Change of the mean after one bottleneck (N=2) plotted against two-locus allele frequencies for epistatic models IVI. Darker zones correspond to 2M 1 >0 (models IIIVI).

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Lopez-Fanjul, Fernandez, and Toro

FIG. 2. Ancestral additive variance (a), ratio of derived to ancestral additive variances after 1 bottleneck (b, N=2), and between-line variance after 1 (c) or 10 (d) bottlenecks (N=2), plotted against two-locus allele frequencies, for models I and II. Darker zones in (b) correspond to variance ratios greater than one.

Epistasis at Population Bottlenecks

55

FIG. 3. Ancestral additive variance (a), ratio of derived to ancestral additive variances after 1 bottleneck (b, N=2), and between-line variance after 1 (c) or 10 (d) bottlenecks (N=2), plotted against two-locus allele frequencies, for models III and IV. Darker zones in (b) correspond to variance ratios greater than one.

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Lopez-Fanjul, Fernandez, and Toro

FIG. 4. Ancestral additive variance (a), ratio of derived to ancestral additive variances after 1 bottleneck (b, N=2), and between-line variance after 1 (c) or 10 (d) bottlenecks (N=2), plotted against two-locus allele frequencies, for models V and VI. Darker zones in (b) correspond to variance ratios greater than one.

57

Epistasis at Population Bottlenecks

values of VA . Milder bottlenecks (N>2, not shown) gave the same qualitative results, but the variance ratio decreased as the size of the bottleneck increased. With successive bottlenecks of equal size, the shape of the V* At VA surface did not change, but the corresponding values decreased (not shown). When the ancestral frequencies of the positive allele at each locus are high (A 1 , models I, II, V, VI, IX, and X; A 2 , models III, IV, VII, VIII, XI, and XII; B 1 , models IVI; B 2 , models VIIXII), an excess of the derived additive variance over its ancestral value was always observed; however, it was invariably accompanied by inbreeding depression. On the other hand, low ancestral frequencies of the positive allele resulted in both an excess of the additive variance and an enhancement of the mean after bottlenecks. The only exceptions to that general pattern were models I and II, where no excess of the additive variance was detected at low frequencies, and complementary models VII and VIII, where no depression of the mean occurs for any combination of allele frequencies.

Between-Line Variance For each type of epistasis considered, the surfaces giving the ancestral additive variance (Figs. 2a4a) and the between-line variance after one bottleneck (N=2, Figs. 2c4c) have the same shape but, in all cases, the values of the latter were, approximately, one-half those of the former. Thus, at low levels of inbreeding, the betweenline variance with epistasis behaves similarly to that with pure additive gene action as, at that stage, the contribution of the ancestral non-additive variance to the between-line variance is always small. For models IV, the shape of the between-line variance surface did not change much after consecutive bottlenecks, but the dynamics of the redistribution process was much slower than expected with additive gene action (Figs. 2d4d). Thus, after 10 bottlenecks (N=2), the between-line variance was about one-half of the additive expectation V(M t )=2(1&*$2 ) VA , giving V(M 10 ) = 1.89 VA . Model VI differs from the rest in having: (1) a two-peaked ancestral mean surface of the saddle type (single peaked in all other models), and (2) minimum VA and maximum V* At VA for the allele frequencies determining the saddle (q1 =q2 =34). In this instance, after 23 bottlenecks (N=2, not shown) the only important change of the between-line variance surface was the conversion of the initial minimum to a level top. Thereafter, the shape of the between-line variance surface did not change substantially (Fig. 4d).

DISCUSSION For all models considered, an increase in the additive variance after consecutive bottlenecks of equal size will occur only if its ancestral value is very small. In this instance, the difference between the additive and nonadditive ancestral components of variance is large and, therefore, the potential for conversion of the second to the first is also large. However, the magnitude of the excess was inversely related both to bottleneck size and to the number of bottlenecks. There are two situations of interest, characterized by the frequency values of the positive allele at each locus. At high frequencies, an increase of the derived additive variance relative to the ancestral value was generally found, but it was invariably accompanied by strong inbreeding depression. In practice, these cases reduce to: (1) one (model IV) or two simple dominant loci (models I, V, and VI), with the recessive negative allele(s) segregating at low frequencies; (2) one (model X) or two simple dominant loci (models VII, XI, and XII), with the recessive positive allele(s) segregating at high frequencies; (3) all genotypes with appreciable frequencies having equal values and, therefore, generating very little variance (models II, III, VIII, and IX). Thus, dominance can be considered the primary cause of the increase of additive variance after bottlenecks, as previously shown by Robertson (1952) and Willis and Orr (1993) for one-locus models under a broad range of dominance coefficients. In the symmetrical case of low frequencies of the positive allele at each locus, an excess of the derived variance over its ancestral value was always associated with an enhancement of the mean after bottlenecks. Strictly, our analysis assumes neutrality, but the conclusions can be qualitatively extended to fitness. In this situation, it is difficult to conceive simultaneous segregation of low-frequency favourable recessives at a number of loci, except with strong genotypeenvironment interaction converting harmful alleles to beneficial ones. In parallel, for recessive deleterious alleles at low frequencies, the increase of the additive variance after bottlenecks will always be penalised by strong inbreeding depression. Moreover, models IIIVI (and complementary models IXXII) are highly unrealistic, as low-frequency values of the favourable alleles at both loci result in an enhancement of the mean after bottlenecks. Models VII and VIII are extreme cases in which inbreeding depression is not observed for any allele frequency combination. This undesirable property contradicts the ubiquitous observation of fitness-related traits being subjected to inbreeding depression (Charlesworth and Charlesworth, 1987). Thus, the above-mentioned models

58 are unsuitable for the genetic analysis of disorders which do not follow simple Mendelian single-locus inheritance (see Tiwari and Elston, 1998, for references). For additive gene action within and between loci, the between-line variance after a bottleneck (N =2) is equal to VA 2. This prediction holds approximately for the epistatic models studied. The reason for that is that only small fractions of the non-additive components of variance contribute to the between-line variance, those components being generally smaller than the additive one. Thus, at low levels of inbreeding, one can safely make the generalization that the behaviour of the between-line variance will not be greatly affected by the type of gene action of the loci involved. After several bottlenecks, however, differentiation proceeds at a much slower rate than predicted by the additive model in all cases. The behaviour of the additive variance after bottlenecks has also been studied by Cheverud and Routman (1996) for specific two-loci models in which genotypic values equal the epistasis values, concluding that the derived additive variance will always exceed the ancestral variance after one or several bottlenecks. However, these models are very restrictive, implying: (1) minimum ancestral additive variance with intermediate frequencies at both loci, and (2) underdominance or overdominance at one or both loci considered, with additive_dominance or dominance_dominance epistasis, respectively. Those models are also unrealistic, as they imply an enhancement of the population mean with inbreeding (additive_dominance epistasis, for q 1 >12; dominance_dominance epistasis, for extreme frequencies at any one locus). Moreover, Cheverud and Routman considered only the special case of an ancestral population segregating with intermediate frequencies at both loci, i.e., the case of maximum potential for conversion of non-additive to additive variance. These limitations have a marked influence on the dynamics of the additive variance in bottleneck populations. As we have shown, their conclusion of the additive variance invariably increasing after bottlenecks cannot be extended to other epistatic models, where an excess of the additive variance was only detected for specific combinations of allele frequencies, which may be extreme (models I, IV, V, and VI, and complementary models VII, X, XI, and XII) or intermediate at one (model III and complementary model IX) or both loci involved (model II and complementary model VIII). For other epistatic models (additive_additive, multiple dominant genotype favoured, and Dobzhansky-Muller type), similar conclusions have been obtained (Lopez-Fanjul et al., 1999). The effect of linkage on the ratio V* At VA has been studied for additive_additive epistasis (Goodnight,

Lopez-Fanjul, Fernandez, and Toro

1988; Tachida and Cockerham, 1989) and rare nonepistatic recessives (Wang et al., 1998). In both instances, linkage equilibrium in the ancestral population has been assumed: i.e., disequilibrium is only due to sampling. For additive_additive epistasis, recombination does not affect the contribution of the ancestral additive variance to either the derived additive variance or the betweenline variance. In parallel, large recombination rates increase the contribution of the ancestral epistatic variance to the derived additive variance, but decrease it to the between-line variance. Nevertheless, both effects were shown to be small. With rare recessives, linkage disequilibrium can lead only to a small increase in V* At VA , above that expected for drift alone. In this case, the between-line variance is not affected by linkage disequilibrium generated by sampling (Avery and Hill, 1979). Interestingly, the excess in the additive variance after bottlenecks induced by additive_additive epistasis declines with decreasing recombination rates, but the reverse was found for rare recessives. Thus, for more complex systems, as those considered in this paper, those effects will tend to cancel each other. Moreover, linkage disequilibrium generated by drift will be transient, so that the outcome some time after a bottleneck depends only on the distribution of allele frequencies. Therefore, it is unlikely that linkage can qualitatively affect our conclusions. An extension of our results to multilocus systems requires a complete specification of genotypic effects and allele frequencies, as differences in any of these factors can even change the sign of the contribution of specific loci to the total additive variance after bottlenecks. Thus, inferences from theory to experimental data can only be made if individual loci show the same type of gene action and segregate with similar frequencies. A review of pertinent Drosophila and Tribolium results can be found in Lopez-Fanjul et al. (1999). It was concluded that, for morphological traits such as bristle number, where most of the genetic variance in natural populations has been shown to be contributed by quasi-neutral additive alleles segregating at intermediate frequencies, no inbreeding depression was detected and the behaviour of the withinline additive variance after bottlenecks very closely approached the expectation under the pure additive model. Recent data on wing size and shape traits in D. melanogaster also conform with those predictions (Whitlock and Fowler, 1999). For these traits, inbreeding depression after a single bottleneck (N=2) was less than 1 0 of the outbred mean, and the additive variance decreased proportionately to the inbreeding coefficient. This result is in agreement with an analysis of polygenes affecting wing shape in chromosome 3 in D. melanogaster (Weber et al., 1999), showing that the vast majority of

59

Epistasis at Population Bottlenecks

the trait variation in the population considered was explained by additive effects, epistatic variation being minor. It is also consistent with spontaneous mutations affecting wing length and width showing predominant additive gene action (Santiago et al., 1992). Data on mice populations passing through four consecutive bottlenecks (N=4) have been reported for 10-week body weight (Cheverud et al., 1999). No inbreeding depression nor heterosis was detected, suggesting additive gene action at most loci involved. However, the average within-line full-sib component of variance did not significantly differ from that of an outbred control, and was significantly larger than expected under a purely additive model. This result has been considered evidence for epistatic gene action, reducing the loss of additive genetic variance under inbreeding. Notwithstanding, the full-sib component of variance not only contains one-half of the additive variance, but also part of the non-additive variance, as well as the whole component of variance due to environmental effects common to full-sibs (including maternal effects). Thus, a more parsimonious interpretation of the behaviour of the full-sib component of variance after bottlenecks, is an increase of the commonenvironment component of variance in the inbred lines, relative to that of the control (see review by Falconer and Mackay, 1996, pp. 267269). On the other hand, fitnesscomponent traits as viability, where most genetic variance in natural populations is due to partially (or totally) recessive deleterious alleles segregating at low frequencies, show strong inbreeding depression and the additive variance after bottlenecks exceeded the ancestral value. For both kinds of traits, the between-line variance conformed with the additive predictions as indicated by our analysis and Lopez-Fanjul et al. (1999) results. A review covering many species (Fowler and Whitlock, 1999) also points out that fitness-component traits are much more likely to increase in phenotypic variance after bottlenecks than morphological traits. Summarizing, for the epistatic models considered, although occasional increases in the derived additive variance can be observed, it is unlikely that the rate of evolution may be accelerated after population bottlenecks, unless unrealistic parameter values are assumed. This conclusion is in agreement with the results obtained by Lopez-Fanjul et al. (1999) for other two-loci epistatic systems.

ACKNOWLEDGMENT This study was supported by a grant from the Direccion General de Investigacion Cient@ fica y Tecnica (PB98-0814-C03-01).

REFERENCES Avery, P. J., and Hill, W. G. 1979. Variance in quantitative traits due to linked dominant genes and variance in heterozygosity in small populations, Genetics 91, 817844. Charlesworth, B., and Charlesworth, D. 1987. Inbreeding depression and its evolutionary consequences, Annu. Rev. Ecol. Syst. 18, 237268. Cheverud, J. M., and Routman, E. J. 1996. Epistasis as a source of increased additive genetic variance at population bottlenecks, Evolution 50, 10421051. Cheverud, J. M., Vaughn, T. T., Pletscher, L. S., King-Ellison, K., Bailiff, J., Adams, E., Erickson, C., and Bonislawski, A. 1999. Epistasis and the evolution of additive genetic variance in populations that pass through a bottleneck, Evolution 53, 1009 1018. Crow, J. F., and Kimura, M. 1970. ``An Introduction to Population Genetics Theory,'' Harper 6 Row, New York. Falconer, D. S., and Mackay, T. F. C. 1996. ``Introduction to Quantitative Genetics,'' Longman, New York. Fowler, K., and Whitlock, M. C. 1999. The distribution of phenotypic variance with inbreeding, Evolution 53, 114356. Goodnight, C. 1988. Epistasis and the effect of founder events on the additive genetic variance, Evolution 42, 441454. Lopez-Fanjul, C., Fernandez, A., and Toro, M. A. 1999. The role of epistasis in the increase in the additive genetic variance after population bottlenecks, Genet. Res. 73, 4559. Phillips, P. C. 1998. The language of gene interaction, Genetics 149, 11671171. Robertson, A. 1952. The effect of inbreeding on the variation due to recessive genes, Genetics 37, 189207. Santiago, E., Albornoz, J., Dom@ nguez, A., Toro, M. A., and LopezFanjul, C. 1992. The distribution of effects of spontaneous mutations on quantitative traits and fitness in Drosophila melanogaster, Genetics 132, 771781. Tachida, H., and Cockerham, C. C. 1989. Effects of identity disequilibrium and linkage on quantitative variation in finite populations, Genet. Res. 53, 6370. Tiwari, H. K., and Elston, R. C. 1998. Restrictions on components of variance for epistatic models, Theor. Pop. Biol. 54, 161 174. Wade, M., and Goodnight, C. 1998. Perspective: The theories of Fisher and Wright in the context of metapopulations: When nature does many small experiments, Evolution 52, 15371556. Wang, J., Caballero, A., and Hill, W. G. 1998. The effect of linkage disequilibrium and deviation from HardyWeinberg proportions on the changes in genetic variance with bottlenecking, Heredity 81, 174186. Wang, J., Caballero, A., Keightley, P. D., and Hill, W. G. 1999. Bottleneck effect on genetic variance: Theoretical investigation of the role of dominance, Genetics 150, 435447. Weber, K., Eisman, R., Morey, L., Patty, A., Sparks, J., Tausek, M., and Zeng, Z-B. 1999. An analysis of polygenes affecting wing shape on chromosome 3 in Drosophila melanogaster, Genetics 153, 773786. Whitlock, M. C., and Fowler, K. 1999. The changes in genetic and environmental variance with inbreeding, Genetics 123, 345 353. Willis, J. H., and Orr, H. A. 1993. Increased heritable variation following population bottlenecks: The role of dominance, Evolution 47, 949957.