Abstract
Genetic effects on an index of wing shape on chromosome 2 of Drosophila melanogaster were mapped using isogenic recombinants with transposable element markers. At least 10 genes with small additive effects are dispersed evenly along the chromosome. Many interactions exist, with only small net effects in homozygous recombinants and little effect on phenotypic variance. Heterozygous chromosome segments show almost no dominance. Pleiotropic effects on leg shape are only minor. At first view, wing shape genes form a rather homogeneous class, but certain complexities remain unresolved.
THE genetic control of organ shapes in development can be analyzed using metrics that eliminate the allometric effect of size. For example, wing shape in Drosophila melanogaster can be quantified by the angular deviations of individual wings from baselines that represent the average shape in a base population. Each baseline approximates the centerline of a scatterplot of two dimensions in wildtype flies over a nearly twofold range of body size caused by temperatures from 18° to 30° and by starvation (Weber 1990). Angular offsets from these baselines are independent of most environmental body size effects. Analyses using this method have demonstrated many intriguing features of wing shape genes.
Wing shape is highly selectable. Lines selected divergently for various wing angular offsets showed a mean realized heritability of ~0.35 and a final mean divergence of ~15 phenotypic standard deviations (Weber 1990). Wing shape is controlled by numerous genes, as shown by statistical analysis of selection lines (Weber 1990) and by mapping of quantitative trait loci (QTL; Weberet al. 1999). Some of these genes act within separate localized regions of the wing. When any two dimensions were selected antagonistically to each other, it was possible to change them both oppositely at the same time, in either direction, thus allowing rather plastic deformation of the wing (Weber 1990). Moreover, some genes can affect the shapes of surprisingly small, localized regions of the wing without having proportional effects outside those regions (Weber 1992). Selected lines show changes both in the sizes of intervein areas and in the locations of vein intersections (Weber 1990, 1992). The effects of wing shape genes are strikingly independent of wing size in that flies of a defined genetic line retain the same approximate mean trait value over at least a 20% range in wing dimensions. Even lines with highly selected wing shapes that are many standard deviations outside the range of wild type show this constancy of shape regardless of size (Weber 1990; Weberet al. 1999). Wing shape genes also act largely independently of sex; that is, when shape is measured relative to baselines derived separately for each sex, the offsets of selected males and females from their respective baselines remain almost identical in every selected shape trait (Weber 1990). Wing shape is strongly resistant to environmental influences that cause intergenerational variation, and this resistance is also preserved in extreme selected genotypes (Weber 1990). Probably only Castle's results on rat fur color (Castle and Phillips 1914; Castle 1951) show selected changes freer of intergenerational environmental influences. Wing shape genes show little dominance in that hybrids between different genotypes have nearly intermediate phenotypes (Weberet al. 1999). They are also highly additive in that the phenotypic variance is nearly equal to the sum of the additive genetic variance, as measured in selection experiments, and of the environmental variance, as measured in longinbred lines (Weber 1990). Wing shape genes have only minor interactions with fitness: Highly selected lines can be made isogenic with no loss of the phenotype and little loss of viability (K. Weber, unpublished data), and highly inbred lines show high betweenline variance in wing shape (Weber 1990). Wing shape genes also have little correlated effect on the shapes of other appendages (Weberet al. 1999).
These general observations have exceptions and should be qualified. For example, in one selection line a major shape gene with a large correlated effect on body size appeared (Weber 1990), and small correlated effects on size appeared in several other lines. In another line, a major shape gene appeared that also caused female sterility (Weber 1990). There are significant sources of environmental variance in nature in that phenotypic variances in wing shape are on average 47% higher in wildcaught flies than in their labreared offspring (Weber 1990). Multiple interval mapping (MIM) analysis of chromosome 3 provides evidence of possibly strong pairwise QTL interactions (Weberet al. 1999). Finally, wing shape does not respond to selection with perfect plasticity but shows some correlations among dimensions (Weber 1990; see also Guerraet al. 1997).
Several of these observations have since been confirmed by others using different approaches. Birdsall et al. (2000) performed shape analyses of three intervein regions in D. melanogaster using the morphometric method of relative warps (Bookstein 1991), comparing inbred lines at 18° and 25°. Each wing region varied independently of the others in a genotypespecific way, confirming the existence of shape control by locally acting genes. The two temperatures caused differences in wing area of up to 20%, but the shape parameters of each wing region tended to remain constant within each inbred line at both wing sizes, confirming the wingsize independence of shape genes. The effects of sex also tended to be constant across genotypes. Some shape parameters in some wing regions showed departures from perfect size and sex independence. Zimmerman et al. (2000) followed Birdsall et al. (2000) with QTL analyses of wing shape, using the same intervein regions, shape parameters, and temperatures. This confirmed again the presence of locally acting genes and the relative independence of wing shape gene effects from sex and size. Dominance among QTL was assessed for one intervein region and found to be mostly negligible. These studies indicate that the analysis of shape genes will result in similar conclusions when any sizeindependent shape metric is used.
Wing shape is an ideal trait for quantitative genetic analysis, and angular offsets are convenient for their simplicity. Angular offsets of wing shape have normal distributions and nearly constant variances, even when selection greatly changes the mean (Weber 1990). Selected phenotypes remain stable for many years. The most convenient feature is the remarkable resistance to environmental fluctuations, seen in the smooth graphs of selection response (Weber 1990). It is also convenient that both sexes always have nearly identical angular offsets (from slightly different baselines) so that comparisons can be made using a single sex.
With the publication of a QTL analysis of the third chromosome (Weberet al. 1999), wing shape became the third morphological trait in D. melanogaster, after bristle number (Longet al. 1995) and features of the male genital arch (Liuet al. 1996), to be analyzed by current methods of QTL mapping. Here we extend our investigation to chromosome 2. With these two studies, we have now covered ~80% of the genome and 90% of the total phenotypic difference between the “high” and “low” lines produced by 20 generations of divergent selection on this representative wing shape trait, arbitrarily designated “index F” (Weber 1990). The remaining 10% of the phenotypic difference arises from chromosome 1 (the X chromosome), which is still being analyzed.
This study attains a higher resolution of small genetic effects than was possible for chromosome 3. The data set of 35,050 measured wings from 701 recombinant isogenic lines is 35% larger than the data set for chromosome 3 (Weberet al. 1999). Moreover, the variance among line means with the same marker genotype is ~40% lower for chromosome 2 than it was for chromosome 3, probably due to improved consistency among measurement technicians. The increases in accuracy and number of measurements and the high density of markers (110 cM/47 markers) permit a resolution of effects that is powerful for this type of study. The outlines of the system of polygenic shape control are revealed. Nevertheless, we find that because of the high density of genetic effects, even higher resolution would be required to quantify the numbers and effects with acceptable confidence.
MATERIALS AND METHODS
This study used the same experimental methods as the previous study of chromosome 3 (Weberet al. 1999). Second chromosomes were derived from lines previously selected for high or low values of an index of wing shape. A line that was isogenic for the high (H) second chromosome and for the low (L) first and third chromosomes (LHL) was crossed to another line that was isogenic low in all three chromosomes (LLL). Random recombinants between the high and low second chromosomes were then extracted and made homozygous again in the same background of isogenic low first and third chromosomes by using a third stock with isogenic low first and third chromosomes and a balancer second chromosome (LBL). The balancer chromosome was SM5 (Lindsley and Zimm 1992) with Curly^{O}/Sternopleural.
The high and low second chromosomes could be differentiated by the linespecific insertion sites of the transposable element roo, identified by in situ labeling of salivary gland chromosome squashes. Three additional marker sites in one region were gained by using the transposable element jockey. The sites of secondchromosome insertion markers are listed in Table 1.
One left or right wing at random was measured from each of 50 males per line. Only males were measured because male and female offsets (from different baselines) are nearly identical and respond equally to selection (Weber 1990). Wing measurements were made on whole flies by projecting wings onto a digitizer pad with a microprojector (Weber 1988; Weberet al. 1999).
The phenotypic scale: As in previous studies, wing (and leg) shapes were quantified using indexes based on two dimensions, D_{1} and D_{2}. In the wing shape index used here (designated index F), D_{1} is the width across the middle of the wing and D_{2} is the width across the base, using vein intersections as
landmarks (See Weber 1990, Figure 1). To derive a shape index, two dimensions from each individual are first transformed into polar coordinates [θ = arctan(D_{2}/D_{1}) and
Measurement of legs: Legs of lines LHL and LLL were mounted on slides in glycerol under coverslips and sealed with fingernail polish. Left legs were mounted with anterior side upward and flexed femorotibial joint. On legs thus mounted, D_{1} is the width of each segment at its widest point, and D_{2} is the width of the same segment at its narrowest point, which is more proximal. These were essentially the same dimensions used previously (Weberet al. 1999) but were very slightly redefined for this study. Therefore, the measurements of LLL leg segments were repeated from a new sample for this comparison of LHL to LLL. Mounted legs were projected and digitized in the same way as wings.
Multiple interval mapping analysis: The method of MIM analysis follows Kao et al. (1999), Zeng et al. (1999), and Weber et al. (1999) with some modifications. Briefly, a multiple QTL model is used to search for the number, positions, and epistatic terms of QTL in a stepwise fashion. In each step, the analysis first searches for the position of a new putative QTL and tests it for significance using the residual permutation test (Zenget al. 1999); it then updates the estimate of position for each selected QTL conditional on other QTL in a sequential way. This process is repeated until no new QTL is detected on the basis of the residual permutation test. The initial model for the MIM analysis is provided by a backward stepwise regression on markers with α = 0.01 as the critical level for the test of partial regression coefficients.
The residual permutation test compares a model with k QTL, regarded as the null hypothesis, and a model with k + 1 QTL that contains the k QTL model, regarded as the alternative hypothesis. We first obtained the estimated genotypic value for each individual under the null hypothesis (the k QTL model). We then randomly shuffled the residuals (the difference between the observed phenotypic value and the estimated genotypic value under the null hypothesis) among individuals to obtain a residual permutation sample. Next we searched for a putative QTL conditional on the k QTL in the residual permuted sample and recorded the maximum test statistic. This resampling and search was performed a number of times (500) to obtain an empirical 95% significance threshold. To decide whether to accept the new QTL, we compared this threshold to the test statistic for the (k + 1)th QTL in the original sample.
After selecting the QTL number and positions, a backward stepwiseselection procedure was used to select a subset of significant QTL additivebyadditive interaction effects. In each step, an epistatic effect that is the least significant by a likelihoodratio test is dropped from the model. The process is repeated until each remaining epistatic effect is significant by the likelihoodratio test conditional on other QTL effects.
RESULTS
The data set comprises the phenotypic means and recombination breakpoints of 701 isogenic recombinant secondchromosome lines with the same background of isogenic low first and third chromosomes. The measured lines are distributed among single recombinants (217 HL and 256 LH), double recombinants (79 HLH and 99 LHL), triple recombinants (4 HLHL and 6 LHLH), and nonrecombinants (20 H and 20 L). (Many more nonrecombinants were recovered but not measured.)
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Reduction in variance between lines: Variances between line means with the same marker genotype were higher for the third chromosome (Weberet al. 1999) than for the second chromosome (this study). For the third chromosome, the mean of the variance between nonrecombinant high lines and the variance between nonrecombinant low lines was 1.6 × 10^{−5}, and the weighted mean variance between single recombinant lines of the same marker genotype, among all genotypes with more than two lines, was 2.9 × 10^{−5}. For the second chromosome, the mean variance between nonrecombinant lines was 1.0 × 10^{−5}, and the weighted mean variance between single recombinant lines of the same marker genotype, among all genotypes with more than two lines, was 1.8 × 10^{−5}. Thus, in this study, samegenotype variances declined by 37.5% among nonrecombinants and by 37.9% among single recombinants. This would not be expected because the mean map distance between markers for chromosome 2 is actually a little larger (2.3 cM/marker) than it was for chromosome 3 (1.8 cM/marker). The greater consistency of samegenotype measurements for chromosome 2 probably resulted from our intensive efforts to eliminate small differences among operators in the exact digitizing of landmarks.
Single recombinants: Figure 1 shows the phenotypes of all single recombinant and nonrecombinant lines as a function of recombination breakpoint. Both graphs show the incremental accumulation of genetic effects along the chromosome. The two profiles of cumulative effect can be directly compared by using the mean absolute value of their increasing deviation from the leftend phenotypic mean (Figure 2), i.e., as values of HLL and HLH. Changes in slope differentiate regions with larger or smaller effect. Because the two profiles in Figure 2 follow nearly the same pattern, it is apparent that switching a segment between H and L usually has about the same effect, whether the segment is flanked by H on the left and L on the right (HXL) or the reverse (LXH). In other words, these genes appear to act largely independently of each other.
A series of ttests on the set of all single recombinants and nonrecombinants separates the chromosome into nine segments with significant effects (Figure 2 and Table 2). This is a minimum estimate of the number of genes. The segments are somewhat alike in size but smaller at the ends of the chromosome. They do not correspond to distinct steps in the profiles of cumulative effect.
The net effect of interactions across any point of the chromosome can be quantified by subtracting the HL profile from the LH profile in Figure 2. This net effect is always negative, with a roughly constant value and a mean of −0.0042. Considering the nearly parallel profiles along most of the chromosome, the simplest model would be a single pairwise interaction between loci near 2 and 105 cM, with an effect of −0.0042. However, examination of the double recombinants shows that the situation is more complicated.
Double recombinants: The double recombinants are numerous enough to merit a separate graphic analysis. Double recombinants can be graphed as a surface, where x is the first breakpoint, y is the second breakpoint, and z is the mean phenotype of double recombinant lines with breakpoints x and y (Figure 3). All possible double recombinants fall on the surface above the triangle (0 ≤ x ≤ 110, 0 ≤ y ≤ 110, x ≤ y). There is one surface for HLH double recombinants and another for LHL. The two surfaces can be compared best when represented as positive deviations from opposite ends of the phenotypic range: HLHs as HHLH, and LHLs as LHLL. This inverts the HLH surface so that both surfaces have the same basic shape.
This convention represents not only double recombinants but also single recombinants and nonrecombinants, on a surface of absolute effect. Single recombinants appear in Figure 3 in the two vertical planes along the sides. For example, an LH single recombinant can be regarded either as an LHL double recombinant with the second breakpoint at the right end (110 cM) or as an HLH double recombinant with the first breakpoint at the left end (0 cM). Each single recombinant line appears on both these planes. Thus, the plots of single recombinant means shown in Figure 2 are visible again in Figure 3 in the plane x = 0 on the left side and in inverted orientation in the plane x = 110 on the right side. Nonrecombinants also appear in Figure 3 at the three vertices where the two surfaces converge. For example, H nonrecombinants are equivalent to LHL double recombinants with the first breakpoint at 0 cM and the second breakpoint at 110 cM; and they are also equivalent to HLH double recombinants with both breakpoints at 0 cM or both at 110 cM. The phenotypic surface of zvalues descends from a vertex at x = 0, y = 110 to 0 at all points along the diagonal x = y (where LHL = L and HLH = H). Thus all nonrecombinants and all single recombinants in the data set appear on both the LHL and the HLH surfaces, along with all the double recombinants of either one kind or the other. In fact, the two surfaces in Figure 3 present a natural topography of the whole data set, excluding only the few triple recombinants.
If there were no interactions among loci on the second chromosome, the two surfaces in Figure 3 would be identical. This is the geometrical way of stating that if the genes in the segment between any two breakpoints have no interactions with genes in either flanking segment, then the quantities HHLH and LHLL are equal. That is, if any part of the high chromosome is replaced by low, the absolute change in phenotype is the same as if the same part of the low chromosome were replaced by high, as long as no interacting loci change their linkage phase in the exchange. Thus the vertical separation between the HLH and LHL surfaces (subtracting the latter from the former) at any point (x, y) gives the sign and magnitude of the net of all those interactions where genes located between the breakpoints (x and y) interact with genes located outside the breakpoints.
Most double recombinants have one breakpoint in each chromosome arm; very few have both in the same arm. Therefore, the data in Figure 3 are mainly confined to the square central area of the xyz surface where x < 55 cM (the approximate centromere) and y > 55 cM. Figure 3 shows that in this area the surfaces are separated almost everywhere by roughly constant intervals. This is revealed more clearly by Figure 4.
The graphed double recombinants show that a welter of interactions exists, with mostly negative net effects. They also agree with the single recombinants, in that all detectable net effects of interactions in homozygous recombinants are small; i.e., most gene action is independent of the effects of other genes. This is implied by the demonstration that the HLH and LHL surfaces are nearly congruent, as far as the data can show.
Results of multiple interval mapping: Backward stepwise regression selected 11 markers that together explained 93% of the total variance. On the basis of this initial model, the positions of the putative QTL were first scanned and updated sequentially under the hypothesis of 11 QTL. The residual permutation test was then performed for the leastsignificant QTL. This putative QTL has a LOD score of 1.6 and the 95% residual permutation threshold with a model of the 10 QTL is 3.6 LOD. [This residual permutation threshold is different from and higher than the regular permutation threshold of Churchill and Doerge (1994), which is 1.7 LOD for the experiment. This is because the residual permutation test is conditional on other QTL and the test statistic in different regions separated by QTL is more or less independent.] This candidate QTL was dropped from the model and the positions of the remaining 10 QTL were optimized again. All these QTL were tested to be significant. A forward search for a new QTL was then attempted, and no significant QTL was found.
Given the 10 QTL and the estimated positions, the search for epistatic terms was performed by a backward stepwiseselection procedure to select a subset of significant QTL additivebyadditive interaction effects. The model started with the 10 additive effects and 45 possible additivebyadditive interaction effects, and the selection proceeded to remove nonsignificant interaction effects one at a time. In most of this search process, not many interaction terms in the models were statistically significant. However, after dropping the 31st leastsignificant interaction term, all the 14 remaining interaction terms became very significant, conditional on others. The selection process was stopped here, and the final model contains 10 additive effects and 14 additivebyadditive interaction effects. The estimates of positions, additive effects, and additivebyadditive interaction effects of QTL are given in Table 3. The partition of variances and covariances explained by QTL additive effects is given in Table 4, and that explained by QTL epistatic effects is given in Table 5.
Figure 5 gives the likelihoodratio test statistic profile in LOD score for each QTL conditional on the other QTL effects, including the epistatic effects. This profile shows the strength of statistical evidence for mapping each QTL. It is very difficult to obtain accurate estimates of confidence intervals for QTL positions with multiple QTL (Kao and Zeng 1997). We plot the 1LOD support interval for each QTL (conditional on the estimates of other QTL), which is the genome region covered by the drop of 1 LOD from the peak in the likelihood profile, to indicate the likely region where the QTL may be located. It is not clear how much confidence the 1LOD support interval may have in this multidimensional inference.
The estimated additive effects of the 10 QTL are all in one direction, ranging in magnitude from 5 to 21% of the phenotypic difference between the two parental genotypes (Table 3) and summing to 99.1% of the difference. There is good agreement between the observed sum and the expected sum, as the sum of additive effects in the model is assumed to equal the phenotypic difference between the two parental genotypes and is not affected by interactions according to the model.
The additive effects together explain 95.1% of the total phenotypic variance (Table 4), and the epistatic effects together explain only 0.3% of the total variance (Table 5). The covariances between the additive effects and epistatic effects, expected to be zero (Kaoet al. 1999), are very small (almost zero). The model explains 95.4% of the total variance.
To further support the selection of the model given in Table 3 with the 14 epistatic effects, we show in Table 6 the log likelihoods for a series of models in the backward stepwise elimination process for epistasis. Table 6
includes a few models for comparison, such as the model with the 10 additive effects and 45 additivebyadditive interaction effects (M_{45}, the starting model in the elimination process), one with the 14 epistatic effects (M_{14}, the selected model), and one with no epistatic term (M_{0}). Since the likelihood does not decrease as the number of parameters fitted in a model increases, the likelihood for a nested model with more parameters is certainly higher than, or at least as high as, that with fewer parameters. However, when comparing two models, we usually use the likelihoodratio (LR) test statistic, which is two times the difference between the log likelihoods of the two models (see Table 6), to test for significance. Under nested models, as in this backward elimination process, LR for two models (one with more parameters, usually regarded as the alternative model, and one with fewer parameters, regarded as the null hypothesis) tends to be asymptotically distributed with a χ^{2} distribution under the null hypothesis, with the degrees of freedom (d.f.) being the difference in number of parameters. Thus, we may compare the observed LR with
It is clear that the likelihood ratio between M_{45} and M_{14} [the difference between the two log likelihoods denoted as LR_{M}_{45:}_{M}_{14}, which is 2 ln(L_{M}_{45}) − 2 ln(L_{M}_{14}) = 7094 − 7088 = 6 ⪡
Although only pairwise interaction terms were considered in the search process, higherorder interaction terms, such as the third order involving three QTL, can also be considered in principle. However, to fit a thirdorder interaction term for three QTL, double recombinants between the three QTL are required for the analysis; otherwise, the analysis will be singular (indicating an overfit of parameters). Although the current data contain double recombinant lines, the number of these lines is still not large enough and the double recombinants are not widespread enough for an effective search for thirdorder interactions. A search for thirdorder interactions can easily run into the singularity problem, and even if the analysis is not singular, the test for thirdorder interaction is unlikely to be significant because of too few double recombinant lines. For these reasons, a search for higherorder interaction terms was not attempted.
Dominance: To assess dominance at loci affecting this trait, males from representative single recombinant lines were crossed to nonrecombinant high or lowline females in 14 crosses. The single recombinant line nearest the center of the chromosome was crossed to both high and low. The other single recombinant lines were crossed to only high or low. In the resulting hybrids, all loci to one side of the breakpoint are heterozygous and all loci to the other side are homozygous. The phenotypes of parents and their hybrids show the average dominance within blocks of genes (Figure 6). Most heterozygous combinations showed nearly zero dominance, except at one breakpoint on the left arm at ~17 cM in the LH × L crosses.
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The original LH line used at this breakpoint was lost, so the measurement could not be repeated. Instead, three other LH lines with the same breakpoint and nearly the same mean were crossed to the same low nonrecombinant. Phenotypes of these hybrids are also shown in Figure 6B. The mean of the four hybrid means at this breakpoint reinforces the conclusion that lack of dominance in heterozygotes is typical. There may be a source of atypical variation that affects some recombinants in this interval, however.
High and low nonrecombinant lines were also crossed reciprocally. This produced indistinguishable hybrid phenotypes with mean and SD of −0.0387 ± 0.008 when sires were high and −0.0384 ± 0.007 when sires were low.
Correlated effects: The previous study (Weberet al. 1999) showed that whole H and L third chromosomes carry genetic differences with small but significant effects on leg segment shape. In the present study, whole H and L second chromosomes were tested for effects on leg segment shapes, as before. As shown in Table 7, the H and L second chromosomes also have significantly different effects on leg segment shape. However, the leg shape differences caused by the second chromosome are even smaller than those reported previously for the third chromosome. [The LLL offset for wings is given as −0.0697 here but as −0.0689 in Weber et al. (1999), because each value is the mean of LLL nonrecombinant lines arising from a different set of recombination crosses.]
DISCUSSION
Chromosomes 2 and 3 (Weberet al. 1999) appear to be virtually identical in their index F quantitative genetics. Both chromosomes carry many genes affecting this trait. These can be modeled as ~10 factors on each chromosome with small subequal effects dispersed fairly evenly along the recombinational maps. On each chromosome, most of the total effect on genetic variance is due to additive gene action, although many interactions among loci do occur. The net effect of interactions in homozygous recombinants is small and negative across almost every point on both chromosomes. On both chromosomes, heterozygotes are almost exactly intermediate between homozygotes when effects are assayed on a segmentbysegment basis. Finally, each chromosome has detectable effects on leg shape when the whole chromosome is substituted high for low, which might be pleiotropic effects of wing shape genes; but these effects are small.
Chromosome 1 is still being analyzed. With its sexspecific regulation and transmission, it may not fit this picture. We also have no information yet on interactions between chromosomes. Nevertheless, because chromosomes 2 and 3 account for 90% of the phenotypic difference between high and low lines, and in view of the similarity between them, we can attempt to generalize about the quantitative genetics of this wing shape trait.
Number of loci: Multiple interval mapping finds 10 QTL on chromosome 2 and 11 on chromosome 3, while serial ttests find a minimum of nine genes on chromosome 2 and eight on chromosome 3. Thus the original high and low lines differ at ~20 sites affecting this trait. This is a minimum estimate; there is no maximum estimate because of the potential for blocks of closely linked genes. The rather linear relation between cumulative effect and chromosome segment length (Figure 2), and the large numbers of evenly dispersed QTL (Figure 5), could both be manifestations of an underlying continuum of more numerous effects too close together to resolve.
Dominance: Alleles affecting this trait show almost no dominance in heterozygous chromosome segments along both the second chromosome (this study) and the third (Weberet al. 1999). This would also be the expected result if there are large numbers of loci with random amounts of dominance, especially if they tend more often to be semidominant, as is usually the case in quantitative traits (Tanksley 1993). Thus our lack of certainty that we have an exhaustive count of loci also affects somewhat our conclusion that alleles are acting with nearly perfect additivity in heterozygotes.
Consistently negative net interactions: In this study and in Weber et al. (1999), the net effect of interactions between loci in homozygous recombinants is negative across almost every point along both major chromosomes. Eshed and Zamir (1996) also found widespread negative interactions (“lessthanadditive” interactions) in tomatoes among QTL for four fitness traits. They suggested that these interactions might reflect a partly canalized system that helps to maintain fitness in many diverse genotypes.
Wing shape traits are not strongly canalized against genetic change. They respond to divergent selection in the same way as most quantitative traits—with an immediate response in both directions that is most rapid at the outset (Weber 1990). However, there could still be weak buffering toward wild type that shows up in highly selected genotypes. In our system, where both chromosomes were measured in a low background and all phenotypes were therefore below the presumably optimal range of wild type, the appearance of consistently negative interactions along the whole chromosome would also be generated if gene effects became increasingly less than additive the more they approached wild type. Such a buffering system might result if wing shape were determined by multiple, independent, partially redundant parallel systems, all with selectable variation. At present one can say only that directional asymmetry in the net effects of genetic interactions is strikingly consistent in our studies of wing shape and also in the report of Eshed and Zamir (1996).
Mean effects and net interactions: Figure 7A compares two ways of showing the cumulative effect of genes. The step function (dashed line) shows the cumulative main effects of QTL from Table 3, as computed by MIM using the whole set of recombinant lines. The other curve (solid line) was computed by averaging the two curves of single recombinant lines (HL and LH) from Figure 2. Figure 7B shows net interactions along the chromosome in two ways. Again, the step function shows the net pairwise QTL interactions from MIM (Table 3), while the solid line is the difference between the LH and HL profiles in Figure 2. The fit between the direct empirical averages and the QTL model values in Figure 7 is close and could be improved only by increasing the number of point effects in the QTL model. These comparisons are a useful check on the whole investigation because the two analyses, using different approaches and partly based on different components of the data, were prepared completely independently by K.W. and Z.B.Z. The graphical and statistical analyses do not yield completely equivalent views, but there are close correspondences between them in the distributions of mean effects and net interactions when these are considered separately.
The weakness of net interactions still allows for hidden interactions that are strong. Although the intervals between markers are small, there could conceivably be large interactions between sites so tightly linked that they have no recombinants in this data set. There could also be large interactions that are balanced by other large interactions of equal magnitude and opposite sign if these are distributed in a nonrandom way so that they cancel each other out.
Symmetric pattern of QTL interactions: The locations of the pairwise QTL interactions listed in Table 3 have to be considered along with the magnitudes of their effects. Figure 8 shows how the interactions in Table 3 are arranged along the chromosome. There is a pattern of large, balanced positive and negative effects that nearly cancel each other at every point along the chromosome, leaving only the small residues of net negative effect seen in Figure 7B. The same type of pattern was found in the QTL analysis of chromosome 3. Nevertheless, the phenotypic variation in chromosome 3 could also be explained fairly well (r^{2} = 0.93) with a model assuming an indefinitely large number of genes acting only independently, i.e., with no interactions (Weberet al. 1999).
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The apparent balancing of pairwise QTL interactions, found in the MIM analyses of both chromosomes 2 and 3, is highly nonrandom and must be either an effect of natural selection or else perhaps a statistical effect arising in the MIM analysis. If the balanced arrangement is a real effect of natural selection, perhaps it is favored as the pattern that creates the fewest extreme phenotypes, i.e., generates the least phenotypic variance. But this would be a remotely secondorder effect. Obviously, we would like more certain confirmation of this phenomenon before venturing too far in search of explanations.
Epistasis in the MIM analysis: Tables 4 and 5 show the partitioning of the genetic variance in the current population of recombinants with a significant amount of linkage disequilibrium among QTL. In this population, the total variance explained by the significant pairwise epistasis is very small compared to the total variance explained by the additive effects (0.3 vs. 95.1). This may be somewhat expected because the current recombinant population is created from a cross between high and low selection lines. After intensive divergent selection for 20 generations in large populations, it is likely that most QTL alleles are appropriately fixed in the selection lines; i.e., most plus alleles are in the high line and most minus alleles are in the low line. This is confirmed by the QTL mapping data reported here and in Weber et al. (1999). However, epistatic effects as defined in the model are not necessarily in one direction in this recombinant inbred population. This is because QTL alleles are not necessarily fixed together during the divergent selection. Also, with strong additive effects, some genotype combinations are still favored by selection even with some negative epistatic effects. Furthermore, the genetic structures in the high and low populations during selection are not the same as those in the current recombinant inbred population.
In the population of recombinants in this data set, there is a significant amount of epistatic variation hidden by linkage disequilibrium. As linkage disequilibria among QTL decrease, this component of epistatic variation will be released into the population. For example, in a hypothetical population that has the same genetic structure as the current population except that QTL are in linkage equilibrium, and thus the covariance between different QTL effects reduces to zero, the ratio of the epistatic variation vs. the additive variation would be 14.5 vs. 45.1 (sums of the diagonals of Table 5 and Table 4 as a percentage of the phenotypic variance in the current population). In this hypothetical population, the majority of the genetic variation would still be due to additive effects of QTL, but the epistatic variation would no longer be negligible.
Pleiotropic effects on shape: In their high and low selected versions, chromosomes 2 and 3 have ~10 times as much effect on wing shape as on leg shape. As shown in Table 7 of this article and in Table 8 of Weber et al. (1999), the wing shape effect in basepopulation phenotypic standard deviations is 10.22 for chromosome 2 and 9.78 for chromosome 3. By contrast, the mean of all measured shape effects on leg segments (again in basepopulation standard deviations) is only 0.37 for chromosome 2 and 1.29 for chromosome 3. If one considers only the effects on the mesothoracic leg (of particular interest because it is on the same body segment as the wing) these numbers are even a bit lower, being 0.23 for chromosome 2 and 1.21 for chromosome 3. We conclude from our studies of both chromosomes that, for the traits assayed here, the genetic controls of wing shape and leg shape are largely decoupled from each other. The small, correlated leg shape differences that are found may be pleiotropic effects of the wing shape genes, or they may simply arise from random fixation at other loci, just as inbreeding in isofemale lines leads to significant differences among lines in wing shape (Weber 1990).
Caveats: These studies tell us much about the genetics of wing shape, but the details are still poorly resolved. We have no upper limit on the number of effects, inadequate constraints on their locations, and great uncertainties about their interactions. We hope our analyses are impeccable as far as they go, but our conclusions are somewhat tentative. If only we could measure an infinite number of lines, the empirical surface of effects would become a step function like the theoretical one, leaving no doubts.
Acknowledgments
We thank Peter Keightley and two anonymous reviewers for comments. This work was supported by grants from the National Science Foundation (DEB9407005) to K.W. and from the Public Health Service (GM45344) to ZB.Z.
Footnotes

Communicating editor: P. D. Keightley
 Received February 22, 2001.
 Accepted August 3, 2001.
 Copyright © 2001 by the Genetics Society of America