An Analysis of Polygenes Affecting Wing Shape on Chromosome 3 in Drosophila melanogaster
 Kenneth Weber*,
 Robert Eisman*,
 Lisa Morey*,
 April Patty*,
 Joshua Sparks*,
 Michele Tausek* and
 ZhaoBang Zeng†
 ^{*} Department of Biological Sciences, University of Southern Maine, Portland, Maine 041049300
 ^{†} Department of Statistics, North Carolina State University, Raleigh, North Carolina 276958203
 Corresponding author: K. Weber, Department of Biological Sciences, University of Southern Maine, Box 9300, Portland, ME 041049300. Email: keweber{at}usm.maine.edu
Abstract
Loci on the third chromosome of Drosophila melanogaster that affect an index of wing shape were mapped, using recombinant isogenic lines, with transposable elements as markers. Many genes with small subequal effects are dispersed along the whole chromosome. Their alleles act nearly additively in heterozygotes. They have small correlated effects on leg shape, but no detectable effects on halteres. Small negative net interactions occur over most of the chromosome. The data set of 519 recombinant isogenic lines can be explained reasonably well by two models. One model posits an indefinitely large number of loci with no interactions. The other model posits 11 loci with additive effects whose sum equals the total phenotypic range and with large positive and negative interactions that nearly cancel each other.
TWO recent studies of wing shape traits in Drosophila melanogaster have indicated a highly polygenic basis when shape is defined by metrics that remove the allometric effect of body size (Weber 1990, 1992). In the first study, wingshape traits showed high realized heritabilities, symmetrical responses to divergent selection, and high ratios of longterm response to initial heritability (Weber 1990). These results show that many loci with small, localized effects on wing shape are segregating in the wild base population at intermediate frequencies. The second study showed that selection can change extremely small subregions of the wing while hardly affecting its major dimensions (Weber 1992). This adds to the evidence that wing shape is influenced by many commonly segregating loci with small and localized effects.
Here we advance to the next stage in the genetic analysis of wing shape—the attempt to map loci, estimate their magnitudes of effect, and describe their modes of action. Two strains of D. melanogaster with highly differentiated wing shapes were produced by selection for high and low values of a shape index. Isogenic crossing stocks were extracted, with third chromosomes high in one stock and low in the other, and with first and second chromosomes identical and low. Homozygous thirdchromosome recombinants of these crossing stocks were analyzed using in situlabeled transposable elements as markers. Recombination breakpoints were determined by comparison to the transposable element insertion sites in the parent high and low third chromosomes. The pattern of wing phenotypes among recombinant lines yields a map of genetic effects along the chromosome (Longet al. 1995; Nuzhdinet al. 1997). Results are also presented here from crosses made to assess dominance and from measurements of the correlated effects of wingshape genes on other appendages.
MATERIALS AND METHODS
The phenotypic scale: The trait in this study is trait F, defined in Weber (1990) as an index of wing shape incorporating two dimensions. The width of the wing across the middle is dimension D_{1}, and the width across the base is dimension D_{2}. On a scatterplot of D_{1} vs. D_{2}, each wing has some offset from a reference baseline. This offset is the phenotype, measured in radians as an angle of rotation about the origin. The reference baseline is empirically derived as a regression line through the scatterplot of D_{1} vs. D_{2} in a large sample of wildtype flies. The equation of the baseline is of the form θ = βr^{α}, derived by the regression of log(θ) on log(r), after conversion of each point (D_{1}, D_{2}) to polar coordinates [where θ = arctan(D_{2}/D_{1}) and
Angular offsets from control population baselines were also used to quantify correlated aspects of leg shape. Shape indexes for the femurs and tibias of each leg were constructed by letting D_{1} equal the widest distal width of each segment and D_{2} the narrowest proximal width. Table 1 shows the empirical constants for the baseline equations of the control population for both wing and legshape indexes, plus standard deviations of control population offsets from each baseline.
Wing measurement: The “morphometer” system (Weber 1988) utilizes a small, handheld suction device, attached to an air pump, to hold an anesthetized fly in a transparent frame that fits on the stage of a microprojector so that a large image of the wing can be focused on a digitizer pad. Wing landmarks are recorded as x and y pad coordinates. Measurements of wing dimensions obtained by this system have the same coefficients of variation as published measurements of wings that have been dissected and mounted on microscope slides (Cowleyet al. 1986; Cowley and Atchley 1988; cf. Weber 1990, 1992). For each recombinant line, one right or left wing at random was measured from each of 50 male flies. The 50 males from each recombinant line were measured in two separate samples of 25, taken from two replicate culture vials, cultured on potato flake medium at 26°.
The crossing stocks: Starting from a large laboratory population that was founded from wildcaught flies, pairs of stocks with divergent wing shapes were created by 15 generations of mass selection of the extreme 20 individuals of 100 measured in each sex (Weber 1990). Five different angularoffset indexes of wing shape were used (designated as M, S, F, G, and R). After 3 yr in small vial cultures, the high F and low F lines were selected again for 5 more generations, then held in small vial cultures again for another 3 yr. Multiple isogenic lines were then extracted, and one high indexF line (HHH) and one low indexF line (LLL) with extreme phenotypes and excellent viability were chosen for mapping. From these stocks, three isogenic crossing stocks for the third chromosome were derived, with the three major chromosomes substituted as LLL, LLH, and LLB—where the letters L, H, and B indicate homozygous low indexF, homozygous high indexF, or balancer chromosomes, respectively. The balancer chromosomes used throughout this study were FM7 (with Bar) for chromosome 1, SM5 (with Curly^{O})/Sternopleural for chromosome 2, Andy TM6 (with Serrate)/Stubble, brown^{D} for chromosome 3 (see Ashburner 1989). These stocks were all obtained from the MidAmerica Drosophila Stock Center (Bowling Green, OH).
The markers: Insertion sites of the transposable element roo or B104 (Schereret al. 1982; Lindsley and Zimm 1992, p. 1106) were labeled by a standard in situ protocol (cf. Longet al. 1995). The treatments (separated by appropriate transitional baths) were 20 min 2× SSC at 65°, 3 min 0.14 m NaOH, airdrying, hybridization with biotinylated probe overnight, 20 min streptavidin/biotin treatment (Vectastain; Vector Laboratories, Burlingame, CA), 30 min in diaminobenzidine/H_{2}O_{2} solution, and light Giemsa counterstaining. The breakpoints for each recombinant line were confirmed by chromosome squashes of two larvae for every line. No recombinant lines showed segregating in situ band patterns except for occasional single bands in new locations due to lowfrequency spontaneous transpositions of roo.
The low third chromosome has insertions at 29 sites with 15 on the left arm and 14 on the right arm. The high third chromosome has insertions at 35 sites with 18 on the left arm and 17 on the right arm. These roo sites form rather complementary patterns, and only two or three markers are identical or ambiguous between high and low lines. Additional transposable elements 412 and 297 were used to resolve a gap in the distribution of roo sites on the right arm. The extra bands located with these markers were at 90D1, 90E3, and 91D1. In all, 62 markers were located at sufficiently differentiable positions to be easily referable to one line or the other in recombinants. Thus each useful marker represents on average 111/62 = 1.8 cM. Table 2 shows the cytological positions of all 62 useful markers, with estimated genetic distances in centimorgans from the left end. Distances were estimated from a graph of centimorgans as a function of band location, based on the tables of gene locations by Ashburner in Lindsley and Zimm (1992, p. 1117). The first three markers on the left tip of the chromosome are not separable on the linkage map.
Crossover breakpoint mapping: The crossing stocks were used to produce recombinant third chromosomes by crossing LLL × LLH to create females heterozygous for the third chromosome, crossing these to males of the LLB balancer stock, and crossing the male progeny of these back to the balancer stock again, to extract isogenic lines with L × H recombinant third chromosomes. Also at one point a LH singlerecombinant line with breakpoint at 67F169A1 was crossed to an H nonrecombinant line to gain resolution on the left arm. This produced a total of 34 LH single recombinants with breakpoints left of 67F169A1, and 23 HLH double recombinants with a second breakpoint at 67F169A1. In the recombinant isogenic lines, the X and the second chromosome are isogenic among all lines, the Y chromosome must always be from the original low line, and the fourth chromosomes could be from the parent high, low, or original balancer stocks. The fourth chromosome carries ∼1% of the euchromatic genome (Hartl and Lozovskaya 1995) and thus can be ignored.
Measurement of correlated effects on appendages: Legs of lines LLH and LLL and of control lines were mounted on slides in glycerol under coverslips sealed with fingernail polish. Left legs were positioned with anterior side upward and partly flexed femorotibial joint. Leg images were projected from the slides and measured in the same way as wings. Comparisons between LLH and LLL leg shapes were made in terms of their angular offsets from control population baselines for femurs and tibias of prothoracic, mesothoracic, and metathoracic legs.
Detached and mounted halteres cannot be measured with precision, being compressible and round with no reproducible plane of mounting. For this reason it was necessary to measure unmounted halteres still attached to live anesthetized flies. All were measured with bodies positioned in the morphometer in the same viewing plane. Halteres were not compared by their offsets from control baselines; instead dimensions of LLL and LLH halteres were compared directly. This removed the necessity of also measuring a large sample of control halteres. To avoid bodysize complications in haltere measurements, flies of the LLH and LLL lines were cultured at similar density and matched carefully for body size.
Statistical separation of chromosome regions: To estimate the minimum number of segments with effects on the trait, the data set was first simplified by excluding all lines except (1) singlerecombinant lines at breakpoints having at least four lines in both the HL and LH directions and (2) the H and L nonrecombinant lines. Starting from the left end of the chromosome, singlerecombinant means at successive breakpoints were compared to the leftend nonrecombinant mean until the combined probability of the two phenotypic differences (HL  L, and LH  H) reached P < 0.01. Starting at this breakpoint, the same procedure was applied again until the whole chromosome was separated into significantly different intervals.
Individual and combined probabilities per interval were calculated as follows. The probability of the individual phenotypic difference at each interval was first evaluated for each direction (HL or LH) with a ttest based on the means and variances of line means at each breakpoint. Only phenotypic differences in the same direction as the total chromosome (HL or LH) were considered, thus all ttests were onetailed. Exact probabilities for each ttest were calculated using a computer program (K.W.) of the series expansions given in Abramowitz and Stegun (1970, p. 948, items 26.7.3 and 26.7.4), which exactly reproduces all tabulated ttest probabilities across the whole experimental range. The separate ttest probabilities for each direction (HL or LH) were then combined for each interval by calculating for each pair of probabilities the sum 2R ln(P), which is distributed as chisquare with 2k d.f., where k is the number of separate tests (Sokal and Rohlf 1969, p. 623). Combined probability thresholds for each interval were then taken from a standard table of chisquare values using 4 d.f.
Multiple interval mapping analysis: Quantitative trait loci (QTL) analysis was performed using the multiple interval mapping (MIM) method of Kao et al. (1999). This is a multiple QTLoriented method that combines QTL mapping with analysis of genetic architecture through maximumlikelihood analysis and a search algorithm to determine number, positions, effects, and interactions of significant QTL. The model of multiple interval mapping for m QTL is specified as
The likelihood function of the data is
Analysis of likelihood is by an EM algorithm (Kao and Zeng 1997). The test for each QTL effect is performed by a likelihood ratio (in LOD) conditional on other selected QTL effects. For testing a null hypothesis of H_{0}: E_{r} = 0 against H_{1}: E_{r} = 0, LOD = log_{10}[L_{1} (E_{r} = 0)/L_{0}(E_{r} = 0)], where L_{0} and L_{1} are maximum likelihoods under H_{0} and H_{1}, respectively, conditional on all other selected QTL effects.
A search for genetic models for likelihood evaluation was performed in two steps. First, several simplified procedures were used (see below) to select a premodel, to reduce the numerical burden of MIM analysis in model selection, and to combine information from different analyses to arrive at an appropriate starting model.
Given an initial model, the following stepwise selection analysis was performed to finalize the search for a genetic model under MIM:
Begin with a model that contains m QTL and t epistatic effects.
Scan the genome to search for the best position of an (m + 1)th QTL, and then perform a likelihoodratio test for the marginal effect (α_{m}_{+1}) of this putative QTL. If the test statistic exceeds the critical value (see below), α_{t}_{+1} is retained in the model.
Search for the (t + 1)th epistatic effect (β_{t}_{+1}) among those pairwise interaction terms not yet included in the model, and perform the likelihood ratio test on β_{t}_{+1}. If LOD exceeds the critical value (see below), β_{t}_{+1} is retained in the model. Repeat the process until no more significant epistatic effects are found.
Reevaluate the significance of each QTL effect currently fitted in the model. If LOD for a marginal or epistatic effect falls below the significance threshold conditional on other fitted effects, the effect is removed from the model. However, if the marginal effect of a QTL that has significant epistatic effect with other QTL falls below the threshold, this marginal effect is still retained. This process is repeated until the test statistic for each effect is above the significance threshold.
Optimize estimates of QTL positions based on the currently selected model. Instead of performing a multidimensional search around the regions of current estimates of QTL positions (which is an option), estimates of QTL positions are updated in turn for each region. For the ith QTL in the model, the region between its two neighbor QTL is scanned to find the position that maximizes the likelihood (conditional on the current estimates of positions of other QTL and QTL epistasis). This refinement process is repeated sequentially for each QTL position until there is no change in estimates of QTL positions.
Return to step 2 and repeat the process until no more significant QTL effect can be added into the model and estimates of QTL positions are optimized.
It is not clear how to determine an appropriate stopping rule for this method, with multiple locus main effects and epistatic effects combined in a single model. For this analysis, LOD = 2 was adopted as the critical value for retaining a QTL effect in the model. Permutation resampling was performed (Churchill and Doerge 1994) at the request of a reviewer, and a 5% permutation threshold of LOD = 1.89 was obtained for the data set. However, it should be pointed out that this permutation threshold, simulated at the null hypothesis of no QTL, is suited for the test of zero vs. one QTL and does not address the issue of model selection.
The analysis with LOD = 2 threshold picked up 10 QTL. The next putative QTL has the support of LOD = 1.2 and has a negative effect. Considering the strong appearance of repulsion linkage at this site in the raw data shown in Figure 2, we decided to retain this QTL in the model and correspondingly lowered the threshold for admitting epistatic effects to LOD = 1 for the purpose of estimating the relative contribution of epistatic effects to the total variance, as there are not many very significant QTL epistatic effects on the trait.
Given a selected model that contains the number and positions of QTL and QTL pairs interacting significantly, the estimation of QTL effects and the variance explained by selected QTL effects can be obtained from the likelihood analysis. This explained variance, σ^{2}_{g}, can be further partitioned into individual variances and covariances of QTL effects as
RESULTS
Figure 1 shows the index F baseline and samples of the isogenic parent lines. In one sample all three main chromosomes are high (HHH) and the angular offset is clockwise; in the other sample all three are low (LLL) and the offset is counterclockwise. The mean offsets of these lines differ by ∼0.13 rad, or 20 standard deviations of the base population. The figure illustrates three important facts about the trait. First, in the range of body size of these studies, wing shape is essentially independent of size when quantified by offsets from natural baselines. Second, the phenotypic correlation of D_{1} and D_{2} is extremely positive even though selection makes both dimensions change in a negatively correlated way. Third, selection on angular offset in wings causes an orderly rotation of the baseline about the origin, not a random realignment of the long axis.
The mean offset of 15 nonrecombinant L thirdchromosome lines (LLL) derived from these crosses was 0.0689 rad, and the mean of 15 nonrecombinant H thirdchromosome lines (LLH) was 0.0073 rad. The phenotypic difference of 0.0616 radians is the total effect of the third chromosome mapped in this study. In all, 519 lines were measured, or 25,950 wings. The third chromosomes of measured lines included 194 LH single crossovers, 149 HL single crossovers, 70 HLH double crossovers, 58 LHL double crossovers, 7 HLHL triple crossovers, 11 LHLH triple crossovers, 15 H nonrecombinants, and 15 L nonrecombinants. (Many more nonrecombinant lines were found but not measured.)
Singlerecombinant lines: Figure 2 shows the phenotypes of singlerecombinant lines as a function of recombination breakpoint. The profiles show clearly the cumulative effect of short sequentially added chromosome segments, starting from either a low or a high nonrecombinant chromosome. Effects occur all along the chromosome, and there is a strong appearance of repulsion linkage in one or more regions where the profiles are nonmonotonic, for example, between 70 and 90 cM. The graphs of singlerecombinant phenotypes convey, without further analysis, several of the most interesting qualitative results of this study. Many intervals along the third chromosome show discernible effects on the shape index, indicating genes located diffusely along most of the chromosome. The effects do not appear to fall into distinct categories of major and minor, and compared to the wing as a whole the effects are all very small.
The minimum number of genes: In analyses of QTL, minor QTL may crop up that are only statistical artifacts, especially if a large number of QTL are claimed. With the present large data set, one can get a preliminary idea of the distribution of effects along the chromosome by differentiating segments with a series of ttests, using the singlerecombinant data of Figure 2. Only breakpoints with four or more lines in both directions (LH and HL) were considered. Starting at the left end of the chromosome and moving right, the mean at each breakpoint was compared to the leftend mean until reaching a breakpoint where the difference, in both the HL and LH directions, attained a combined significance of P < 0.01 (see materials and methods). This defined the first segment with significant effect. Continuing from this point, a second breakpoint was reached, where the difference from the last starting point again reached a combined significance of P < 0.01. The objective was not to assess each fluctuation that might be due to a gene, but to estimate the minimum number of segments contributing to the total phenotypic difference of 0.0616 rad. Thus only differences in the overall forward direction were considered. Proceeding to the right end of the chromosome, this method separates it into eight segments (Table 3). This analysis does not incorporate the information available from the 146 multirecombinant lines or the 212 singlerecombinant lines at intermediate or excluded breakpoints. This is, however, enough to show that at least eight loci contribute effects.
Independent gene action vs. interaction: The profiles of HL and LH singlerecombinant means are nearly the same when one is inverted on the other (Figure 3A). This shows that the effects of individual loci are not much different whether they are sequentially combined from left to right or from right to left. Figure 3B shows the weighted mean of the two graphs in Figure 3A for all breakpoints with two or more single recombinants in both directions (there are 21 such breakpoints). This shows the cumulative mean genetic effect of short segments.
Wherever two loci interact, the profiles in Figure 3A must be separated between the sites involved by a distance equal to the magnitude of the interaction. If multiple pairwise interactions exist, the sites involved may be arranged in various patterns, overlapping each other, embedded, or endtoend, and only the net interaction across each breakpoint would be revealed in Figure 3A. Figure 3C shows the difference between the HL and LH means in Figure 3A for breakpoints with two or more single recombinants in both directions. This estimates graphically the magnitude and sign of the net interaction between loci to the left and right of each point. Interactions across the central region of the chromosome have a net positive effect; the net effect along the arms is negative. These could represent many small local interactions, or a few small widely separated interactions, or strong overlapping interactions that cancel each other. This also does not rule out strong interactions between closely linked loci that did not recombine nor interactions with loci on other chromosomes.
A model with many independently acting polygenes: If one assumes an indefinitely large number of loci, all acting independently (additively), Figure 3B becomes a model that can be used to predict the phenotype of any recombinant line. If the breakpoints of a chromosome are projected upward to intersect the graph, the predicted effect of each L or H segment can be read off on the yaxis. A computer program was written to estimate the phenotype of any recombinant line by interpolation from Figure 3B. This simple model is reasonably successful in explaining the phenotypic variation of the whole data set, with an R^{2} of 0.93 (Figure 4).
The model can also be checked by considering the variance within each recombinant category along with its necessary increase in error. The error variance must increase with each additional breakpoint among single, double, and triplerecombinant lines, because when a gene for the trait falls in any interval, crossovers can be either right or left of it. The mean increase in error variance among line means per breakpoint can be estimated empirically from the difference between single recombinants and nonrecombinants. Thus the variance among nonrecombinant lines of identical roo genotype (mean of S^{2}_{H} and S^{2}_{L}, total of 30 lines) is 0.000016; and the variance among singlerecombinant lines of identical roo genotype (weighted mean variance of line means at 22 HL and 21 LH breakpoints having >2 lines, total of 314 lines) is 0.000029; the difference is +0.000013 or the effect of one breakpoint. Therefore the mean variance among line means of identical roo genotype in any recombinant class, equal to the error variance among line means for the whole recombinant class, can be estimated as 0.000016 plus 0.000013 times the number of breakpoints. The resulting estimates of total phenotypic variance are compared with observed phenotypic variance in each recombinant class in Table 4 and Figure 5.
Results of multiple interval mapping: Several analyses were first performed to select a premodel for MIM analysis. The first analysis was a backward stepwise regression analysis on markers based on Ftodrop statistic with α = 0.01. This analysis selected 10 markers, with R^{2} = 0.93. The second analysis was by composite interval mapping (Zeng 1994) using QTL Cartographer (Bastenet al. 1998). The markers selected by backward stepwise regression were used as background markers in model 6 of QTL Cartographer with a window size of 3 cM (after trying a few other values). This analysis identified 10 significant positions based on a LOD = 2 criterion. The positions identified here, though generally agreeing with those marker positions picked by backward stepwise regression, differ significantly in a few critical major regions.
Based on these analyses, a premodel of 10 putative QTL positions was selected for MIM analysis, and the search process was started using the stepwise selection procedure. After several cycles of adding, dropping, and optimizing, the model quickly stabilized at 10 QTL and 5 significant epistatic effects, based on the LOD = 2 criterion. Given this model, the next QTL that may be considered is the one near 78 cM, showing repulsion linkage with other QTL. Although the support for this QTL (LOD = 1.2) is below the subscribed threshold, there is enough evidence from Figure 2 to include this QTL in the model. Subsequently, the threshold for epistatic effects was also lowered to LOD = 1, and 4 more epistatic effects were picked up. The final model contains 11 QTL main effects and 9 epistatic effects, with R ^{2} = 0.96.
A search was made for significant epistatic effects between selected QTL positions and unselected positions to identify potential QTL that show significant epistatic effects but not significant marginal effects. This was done in a stepwise manner by seeking the largest epistatic effect between a given QTL position and an unselected position at 1cM intervals and testing it for significance. However, the current model fits the data very well, and hardly any new component can improve the fit. This is not to say that no potential candidate pairs can be picked up. When they are picked up, the number of model parameters is increased and the fit of the model is not improved significantly.
The marginal likelihood profile (in LOD score) for each QTL position under the final model is shown in Figure 6. The estimates of positions and effects of QTL are given in Table 5. The estimated additive effects of the 11 QTL add up to 102.9% of the observed phenotypic difference between nonrecombinant high and low thirdchromosome lines.
Tables 6 and 7 give the estimated variances and covariances of individual QTL additive and epistatic effects. Together, 11 QTL additive effects explain 94.7% of the total phenotypic variance (27.4% is due to the variances of additive effects and 67.3% is due to the covariances between additive effects), and nine epistatic effects explain 1.2% of the total variance (7.2% is due to the variances of epistatic effects and 6.0% is due to the covariances between epistatic effects). The covariances between additive and epistatic effects, expected to be 0 (Kaoet al. 1999), account for 0.4% due to sampling. Thus the model explains 95.5% of the total phenotypic variance. It is clear that the vast majority of the trait variation in the population is explained by additive effects, and epistatic variation is minor.
Heterozygous effects of wingshape alleles: To assess dominance, representative singlerecombinant isogenic stocks were crossed either to a nonrecombinant isogenic high thirdchromosome (LLH) stock or to a nonrecombinant isogenic low (LLL) stock, according to the scheme in Figure 7. Hybrid lines created in this way are homozygous L or H for the whole chromosome segment on one side of the breakpoint and heterozygous on the other side. The phenotypes of these hybrid lines were mostly near the midpoint between the parental phenotypes, showing near additivity of alleles in the heterozygous part. In every cross there was a slight dominance of H alleles, which in this case means dominance in the direction of the wildtype phenotype. This pattern is consistent across the whole chromosome in either direction of the crossing scheme.
Correlated effects on leg shapes: The leg shapes of LLH and LLL flies cultured under identical conditions were compared to test the third chromosome as a whole for correlated effects of wingshape genes. Legshape indexes were constructed for the femur and the tibia, the two main leg segments. D_{1} was the widest part of each segment and D_{2} was the narrowest (and more proximal) part of the same segment, so these shape indexes were somewhat analogous to the one used for the wing. Baselines for these indexes were derived from measurements of control flies, and the angular offsets of LLH and LLL flies from these baselines were then compared with each other exactly as in the wing measurements. In all three legs, the differences between LLH and LLL are always negative for the femur and positive for the tibia (Table 8). These differences are also all highly significant (P < 10^{6}). However, they are small compared to the difference in wing shape. The mean legshape difference is only 1.29 SD of the base population, while the wingshape difference is 9.78 SD.
Correlated effects on haltere shapes: The halteres of LLH and LLL flies were compared in five linear dimensions (Figure 8 and Table 9). This comparison is not as sensitive to shape differences as comparisons of angular offsets from control baselines would be. This method was used because of the extreme difficulty of obtaining accurate haltere measurements, as it eliminated the need to measure a large sample of control halteres to create the baseline (see materials and methods). No significant differences in haltere dimensions were found between lines LLH and LLL. The statistical power of each comparison was estimated by the formula z_{power} = Δ_{H}/SED  t, where Δ_{H} is the hypothetical difference of 5 μm, SED is the standard error of the difference in means, t_{0.05} = 1.984 for d.f. = 98, and z_{power} has been tabulated with the corresponding values of statistical power (Motulsky 1995, p. 215). Halteres, though serially homologous to wings, show no detectable correlated effects of the wingshape alleles that differ on the H and L third chromosomes.
DISCUSSION
The genetic control of wing shape in D. melanogaster involves many loci with small effects. These loci recombine readily due to their dispersion along the chromosomes; they segregate at intermediate frequencies in wild populations; their effects are mostly additive; and they have generally only minor pleiotropic effects on other aspects of body form or on reproductive fitness. These facts were inferred from quantitative studies of natural variation, selection response, and inbreeding effects on shape (Weber 1990, 1992) that preceded this study. The present results confirm several of these features and add some details, in particular for one shape trait on one chromosome. There are also important questions that remain unresolved.
Unlimited polygenes vs. QTL: Statistical analyses of QTL have certain inherent ambiguities. Effects are explained with the smallest defensible number of QTL, and often only QTL with larger effects may be interesting. However, any factor identified by these methods may actually represent a cluster of loci. Thus there is always the potential for an undercount of the true number of loci, especially when statistically supported QTL are many and dense. It is also clear that multiple alternative sets of QTL effects can be derived to account for the same genetic variance. The possible ways to do this tend to mount as the number of QTL increases. These facts should motivate an interest in supplemental ways of looking at effects.
If markers are dense enough to eliminate hidden double crossovers between them, then the H or L character of any interval between markers is only ambiguous if it spans a detected crossover. Therefore the genotypes of single recombinants are known with greater accuracy than the genotypes of double or triple recombinants. If singlerecombinant lines are numerous, the graphs of their grand means vs. their breakpoints provide a robust picture of mean effects. Stepwise changes show the locations and magnitudes of effects. Significantly nonmonotonic points are reasonably interpreted as repulsion linkage.
The reciprocal profiles of HL and LH single recombinants check each other (Figure 3A). Matching increments, of equal magnitude in both profiles, indicate genetic effects that are the same in different combinations, i.e., not interacting with other loci on the chromosome. Unequal increments, which separate parts of the profiles when one is inverted on the other, indicate genetic interactions. Simple pairwise interactions should stand out because one recombination between the two loci turns the effect on or off. This separates the profiles between the loci by a distance equal to the interactive effect. The type of interaction can be partly worked out from the direction of the split.
Figure 3A can be contrasted with Figure 4 of Shrimpton and Robertson (1988), showing similar plots of singlerecombinant lines that were used to map loci for sternopleural bristle number in D. melanogaster on the third chromosome. In that case, the plots do not have matching profiles, revealing prominent interactions among genes on the right arm. In other systems, QTL have shown various amounts of interaction. A recent study of QTL for abdominal and sternopleural bristle number (Longet al. 1995) found significant epistasis in both traits. A review by Tanksley (1993) reports that strong interactions among QTL are not common. However, the resolution achievable in QTL studies is usually not high enough to exclude interactions within complexes of tightly linked loci (Cabotet al. 1994; Palopoli and Wu 1994), which would behave like independent factors in a typical QTL study.
The graphic analysis of chromosome 3 presented here avoids any attempt to enumerate or characterize loci. The underlying model assumes large numbers of polygenes, whose effects approximate a continuous but not linear function of chromosome length. The other assumption is that there are no interactions, so that any segment of the chromosome has the same effect in any combination. Thus the small net interactions revealed along the chromosome in Figure 3C are averaged out as if they were only error among line means. The resulting predictions explain most of the variation (R^{2} = 0.93).
Independent analysis by MIM confirms that many subequal additive effects are present. Another result is that important interactions may exist that cannot be resolved by the graphic analysis. Out of the 11 QTL, all but one (QTL 7) have at least one significant pairwise interaction. The two largest interactions (0.0138 and 0.0129) are larger than the largest main effect (0.0115). The sum of absolute magnitudes of the nine detected pairwise interactions is 98% as large as the sum of main effects. However, interactions are approximately balanced between positive and negative effects. These balanced interactions occur in QTL pairs that span nearly the same regions, so that their effects are also geometrically balanced along the chromosome. This symmetry is revealed by the four largest pairwise interactions from Table 5:
The main effects of the 11 QTL from Table 5 give a good fit to the graph of cumulative effects based on single recombinants (Figure 9A). The interactions among QTL also match the graph of net interactions based on single recombinants (Figure 9B). The graphic and statistical analyses were done independently by K.W. and ZB.Z., except that QTL 10 with its marginal LOD score was accepted into the QTL model because of K.W.’s insistence that the single recombinants show repulsion linkage in that region, implying the existence of an effect. Otherwise the two ways of looking at the data were developed entirely separately. The close fit between them in Figure 9, A and B, was only noted afterward.
The simple graphic analysis and the complex statistical one are in general agreement. Whether effects are concentrated into the minimum possible number of sites (∼11) or divided among some larger number, their number is large, their effects are subequal, and their distribution is diffuse and proportionate to Figure 3B. The complexity of the data is apparent in the attempts to evaluate interactions. In the QTL analysis, interactions are seen to be nearly as large as main effects (though with generally lower LOD scores). Among interacting QTL pairs, symmetries of location, magnitude, and sign appear to reduce large interactions to small net effects along the chromosome. The predominance of negative interactions could be explained most simply by frequent redundancies among loci that act by similar mechanisms; but this might not accord with the generally additive action of alleles in heterozygotes.
In any case, when all is said and done, we must admit that we are still left with the basic questions of quantitative genetics. Exactly how many genes are there, where are they, and what are their individual effects?
Advantages and limitations of MIM: Compared with other current QTL mapping methods, MIM has several advantages including increased statistical power and resolution (Kaoet al. 1999). MIM also provides appropriate estimation of individual QTL effects (including epistatic effects) and their contribution to variances and covariances.
However, it must be noted that the estimation of genetic parameters and interpretation of mapping results depend critically on model selection. Model selection in a high and unknown dimension is very complicated and difficult, particularly when it is based on genetic map positions, not just on markers. There could be numerous peaks separated by valleys or connected by ridges in a likelihood landscape with different dimensions. Any model selected, including the current one, may well be just a local peak, and there is no guarantee that a global peak can be found. It is possible that the number of effect components is underestimated; that is to say, that the 20 effects estimated here could be composed of more underlying biological components.
Not all estimates of QTL positions are sensitive to model selection. Some QTL position estimates are relatively consistent in different analyses and competing models and are largely independent of estimates of positions of other QTL and epistatic components. Other QTL estimates are, however, very sensitive to model selection and estimates of other QTL parameters. Among the 11 QTL mapped, position estimates of 8 (QTL 1, 2, and 611) are very consistent and relatively independent of mapping of other QTL and selection of epistatic components. These positions are consistently picked up in different analyses and in different models. Position estimates of QTL 3, 4, and 5 are, however, somewhat sensitive to model selection; i.e., estimates may shift locally if some other QTL effects are added to or dropped from the model.
The nature of wingshape genes: The loci of D. melanogaster listed by Lindsley and Zimm (1992) include many with major effects on the wing or with visible pleiotropic effects on the wing. Perhaps other alleles at such loci have small quantitative effects like those detected here (see Falconer and Mackay 1996). It seems unlikely that a class of genes as numerous as wingshape genes appear to be, in this and previous studies (Weber 1990, 1992), would not include any known loci. On the other hand, it will also be rather amazing if the detailed sculpting of the wing is achieved just by tuning the regulation of mostly known genes that often act over broad areas and in different domains during development. This would apply especially to the complex base (Weber 1992) and hinge regions. It remains to be seen whether any wingshape polygenes will turn out to be familiar loci.
Acknowledgments
Thanks to Trudy Mackay for explaining the method, to Cathy Laurie and Tony Long for advice, to Lynn Stam for teaching in situ labeling, to Burke Judd and Karl Suter for providing roo, and to Brian Charlesworth and Stavroula Assimacopoulos for the transposable elements 412 and 297. Thanks also to Andy Clark and two anonymous reviewers for suggestions on the manuscript. This work was supported by a grant from the National Science Foundation (DEB9407005) to K.W. and by U.S. Public Health Service grant GM45344 to ZB.Z.
Footnotes

Communicating editor: A. G. Clark
 Received July 6, 1998.
 Accepted June 8, 1999.
 Copyright © 1999 by the Genetics Society of America