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Institute of Plant Breeding, Seed Science, and Population Genetics, Univ. of Hohenheim, 70593 Stuttgart, Germany
* Corresponding author (melchinger{at}uni-hohenheim.de)
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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, 
, F1, F2, F2–Syn1, F2–Syn2, and F2–Syn3 did not significantly differ from each other for grain yield and grain moisture, indicating that epistasis between unlinked and moderately linked loci was negligible in its net effect. Depending on the cross, QTL mapping for per se and testcross performance with the dent tester was conducted with 71 to 344 lines (F3 to F6) grown in four environments. In genome-wide two-way ANOVAs, significant epistatic interactions were found with only a few marker pairs that did not improve the fit of the model after including main-effect QTLs previously detected by composite interval mapping. Poor correspondence of the results from per se and testcross analyses reflects dominance and epistatic interactions between parental and tester alleles. Our results suggest that epistasis is of minor importance for both traits with regard to the optimum type of population (F2 vs. BC) in recycling breeding of elite maize inbreds. Estimates of digenic epistasis detected with genome-wide tests must be treated with caution because of the problems associated with model selection in QTL mapping with the sample sizes commonly used.
Abbreviations: ANOVA, analysis of variance • BC1, BC2, first backcrosses of generation F1 to parents 1 and 2, respectively • BIC, Bayesian information criterion • P1, parent one • P2, parent two • QTL, quantitative trait locus/loci
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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Traditional approaches to assess the importance of epistasis have relied on the analysis of first- and second-degree statistics by using either generation means analysis (Mather and Jinks, 1982) or estimation of variance components from covariances of relatives generated via special mating designs (Hallauer and Miranda, 1981). Nevertheless, the underlying reference populations were in most studies not representative of elite hybrids because crosses within heterotic groups were mainly employed.
To overcome this problem, Melchinger (1987) proposed the testcross generation means analysis. Hereby, the basic generations are not evaluated for their per se performance but for their performance in testcross to a tester from the opposite heterotic pool. This formally eliminates dominance effects from the model, which otherwise tend to override estimates of epistatic effects. Furthermore, by testing interpool hybrids, the results are of direct relevance for hybrid breeding.
First experimental results from a testcross generation means analysis were reported by Melchinger et al. (1988) on a cross of European dent lines. Epistasis was generally of minor importance but significant for grain and forage dry matter content as well as root lodging resistance. In U.S. dent germplasm, Lamkey et al. (1995) found significant epistatic effects for grain yield and grain moisture explaining 21 and 18% of the variation among testcross generation means, respectively. In a follow-up study with 40 hybrid combinations, only five crosses yielded significant additive x additive epistatic effects for grain yield (Hinze and Lamkey, 2003). Hitherto, no study is available on the importance of epistasis in elite lines of European flint maize germplasm.
With traditional generation means analysis, significant epistatic effects have been detected for important agronomic traits of maize (Hayman, 1958; Gamble 1962a, 1962b; Melchinger et al., 1986). Positive additive x additive and negative dominance x dominance epistatic effects were small compared with additive and dominance effect (Melchinger et al., 1986).
Both testcross generation means analysis and ordinary generation means analysis estimate only net effects of genes or gene combinations summed over loci. Thus, positive and negative epistatic effects among individual quantitative trait loci (QTL) may cancel each other. QTL analyses allow dissecting quantitative traits into the effects of individual factors. In most instances, they revealed little or no evidence for epistasis (Stuber et al., 1992; Xiao et al., 1995; Liu et al., 1996; Lu et al., 2004). However, when individual QTL were isolated in isogenic backgrounds, epistasis was commonly observed (Doebley et al., 1995; Long et al., 1995; Eshed and Zamir, 1996; Laurie et al., 1997).
With composite interval mapping, we rarely found significant digenic epistatic effects among the detected QTL for testcross and per se performance of lines derived from three crosses of European flint maize (Mihaljevic et al., 2004, 2005). However, with genome-wide tests for epistasis, many important epistatic interactions were detected even among marker loci that did not show significant main effects (Damerval et al., 1994; Li et al., 1997; Holland et al., 1997).
The major goal of the present study was to assess the importance of epistasis for grain yield and grain moisture in four crosses of elite European flint maize with different approaches. Our objectives were to (i) estimate the relative importance of aggregate epistatic effects by generation means analyses of per se and testcross performance, (ii) perform genome-wide tests for significant epistatic effects between individual marker loci, and (iii) compare the results of each analysis and previous QTL analyses for both per se and testcross performance.
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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Field Experiments
Testcross Generation Means Analysis
Testcross progenies of generations P1, P2, F1, F2, F2–Syn1, F2–Syn2, F2–Syn3, BC1, and BC2 were evaluated in a 5 x 10 
-design (Patterson and Williams, 1976) at four environments (Eckartsweier, Bad Krozingen, Zell, and Stuttgart-Hohenheim) in Germany with three replications. Testcrosses of P1 and P2 were included as duplicate entries.
Generation Means Analysis
The generations P1, P2, F1, F2, BC1, and BC2 derived from each of the four crosses were evaluated for per se performance in a split-plot design with generations comprising the main plots and crosses comprising the subplots. The trials were grown at four environments (Eckartsweier, Bad Krozingen, Zell, and Hochburg) in Germany with four replications.
For all experiments, plots consisted of two rows, 4.0 m long and 1.5 m wide with 0.7 m between rows. Two-row plots were overplanted and later thinned to reach a final stand of 90000 plants ha–1. All experiments were machine planted and harvested as grain trials with a combine. Data were analyzed for grain moisture (g kg–1) at harvest and grain yield (Mg ha–1) adjusted to 155 g kg–1 grain moisture.
Agronomic Data Analyses
Lattice and split-plot analyses of variance for testcross and per se data, respectively, were performed for each environment. Adjusted entry means and effective error mean squares from the lattice analyses as well as means and error mean squares from the split-plot analyses were then used to compute the combined analyses of variance across environments (Cochran and Cox, 1957). Generation means across environments were further used in the quantitative genetic analyses.
Testcross Generation Means Analysis
Two genetic models were fitted to the testcross generation means (Melchinger, 1987). Model 1T accounts for additive effects only. Model 2T allows for epistatic effects between unlinked pairs of loci but ignores linked epistatic pairs. The superscript T in the following models indicates that these values pertain to testcross effects.
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T) = additive effect summed over loci (equivalent to one-half the average effect of a gene substitution (T) at a single locus with a positive sign if P2 contains the favorable allele); (T) = additive x additive digenic epistatic effect summed over locus pairs.
Generation Means Analysis
Two genetic models were fitted to the per se performance data of the six generations. Model 1 includes only additive and dominance effects. Model 2 allows for epistatic effects between unlinked pairs of loci but ignores linked epistatic pairs. All effects were defined according to the F2 metric (Hayman, 1958).
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The formulas for the genotypic means of the various generations are
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Estimation of Effects and Model Fit
The genetic parameters for all four models were estimated using weighted least squares:
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denotes the column vector of estimated genetic effects; X the matrix with elements that are a function of the generation; W the weight matrix with the inverse of the variances of the generation means on the diagonal and zero on the off-diagonal; and y the column vector Y or YT, respectively. Weighted estimates were calculated because the parental generations were tested as duplicate entries. Standard errors for the genetic parameters were estimated as the square root of the diagonal of the (X'WX)–1 matrix. The coefficient of determination (R2) was calculated to estimate the proportion of the variation among generation means accounted for by each model.
For both testcross and per se performance data, the goodness-of-fit of a model was tested with a weighted Chi-square (Mather and Jinks, 1982), 
2 = 
[(O – E)2 x W], where O = the observed generation mean, E = the expected generation mean, and W = the inverse of the variance of the generation mean.
QTL Experiments
QTL analyses for testcross and per se performance of the crosses AxB, AxC, and CxD were published previously (Schön et al., 1994; Melchinger et al., 1998; Mihaljevic et al., 2004, 2005). No QTL analysis was performed for the cross AxD because the population size was too small (N = 42) to obtain meaningful results. Briefly, four populations, AxBI (344 F2:3 lines for testcross and 280 F2:3 lines for per se performance), AxBIII (71 F4:5 for testcross and 120 F4:6 for per se performance), AxC (109 F3:4 lines for testcross and 131 F3:4 lines for per se performance), and CxD (84 F3:4 lines for testcross and 135 F3:4 lines for per se performance) were employed in QTL analyses. Here, AxBI and AxBIII represent different samples of the same cross, the notation being in accordance with Mihaljevic et al. (2004)( 2005). All these populations were reanalyzed here with a genome-wide test for epistatic effects to detect interactions among QTL which do not necessarily have a significant main effect. The number of markers employed ranged from 73 to 95 depending on the population. Only those markers used for constructing the joint map across populations described by Mihaljevic et al. (2004)(2005) were employed herein for further analyses. The average marker density on the joint map ranged from 10.2 cM in CxD to 15.0 cM in AxBIII.
Digenic epistatic effects, (aa) for per se performance and (T) for testcross performance, between all pairs of marker loci were tested by EPISTACY, a two-way ANOVA routine in SAS based on the F2 metric (Holland, 1998). Epistatic interactions were declared significant if they exceeded the threshold of P < 0.001. This threshold was determined because 45 independent combinations exist among the ten linkage groups of maize. A comparison-wise error rate of 10–3 would correspond approximately to an experiment-wise error rate of 0.05. This seems a liberal estimate of the genome-wise error rate for epistatic interactions (Holland et al., 1997).
The Bayesian information criterion (BIC; Piepho and Gauch, 2001) implemented in software PLABQTL (Utz and Melchinger, 1996) was used to compare the model including only positions of main-effect QTL estimated by standard composite interval mapping with an extended model, which included the position of the main-effect QTL plus those marker pairs with significant epistatic effects detected by EPISTACY.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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, backcross mean , and F1 and F2 generations in any cross for both traits. Likewise, no significant changes were observed between testcrosses of generations F1, F2, F2–Syn1, F2–Syn2, and F2–Syn3 for all crosses and both traits. Model 1T explained over 78% of the variation among generation means for grain yield in all crosses except CxD (Table 2). The 2 goodness-of-fit test for Model 1T was not significant in any of the four crosses. Inclusion of epistatic effects in Model 2T resulted in a substantial increase of R2 values for AxB and AxC, with estimates of (T) being significant. For grain moisture, the 2 goodness-of-fit test for Model 1T was significant (P < 0.05) in AxD. R2 values of Model 1T varied between 57.5 and 73.0% for grain moisture and increased substantially for Model 2T in AxB, AxC, and AxD. In all three crosses, estimates of additive effects (T) were highly significant (P < 0.01) for both traits. Estimates of epistatic effects were negative in all crosses for grain yield. For grain moisture, estimates of (T) were significant in two crosses and of positive sign. CxD deviated from the other three crosses in that R2 values were low (
21.5%) for both models and estimates of (T) and (T) were nonsignificant for both traits.
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; the F2 generation means were significantly smaller than the means in AxD and CxD for grain yield. For grain moisture, no significant differences existed among these generations in three of the four crosses (Table 3).
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2 values for crosses AxD and CxD (Table 4). Estimates of epistatic effects (aa) were significant only for CxD. The 2 goodness-of-fit test of Model 2 was nonsignificant for all crosses except AxD. For grain moisture, R2 values for Model 1 were lower and ranged between 63.1 and 90.5%. Inclusion of epistatic effects in Model 2 improved the fit, but estimates of epistatic effects (aa) were not significant for either cross.
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Digenic Epistatic Interactions
Testcross Performance [(T) Type of Epistasis]
The number of marker pairs with significant (P < 0.001) epistatic interactions for grain yield was two for AxBI, zero for AxBIII, and one for AxC and CxD (Table 5). The absolute size of the (T) effects for grain yield ranged from 0.21 to 0.31 Mg ha–1 across populations. The sum of absolute values of the two (T) effects in AxBI of opposite sign was about half the sum of absolute additive QTL effects estimated in the same population. In AxC and CxD, the absolute (T) effect was by far less than half of the sum of the absolute additive QTL effects.
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T) effects were detected in the largest population AxBI. Three epistatic marker pairs were detected in AxC and CxD, respectively. All three had a positive sign in AxC, but one showed a negative sign in CxD. The absolute size of (T) effects ranged from 1.2 to 4.4 g kg–1 across populations. For grain moisture, the sum of absolute (T) effects was about one-third of the sum of absolute additive QTL effects in AxC and about one-fourth in CxD but comparatively small in AxBIII (Table 5). Of the 11 epistatic marker pairs detected across populations and traits, no marker was flanking a QTL with main effects. The effects estimated with PLABQTL by the model including only epistatic marker pairs from EPISTACY were reduced in size only, apart from two changes in sign when the model included both the epistatic marker pairs and the main-effects QTL previously detected by composite interval mapping (Table 5). According to the BIC, the model extended for epistatic marker pairs was not superior to the basic model, including only main-effect QTL in each population except for grain moisture in CxD.
Per Se Performance [(aa) Type of Epistasis]
Between one and three marker pairs per population showed significant (P < 0.001) epistatic interactions for grain yield (Table 6). The absolute size of the (aa) effects for this trait ranged from 0.29 to 0.69 Mg ha–1 across populations. The sum of absolute (aa) effects was comparable with the sum of absolute additive QTL effects in AxBIII, AxC, and CxD (Table 6). The (aa) effects were negative in cross AxB, but of opposite sign in AxC and CxD.
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The effect size estimated with PLABQTL by the model including only the marker pairs detected with EPISTACY was mostly larger compared with the model that included these marker pairs plus the positions of main-effect QTL detected previously by composite interval mapping (Table 6). According to the BIC, the latter model was consistently not superior to the basic model including only main-effect QTL.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONREFERENCES |
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In the testcross generation mean analysis, epistasis between unlinked loci can alter only the means of generations prior to the F2 (i.e., , , and F1) because the gametic array produced by the F1 (or any generation derived from it by random mating) is expected to be in linkage equilibrium. The contribution of positive epistasis between linked loci should therefore decline monotonically in the order > > F2 > F2-Syn1 > F2-Syn2 > F2-Syn3 as a function of the recombination frequency (Melchinger, 1987). Since the parental lines of this study were developed by recycling breeding of elite lines, we expected to find epistasis in generation means analyses for both per se and testcross performance.
Epistasis in Testcross and Per Se Generation Means
In the testcross generation means analysis, contrasts vs. F1, vs. , or vs. F1 were not significant for grain yield and grain moisture and, thus, provided no evidence for epistasis among unlinked loci (Table 2). Likewise, generations F2 to F2-Syn3 were not significantly different in their testcross means. In contrast, four of the eight estimated additive x additive epistatic effects (T) were significant. Following the common procedure in the literature (e.g., Hinze and Lamkey, 2003), the latter were tested against the deviation means squares, which are often smaller than the estimated error variance of generation means corresponding to SE in Table 1. With the latter error term in this cases, no significant (T) effects were detected. Given the low number of degrees of freedom for the residual error or test environments in the generation means analyses, the power of these tests is relatively poor. Therefore, the relative importance of epistasis for testcross performance will be briefly assessed by considering the ratio AAT% = (T)/mT x 100.
Averaged across all four crosses in our study, AAT% amounted to –2.7% for grain yield and 0.4% in grain moisture. Thus, for grain yield there was no indication for positive epistasis. By comparison, Lamkey et al. (1995) observed a high reduction in testcross performance for grain yield after eight generations of random mating in cross B73xB84, corresponding to AAT% of 6.3% for grain yield and 1.8% for grain moisture. In a more extensive study with 40 crosses of current U.S. elite lines, Hinze and Lamkey (2003) found epistasis to be unimportant for grain yield with an average estimate of AAT% of –0.8% and a range between –5.2% and 5.2%, depending on the cross. Thus, the net effect of epistasis on testcross generation means seems to be generally of minor importance, but higher values for individual crosses and environments cannot be ruled out.
A clear distinction of the contribution of unlinked vs. linked locus pairs or epistasis of higher order than (T) is complicated by the fact that (i) the effects of epistasis cannot be completely separated from those of linkage (Melchinger, 1987), (ii) epistatic effects are partly contributing to additive effects and higher-order epistatic effects are contributing to estimates of lower order effects (Cheverud and Routman, 1995), (iii) (T) effects are confounded with additive x dominance and dominance x dominance interactions between parental and tester alleles (Eta-Ndu and Openshaw, 1999), and/or (iv) maternal effects are confounded with epistatic effects. In the present study, maternal effects cannot be ruled out, because the testcross seed was produced in an isolation plot with the dent tester line as pollen parent and the various generations from the four crosses as seed parents. In reciprocal crosses of three-way hybrids, Schnell and Singh (1978) reported an average yield advantage of 3.1% for hybrids produced on a vigorous F1 seed parent as compared to those produced on an inbred line seed parent, which have poorer early vigor owing to their smaller seed weight. Obviously, this type of maternal effect would be present in the comparison of with other generations.
In cross CxD, it was striking that the estimate of (T) was fairly small (Table 2) even though the two parents showed pronounced differences in their per se performance (Table 3). Thus, the weak line D expressed strong dominance with the tester. The influence of dominance with the tester on estimates of (T) were discussed in detail by Melchinger et al. (1998). Hence, it seems plausible that in crosses AxD and CxD the correspondence of estimates of (T) with (a) and (T) with (aa) was relatively poor. The agreement between both types of estimates was much better in crosses AxB and AxC. In all instances, the large dominance effect for per se performance reflected the substantial heterosis for grain yield in maize even in crosses within heterotic groups (Table 3, last line).
In conclusion, our study confirms the limitations of generation means analysis for an assessment of the importance of epistasis for quantitative traits. Recognizing these difficulties, marker-based analyses of epistatic effects have been suggested to be more powerful (Damerval et al., 1994; Li et al., 1997; Holland et al., 1997).
Mapping of Epistatic QTL
We found several marker pairs showing significant two-locus epistasis in addition to main-effects QTL. In general, these marker pairs were not flanking main-effect QTL. The sum of the absolute epistatic effects was often half or more of the sum of the absolute additive QTL effects. Thus, at first glance epistasis seems to be important in the analysis of QTL, which is in agreement with experimental results from other plants (Li et al., 1997; Holland et al., 1997; Kearsey et al., 2003). However, when the position of main-effect QTL previously identified in each cross by Mihaljevic et al. (2004)(2005) were included in the model, epistatic effects did not improve the model fit measured by the Bayesian information criterion. As for the generation means analyses, we therefore discuss the limitations of estimation of two-locus epistasis.
The first problem is that the true number and position of QTL, which correspond to the correct statistical model for estimating the gene effects, are unknown and must be determined by model selection (Zeng et al., 1999). The general procedure is to identify among a large number of regressor variables (markers) those that account for the largest proportion in the variance of the response variable (phenotypic values). Subsequently, these genome positions are used for estimation of QTL effects and the proportion p of the genotypic variance explained by the detected QTL. With a limited sample size, model selection leads to an overestimation of QTL effects and p because of sampling effects and, consequently, to a biased assessment of the prospects of marker-assisted selection (Melchinger et al., 1998; Utz et al., 2000). A genome-wide search for epistatic effects among QTL aggravates the problems associated with model selection, because the number of regressor variables (marker pairs) increases tremendously (for two-locus epistasis in quadratic progression, for three-locus epistasis in cubic progression, etc.). Furthermore, collinearity of dummy marker variables in the selected model disturbs the estimation of additive and epistatic QTL effects, especially with dense marker maps. Moreover, epistatic pairs of QTL are fit directly at marker locus positions rather than in intervals, which may reduce the power of QTL epistasis tests compared to the additive effect tests. Determining the appropriate experiment-wise error rate is therefore of crucial importance (Holland, 1998; Holland et al., 1997).
Similar to mapping of a QTL, in which the effect of other QTL should be taken into account for example by including cofactors in the model, the same principle applies to the search for epistatically interacting pairs of QTL (Zeng et al., 1999). In our study, the size of epistatic effects for line per se and testcross performance was reduced, when all previously detected main-effect QTL were added to the model (Tables 5 and 6). Bogdan et al. (2004) reported similar results from simulations. They recommended a larger penalty in the BIC for epistatic terms than for main effects. Even with the ordinary BIC, no epistatic terms remained in the model in our study.
The need for validation with an independent sample or cross validation (Utz et al., 2000) is even more compelling for epistatic than for main effects of QTL. An ultimate proof for the presence of an epistatic pair of QTL and an unbiased estimation of its gene effects requires the isolation of the pair of QTL in a homogeneous background by means of near isogenic lines (NILs) or similar approaches (Doebley et al., 1995). Moreover, we strongly recommend using larger populations at least of the sample size of our biggest experiment (N = 344) for detection and mapping of epistatic QTL for complex traits such as grain yield and grain moisture. The presence of minor biological epistasis, however, cannot be ruled out at least for the cross AxBI, where evidence for weak epistasis was detected with a relatively large number of progenies.
Conclusions and Consequences for Breeding
In agreement with the findings of Hinze and Lamkey (2003) for U.S. dent germplasm, our results indicate that epistasis hardly influences the testcross means of F2 or BC populations produced from elite European flint lines. Consequently, epistasis can be ignored with regard to the choice of the type of base population to be preferably used in recycling breeding (Melchinger et al., 1988). Moreover, we conclude that epistasis generally does not benefit single crosses over other types of hybrids and can safely be ignored in predicting the performance of three-way or double-cross hybrids from the means of their nonparental single crosses (Melchinger et al., 1987).
Our QTL analyses demonstrate that for complex traits such as grain yield, it is extremely difficult to map epistatic QTL with high fidelity and separate their effects from those of main-effects QTL. This is due to the problems associated with model selection, even when relatively large sample sizes are used. A promising way out of this deadlock could be novel approaches such as "genetical genomics" (Jansen and Nap, 2001), in which genome-wide expression data obtained from genomics, proteomics, and metabolomics are analyzed in parallel with phenotypic and marker data to unravel the basis of metabolic, regulatory and developmental pathways. Altogether, the novel approaches currently invented and used in systems biology to study the function and control of genetic networks promise to contribute to a better understanding of epistasis at the molecular level, and in turn also at the level of the entire genotype (Carlborg and Haley, 2004).
It is anticipated that the relative importance of epistatic effects in hybrid maize breeding may strongly increase with the current paradigm shift in line development from recurrent selfing to the production of doubled haploids (Seitz, 2004, pers. communication). This is because the variance of epistatic effects of order m among unlinked loci contributes to the genetic variance among Sn lines (n 
1) only with a coefficient (1 – 0.5n)m (Cockerham, 1963). Hence, with early generation testing in traditional line development, digenic epistasis and even more so higher-order epistasis contribute only marginally to the genetic variance among S1 or S2 lines compared with additive effects. In contrast, with doubled haploid lines (corresponding to S
lines), the coefficients of all epistatic variance components are equal to one and, hence, epistasis contributes fully to the genetic variance from the very beginning of the selection process. Moreover, because recombination is limited to a single meiosis for each breeding cycle, doubled haploids minimize recombination between linked loci and, thus, should be very effective in conserving tightly linked complexes of genes with positive epistasis. The drawback of restricted recombination is, however, the low chances to identify positive complexes of genes if these occur in repulsion phase in the parents. This requires either extremely large population sizes or several generations of intermating before producing the doubled haploid lines. It will therefore be of interest to investigate the importance of epistasis after several cycles of recycling breeding with doubled haploid lines have been completed.
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ACKNOWLEDGMENTS |
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Received for publication December 25, 2004.
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