Design III with RILs:

Let us assume a random population of RILs derived from the cross between the two homozygous lines P1 and P2. Further, we assume that the RILs are backcrossed to their parental lines, yielding testcross progenies H₁ and H₂. Gene effects of the design III testcross progenies are expressed with regard to the corresponding gene-orthogonal population, i.e., the F₂. The parental line exhibiting superior average testcross performance across both testers is denoted as P2. With testcrosses evaluated in a randomized complete block design we obtain phenotypic trait values Y_tpk of the testcross progeny H_t of RIL p (p = 1,…, n) crossed with tester t (t = 1, 2 for design III) in the kth block (k = 1,…, r). Following Comstock and Robinson (1952), we perform two linear transformations Z_s (s = 1, 2) on the performance data Y_tpk of testcross progenies H_t with pair means Z_1pk = (Y_1pk + Y_2pk)/2 and pair differences Z_2pk = Y_1pk − Y_2pk. Thus, Z_spk denotes the phenotypic value of transformation Z_s for RIL p grown in the kth block and Z_sp the progeny mean value of Z_s for RIL p. Expected mean squares from the ANOVA are given in Table 1.

Marker-based estimation of augmented effects a_i* and d_i*:

The first to present the statistical theory for estimation of the type of gene action at individual QTL with design III were Cockerham and Zeng (1996). They presented genotypic expectations of marker contrasts using single-marker ANOVA and extended the analysis to test for two-locus interactions between linked QTL. They demonstrated that genotypic expectations of QTL mapped with design III were biased with epistasis. Dominance effects were confounded with aa epistasis, additive effects with da interactions regardless of linkage. However, interactions of individual QTL with the entire genetic background were not accounted for and, most importantly, they did not make the connection to genetic expectations of MPH. While Cockerham and Zeng (1996) considered the confounding epistatic effects in their analysis a limitation, we claim that they are favorable for the identification of genomic regions contributing to heterosis. The following derivations show that genotypic expectations of QTL identified with design III precisely equal the augmented dominance effect d_i*.

Cockerham and Zeng (1996) defined four orthogonal single-marker contrasts (C₁, C₂, C₃, C₄) among the means of testcross progenies from F₂ individuals or F₃ lines, estimating additive (C₁) and dominance effects (C₃) as well as digenic epistasis of linked QTL (C₂, C₄). Here, we extend their methods to the analysis of heterosis and to the analysis of RILs. We defined two orthogonal single-marker contrasts on the basis of progeny mean values Z_sp for pair means (Z₁(m)) and pair differences (Z₂(m)) (see the appendix for a generalized derivation). While contrasts C₁ and C₃ in the notation of Cockerham and Zeng (1996) correspond to Z₁(m) and Z₂(m) in our notation, contrasts C₂ and C₄, which provide tests for epistasis among linked QTL, cannot be calculated for RILs because they rely on comparisons of the heterozygous vs. homozygous marker classes.

With epistasis restricted to digenic effects, we obtain(8)(9)D_mi being the linkage disequilibrium between QTL i and marker m (Weir 1996). For RILs, D_mi can be calculated from the recombination frequency r_mi between QTL i and marker m as (for further details see the appendix).

As can be seen from Equation 9, the expectation of Z₂(m) is a multiple of the sum of the augmented dominance effects d_i* for all QTL with D_mi > 0, weighted by their linkage disequilibrium D_mi to the marker m. Thus, estimation of dominance effects of QTL linked to marker m is confounded with digenic epistatic interactions of type additive × additive with the entire genetic background. Analogously, the linear contrast Z₁(m) is a multiple of the sum of augmented additive effects a_i* for all QTL with D_mi > 0, weighted by their linkage disequilibrium D_mi to the marker.

Expected mean squares for an ANOVA that includes contrasts Z₁(m) and Z₂(m) for design III evaluated in a randomized complete block design are presented in Table 1. Appropriate statistical tests can be derived by standard statistical theory. From expected mean squares, it becomes evident that the power of detecting significant QTL effects linked to marker m depends mainly on segregation variances and of the variables Z₁ and Z₂ with respect to QTL effects not accounted for by the marker m, i.e., the genetic background variation.

With single-marker ANOVA the QTL position (here reflected by parameter D_mi) and QTL effects (here a_i* or d_i*) are confounded. Hence, the effects of linked QTL cannot be separated, and effects of other unlinked QTL, which cause background noise, are not accounted for. The above shortcomings have been overcome by CIM (Zeng 1994; Jansen and Stam 1994), which allows separate estimation of the QTL position and the QTL effect as well as separation of linked QTL and control of genetic background variation. We have adopted this method for identification of genomic regions affecting heterosis. Using the model(10)where Z_sp is the linear function Z_s calculated for the pth RIL, b_0s is the mean effect of the model for Z_s, b_s* is the augmented effect of the putative QTL on Z_s, x_p* is the conditional probability of the dummy variable θ_m given the observed genotypes at the marker loci m and (m + 1) flanking the putative QTL (θ_m takes value 0 or 2 if the genotype of the RIL at the QTL is homozygous P1 or P2, respectively), b_ls are partial regression coefficients of Z_sp on the lth marker assuming h markers included in the model as cofactors, x_lp is an indicator variable taking value 0 or 2 depending on the genotype at marker l, and is the residual effect on Z_s for the pth RIL. CIM analyses can be individually performed for the two linear functions Z_s from design III. Tests of significance are straightforward as described for CIM (Zeng 1994) with the null hypothesis H₀: b_s* = 0 and the alternative hypothesis H_A: b_s* ≠ 0.

If we assume complete linkage between the marker m and one of the QTL (r_mi = 0), we obtain the following genotypic expectations for the contrast of the two homozygous marker classes at marker m:(11)(12)

The advantages of CIM in comparison to single-marker ANOVA become obvious immediately: (i) the position of the QTL i can be estimated and (ii) the effect of the linked QTL j should be blocked by the use of cofactors (Jansen and Stam 1994; Zeng 1994). Thus, it is possible to test contrasts for QTL (not markers) and the genotypic expectation for Z₂(i), the contrast of Z₂ between the two unobservable homozygous genotype classes at QTL i, reduces to Consequently, a genome scan with Z₂ localizes genomic regions affecting MPH and b₂*/2 = d_i*. Accordingly, a_i*, the contribution of QTL i to the parental difference, is equal to b₁*. The advantage of design III is that genotypic expectations of QTL mapped with Z₂ precisely equal their net contribution to MPH. However, the contribution of the main dominance effect d_i cannot be assessed independently from the sum of aa epistatic interactions of QTL i with the genetic background, which is a limitation of design III.

Extension of heterotic QTL analyses:

In addition to separate CIM of Z₁ and Z₂ we propose to apply the theory of a joint analysis for multiple traits as suggested by Jiang and Zeng (1995) to the simultaneous mapping of augmented QTL effects a_i* and d_i*. Depending on the correlation structure of data on multiple traits a joint analysis may increase the power of the likelihood-ratio (LR) tests for QTL detection and may allow us to distinguish between pleiotropy and close linkage of QTL for individual traits. Following Jiang and Zeng (1995), we regard Z₁ and Z₂ as two different traits and rewrite the model from Equation 10 in matrix notation,(13)where Z is a matrix of Z_sp, x* is a column vector of x_p*, b* is a row vector of b_s*, X and B are two matrices controlling the genetic background variation, and ℰ is a matrix of The power of QTL detection using the model from Equation 13 is significantly increased by a joint LR test if the product of the effects b₁* (corresponding to a_i*) and b₂* (corresponding to 2d_i*) and the correlation ρ₁₂ between residuals and have different signs (Jiang and Zeng 1995). In the joint QTL analysis, the hypotheses to be tested are: H₀: b₁* = 0, b₂* = 0 vs. H_A: b₁* ≠ 0 or b₂* ≠ 0.

On the basis of experimental results from the analysis of complex traits it seems reasonable to assume that marker–trait associations detected with the model in Equation 10 will rarely explain >50% of the genotypic variance even with large sample sizes (Schön et al. 2004). Thus, the residual component must be subdivided into two components; i.e., = g_sp + e_sp, with g_sp reflecting the genetic component in not accounted for by the putative QTL and the cofactors in the model, and e_sp being the experimental error. As a consequence, residual components and can be correlated within RILs but are independent among RILs. The correlation ρ₁₂ between residuals and can be approximated by Using the derivation of covariances presented in the appendix and assuming digenic epistasis and absence of linkage,(14)Hence, it follows that ρ₁₂ depends mainly on the variation in sign and magnitude of a_i* and d_i*. When P1 and P2 are elite inbred lines, we expect that about half of the trait-increasing alleles are contributed by P1 (i.e., a_i* < 0) and half by P2 (i.e., a_i* > 0). With directional dominance (d_i* > 0), the correlation of residuals is in this case expected to be close to zero. Thus, the joint LR test statistic will approximate the sum of the individual LR test statistics and no increase in the power of QTL detection can be expected from a genome scan using the multivariate model from Equation 13. However, when analyzing progenies from crosses between an elite parent (P2) and an exotic donor line (P1) with a much smaller proportion of positive alleles than the elite parent the joint LR test statistic can be advantageous for finding heterotic QTL contributed by P1. If P2 contributes the majority of positive dominant alleles, i.e., a_i* > 0 and d_i* > 0, then ρ₁₂ > 0. In genome regions where a positive dominant allele originates from P1 (a_i* < 0; d_i* > 0), the product of effects a_i* and d_i* will be negative, while ρ₁₂ > 0. Thus, the power of detecting heterotic QTL from exotic donor parents will be increased with the joint LR test.

In addition to increasing the power of QTL detection with multiple traits, the statistical method suggested by Jiang and Zeng (1995) also provides a test for separating pleiotropy from close linkage. In our case, the analogous test can be applied to distinguish between the hypothesis of dominance vs. overdominance in genomic regions with significant effects a_i* and d_i*. LR tests for the null hypothesis H₀: z(1) = z(2) vs. H_A: z(1) ≠ z(2), where z is the position of the QTL affecting Z₁ and Z₂, respectively, have been described by Jiang and Zeng (1995). Under the null hypothesis, the same genetic locus contributes to the augmented effects a_i* and d_i*. If both effects are of similar magnitude, acceptance of H₀ would imply that dominant gene action is prevalent. In contrast, acceptance of H_A would imply that one locus shows additive gene action (a_i* > 0; d_i* = 0) and at the second locus the heterozygote outperforms both homozygotes, thus exhibiting overdominance (a_j* = 0; d_j* > 0; ).

Genotypic components of trait performance and heterosis:

Elucidating the genetic basis of heterosis has been the aim of a number of studies making use of advances in molecular biology. Some studies compared specific molecular traits such as differential gene expression in the F₁ hybrid and the parental inbred lines (e.g., Guo et al. 2006; Swanson-Wagner et al. 2006). A number of authors used testcross progenies of F₃ lines or RILs with the parental inbred lines as testers for QTL studies on heterosis (Stuber et al. 1992; Xiao et al. 1995; Li et al. 2001; Luo et al. 2001). Estimation of QTL effects was separately performed for the backcrosses to each parental line, except for a recent study with triple testcross progenies in maize, which used also 2Z₁ and Z₂ for QTL mapping (Frascaroli et al. 2007). For the backcrosses to the better-performing parent, this corresponds to the marker contrast −H₂(m) defined in Equation 15. Thus, with interval mapping (Lander and Botstein 1989) their analyses yielded estimates for QTL contributing to BPH (d_i* − a_i*) for the progenies backcrossed to one parent and for poorer parent heterosis (d_i* + a_i*) for the progenies backcrossed to the other parent. Consequently, conclusions on gene action at individual QTL can be made only within this context.

Semel et al. (2006) used a set of tomato introgression lines to exclude the confounding effects of genetic background variation and epistasis from the analysis of heterosis. Parental genotypes and the F₁ hybrid differed exclusively in one defined chromosome segment while the entire genetic background originated from the elite parent. For a multitude of traits, the authors compared the phenotypic means of the elite parent, the homozygous introgression line, and the hybrid between the two. On the basis of these comparisons, the type of gene action at QTL was determined. However, as pointed out earlier, trait performance and heterosis have different genotypic expectations. Employing the quantitative genetic theory derived in this article, it can be shown in terms of the F₂ metric that genotypic expectations of QTL effects contributing to MPH and BPH comprise both main and epistatic effects, despite the fact that the entire genetic background originates from the elite parent (Melchinger et al. 2007).

The use of an immortalized F₂ population was proposed by Hua et al. (2003) for identifying genomic regions that contribute to MPH. RILs were derived from a heterotic rice cross and intermated for construction of an immortalized F₂ population. Heterosis was calculated as the phenotypic deviation of each immortal F₂ from the mean of its two RIL parents. Digenic interactions were estimated on the basis of the interaction effect of two marker loci. Genotypic expectations of the QTL main effects and interactions were not given by the authors. On the basis of our findings, we postulate that the immortalized F₂ design has great value for estimating the dominance effect d_i and certain types of digenic epistatic interactions but does not identify QTL with genotypic expectations that equal precisely d_i*, their contribution to MPH (derivations not shown).

We conclude that none of the available experimental designs of quantitative genetics has the potential to separate the dominance effect d_i and half the sum of aa epistatic interactions confounded in the augmented dominance effect d_i* of heterotic QTL. Design III identifies heterotic QTL but does not allow separation of dominance and aa epistasis. For the time being, the only solution to this problem is to identify QTL contributing to heterosis with design III and to estimate the augmented QTL effect d_i*. In genomic regions exhibiting significant heterosis, the augmented dominance effect d_i* can be dissected into its components (d_i and ) by employing additionally the immortal F₂ design and estimating the magnitude of dominance effect d_i. With RILs, the same lines can be used for generating the progenies for design III and the immortal F₂ population, and marker data generated for the RILs can readily be employed in both types of analysis. The optimal dimensions of experimental studies applying our approach with respect to population size and population type as well as marker density have yet to be determined. Epistatic interactions of QTL i with the genetic background are expected to vary considerably across RILs. Implications of this biological variation on estimates of d_i* will be addressed in a separate study.

Epistasis and heterosis:

As has been shown in this study, even small epistatic interactions can be important for the expression of heterosis because their contribution to the genotypic expectation of augmented QTL effects sums up over many effects. On the basis of results from classical quantitative genetic experiments, we have reason to assume that epistasis plays an important role in the inheritance of quantitative traits and heterosis (e.g., Jinks and Jones 1958) but results from QTL studies on the importance of epistasis have been rather ambiguous (e.g., Stuber et al. 1992; Cockerham and Zeng 1996; Frascaroli et al. 2007). In QTL analyses, the power of detecting QTL with epistatic effects is generally low mainly due to the problem of multiple testing in two- or multidimensional genome scans (Lander and Botstein 1989) or due to the necessity of a priori model selection with one-dimensional scans (Kao et al. 1999). We need to keep in mind that QTL with significant epistatic interaction effects might not be representative for the majority of QTL with small effects contributing to gene networks that control the expression of quantitative traits. Hence, we are likely to introduce an ascertainment bias as pointed out by Kroymann and Mitchell-Olds (2005).

Type of epistasis:

When estimating the most prominent type of epistatic interactions in the expression of quantitative traits and heterosis in rice, a preponderance of aa epistatic effects was identified compared with ad or dd interactions (Yu et al. 1997; Hua et al. 2003). In self-pollinated crops, it is well known that coadapted gene complexes are favored by selection. As a consequence, half the sum of mainly positive aa interactions enters d_i* with a negative sign, thus decreasing MPH and the power for detection of heterotic QTL. It is questionable if the results on heterosis from self-pollinated crops are directly applicable to cross-pollinated crops (Frascaroli et al. 2007). Economic seed production, however, requires the development of inbred lines with high grain yield. Combined with the management of separate heterotic pools, it is highly probable that coadapted gene complexes are selected during inbred line development. As a consequence, if the sum of aa epistatic interactions in the parents increases due to selection, MPH should decrease over time unless the sum of dominance effects at QTL influencing heterosis increases proportionally. A decrease in relative superiority of hybrids compared with their inbred lines has been described for maize (Duvick 1999). This decrease in relative heterosis can be the result of the accumulation of favorable dominant alleles at individual QTL, but it can also be explained by overdominance in the presence of aa epistatic effects contributed by the parents. If this is the case, the outcome of marker-assisted selection programs aiming at the transfer of QTL for maximization of heterosis will strongly depend on the presence of favorable epistatic interactions with the genetic background in the respective germplasm and will be difficult to predict.

In conclusion, the results presented here are important in two ways. First, they provide the quantitative genetic theory to express heterosis as the sum of individual QTL effects. Second, they allow the assessment of epistatic interactions of individual QTL with the entire genetic background, thus extending the concept of epistasis from single-gene to system-level interactions. We suggest the use of CIM and design III with RILs to identify QTL expressing maximum heterosis (i.e., maximum d_i*). All analyses can be performed with an extended version of the software PLABQTL (Utz and Melchinger 1996; http://www.uni-hohenheim.de/plantbreeding/software/index.html). Permutation tests for determining the significance threshold (Doerge and Churchill 1996) and cross-validation for unbiased QTL estimation (Utz et al. 2000) can be readily applied. Using the joint likelihood-ratio test for augmented effects a_i* and d_i* will improve identification of heterotic QTL from elite × exotic crosses and provide a first test to distinguish between dominance and overdominance. Applying models accounting for multilocus epistasis and using molecular tools to finely dissect genomic regions contributing to heterosis will allow an assessment of the relative contribution of epistatic interactions in the manifestation of heterosis.