Analysis of a Triple Testcross Design With Recombinant Inbred Lines Reveals a Significant Role of Epistasis in Heterosis for Biomass-Related Traits in Arabidopsis

Abstract

Primary causes of heterosis are still unknown. Our goal was to investigate the extent and underlying genetic causes of heterosis for five biomass-related traits in Arabidopsis thaliana. We (i) investigated the relative contribution of dominance and epistatic effects to heterosis in the hybrid C24 × Col-0 by generation means analysis and estimates of variance components based on a triple testcross (TTC) design with recombinant inbred lines (RILs), (ii) estimated the average degree of dominance, and (iii) examined the importance of reciprocal and maternal effects in this cross. In total, 234 RILs were crossed to parental lines and their F₁'s. Midparent heterosis (MPH) was high for rosette diameter at 22 days after sowing (DAS) and 29 DAS, growth rate (GR), and biomass yield (BY). Using the F₂-metric, directional dominance prevailed for the majority of traits studied but reciprocal and maternal effects were not significant. Additive and dominance variances were significant for all traits. Additive × additive and dominance × dominance variances were significant for all traits but GR. We conclude that dominance as well as digenic and possibly higher-order epistatic effects play an important role in heterosis for biomass-related traits. Our results encourage the use of Arabidopsis hybrid C24 × Col-0 for identification and description of quantitative trait loci (QTL) for heterosis for biomass-related traits and further genomic studies.

HYBRID vigor or heterosis is the superior performance of an F₁ hybrid compared with the mean of its parental lines. In breeding of allogamous crops, the phenomenon of heterosis has been heavily exploited in all breeding categories except line breeding (Simmonds and Smartt 1999), mainly because of its large effect on grain and biomass yield (Becker 1993). However, the causes underlying this important phenomenon have remained unknown, even though many genetical, physiological, and molecular explanations have been suggested. Genetic hypotheses, such as dominance, overdominance, and epistasis were the first to be put forward and are still prevailing (cf. Lamkey and Edwards 1999).

Since there is strong evidence that heterosis is a polygenic phenomenon, its causes have been investigated primarily by quantitative genetics research, using either first- or second-degree statistics (cf. Hallauer and Miranda 1981). The former approach estimates the net contribution of dominance and epistatic effects of loci affecting heterosis of a trait by generation means or diallel analyses. In maize and other species, most studies demonstrated that dominance and/or overdominance are of major importance in heterosis for grain yield (Melchinger et al. 1986). Epistasis cannot be ruled out entirely, but its net effect on testcross generation means seems to be of secondary importance in most crosses of elite lines (Hinze and Lamkey 2003; Mihaljevic et al. 2005).

Alternatively, second-degree statistics estimate variance components attributable to additive, dominance, and epistatic effects from covariances of relatives. In particular, design III has played a prominent role in estimating the average degree of dominance over loci from the ratio of dominance to additive variance (Comstock and Robinson 1952). However, linkage between quantitative trait loci (QTL) in repulsion phase may mimic pseudooverdominance and, thus, result in biased estimates of the average squared degree of dominance. Moreover, the effects of epistasis are generally ignored in estimates of additive and dominance variance components obtained by design III. The triple testcross (TTC) design, proposed by Kearsey and Jinks (1968) as an extension of design III, provides a test for epistatic effects (A. E. Melchinger, H. F. Utz, H. P. Piepho and C. C. Schön, unpublished results).

Maize has long served as a primary model species for heterosis research due to the huge amount of heterosis displayed for grain yield. However, the completely sequenced genomes and most advanced genetic tools and resources available in Arabidopsis and rice recommend these two species as experimental model organisms in plant genetics and functional genomics. Up to now, several studies are available on heterosis in rice (Xiao et al. 1995; Yu et al. 1997; Li et al. 2001; Luo et al. 2001; Hua et al. 2002, 2003), whereas the information on heterosis in Arabidopsis is scarce. Kearsey et al. (2003) employed the TTC design to study the genetics of 22 quantitative traits related to plant size and development in the Arabidopsis hybrid Columbia (Col) × Landsberg (Ler). However, inferences on heterosis could not be made with the genetic material studied, because significant heterosis was observed only for one of the 22 traits. On the contrary, the extent of midparent heterosis (MPH) and best-parent heterosis (BPH) for biomass yield (MPH = 60.3%, BPH = 32.9%) and rosette diameter (MPH = 49.4%, BPH = 34.8%) averaged across five Arabidopsis hybrids derived from five ecotypes (Barth et al. 2003) was surprisingly high. The values of MPH and BPH of individual hybrids varied substantially, with even 140.1% MPH for biomass yield in one hybrid (C24 × Aa-0). Another systematic survey of 63 Arabidopsis accessions crossed with three reference lines (Meyer et al. 2004) also reported significant MPH for biomass yield, with hybrid Col-0 × C24 displaying significant hybrid vigor.

The goals of our study were to determine the extent of heterosis for biomass-related traits in Arabidopsis hybrid C24 × Col-0 and investigate its underlying genetic causes by quantitative genetic analyses. In particular, our objectives were to (i) investigate the relative contribution of dominance and epistatic effects to heterosis in this cross by generation means analysis and estimates of variance components calculated from a TTC design with recombinant inbred lines (RILs), (ii) estimate the average degree of dominance, and (iii) examine the importance of reciprocal and maternal effects.

MATERIALS AND METHODS

Plant materials:

Seeds from the Arabidopsis thaliana accessions C24 (provided by J. P. Hernalsteens, Vrije Universiteit Brussels, Belgium) and Col-0 (provided by G. Rédei; University of Missouri, Columbia, MO), subsequently referred to as parents P1 and P2, respectively, were used to establish the plant materials employed in this study. The F₁ generation was produced in two reciprocal forms: P1 × P2 (F₁-a) and P2 × P1 (F₁-b), where the first parent refers to the seed parent. The F₂ generation originating from F₁-a is further referred to as F₂-a and that originating from F₁-b as F₂-b. Two sets of RILs were derived as described by Törjék et al. (2006). RILs-a comprised 214 RILs derived from F₁-a, and RILs-b comprised 209 RILs derived from F₁-b. For this study, we used random subsets of 111 RILs-a and 123 RILs-b. The lines were propagated via single-seed descent to generation F₇. Testcross progenies were produced using the TTC design (Kearsey and Jinks 1968). RILs were used as pollen parents and mated with P1 and P2. In addition, RILs-a were crossed with F₁-a and RILs-b with F₁-b. In all instances, one representative plant of each RIL was used to pollinate three plants of each tester (P1, P2, F₁-a, or F₁-b) and, apart from six siliques per mother plant, all others were removed to warrant a homogeneous seed size.

Experimental design:

The entire set of 234 RILs was subdivided into three experiments, each with 78 RILs and six checks. Experiments were arranged in a split-plot design with three replicates. Main plots were arranged in a 12 × 7 α-design. Each main plot comprised four entries: one RIL and its three testcross progenies produced by the TTC design. The main plots of checks also comprised four entries: parents P1 and P2, as well as the F₁ and F₂ generations either from F₁-a or from F₁-b. In all instances, the entries within each main plot were randomly assigned to the subplots. Each subplot consisted of a row of 10 plants per entry. For the subsequent statistical analyses with SAS PROC MIXED (SAS Institute 2004) described below, treatment effects were considered as fixed for checks as well as all other generations listed (Table 1) and considered as random for entries nested within generations (RILs and their TTC progenies). Apart from these treatment effects, the model contained fixed effects for experiments, replicates within experiments, and incomplete blocks and contained random effects for main plots and subplots.

TABLE 1.

Genetic expectations for the generation means of generations P1, P2, F₁, and F₂ in both reciprocal forms (a = P1 × P2; b = P2 × P1), corresponding RILs from F₁-a and F₁-b, as well as triple testcross progenies with testers P1, P2, and F₁

	Genetic parametera
Generation	Nb	μ	c	m	[a]	[d]	[aa]	[ad + da]	[dd]
P1	18	1	1	0	−1	−	1
P2	18	1	0	0	1	−	1	−
F1-a	9	1	1	0	0		0	0
F1-b	9	1	0	0	0		0	0
F2-a	9	1	1	1	0	0	0	0	0
F2-b	9	1	0	1	0	0	0	0	0
RILs-a	123	1	1	0	0	−	0	0
RILs-b	111	1	0	0	0	−	0	0
P1 × RILs	234	1	1	0	−	0	1/4	0	0
P2 × RILs	234	1	0	0		0	1/4	0	0
F1-a × RILs-a	123	1	1	1	0	0	0	0	0
F1-b × RILs-b	111	1	0	1	0	0	0	0	0
General coefficient		1	u	v	x	z	x2	xz	z2

^aFor a detailed explanation of the terminology and definition of genetic effects, see materials and methods.

^bNumber of entries.

Plant cultivation:

Seeds were sown in petri dishes under sterile conditions on Murashige–Skoog medium. At the two-leaf stage, ∼10 days after sowing (DAS), each seedling was transferred to sterilized soil (Euflor GmbH: 90% peat, 7% perlite, and 3 vol% sand at pH of 5–6, salt content <1.5 g/liter, nitrogen availability <300 mg/liter N, phosphate availability <300 mg/liter P₂O₅, and calcium oxide availability <400 mg/liter K₂O). All plants in a replicate were grown on a large bench of sterilized soil. Distance between plants inside the row as well as the distance between rows was 8 cm. Plants in soil were irrigated with tap water. Standard light and temperature regime under greenhouse conditions was 16 hr light (20,000 lux) at 21° and 8 hr dark at 18°.

Traits measured:

Rosette diameter (RD) (in millimeters) was recorded on a single-plant basis 22 days after sowing (RD22) and 29 days after sowing (RD29). The absolute growth rate (GR) per day (in millimeters per day) was determined as (RD29 − RD22)/7; this provides an approximation of the expansion of the rosette diameter under an exponential growth model. All plants of a subplot were harvested without the root system at 29 DAS and bulked into a plastic jar. Biomass yield (BY) (in milligrams) was recorded after drying in an oven to practically 0% moisture content. Dry matter content (DMC) (in percent) was calculated as the ratio between dry and fresh biomass, ×100.

Generation means:

Means and standard errors of all generations were calculated over the series of three split-plot experiments with SAS PROC MIXED (SAS Institute 2004). Midparent heterosis was calculated as MPH = 100 × , where = (F₁-a + F₁-b)/2 and . Means of generations P1, P2, F₁-a, F₁-b, F₂-a, F₂-b, RILs-a, RILs-b, P1 × RILs, P2 × RILs, F₁-a × RILs-a, and F₁-b × RILs-b, averaged over the three experiments, were used to estimate the genetic parameters in two genetic models fitted to the data. All parameters in the models were defined according to the F₂-metric (Cockerham 1954; Yang 2004). Formulas for the generation means are given in Table 1.

Model 1 included cytoplasmic, maternal, additive, and dominance effects,

where Y is the mean of the generation considered; μ is the mean of the F₂ generation over the three experiments, in the absence of cytoplasmic and maternal effects; c is the cytoplasmic effect attributable to seed parent P1 vs. seed parent P2; m is the maternal effect attributable to a heterozygous vs. a homozygous seed parent; [a] = Σ_jθ_ja_j, with θ_j = −1 if P1 carries the favorable allele at locus j and +1 otherwise, and a_j is the additive effect of locus j; [d] = Σ_jd_j, with d_j the dominance effect of locus j; and u, v, x, and z are generation-dependent coefficients given in Table 1.

Besides the effects defined for model 1, model 2 included epistatic effects between unlinked pairs of loci,

where [aa] = Σ_j<kθ_jθ_kaa_jk, with aa_jk the additive-by-additive (a × a) epistatic effect between loci j and k; [ad + da] = Σ_j<k{θ_jad_jk + θ_kda_jk}, with ad_jk the additive-by-dominance (a × d) epistatic effect between loci j and k and da_jk the dominance-by-additive (d × a) epistatic effect between loci j and k; and [dd] = Σ_j<kdd_jk, with dd_jk the dominance-by-dominance (d × d) epistatic effect between loci j and k.

In addition to the genetic effects in model 1 and model 2, we included in the generation means analyses a correction term τ (data not shown) to account for differences in the overall performance level of checks vs. RILs and TTC progenies, which were most likely attributable to differences in seed quality between the two groups of materials.

The genetic parameters for both models were estimated by weighted least squares: = (X′WX)⁻¹ (X′WY), where is the column vector of estimated genetic parameters, X is the design matrix with elements determined by model 1 or model 2, W is the matrix with the inverse of the variances of the generation means on the diagonal and zero on the off-diagonal, and Y is the column vector of the generation means.

Standard errors of the genetic parameter estimates were calculated as the square root of the diagonal elements of matrix (X′WX)⁻¹. Adequacy of each model was tested with a chi-square (χ²) test (Mather and Jinks 1982): χ² = (O − E)² × w], where O is the observed generation mean, E is the expected generation mean based on the underlying model, and w designates the diagonal elements of matrix W, with summation over all generations considered in the model. The coefficient of determination (R²) was calculated to estimate the proportion of the variation among generation means accounted for by each model.

Genetic variances and correlations:

Adjusted entry means of each RIL_n (n = 1, 2, . . ., 234) and its TTC progenies with testers P1, P2, and F₁ (subsequently denoted H_1n, H_2n, and H_3n) were used for estimating the genotypic variance () and associated standard errors. Since the difference between the means of populations RILs-a and RILs-b was not significant, we ignored subsets that referred to different reciprocal crosses and treated the entire set of RILs as one single population in the further analyses. The error variances for the main and subplots were calculated as combined estimates over the four types of progenies. Heritability (h²) on an entry-mean basis was estimated as h² = /(/r), where r corresponds to the number of replicates, and = main-plot error + subplot error. In the case of TTC progenies, we calculated only one estimate for h², by averaging across all three types of testcross progenies.

Additive, dominance, and epistatic variance components as defined for the F₂-metric by Yang (2004) were estimated from the TTC progeny data by equating genotypic variances and covariances of H₁, H₂, and H₃ to the corresponding linear combination of additive, dominance, and epistatic variance components given in Table 2. Six models were estimated directly with PROC MIXED, containing: , (model A); , , (model B); , , (model C); , , , (model D);, , , (model E); and , , , , (model F), where represents the covariance between additive and dominance effects. Other covariances among effects of different types as well as linkage were ignored in all models. The Bayesian information criterion (BIC) (Burnham and Anderson 2004) was applied to select among the six models. All necessary computations were performed with SAS PROC MIXED with the REML option (SAS Institute 2004).

TABLE 2.

Variance and covariance matrix for triple testcross progenies H₁, H₂, and H₃ under digenic epistasis, ignoring linkage and covariance terms involving epistatic effects

Statistic
Var(H1)		1			1	1
Var(H2)		1	−		1	1
Var(H3)		0	0		0	0
Cov(H1, H2)		−1	0	−	0	1
Cov(H1, H3)		0			0	0
Cov(H2, H3)		0	−		0	0

For description of triple testcross progenies and definition of genetic variance components see materials and methods.

The average degree of dominance was estimated as D = √ (2)/() (Cockerham and Zeng 1996), and its standard error is according to Mood et al. (1974). Alternatively, we estimated the overall dominance ratio DR = from first-degree statistics as

where Range(RILs) refers to the range across the entire set of n = 234 RILs, with regard to their per se performance. The denominator was chosen because in the absence of epistasis and linkage, the genetic expectation of Range(RILs) converges toward for n → ∞.

Genotypic correlations among traits were estimated in RILs as well as their TTC progenies averaged over the three testers and between RILs and the mean of their TTC progenies. These computations were performed on the basis of adjusted entry means and with software PLABSTAT (Utz 2000).

RESULTS

Generation means:

Parental lines P1 and P2 differed significantly (P < 0.01) in RD29, GR, and DMC (Table 3). Significant (P < 0.01) differences were also observed between F₁-a and F₁-b in RD22, RD29, and GR. Differences between F₂-a and F₂-b, as well as between the means of RILs-a and RILs-b were not significant in any of the traits (data not shown). Generation F₁ surpassed significantly (P < 0.01) both parents in RD22, RD29, GR, and BY. MPH was highest (49%) for BY and exceeded 23% in all traits but DMC.

TABLE 3.

Means and associated standard errors in generations P1, P2, F₁, F₂, RILs and triple testcross progenies with testers P1, P2, and F₁ as well as range of the RILs and dominance ratio for five biomass-related traits in Arabidopsis hybrid C24 × Col-0

Generations	Na	RD22 (mm)	RD29 (mm)	GR (mm/day)	DMC (%)	BY (mg)
Checks
P1	18	22.3 ± 0.75	59.1 ± 1.45	5.27 ± 0.15	7.66 ± 0.11	53.8 ± 4.09
P2	18	24.0 ± 0.75	67.8 ± 1.45	6.26 ± 0.15	8.15 ± 0.11	62.1 ± 4.03
b	18	28.4 ± 0.89	79.9 ± 1.75	7.39 ± 0.18	7.76 ± 0.13	86.6 ± 4.75
b	18	25.3 ± 0.75	71.6 ± 1.45	6.65 ± 0.15	7.85 ± 0.11	69.0 ± 4.03
MPHc		22.7 ± 2.37	25.9 ± 1.21	28.2 ± 4.21	−1.83 ± 3.07	49.4 ± 1.94
Entries
RILs	234	25.8 ± 0.27	66.9 ± 0.53	5.94 ± 0.05	8.34 ± 0.04	76.6 ± 1.37
P1 × RILs (H1)	234	29.9 ± 0.45	76.0 ± 0.82	6.65 ± 0.06	7.51 ± 0.04	93.3 ± 2.30
P2 × RILs (H2)	234	30.5 ± 0.44	78.7 ± 0.74	6.92 ± 0.06	8.09 ± 0.04	103.0 ± 2.43
F1 × RILs (H3)	234	30.7 ± 0.46	78.5 ± 0.69	6.87 ± 0.05	7.83 ± 0.04	97.7 ± 2.31
Ranged
RILs		33.45	61.26	5.25	5.54	165.68
Dominance ratio
DRe		0.51	0.60	0.72	−0.04	0.99

^aNumber of entries.

^bF₁ and F₂ were calculated as means across both reciprocal forms.

^cMidparent heterosis (MPH) = 100 × , where , with standard error calculated according to Mood et al. (1974).

^dRange(RILs) = Max(RIL) − Min(RIL).

^eDR = /(h² × Range(RILs)/2).

Testcross progenies outperformed (P < 0.01) the RILs in all traits except DMC (Table 3). The mean of P2 × RILs significantly surpassed (P < 0.01) that of P1 × RILs in all traits except RD22. Compared with F₁ × RILs, P1 × RILs reached significantly (P < 0.01) lower values in all traits except RD22 and BY. Compared with F₁ × RILs, P2 × RILs reached significantly (P < 0.01) higher values in DMC and P1 × RILs had significantly (P < 0.05) smaller values for GR and DMC. The F₁ × RILs deviated significantly (P < 0.05) from the overall mean of P1 × RILs and P2 × RILs only for RD29 (data not shown). The range in the variation of the RILs exceeded by far the difference of the parents, indicating transgressive segregation in all traits. As expected, the dominance ratio DR was lowest (−0.04) for DMC, highest (0.99) for BY, and intermediate (between 0.51 and 0.72) for the other traits.

Model 1 accounted for 84–98% of the variation among generation means for all traits (Table 4). Nevertheless, the χ²-values for the goodness-of-fit of model 1 were highly significant (P < 0.01) for DMC. Neither cytoplasmic nor maternal effects were significant (P < 0.01) (data not shown). Inclusion of epistatic effects (model 2) considerably improved the fit, explaining from 97 to almost 100% of the variation among generation means for all traits. However, the χ²-value of model 2 was still significant (P < 0.05) for DMC.

TABLE 4.

Genetic effects and associated standard errors for five biomass-related traits estimated by generation means analysis in Arabidopsis hybrid C24 × Col-0

Parameters	RD22 (mm)	RD29 (mm)	GR (mm/day)	DMC (%)	BY (mg)
	Model 1
μ	28.04 ± 0.43**	74.85 ± 0.66**	6.72 ± 0.05**	7.78 ± 0.10**	84.78 ± 1.78**
[a]	0.75 ± 0.69	3.57 ± 1.06*	0.36 ± 0.08**	0.46 ± 0.15*	6.47 ± 2.92
[d]	7.67 ± 1.02**	19.99 ± 1.58**	1.73 ± 0.11**	−0.80 ± 0.25*	37.57 ± 4.19**
a	11.52*	8.03	4.82	40.92**	7.19
R2 (%)	95.3	97.9	98.5	83.6	97.0
	Model 2
μ	27.51 ± 0.46**	74.32 ± 0.93**	6.73 ± 0.06**	7.92 ± 0.08**	82.44 ± 0.87**
[a]	0.59 ± 0.91	2.60 ± 1.71	0.27 ± 0.08	0.58 ± 0.11*	9.75 ± 1.73*
[d]	8.05 ± 0.89*	20.37 ± 1.77**	1.73 ± 0.10**	−0.84 ± 0.15*	39.17 ± 1.65**
[aa]	1.93 ± 1.04	1.92 ± 2.11	−0.05 ± 0.13	−0.43 ± 0.18	8.72 ± 1.97*
[ad + da]	−0.52 ± 2.39	−3.49 ± 4.67	−0.45 ± 0.27	0.67 ± 0.37	11.18 ± 4.55
a	4.22	4.81	1.92	7.35*	0.53
R2 (%)	98.3	98.8	99.4	97.1	99.8

For description of traits see materials and methods. *Significant at P < 0.05 and **significant at P < 0.01.

^aChi-square degrees of freedom are in parentheses.

Both models yielded similar estimates of [a] and [d] (Table 4). Dominance effects were significant (P < 0.01) and had a positive sign for all traits except for DMC. Additive effects were consistently smaller in magnitude and under model 2 significant (P < 0.05) only for DMC and BY. Estimates of [aa] under model 2 were significantly (P < 0.05) positive for BY. The estimates of [ad + da] were not significant for any trait and the same applied to estimates of [dd] (data not shown).

Variances and heritabilities:

Genotypic variances () among RILs were highly significant (P < 0.01) for per se and testcross performance with all three testers for all biomass-related traits (Table 5). Surprisingly, estimates of in RILs were smaller than those in their TTC progenies for RD22 and BY, of similar size for RD29, but significantly (P < 0.01) larger for GR and DMC. Heritability (h²) estimates for the RILs and their TTC progenies were in most instances high and ranged between 76 and 84%, except for GR and DMC in the testcrosses.

TABLE 5.

Variance components (, genotypic variance; , error variance) and heritability (h²) estimates with associated standard errors for five biomass-related traits in RILs and their triple testcross progenies with testers P1, P2, and F₁ in Arabidopsis hybrid C24 × Col-0

Parameters	RD22	RD29	GR	DMC	BY
(RILs)	37.10 ± 4.50**	145.53 ± 17.37**	1.076 ± 0.140**	1.004 ± 0.117**	874 ± 112**
(H1)a	49.51 ± 5.60**	156.03 ± 18.14**	0.682 ± 0.104**	0.237 ± 0.047**	1109 ± 133**
(H2)b	50.76 ± 5.79**	120.75 ± 15.31**	0.535 ± 0.091**	0.219 ± 0.051**	1557 ± 176**
(H3)c	44.32 ± 5.14**	100.87 ± 13.11**	0.421 ± 0.078**	0.160 ± 0.039**	1073 ± 131**
	28.42 ± 0.89	98.77 ± 3.20	1.018 ± 0.033	0.663 ± 0.021	833 ± 26
(%)	79.66 ± 2.13	81.55 ± 1.95	76.02 ± 2.56	81.96 ± 1.89	75.88 ± 2.55
(%)	83.57 ± 1.28	79.27 ± 1.69	61.68 ± 3.06	48.21 ± 4.44	81.78 ± 1.41

* Significant at P < 0.05 and ** significant at P < 0.01.

^aH₁ denotes the testcross progenies of RILs with tester P1.

^bH₂ denotes the testcross progenies of RILs with tester P2.

^cH₃ denotes the testcross progenies of RILs with tester F₁.

Additive () and dominance () variances were highly significant (P < 0.01) for all traits in all six models studied (data not shown). Regarding models without epistasis, model B (data not shown) provided a similar fit to the data as model A did, as reflected by the BIC values and estimates of , which were not significant (P < 0.05) except for RD29. For models including epistasis, model D (including , , , and ) yielded the lowest BIC value for all traits. Hence, presentation of the results in Table 6 is restricted to models A and D.

TABLE 6.

Estimates of additive, dominance, and epistatic variance components with associated standard errors, as well as average degree of dominance for five biomass-related traits in Arabidopsis hybrid C24 × Col-0

Parameters	RD22	RD29	GR	DMC	BY
	Model A
	65.00 ± 7.29**	179.99 ± 20.86**	0.762 ± 0.112**	0.393 ± 0.057**	1479 ± 178**
	11.32 ± 1.57**	34.33 ± 4.69**	0.211 ± 0.034**	0.0832 ± 0.014**	321 ± 45**
Da	0.59	0.62	0.74	0.65	0.66
BICb	14924	16309	6512	5278	22223
	Model D
	71.14 ± 8.79**	169.61 ± 21.79**	0.719 ± 0.113**	0.361 ± 0.058**	1656 ± 223**
	10.48 ± 1.58**	34.90 ± 4.67**	0.216 ± 0.033**	0.080 ± 0.015**	309 ± 45**
	5.25 ± 0.23**	3.89 ± 0.18**	0.000 ± 0.000	0.020 ± 0.001**	152 ± 7**
	25.63 ± 3.37**	39.68 ± 7.00**	0.070 ± 0.037	0.048 ± 0.019**	854 ± 107**
Da	0.54	0.64	0.77	0.67	0.61
BICb	14742	16251	6519	5280	21986

* Significant at P < 0.05 and ** significant at P < 0.01.

^aD = √(2/).

^bThe Bayesian information criterion.

In general, estimates of were approximately four to six times larger than those of under both models (Table 6). Inclusion of epistatic variance components improved the fit of model D, as reflected by the reduced BIC values for all traits but GR and DMC. Estimates of were highly significant (P < 0.01) and up to one-half of those for for all traits except GR. Estimates of ranged between one-quarter and one-half of those for , except for GR and DMC, where was only one-eighth of .

Estimates of the average degree of dominance D were similar for models A and D except for RD22 and BY, where the inclusion of and in the model resulted in reduced values (Table 6). D-values ranged between 0.54 and 0.77 and were of intermediate size for BY.

Correlations:

Genotypic correlations (r_g) among traits were consistently higher in the RILs than in the TTC progenies averaged over the three testers (Table 7). Estimates of r_g among RD22, RD29, GR, and BY were positive and high, ranging from 0.51 to 0.95, with a single exception. DMC was negatively correlated with all other biomass-related traits. Genotypic correlations between RILs and the mean of their TTC progenies ranged between 0.62 and 0.80 for the studied traits.

TABLE 7.

Genotypic correlations among five biomass-related traits in RILs (above diagonal), the mean of their triple testcross progenies averaged over the three testers (below diagonal), and that between both these measures (on the diagonal in italics) in Arabidopsis hybrid C24 × Col-0

	RD22	RD29	GR	DMC	BY
RD22	0.66a	0.94a	0.73a	−0.53a	0.95a
RD29	0.84a	0.73a	0.92a	−0.66a	0.89a
GR	0.51a	0.90a	0.77a	−0.67a	0.67a
DMC	−0.28a	−0.53a	−0.63a	0.80a	−0.39a
BY	0.87a	0.66a	0.38a	−0.08	0.62a

^aGenotypic correlation exceeded twice its standard error.

DISCUSSION

Choice of the experimental design:

Design III and the TTC design are most suitable for investigations of heterosis in the presence of epistasis because they provide estimates of augmented dominance effects in the terminology of A. E. Melchinger, H. F. Utz, H. P. Piepho and C. C. Schön (unpublished results). These effects capture the effects of dominance as well as the sum of a × a epistatic interactions of a QTL with the genetic background contributing to heterosis under the F₂-metric. Instead of F₂ plants or F₃ lines used in design III studies with maize, we used RILs as parents for producing the testcross progenies following Kearsey et al. (2003). Use of homozygous parents maximizes the genetic variance among testcross progenies and leads to an increased power in F-tests as well as smaller standard errors of variance components estimates and D (A. E. Melchinger, H. F. Utz, H. P. Piepho and C. C. Schön, unpublished results). Use of RILs has the further advantage that the effects of linkage are reduced because linkage disequilibrium between tightly linked loci is almost halved compared with that in F₂ plants or double-haploid lines.

Although the experimental design we applied possesses the above-mentioned advantages, we used a considerably larger sample of RILs and number of plants per RIL or TTC progenies than in previous studies to achieve a high level of precision in parameter estimates. For genetic variance components and h², standard errors were generally small in comparison with the estimates, but for D they were of the same order.

Choice of the F₂-metric:

Three different metrics have been proposed in the literature (Van der Veen 1959): (i) the F_∞-metric (Hayman 1954); (ii) the F₂-metric, which is a special case of Cockerham's (1954) model; and (iii) the mixed metric (Hayman and Mather 1955), which is a mixture of Cockerham's model and the F_∞-metric. While the F₂-metric yields orthogonal parameters with regard to an F₂ population, the same applies to the F_∞-metric with regard to homozygous lines. Following the definition and terminology of Melchinger (1987), the progenies of design III and the TTC design can be considered as gamete-orthogonal populations. Gene effects in a gamete-orthogonal population are commonly defined with regard to the respective gene-orthogonal population, which corresponds to the F₂ population in the case of a TTC design.

For this study, we chose the F₂-metric for three reasons. First, it has the advantage that each variance component is proportional to the sum of squares of the corresponding genetic effects (e.g., , , , etc.; Yang 2004) and does not involve any other type of genetic effects that could obscure their interpretation. This would not apply if we had chosen the F_∞-metric, as shown by Cheverud (2000). Second, the F₂-metric underlies the genetic model used in testing for epistatic interactions among QTL by two-way ANOVAs for pairs of marker loci (Cockerham and Zeng 1996). Third, with digenic epistasis, MPH = [d] − [aa], i.e., besides dominance effects it involves only a × a effects, whereas under the F_∞-metric MPH is additionally influenced by d × d effects (Van Der Veen 1959).

Cytoplasmic and maternal effects:

While reciprocal differences in some traits were observed in the F₁ generation, cytoplasmic effects were consistently not significant in the generation means analysis. One simple explanation, that both parents have the same cytoplasm, can be excluded because Forner et al. (2005) found differences in mtDNA between C24 and Col-0.

Maternal effects due to heterozygosity of seed parent were observed for grain and forage yield in maize (Melchinger et al. 1986). We found no such effect in Arabidopsis and, hence, excluded the respective parameter from the model. Possible explanations for this observation are that there is only marginal heterosis for seed weight in Arabidopsis and uniform seed quality in our study was achieved by restricting pollinations and selfing to six siliques per plant.

Interpretation of first- and second-degree statistics:

Our generation means analysis extends the ordinary generation means analysis based on the parents, F₁, selfing, and backcross generations from biparental crosses to the populations of RILs and testcross progenies, underlying the test for epistasis in the TTC design (Kearsey and Jinks 1968). In both analyses, the parameters capture the net contribution of gene effects. However, the consequences of summation over loci are quite different for various parameters. If directional dominance (d_i > 0) prevails, as suggested by most QTL studies on traits related to plant growth, the sum in [d] involves mainly positive terms. In contrast, if the parents have similar performance levels, the sum in [a] involves both positive and negative terms that cancel each other to a substantial amount. Because the range in the RILs surpassed by far the absolute difference |P1 − P2| for all traits, we conclude that θ_i, the sign of the additive effect a_i, was not uniform across the QTL involved in trait expression. Significance of epistatic effects [aa], [ad + da], or [dd] is indicative of their presence but does not provide any clue on their relative importance. Likewise, absence of significant estimates for [aa], [ad + da], or [dd] does not preclude the existence of these types of effects, because positive and negative effects may be canceled in the sum.

Under the F₂-metric, this shortcoming is overcome by estimation of variance components because they represent sums of squared effects at individual loci except in the case of linkage among loci. Under directional dominance, linkage causes an inflation in , whereas is inflated by the covariance of linked QTL in coupling phase and deflated by those in repulsion phase with corresponding carryover for estimates of D (Comstock and Robinson 1952).

In the presence of epistasis, estimates of and obtained from design III are confounded by those of and , respectively, and to a minor extent also influenced by other types of epistatic terms (A. E. Melchinger, H. F. Utz, H. P. Piepho and C. C. Schön, unpublished results). With the TTC design, it is possible to separate epistatic variance components from those of and , but one cannot clearly discriminate between and . This is because coefficients for these two variances are almost identical in the genetic expectation of the (co)variance of H_1n, H_2n, and H_3n (Table 2). Hence, model D and model E yielded similar estimates of and besides almost identical estimates of , , and .

The comparison between RILs and their TTC progenies provides further evidence for the presence of epistasis. In the absence of epistasis, the genotypic correlation between the performance of RILs per se and the means of their TTC progenies is expected to be 1.0. Furthermore, of RILs is expected to be twice as large as of testcross progenies with the F₁ tester (A. E. Melchinger, H. F. Utz, H. P. Piepho and C. C. Schön, unpublished results). Estimates of these parameters were closest to these expectations for DMC and GR, but deviated substantially for all other biomass-related traits, most notably for BY.

We estimated the average dominance ratio D_i = d_i/a_i across all QTL by D and DR. In both statistics, QTL with large a_i and d_i effects receive a higher weight than those with smaller effects. Both statistics are inflated by linked QTL with additive effects in repulsion phase (θ_i ≠ θ_j). In contrast to D, which is always positive and provides a realistic estimate of the average dominance ratio only in the case of directional dominance (d_i ≥ 0), DR takes into account different signs in d_i. This explains the large discrepancy between DR and D for DMC and suggests that almost half of the QTL influencing this trait have negative sign of d_i. Conversely, the good agreement between D and DR for the other traits suggests a prevalence of directional dominance at the majority of QTL. The difference in size between D and DR for BY could be regarded as an indicator that our sample of 234 RILs was too small to represent adequately the extreme genotypes for this complex polygenic trait.

Level of heterosis in Arabidopsis hybrid C24 × Col-0:

In agreement with other studies on Arabidopsis (Barth et al. 2003; Meyer et al. 2004), we detected medium to high MPH for biomass-related traits with differences in the absolute values of MPH due to different light intensities and recording or harvesting times. MPH for BY was about twice as large as for RD. This result is in harmony with the hypothesis of Williams (1959) that multiplication effects are a major cause of heterosis in complex traits. In our study, RD reflects plant growth in just one dimension, namely leaf length, whereas BY depends in multiplicative fashion also on leaf width and leaf number as well as on other features of biomass accumulation, each of which presumably displays a small amount of heterosis.

Growth rate displayed a medium amount of MPH. However, because biomass accumulation follows an exponential curve during the early stages of plant development, a small amount of heterosis for GR translates into much bigger differences among the hybrid and the parental lines for BY.

No significant MPH was observed for DMC in our study. This finding is in contrast to negative MPH observed for DMC of grain in maize (Mihaljevic et al. 2005) due to earlier flowering and a more rapid vegetative and generative growth of hybrids. Although MPH for DMC has not been investigated in other studies with Arabidopsis, Barth et al. (2003) and Kearsey et al. (2003) also reported only a low magnitude of MPH for flowering date.

Genetic basis of heterosis for biomass-related traits in Arabidopsis hybrid C24 × Col-0:

We observed positive estimates of [aa] for all biomass-related traits except for GR and DMC. This indicates that pairs of genes at nonhomologous loci, which show positive epistatic interactions for biomass-related traits, have been accumulated in the parents C24 and Col-0 presumably as a result of natural selection. These gene effects reduce MPH, which is evident from the relationship MPH = [d] − [aa]. Therefore, they cannot be exploited in the hybrid. In the TTC progenies, the term [aa] favors the average of testcross progenies with testers P1 and P2 over the mean of testcross progenies with tester F₁, as detected by the test for epistasis in the TTC analysis devised by Kearsey and Jinks (1968).

Significant estimates of support the presence of epistasis of type a × a for all biomass-related traits except for GR (Table 6). In the case of RD22 and BY, these estimates amounted to half the size of those for , suggesting that the epistasis of type a × a is a major factor influencing heterosis for biomass-related traits in hybrid C24 × Col-0. Estimates of in model D were consistently of greater importance than estimates of and of equal (RD29) or about twice the size (RD22, BY) of estimates of . Epistasis of types a × d and d × d does not contribute to MPH. However, it cannot be precluded that estimates of and account also for higher-order types of epistatic effects that influence heterosis. Hence, digenic and higher-order epistatic effects seem to play an important role in heterosis for biomass-related traits. This is in agreement with the study of Cockerham and Zeng (1996), who reported significant epistasis among linked QTL besides dominance of favorable genes contributing to heterosis for grain yield and other traits in maize hybrid B73 × Mo17. In accordance with the studies on maize, Hua et al. (2002) and Li et al. (2001) detected substantial digenic epistasis in various populations of rice. Li et al. (2001) reported that most of the variance for BY was due to epistatic QTL.

Estimates of D were of medium size for all biomass-related traits. This is in sharp contrast to most reports on design III studies with F₂ populations in maize, where estimates of D generally exceeded 1.0 (Wolf et al. 2000). Possible explanations are that the dominance ratio D_i at individual loci is much smaller in Arabidopsis than in maize and the same may generally hold true for autogamous vs. allogamous crops, as reflected by the different levels of MPH displayed by each group of species (Becker 1993).

The medium estimates of D and DR suggest that gene action at the majority of loci in the cross C24 × Col-0 is in the range of partial to complete dominance. However, because D averages over loci, it cannot be ruled out that gene action is purely additive at the majority of loci and only a minority of loci displays dominance or even (pseudo)overdominance. Discrimination among these various hypotheses on the genetic causes of heterosis in Arabidopsis is not possible by first- or second-degree statistics alone but requires mapping individual QTL involved in the expression of heterosis. This research will also provide information about the potential linkage bias in the estimates of D and DR.

In conclusion, Arabidopsis and particularly its hybrid C24 × Col-0 are an excellent plant resource to satisfy many requirements for analysis of the genetic causes of heterosis. The hybrid displays a substantial amount of biomass-related hybrid vigor. Furthermore, a high-density map has been established (Törjék et al. 2006) and also metabolic profile data have been collected (Meyer et al. 2007). Therefore, several follow-up studies with these materials are underway to gain a deeper understanding of the physiological, genetic, and molecular causes of heterosis by QTL mapping as well as analysis of transcript, protein, and metabolite profiles.