Genetic dissection of main and epistatic effects of QTL based on augmented triple test cross design.

The use of heterosis has considerably increased the productivity of many crops; however, the biological mechanism underpinning the technique remains elusive. The North Carolina design III (NCIII) and the triple test cross (TTC) are powerful and popular genetic mating design that can be used to decipher the genetic basis of heterosis. However, when using the NCIII design with the present quantitative trait locus (QTL) mapping method, if epistasis exists, the estimated additive or dominant effects are confounded with epistatic effects. Here, we propose a two-step approach to dissect all genetic effects of QTL and digenic interactions on a whole genome without sacrificing statistical power based on an augmented TTC (aTTC) design. Because the aTTC design has more transformation combinations than do the NCIII and TTC designs, it greatly enriches the QTL mapping for studying heterosis. When the basic population comprises recombinant inbred lines (RIL), we can use the same materials in the NCIII design for aTTC-design QTL mapping with transformation combination Z1, Z2, and Z4 to obtain genetic effect of QTL and digenic interactions. Compared with RIL-based TTC design, RIL-based aTTC design saves time, money, and labor for basic population crossed with F1. Several Monte Carlo simulation studies were carried out to confirm the proposed approach; the present genetic parameters could be identified with high statistical power, precision, and calculation speed, even at small sample size or low heritability. Additionally, two elite rice hybrid datasets for nine agronomic traits were estimated for real data analysis. We dissected the genetic effects and calculated the dominance degree of each QTL and digenic interaction. Real mapping results suggested that the dominance degree in Z2 that mainly characterize heterosis showed overdominance and dominance for QTL and digenic interactions. Dominance and overdominance were the major genetic foundations of heterosis in rice.

Introduction

Heterosis, or hybrid vigor, describes the superior performance of heterozygous hybrid plants over their homozygous parental inbred lines [1-3]. The development of heterotic crops, especially those for hybrid rice and maize, is one of the most important applications of genetics in agriculture [4-5], but the molecular basis underlying heterosis remains elusive.

Indeed, much of our knowledge regarding heterosis derives from classical genetic studies on maize, during which the fundamental hypotheses for heterosis were defined, with the main competing hypotheses including dominance, overdominance, and epistasis [1,5-6]. The dominance hypothesis explains heterosis by the complementing action of superior dominant alleles from both parental inbred lines at multiple loci over the corresponding unfavorable alleles leading to the improved vigor of hybrid plants [1,5,7-8]. The overdominance hypothesis attributes heterosis to allelic interactions at one or multiple loci in hybrids that result in superior traits compared to the homozygous parental inbred lines [1,9-10]. In addition, the epistasis hypothesis considers epistatic interactions between non-allelic genes at two or more loci as the main factor for the superior phenotypic expression of a trait in hybrids [1,10-12].

To decipher the genetic basis of heterosis, NCIII [13] and TTC [14] are powerful genetic mating designs widely used in maize [12,15-19], rice [12,8,20-26], and Arabidopsis thaliana [27-30]. Rice is the staple food for a large segment of the world’s population. The success of hybrid rice breeding [31], together with its relatively small genome size [32], saturated molecular linkage maps [33], and rapid advances in genome sequencing [34-35], have provided a novel opportunity for dissecting the genetic basis of heterosis.

In Xiao et al.’s study [8], based on the NCIII design, 194 F7 RIL were backcrossed to their parental lines to develop the mapping population, and 37 QTL were detected for 12 quantitative traits by single-point analysis [one-way analysis of variance (ANOVA)] and an interval mapping method. In one of the two BC1F7 populations, 82% of the detected heterozygotes were superior to the respective homozygotes; therefore, Xiao et al. concluded that dominance complementation was the major genetic basis of heterosis in rice. On the other hand, Li et al. [21] and Luo et al. [22] investigated five interrelated mapping populations by an interval mapping method in which 254 F10 RIL were selected as the base population; two BC1F1 populations were derived from the NCIII design and two test cross populations were obtained by test crossing the RIL with two testers (Zhong 413 and IR64). The results suggested that epistasis and overdominance, rather than dominance, were the major genetic bases of heterosis in rice. Yu et al. [20] also pointed out that epistasis played a major role as the genetic basis of heterosis. Hua et al. [23] investigated the genetic components conditioning the heterosis of yield and yield component traits in an elite rice hybrid using an immortalized F2 population with modified composite interval mapping (CIM) and two-way ANOVA methods and found that heterotic effects at the single-locus level and a dominance × dominance interaction at the two-locus level could adequately explain the genetic basis of heterosis. In our previous study [24] based on the NCIII design, two recombinant inbred populations were backcrossed to their respective parents to develop mapping populations (L1 and L2) in which main-effect QTL were detected by the CIM method and epistatic QTL were detected by the mixed linear approach in the RIL population and summation (L1 + L2) and subtraction (L1 − L2) data of two backcross populations. The research demonstrated that heterosis was attributable to the orchestrated outcome of partial-to-complete dominance, overdominance, and epistasis. In addition, based on an ultra-high-density single nucleotide polymorphism bin map constructed with population sequencing, the immortalized F2 population in Hua et al. [23] was reanalyzed by Zhou et al. [26] with an h test in one-locus effects detection and two-way ANOVA in two-locus interactions for the whole genome. The results suggested that relative contributions of the genetic components varied with traits; single-locus dominance had relatively small contributions in all of the traits and the cumulative effects of these different components may adequately explain the genetic basis of heterosis. This conclusion was consistent with our previous study [24].

In summary, most of the mapping populations above derive from the NCIII design, and QTL mapping methods usually employ ANOVA, interval mapping, or the CIM method in one-locus effects detection and two-locus interactions. However, the estimated additive and dominant effects are confounded with epistatic effect if epistasis is present. Kao and Zeng [36] pointed out that a two-way ANOVA-exploiting genetic marker and trait phenotype data from an F2-segregating population was, in principle, inappropriate for testing for pairwise epistasis, even though this approach has been widely used in analyses of such data sets [20-21,23]. In addition, only one variable was involved in the model at one time, which was not able to capture all types of genetic effects, especially epistatic effects, simultaneously on the whole genome. In 2008, Garcia et al. [25] developed a multiple-interval mapping model for the NCIII design that provided a platform to simultaneously estimate the number, genomic positions, augmented additive and dominance effects, and digenic interactions (aa + dd and ad + da) of QTL. This method was used to reanalyze the datasets by Stuber et al. [12], who found that additive × additive effect (aa) epistatic effects of QTL could be the main cause for the heterosis in rice. After this, He et al. [37] proposed a method for mapping epistatic QTL associated with heterosis using the RIL-based NCIII design by a series of simulation studies; however, main or epistatic effects were mixed measured as augment effects.

In 1988, Liu [38] proposed an aTTC design based on the TTC design. In TTC design, base populations are backcrossed to P, P and F to get L, L, and L, i = 1…n, whereas in aTTC design, base populations are simultaneously self-mated to get L. The aTTC design provided several ways to detect epistasis by detecting a variance component. However, there was no report based on aTTC design for QTL mapping on a Mendelian factor level.

In this paper, under aTTC design, based on four data sets (L, L, L, and L), we developed six data set transformations [38]: Z1 = L1 + L2, Z2 = L1 − L2, Z3 = L1 + L2 − 2L3, Z4 = L1 + L2 − L4, Z5 = L1 + L2 + L3, and Z6 = 2L3 − L4. By employing Z1, Z2, and Z3, He and Zhang [39] provided a complete solution for dissecting main and epistatic effects in the F2-based TTC design through a simulation study. Our study utilized different data set combinations (Z1, Z2, and Z4), (Z1, Z2, and Z5), and (Z1, Z2, and Z6), respectively, to provide a two-step approach for estimating, in an unambiguous and unbiased manner, all the main and, especially, epistatic effects of QTL; this method also fits for many types of base populations such as RIL, F2, and Double Haploid (DH). Here, we will take the first combination (Z1, Z2, and Z4) of the RIL-based aTTC design as an instance for QTL mapping to dissect genetic effects. The other combinations (Z1, Z2, and Z5) and (Z1, Z2 and Z6) listed above can also be used to estimate genetic effects. A series of Monte Carlo simulation studies were carried out to confirm the proposed approach. We further applied the proposed method to real data analysis.

Materials and methods

Genetic design

In aTTC design, F2 populations or their offspring (BC, DH, or RIL) derived from the hybridization of two pure lines (P and P) and were selected as the base population. On one hand, n individuals in the base population were crossed to three testers (P, P, and F) to get L, L, and L, respectively (i = 1, 2… n); on the other hand, the n individuals in the base population were self-mated to get L. Therefore, 4n aTTC lines (L, L, L, and L) can be obtained and used for the detection of epistasis.

All 4n families, each with m replications, were planted. Molecular marker information was observed from all of the n base population lines and the testers P1, P2 and F1, whereas quantitative traits were measured for all 4nm aTTC progeny. The phenotypic observations were denoted by y, where t = 1, 2, 3, and 4 for L, L, L, and L; respectively; j = 1, 2… m. The family means were denoted by .

The genetic expectations of six data set transformations, Z1i, Z2i, Z3i, Z4i, Z5i, and Z6i, were obtained from L, L, L, and L. Z1 = L1 + L2, Z2 = L1 − L2, Z3 = L1 + L2 − 2L3, Z4 = L1 + L2 − L4, Z5 = L1 + L2 + L3, and Z6 = 2L3 − L4. Two main metrics were adopted for the 4n aTTC lines: the F∞ and F2 metrics [36,40]; their genetic expectations are listed in S1 Supporting Information.

Genetic models for mapping heterotic QTL in RIL-based aTTC design

The derivation of the expected genetic values of Z1i, Z2i, Z3i, Z4i, Z5i, and Z6i under both the F∞ and F2 metric models is presented in S3 Supporting Information under the assumption that the quantitative trait was determined by two QTL with digenic epistasis and arbitrary linkage. The genetic effect symbols adopted in this research were described by Kao and Zeng [36]. He et al. [37] simulated and estimated main and epistatic QTL in the RIL-based NCIII design under both the F∞ and F2 metrics models and found that QTL mapping results under the F∞ metric were superior to the F2 metric; therefore, this paper simulated QTL under the F∞ metric models.

QTL mapping models in the RIL-based aTTC design under the F∞ metric model

The phenotypic values of Z1i and Z2i in the RIL-based aTTC design are the same as the RIL-based NCIII design. Details can be found in the publication by He et al. [37]. According to the genetic expectations of Z1i under the F∞ metric model (Table A5 in S1 Supporting Information), the phenotypic value of Z1i can be described as where μ is the mean genotypic value of the four homozygotes in the RIL population; a and d are additive and dominance effects of the kth QTL (k = 1, 2); , , , and are additive × additive, additive × dominance, dominance × additive, and dominance × dominance interactions between two QTL, respectively; , , , , , and are dummy variables and are determined by the genotype of the ith RIL line (Table A5 in S1 Supporting Information); and e1 is the residual error with an distribution. According to Table A5 in S1 Supporting Information, and , model (1) can be reduced to where , , , , and . If the quantitative trait was controlled by q QTL, model (2) should be extended to where the model mean ; is the augmented additive effect of QTL k; is the augmented epistatic effect between QTL k and l. Coefficients and are determined by genotypes of the kth and lth QTL for the ith RIL line, as shown in Table 1.

Table 1

Coefficients of genetic parameters for the RIL based aTTC Z1i, Z2i and Z4i data under the F∞metric model.

				F∞ metric model
Genotype ofMarker		Z1i			Z2i		Z4i
	x_a_k^*	x_a_l^*	x_i↔__kl	u_d_k^*	u_d_l^*	u_i˜__kl	w_i→__kl	w_i←__kl
MkMkMlMl	1	1	1	-1	-1	0	-1	0
MkMkmlml	1	-1	0	-1	1	1	1	1
mkmkMlMl	-1	1	0	1	-1	1	1	-1
mkmkmlml	-1	-1	1	1	1	0	-1	0

Similarly, the phenotypic value of Z2i can be described as where , , , , , and are determined by the genotype of the ith RIL line (Table A6 in S1 Supporting Information), e2 is the residual error with an distribution. According to Table A6 in S1 Supporting Information, and , model (4) can be reduced to where , , , , and . If the quantitative trait controlled by q QTL, model (5) should be extended to where the model mean ; is the augmented dominance effect of QTL k, is the augmented epistatic effect between QTL k and l. Coefficients and are determined by genotypes of the kth and lth QTL for the ith RIL line, as shown in Table 1.

Similarly, the phenotypic value of Z3i can be described as where , r is the recombination fraction between two QTL; dummy variables , , and are determined by the genotype of the ith RIL line (Table A7 in S1 Supporting Information). e3 is the residual error with an distribution. Genetic effects can be estimated directly.

In the same way, the phenotypic value of Z4i can be described as where , , , and are determined by the genotype of the ith RIL line (Table A8 in S1 Supporting Information), e4 is the residual error with an distribution. According to Table A8 in S1 Supporting Information, there are and . Therefore, model (8) can be reduced to where , , , and , .

If the quantitative trait was controlled by q QTL, model (9) can be extended to where the model mean , , is the augmented epistatic effect between QTL k and l. Coefficients and are determined by genotypes of the kth and lth QTL for the ith RIL line, as shown in Table 1.

In the same way, the phenotypic value of Z5i can be described as where , , , , , and are determined by the genotype of the ith RIL line (Table A9 in S1 Supporting Information), e5 is the residual error with an distribution. According to Table A9 in S1 Supporting Information, model (11) can be reduced to where the model mean . Genetic effects can be estimated directly.

In the same way, the phenotypic value of Z6i can be described as where , , , and are determined by the genotype of the ith RIL line (Table A10 in S1 Supporting Information), e6 is the residual error with an distribution. According to Table A10 in S1 Supporting Information, model (13) can be reduced to where the model mean μ = μ + d1 + d2. Genetic effects can be calculated directly.

Model parameter components for Z1i, Z2i, Z3i, Z4i, Z5i and Z6i in the RIL-based aTTC design under both the F∞ were listed in Table 2.

Table 2

Model parameter components for Z1i, Z2i, Z3i, Z4i, Z5i and Z6i in the RIL-based aTTC design under both the F∞ model.

Data	F∞ metric models
	Model mean	Main effect	Epistatic effect
Z1i	μ_z_1=2μ+∑k=1qd_k	a_k^*=a_k+12∑l≠kq(i_a_kd_l−i_d_ka_l)	i↔_kl=i_a_ka_l+i_d_kd_l
Z2i	μ_z_2=∑k=1qa_k	d_k^*=d_k−12∑l≠kq(i_a_ka_l−i_d_kd_l)	i˜_kl=i_a_kd_l+i_d_ka_l
Z3i	μ_z3=ri_a_1a_2	-	i_a_1d_2,i_d_1a_2,i_d_1d_2
Z4i	μ_z_4=μ+∑l≠kq(d_k+d_l+i_d_kd_l)	-	i→_kl=i_a_ka_l−i_d_kd_l i←_kl=i_a_kd_l−i_d_ka_l
Z5i	μ_z_5=3(μ+12d_1+12d_2)	a1,a2	i_a_1a_2,i_a_1d_2,i_d_1a_2,i_d_1d_2
Z6i	μz6 = μ + d1 + d2	-	i_a_1a_2,i_a_1d_2,i_d_1a_2,i_d_1d_2

Genetic models for mapping heterotic QTL in F2-based aTTC design

Genetic models for mapping heterotic QTL in the F2-based aTTC design under both F∞ and F2 metric can be found in S4 Supporting Information.

Parameter estimation

For a continuously distributed trait, the observed phenotypic value y of individual i can be described by the linear regression model where q is the number of markers, μ is the overall mean, x denotes the genotype of marker k for individual i and is defined as −1 or 1 for the two genotypes in the mapping population, and x represents the epistatic genotype between the kth and lth QTL of individual i, and is obtained as the element-wise product of x and x. In addition, a and i are the corresponding augmented main and epistatic effects, respectively. is the total number of genetic effects and x and β are the corresponding genotypes and coefficients, including the main and epistatic effects. e is the residual error assumed to follow an N(0,σ2) distribution.

Model (15) can be written as where vectors β and represent the augmented main and epistatic effects of all markers, respectively. X and are corresponding design matrices of different effects and e is the residual error that follows an N(0,σ2) distribution. Defining and , model (16) can be written in a more compact form

Due to the physical linkage or epistatic interactions among multiple QTL, it is rational when taking a large number of loci into consideration simultaneously. However, the total number of genetic effects p is very large because we set each marker as a QTL initially. Typically, we have p >> n. To handle such an oversaturated model, we employed a fast empirical Bayesian LASSO (EBLASSO) algorithm. Simulation studies demonstrate that the EBLASSO method can sharply reduce the computational burden by shrinking small effects into zero, and can detect more true QTL effects without increasing the false-positive rate. Details of the EBLASSO algorithm can be seen by reference to the work of Cai et al. [41]. At last, all remaining markers with are picked up, where is the standard deviation estimation for normal prior . The epistatic model is then established that only includes effects that pass the first round of selection. We also perform a usual likelihood ratio test on the model to obtain significant QTL and epistatic interactions. With transformation combinations (Z1, Z2, and Z4), following the above steps, in the first round, the argument effects a*, , d*, , , and can be obtained; in the second round, main and epistatic effects of each QTL and interaction can be calculated according to equation transformations , , , , , and .

Take any two significant QTL in a model as an example to explain how we obtain genetic effects. After performing QTL mapping, genetic parameters and in Z4, in Z1, and in Z2 can be obtained. For and , so , , while and ; we then get , . Meanwhile, a1*, a2*, d1*, and d2* are obtained after performing QTL mapping in Z1 and Z2 because , , , and , with the estimation value of d1*, d2*, and above , , , and values, main effects can be calculated by , , , and . All main and epistatic effects were dissected by the integration of augmented effects in Z1, Z2, and Z4.

Similarity, with transformation combination (Z1, Z2, and Z5) and (Z1, Z2, and Z6), we can also get the genetic effect of each QTL or interaction by QTL mapping under the aTTC design, respectively. More details are listed in S1 Supporting Information.

Simulation study

We took all the possible types of epistatic interaction patterns into consideration. The simulated genome, covered by 100 evenly spaced markers with a marker interval of 5 cM, was 495 cM in total length and comprised four chromosomes. For data sets Z1, Z2, or Z4, six QTL positions were preset, of which three positions (QTL1, QTL2, and QTL4) had main effects. Pairwise interactions were set between positions with main effects (QTL1 and QTL2), with and without main effects (QTL3 and QTL4), and without main effects (QTL5 and QTL6), respectively. The assumed QTL positions, parameters, and augmented effects (including main and epistatic effects) are listed in Table 3 and Tables A-B in S11 Supporting Information. The sample size (n) was set at three levels: 800, 400, and 200. The broad heritability (h) was also set at three levels: 0.8, 0.5, and 0.2, separately representing high, middle, and low heritabilities. The replication number of offspring (m) was set at two levels: 5 and 10. In three transformations, Z1, Z2, and Z4, each treatment was replicated 100 times. Simulation study also be conducted for Z5 and Z6. The results of Z5 and Z6 were listed in Tables C and D in S11 Supporting Information.

Table 3

QTL mapping results for Z4 in RIL-based aTTC design under the F∞ metric model in simulation study.

n	h	m				QTL1×QTL2								QTL3×QTL4								QTL5×QTL6
			I1a	posc	pos	pd	I2b	pos	pos	p	I1	pos	pos	p	I2	pos	pos	p	I2	pos	pos	p	I1	pos	pos	p
			3.2	20	36		2.4	20	36		-3.6	45	70		2.2	45	70		3	80	95		3.4	80	95
800	0.8	10	3.204	20	36	1	2.399	20	36	1	-3.596	45	70	1	2.194	45	70	1	2.994	80	95	1	3.385	80	95	1
			(0.062)				(0.03)				(0.062)				(0.035)				(0.03)				(0.062)
		5	3.191	20	36	1	2.4	20	36	1	-3.594	45	70	1	2.192	45	70	1	2.989	80	95	1	3.38	80	95	1
			(0.075)				(0.045)				(0.085)				(0.05)				(0.048)				(0.093)
	0.5	10	3.175	20	36	1	2.37	20	36	1	-3.595	45	70	1	2.178	45	70	1	2.989	80	95	1	3.384	80	95	1
			(0.118)				(0.069)				(0.109)				(0.062)				(0.065)				(0.109)
		5	3.17	20	36	1	2.342	20	36	1	-3.538	45	70	1	2.178	45	70	1	2.985	80	95	1	3.371	80	95	1
			(0.195)				(0.098)				(0.168)				(0.093)				(0.085)				(0.168)
	0.2	10	3.146	20	36	1	2.333	20	36	1	-3.531	45	70	1	2.141	45	70	1	2.921	80	95	1	3.336	80	95	1
			(0.233)				(0.127)				(0.265)				(0.141)				(0.134)				(0.232)
		5	3.171	20	36	1	2.278	20	36	0.99	-3.566	45	70	1	2.055	45	70	0.99	2.923	80	95	1	3.338	80	95	1
			(0.354)				(0.192)				(0.417)				(0.255)				(0.193)				(0.342)
400	0.8	10	3.183	20	36	1	2.385	20	36	1	-3.59	45	70	1	2.192	45	70	1	2.989	80	95	1	3.381	80	95	1
			(0.081)				(0.045)				(0.089)				(0.048)				(0.046)				(0.088)
		5	3.19	20	36	1	2.374	20	36	1	-3.569	45	70	1	2.176	45	70	1	2.999	80	95	1	3.399	80	95	1
			(0.141)				(0.061)				(0.138)				(0.065)				(0.069)				(0.130)
	0.5	10	3.16	20	36	1	2.381	20	36	1	-3.58	45	70	1	2.15	45	70	1	2.981	80	95	1	3.345	80	95	1
			(0.156)				(0.092)				(0.158)				(0.085)				(0.099)				(0.173)
		5	3.154	20	36	1	2.313	20	36	1	-3.52	45	70	1	2.097	45	70	1	2.943	80	95	1	3.344	80	95	1
			(-0.263)				(0.147)				(0.245)				(0.141)				(0.147)				(0.291)
	0.2	10	3.118	20	36	1	2.283	20	36	1	-3.478	45	70	1	2.066	45	70	0.96	2.904	80	95	0.98	3.316	80	95	1
			(0.351)				(0.337)				(0.345)				(0.277)				(0.197)				(0.337)
		5	2.899	20	36	0.99	2.247	20	36	0.89	-3.477	45	70	1	1.985	45	70	0.75	2.812	80	95	0.99	2.699	80	95	0.99
			(0.889)				(0.26)				(0.558)				(0.45)				(0.362)				(1.792)
200	0.8	10	3.18	20	36	1	2.386	20	36	1	-3.592	45	70	1	2.192	45	70	1	2.993	80	95	1	3.389	80	95	1
			(0.118)				(0.069)				(0.126)				(0.065)				(0.065)				(0.139)
		5	3.192	20	36	1	2.342	20	36	1	-3.56	45	70	1	2.163	45	70	1	2.947	80	95	1	3.346	80	95	1
			(0.166)				(0.095)				(0.181)				(0.097)				(0.098)				(0.174)
	0.5	10	3.087	20	36	1	2.317	20	36	1	-3.541	45	70	1	2.12	45	70	1	2.943	80	95	1	3.321	80	95	1
			(0.253)				(0.147)				(0.224)				(0.13)				(0.147)				(0.268)
		5	3.142	20	36	1	2.244	20	36	0.99	-3.552	45	70	1	1.862	45	70	0.97	2.94	80	95	1	3.308	80	95	1
			(0.354)				(0.36)				(0.331)				(0.628)				(0.186)				(0.349)
	0.2	10	3.068	20	36	0.97	2.329	20	36	0.76	-3.31	45	70	0.93	2.05	45	70	0.64	2.936	80	95	0.97	3.15	80	95	0.97
			(0.47)				(0.257)				(0.859)				(0.326)				(0.263)				(0.829)
		5	3.591	20	36	0.61	2.834	20	36	0.24	-3.593	45	70	0.61	2.552	45	70	0.25	3.044	80	95	0.49	3.543	80	95	0.59
			(0.612)				(0.494)				(0.624)				(0.332)				(0.359)				(0.527)

Epistatic effects estimated in Z4 were multiplied by 2; n: sample size; h: broad heritability; m: the replication number of offspring.

aI1: i—i.

bI2: i—i.

cpos: position.

dp: power.

QTL mapping in simulation study

All independent variable for Z1, Z2, Z4 in simulation study were simultaneously included in one genetic model, which was much larger than the sample size. Data sets were implemented in R (version 3.0) with the EBLASSO package obtained from Cai et al. [41]. Hyper-parameters a and b were obtained by three-fold cross-validation (by default) in each individual model; after 100 replications, hyper-parameters with minimum predicted errors were fixed to estimate parameters. The time was approximately 5 minutes in each transformation in a stand-alone personal computer (Intel Pentium CPU 2.9 GHz; memory 4 GB); therefore, the EBLASSO algorithm presented high efficiency and saved time.

Real data analysis

Considering the unbiased estimate of coefficients and the excellent detection power in the simulation study, we further applied the proposed approach to a real mapping population and presented a comparison to previous mapping results.

Populations

Two elite rice hybrids, one inter-subspecific between 9024 (indica) and LH422 (japonica) and one intra-subspecific between Zhenshan97 (indica) and Minghui63 (indica), were analyzed, and details were documented in our previous study [24]. For convenience, we designated the two hybrids as IJ and II hybrids, respectively. The RIL were derived from the cross of a random sample of F2 individuals to their parental lines (194 F7 lines for the IJ hybrid and 222 F12 lines for the II hybrid, respectively).

Genetic linkage maps

For the II hybrid, the linkage map comprised 221 marker loci and covered 1796 cM in total [42]. For the IJ hybrid, Xiao et al. [8] constructed a linkage map of the recombinant population in which a subset of 141 polymorphic restriction fragment length polymorphism markers was used.

Phenotypic traits

Nine quantitative traits, including heading date (HD, in days), plant height (PH, in centimeters), tillers per plant, panicle length (PL, in centimeters), filled grains per panicle (FGPP), percentage of seed set, grain density (GD, in grain numbers per centimeter of panicle length), 1000-grain weight (KGW, in grams), and grain yield (YD, in tons/hectare) were investigated in RIL, Z1, Z2, and Z4 respectively. All the materials described above were laid out in a field in a randomized complete block design with two replications (plots) for phenotypic evaluation.

QTL mapping in real data analysis

Data sets Z1, Z2 and Z4 were implemented in R (version 3.0) with the EBLASSO package obtained from Cai et al. [41] for QTL mapping.

Results

Simulation study results

Augmented effects in simulation study

As shown in Table 3 and Tables A-B in S11 Supporting Information, the augmented additive () and epistatic effects () in Z1, augmented dominance () and epistatic effects () in Z2, augmented epistatic effects (, ) in Z4 were rightly and unbiased estimated with a high statistical power in preset positions. The ratio of the number of samples, in which the LOD statistic was greater than 2.5, to the total number of replicates represented the empirical power of this simulated QTL or interaction.

In Z1 transformation (Table A in S11 Supporting Information, S1 Fig), when the sample size was 800 or 400, almost all augmented additive and epistatic effects were detected, except for the detection power of digenic interactions in 400 samples, 0.2 heritability, and 5 replications. This indicated that smaller heritability or less individual replication had little influence on the detection of QTL in a relatively large sample size. When the sample size was reduced to 200, all the preset QTL were detected successfully with a heritability of 0.8 and 0.5; however, detection power decreased sharply to the level of 0.2, which is more true for the preset digenic interactions. When individual replication was 5, the detection power of the augment additive effect of QTL3 was 0.73, whereas it was 0.985 when the individual replication was 10, and the detection power of augmented epistatic effects dropped to 0.59 for interaction QTL5 × QTL6. Similar results could be found in Z2 (Table B in S11 Supporting Information, S2 Fig); all the preset QTL were precisely detected and the QTL effects were estimated in an unbiased manner, even on the level of the smallest sample size (200) or the lowest heritability (0.2). In addition, all the augmented epistatic effects estimated in an unbiased manner in Z4 (Table 3, Fig 1). The poor detection power occurred only on a low heritability level (0.2) with sample sizes 400 or 200.

Fig 1

The mean statistic power of augmented epistatic effect interactions in Z4.

Epis-1 refer the augmented epistatic effect (i—i) interaction, epis-2 refer the augmented epistatic effect (i—i) interaction.

Main and epistatic effects in simulation study

Table 4 shown the main and epistatic effects of QTL1 and QTL2 in the in RIL-based aTTC design using the two-step approach under the F∞ metric model. Other pairs of interactions are listed in Tables 5 and 6 for the interaction between QTL3 and QTL4 and QTL5 and QTL6, respectively. We can see that the main-effect and epistatic effects of QTL were very close to set value when sample size is big (800) and heritability is high (0.8). all the preset QTL were precisely detected and the QTL effects were estimated in an unbiased manner, except on the level of the smallest sample size (200) and the lowest heritability (0.2) with 5 replications in Z1, Z2 and Z4.

Table 4

Dissected main and epistatic effects of QTL1 and QTL2 in the in RIL-based aTTC design using the two-step approach under the F∞ metric model in simulation study.

N	h	m	statistics	QTL1		QTL2		QTL1×QTL2
				a1	d1	a2	d2	a1a2	a1d2	d1a2	d1d2
Parameter values				1.8	3.2	2.8	-1.5	2.7	3.4	1	-0.5
800	0.8	10	mean	1.802	3.204	2.795	-1.501	2.699	3.399	0.999	-0.506
			sd	(0.022)	(0.038)	(0.021)	(0.036)	(0.021)	(0.030)	(0.033)	(0.049)
		5	mean	1.804	3.192	2.794	-1.506	2.698	3.403	1.004	-0.492
			sd	(0.032)	(0.047)	(0.030)	(0.051)	(0.026)	(0.047)	(0.045)	(0.061)
	0.5	10	mean	1.807	3.183	2.789	-1.511	2.686	3.370	1.000	-0.489
			sd	(0.049)	(0.071)	(0.043)	(0.069)	(0.045)	(0.066)	(0.061)	(0.093)
		5	mean	1.825	3.181	2.758	-1.523	2.672	3.356	1.015	-0.498
			sd	(0.058)	(0.124)	(0.064)	(0.120)	(0.068)	(0.095)	(0.097)	(0.155)
	0.2	10	mean	1.830	3.158	2.748	-1.526	2.653	3.350	1.017	-0.493
			sd	(0.084)	(0.154)	(0.089)	(0.154)	(0.092)	(0.131)	(0.141)	(0.177)
		5	mean	1.827	3.142	2.688	-1.492	2.614	3.333	1.067	-0.557
			sd	(0.151)	(0.222)	(0.122)	(0.228)	(0.123)	(0.178)	(0.175)	(0.284)
400	0.8	10	mean	1.808	3.184	2.795	-1.515	2.689	3.388	1.002	-0.494
			sd	(0.035)	(0.052)	(0.030)	(0.051)	(0.031)	(0.042)	(0.044)	(0.064)
		5	mean	1.810	3.187	2.789	-1.506	2.688	3.385	1.011	-0.501
			sd	(0.044)	(0.084)	(0.044)	(0.087)	(0.050)	(0.061)	(0.065)	(0.103)
	0.5	10	mean	1.807	3.156	2.786	-1.525	2.673	3.378	0.998	-0.486
			sd	(0.066)	(0.096)	(0.062)	(0.099)	(0.063)	(0.084)	(0.084)	(0.126)
		5	mean	1.831	3.168	2.744	-1.518	2.660	3.354	1.041	-0.494
			sd	(0.099)	(0.166)	(0.090)	(0.164)	(0.098)	(0.132)	(0.143)	(0.201)
	0.2	10	mean	1.842	3.118	2.698	-1.559	2.625	3.305	1.022	-0.494
			sd	(0.176)	(0.216)	(0.179)	(0.219)	(0.153)	(0.244)	(0.226)	(0.313)
		5	mean	1.946	2.993	2.568	-1.629	2.399	3.206	1.138	-0.471
			sd	(0.251)	(0.513)	(0.257)	(0.491)	(0.461)	(0.311)	(0.375)	(0.687)
200	0.8	10	mean	1.805	3.184	2.785	-1.500	2.688	3.391	1.006	-0.492
			sd	(0.050)	(0.072)	(0.044)	(0.073)	(0.042)	(0.071)	(0.058)	(0.096)
		5	mean	1.816	3.190	2.771	-1.484	2.668	3.360	1.018	-0.525
			sd	(0.059)	(0.113)	(0.069)	(0.099)	(0.059)	0.086	0.083	0.133
	0.5	10	mean	1.834	3.137	2.749	-1.557	2.635	3.355	1.038	-0.451
			sd	(0.098)	(0.153)	(0.092)	(0.151)	(0.088)	0.141	0.135	0.201
		5	mean	1.862	3.129	2.684	-1.510	2.611	3.288	1.054	-0.531
			sd	(0.194)	(0.210)	(0.208)	(0.219)	(0.146)	0.244	0.231	0.329
	0.2	10	mean	1.991	3.057	2.498	-1.588	2.474	3.156	1.239	-0.502
			sd	(0.312)	(0.393)	(0.290)	(0.397)	(0.398)	0.359	0.376	0.602
		5	mean	2.169	2.626	2.099	-1.983	1.830	2.770	1.423	-0.908
			sd	(0.474)	(0.958)	(0.489)	(0.961)	(0.515)	0.569	0.505	1.330

Table 5

Dissected main and epistatic effects of QTL3 and QTL4 in the in RIL-based aTTC design using the two-step approach under the F∞ metric model in simulation study.

N	h	m	statistics	QTL3		QTL4		QTL3×QTL4
				a3	d3	a4	d4	a3a4	a3d4	d3a4	d3d4
Parameter values				0	0	-2	2.1	-0.7	-1.6	-3.8	2.9
800	0.8	10	mean	0.001	0.003	-2.003	2.097	-0.698	-1.605	-3.798	2.898
			sd	(0.022)	(0.038)	(0.020)	(0.042)	(0.022)	(0.035)	(0.029)	(0.048)
		5	mean	0.001	0.002	-2.006	2.100	-0.702	-1.602	-3.794	2.892
			sd	(0.033)	(0.061)	(0.034)	(0.053)	(0.028)	(0.047)	(0.044)	(0.074)
	0.5	10	mean	0.008	0.012	-2.007	2.098	-0.703	-1.606	-3.784	2.892
			sd	(0.044)	(0.073)	(0.047)	(0.075)	(0.038)	(0.062)	(0.066)	(0.086)
		5	mean	0.010	0.021	-2.007	2.126	-0.683	-1.614	-3.792	2.855
			sd	(0.063)	(0.114)	(0.054)	(0.100)	(0.067)	(0.089)	(0.088)	(0.123)
	0.2	10	mean	0.023	0.021	-2.029	2.116	-0.695	-1.620	-3.761	2.836
			sd	(0.090)	(0.158)	(0.093)	(0.150)	(0.096)	(0.136)	(0.153)	(0.206)
		5	mean	0.029	0.002	-2.047	2.111	-0.745	-1.664	-3.708	2.821
			sd	(0.193)	(0.230)	(0.166)	(0.256)	(0.187)	(0.237)	(0.226)	(0.389)
400	0.8	10	mean	0.002	0.006	-2.003	2.104	-0.699	-1.602	-3.794	2.891
			sd	(0.030)	(0.053)	(0.030)	(0.059)	(0.031)	(0.040)	(0.043)	(0.072)
		5	mean	0.008	0.001	-2.008	2.116	-0.698	-1.611	-3.788	2.872
			sd	(0.040)	(0.079)	(0.047)	(0.087)	(0.049)	(0.068)	(0.063)	(0.107)
	0.5	10	mean	0.006	0.008	-2.019	2.101	-0.703	-1.629	-3.778	2.877
			sd	(0.055)	(0.101)	(0.058)	(0.096)	(0.051)	(0.087)	(0.092)	(0.134)
		5	mean	0.025	0.014	-2.047	2.121	-0.687	-1.658	-3.755	2.833
			sd	(0.099)	(0.155)	(0.094)	(0.156)	(0.092)	(0.152)	(0.132)	(0.204)
	0.2	10	mean	0.049	0.032	-2.079	2.164	-0.700	-1.661	-3.664	2.778
			sd	(0.213)	(0.226)	(0.230)	(0.225)	(0.121)	(0.255)	(0.273)	(0.274)
		5	mean	0.244	0.010	-2.333	2.142	-0.727	-1.920	-3.419	2.749
			sd	(0.402)	(0.315)	(0.444)	(0.351)	(0.224)	(0.528)	(0.495)	(0.467)
200	0.8	10	mean	0.002	0.002	-2.004	2.105	-0.697	-1.605	-3.797	2.896
			sd	(0.046)	(0.079)	(0.045)	(0.079)	(0.047)	(0.065)	(0.062)	(0.098)
		5	mean	0.011	0.016	-2.011	2.110	-0.693	-1.615	-3.778	2.867
			sd	(0.065)	(0.116)	(0.064)	(0.095)	(0.066)	(0.096)	(0.089)	(0.137)
	0.5	10	mean	0.031	0.017	-2.037	2.129	-0.692	-1.628	-3.749	2.848
			sd	(0.093)	(0.157)	(0.082)	(0.149)	(0.085)	(0.121)	(0.135)	(0.182)
		5	mean	0.127	0.005	-2.164	2.111	-0.700	-1.751	-3.588	2.852
			sd	(0.307)	(0.202)	(0.313)	(0.198)	(0.131)	(0.336)	(0.352)	(0.262)
	0.2	10	mean	0.332	0.243	-2.373	2.352	-0.588	-2.008	-3.340	2.553
			sd	(0.482)	(0.599)	(0.404)	(0.594)	(0.555)	(0.493)	(0.506)	(0.674)
		5	mean	0.255	0.485	-2.637	2.808	-0.737	-2.363	-3.001	2.351
			sd	(0.825)	(1.049)	(0.614)	(0.960)	(0.909)	(0.497)	(0.559)	(0.955)

Table 6

Dissected main and epistatic effects of QTL5 and QTL6 in the in RIL-based aTTC design using the two-step approach under the F∞ metric model in simulation study.

n	h	m	statistics	QTL5		QTL6		QTL5×QTL6
				a5	d5	a6	d6	a5a6	a5d6	d5a6	d5d6
Parameter values				0	0	0	0	2.8	3.9	0.9	-0.6
800	0.8	10	mean	0.001	0.008	0.003	0.006	2.791	3.897	0.903	-0.594
			sd	(0.019)	(0.038)	(0.022)	(0.039)	(0.022)	(0.033)	(0.029)	(0.049)
		5	mean	0.006	0.010	0.003	0.008	2.791	3.894	0.905	-0.588
			sd	(0.029)	(0.047)	(0.035)	(0.058)	(0.030)	(0.041)	(0.047)	(0.075)
	0.5	10	mean	0.001	0.002	0.006	0.009	2.788	3.892	0.903	-0.597
			sd	(0.046)	(0.075)	(0.045)	(0.067)	(0.038)	(0.059)	(0.063)	(0.088)
		5	mean	0.005	0.006	0.000	0.003	2.772	3.888	0.903	-0.600
			sd	(0.055)	(0.102)	(0.063)	(0.102)	(0.057)	(0.083)	(0.086)	(0.134)
	0.2	10	mean	0.019	0.031	0.005	0.021	2.746	3.847	0.926	-0.590
			sd	(0.105)	(0.136)	(0.086)	(0.136)	(0.083)	(0.123)	(0.120)	(0.183)
		5	mean	0.016	0.022	0.011	0.011	2.721	3.841	0.918	-0.618
			sd	(0.122)	(0.221)	(0.129)	(0.212)	(0.109)	(0.201)	(0.183)	(0.290)
400	0.8	10	mean	0.001	0.002	0.005	0.007	2.788	3.897	0.908	-0.593
			sd	(0.030)	(0.058)	(0.030)	(0.057)	(0.033)	(0.045)	(0.047)	(0.071)
		5	mean	0.001	0.002	0.002	0.000	2.789	3.897	0.899	-0.610
			sd	(0.046)	(0.079)	(0.047)	(0.077)	(0.046)	(0.059)	(0.068)	(0.100)
	0.5	10	mean	0.002	0.011	0.005	0.030	2.764	3.869	0.889	-0.581
			sd	(0.067)	(0.115)	(0.059)	(0.102)	(0.059)	(0.097)	(0.095)	(0.141)
		5	mean	0.020	0.016	0.005	0.009	2.736	3.859	0.916	-0.608
			sd	(0.095)	(0.182)	(0.084)	(0.180)	(0.097)	(0.142)	(0.138)	(0.225)
	0.2	10	mean	0.008	0.018	0.044	0.010	2.721	3.833	0.928	-0.595
			sd	(0.142)	(0.217)	(0.158)	(0.234)	(0.117)	(0.185)	(0.155)	(0.269)
		5	mean	0.089	0.316	0.068	0.282	2.343	3.791	0.994	-0.329
			sd	(0.264)	(0.905)	(0.255)	(1.015)	(0.879)	(0.344)	(0.338)	(1.027)
200	0.8	10	mean	0.003	0.005	0.000	0.003	2.787	3.896	0.902	-0.603
			sd	(0.044)	(0.080)	(0.041)	(0.076)	(0.049)	(0.061)	(0.062)	(0.106)
		5	mean	0.018	0.018	0.008	0.029	2.768	3.871	0.924	-0.578
			sd	(0.067)	(0.111)	(0.065)	(0.107)	(0.061)	(0.086)	(0.094)	(0.138)
	0.5	10	mean	0.023	0.011	0.001	0.031	2.740	3.846	0.903	-0.581
			sd	(0.099)	(0.151)	(0.097)	(0.169)	(0.085)	(0.136)	(0.140)	(0.209)
		5	mean	0.010	0.021	0.004	0.027	2.723	3.880	0.939	-0.586
			sd	(0.129)	(0.232)	(0.141)	(0.225)	(0.123)	(0.163)	(0.194)	(0.281)
	0.2	10	mean	0.039	0.132	0.060	0.114	2.535	3.806	0.958	-0.520
			sd	(0.276)	(0.532)	(0.265)	(0.495)	(0.469)	(0.294)	(0.368)	(0.696)
		5	mean	0.311	0.573	0.314	0.517	1.837	3.201	1.439	-0.776
			sd	(0.646)	(0.963)	(0.672)	(1.050)	(0.510)	(0.719)	(0.805)	(1.370)

Real data analysis results

QTL mapping in II and IJ hybrid

In the II hybrid, all were simultaneously included in the genetic model, about 115 times as large as the sample size, while in IJ hybrids, , which was about 45 times bigger than the sample size. QTL effect-explained 1% phenotypic variation was set as a threshold for declaring the presence of QTL. QTL mapping results for the II and IJ hybrids are listed in Tables A and B in S12 Supporting Information, respectively.

Augmented effects in II and IJ hybrid

As shown in Table A in S12 Supporting Information, 14 QTLs and 36 digenic interactions were detected in the II hybrid, and the explained variation of a single QTL or interaction varied from 1.13% to 7.67%. In the RIL mapping population, 8 QTL (16%) were revealed, of which 2 QTLs (25%) were detected in trait GD with relative small phenotypic variation. In trait YD, a digenic interaction of marker C1016 and C483 explained the maximum (7.49%) phenotypic variation. In Z1, 11 QTL (22%) were detected, and one or more QTL were revealed in each trait. The explained variation in Z1 varied from 1.21% to 7.67%. The interaction between marker R3166 and RZ667 was also detected in traits FGPP and GD, and explained 7.67% and 7.33% variation, respectively. In Z2, 10 QTLs were identified. There was no QTL detected in trait PL. The explained variation in Z2 varied from 1.13% to 5.91%. In Z4, 11 interactions were found, and at least one QTL or interaction was revealed in each trait. The maximum explained variation was in trait HD (6.05%). Ten interactions were also dissected, and the explained variation of a single interaction varied from 1.69% to 7.09%.

In the IJ hybrid, as shown in Table B in S12 Supporting Information, a total of 46 QTL and 75 interactions was detected. Of the detected QTL, the majority was detected in RIL (39.37%) and Z4 (41.73%). The explained variation of a single QTL in the RIL ranged from 1.09% to 27.48%. In Z4, the detected QTL-associated marker RG333 affected HD in chromosome 8 explained 27.48% of phenotypic variation. It was also found simultaneously in Z1 and Z2 data sets. Eighteen QTLs in Z1 and 9 QTLs in Z2 were identified. In Z1, QTL-associated marker RG333 influenced HD, accounting for 36.58% of variation, which also explained 8.47% of variation for PH. In Z4, the majority of interactions were detected in trait FGPP in which 17 (58.6%) and 10 (41.7%) marker pairs were found in and .

Dissection of main and epistatic effects

Integrated in the QTL mapping result of Tables A and B in S12 Supporting Information, main and epistatic effects were dissected by the proposed approach distributed previous part 3.4. The results of the II and IJ hybrid are presented in Tables 7 and 8, respectively. For main effect QTL, we dissected the additive and dominance effects, whereas for interactions, additive × additive (aa), additive × dominance (ad), dominance × additive (da), dominance × dominance (dd), effects were dissected. The dominance degree of each QTL or interaction was separately calculated by |d/a| and |dd/aa|, respectively. Where d, a, dd, aa denote the dissected dominance effect, additive effect, dominance × dominance epistatic effect, additive × additive epistatic effect, respectively. According to Stuber et al. [12], main effect QTL can be classified as additive (|d/a|<0.2), partial dominance (0.2≤|d/a|<0.8), dominance (0.8≤|d/a|<1.2), and overdominance (|d/a|≥1.2). Epistatic QTL can be classified as additive (|dd/aa|<0.2), partial dominance (0.2≤|dd/aa|<0.8), dominance (0.8≤|dd/aa|<1.2), and overdominance (|dd/aa|≥1.2).

Table 7

Dissected main and epistatic effects in II hybrid.

Trait	Chra	Marker	Chra	Marker	QTLi		QTLj		QTLi×QTLj				Dominance degreeb
					a1	d1	a2	d2	a1a2	a1d2	d1a2	d1d2
HD	12	C996			2.09	0.00							A
HD	8	C347	4	C56	0.46	-1.84	-0.46	-1.84	0.79	-0.46	0.46	4.47	OD
HD	1	R2632	10	RM258	-0.39	0.33	0.39	0.33	0.33	2.42	1.64	-0.33	D
HD	2	RZ599	11	R3203	0.04	-1.89	-0.04	-1.89	-1.89	-0.04	0.04	1.89	D
HD	3	RM227	8	C1121	-0.27	-1.56	0.27	-1.56	-1.56	0.27	-0.27	1.56	D
HD	6	Waxy	6	R2549	-0.83	-0.15	0.83	-0.15	-0.15	0.83	-0.83	0.15	D
PH	7	RM70			1.92	0.00							A
PH	3	R1925			0.00	1.86							OD
PH	5	RG360	8	R1629	0.24	-2.44	-0.24	-2.44	-2.44	-0.24	0.24	2.44	D
PH	6	RM204	6	C962	1.67	-0.26	-1.67	-0.26	-0.26	-1.67	1.67	0.26	D
TP	2	RM53			0.00	0.38							OD
TP	4	R78	12	C909B	-0.07	0.33	0.07	0.33	-0.27	0.07	-0.07	-0.92	OD
TP	4	C2807	6	P	-0.01	0.60	0.01	0.60	0.60	0.01	-0.01	-0.60	D
TP	2	RG634	2	R1738	-0.81	0.09	0.81	0.09	0.09	0.81	-0.81	-0.09	D
PL	6	G342			0.30	0.00							A
PL	4	G235	7	RM70	-0.19	-0.46	0.19	-0.46	0.13	0.19	-0.19	1.06	OD
PL	3	C316	8	RM25	-0.14	-0.41	0.14	-0.41	-0.41	0.14	-0.14	0.41	D
PL	6	RM204	12	C909B	-0.28	0.20	0.28	0.20	0.20	0.28	-0.28	-0.20	D
FGPP	5	R3166	6	RZ667	-1.39	5.65	1.39	5.65	-2.64	1.39	-1.39	-13.94	OD
FGPP	8	RM223	9	RG570	-0.78	0.17	0.78	0.17	0.17	-2.92	-4.48	-0.17	D
FGPP	2	RM48	9	R1687	1.03	3.97	-1.03	3.97	3.97	-1.03	1.03	-3.97	D
FGPP	8	C1121	8	RG978	-3.82	0.00	3.82	0.00	0.00	3.82	-3.82	0.00	D
SS	12	RM20b			2.62	0.00							A
SS	6	RG424	13	C933	-0.76	-0.97	0.76	-0.97	-0.97	3.13	1.61	0.97	D
SS	2	G1314a	6	R2549	-1.00	2.94	1.00	2.94	2.94	1.00	-1.00	-2.94	D
SS	9	C153B	14	C477	-7.01	-1.55	7.01	-1.55	-1.55	7.01	-7.01	1.55	D
GD	5	R3166	6	RZ667	-0.05	0.19	0.05	0.19	-0.12	0.05	-0.05	-0.50	OD
GD	1	C161	1	RM243	0.05	0.22	-0.05	0.22	0.22	-0.05	0.05	-0.22	D
GD	2	RM48	9	R1687	0.03	0.17	-0.03	0.17	0.17	-0.03	0.03	-0.17	D
GD	2	RZ324	7	RM234	-0.12	0.08	0.12	0.08	0.08	0.12	-0.12	-0.08	D
KGW	3	C1176			0.00	0.33							OD
KGW	8	L363A			0.00	0.22							OD
KGW	5	RG360	8	C347	0.02	-0.01	-0.02	-0.01	-0.01	-0.41	-0.37	0.01	D
KGW	8	RM25	9	RZ404	0.18	0.40	-0.18	0.40	-0.60	-0.18	0.18	-1.39	OD
KGW	1	RG101	4	RM241	-0.13	0.33	0.13	0.33	0.33	0.13	-0.13	-0.33	D
KGW	1	G393	1	R2201	0.75	-0.87	-0.75	-0.87	-0.87	-0.75	0.75	0.87	D
YD	10	C153A			1.50	0.00							A
YD	7	R1789			0.00	1.67							OD
YD	4	C2807	14	C477	-0.08	0.32	0.08	0.32	0.32	-1.86	-2.02	-0.32	D
YD	8	R1394	9	RM215	-0.77	1.99	0.77	1.99	1.99	0.77	-0.77	-1.99	D
YD	9	R1952b	10	C153A	1.37	0.01	-1.37	0.01	0.01	-1.37	1.37	-0.01	D
YD	11	L1044	11	Y6854L	4.53	1.02	-4.53	1.02	1.02	-4.53	4.53	-1.02	D

a Chromosome where the detected QTL located in.

bMain effect QTL can be classified as additive (A) (|d/a|<0.2), partial dominance (PD) (0.2≤|d/a|<0.8), dominance (D) (0.8≤|d/a|<1.2), and overdominance (OD) (|d/a|≥1.2). Epistatic QTL can be classified as additive (A) (|dd/aa|<0.2), partial dominance (PD) (0.2≤|dd/aa|<0.8), dominance (D) (0.8≤|dd/aa|<1.2), and overdominance (OD) (|dd/aa|≥1.2).

Table 8

Dissected main and epistatic effects in IJ hybrid.

Trait	Chra	Marker	Chra	Marker	QTLi		QTLj		QTLi×QTLj				Dominance degreeb
					a1	d1	a2	d2	a1a2	a1d2	d1a2	d1d2
HD	1	RG811			0.73	0.00							A
HD	3	XNPB249			0.83	0.00							A
HD	3	CDO1081			0.72	0.00							A
HD	7	RG711			0.60	0.00							A
HD	8	RG333			2.73	0.96							PD
HD	5	RZ556	8	RZ562	0.80	0.55	0.80	0.55	0.00	0.80	0.80	1.10	OD
HD	8	RZ562	9	RG667	0.85	1.01	0.85	1.01	1.01	0.85	0.85	1.01	D
HD	2	TW500	5	RG480	0.43	0.24	0.43	0.24	0.24	0.43	0.43	0.24	D
HD	6	RZ450	6	CDO204	6.66	0.00	6.66	0.00	0.00	6.66	6.66	0.00	D
HD	8	RG333	11	CDO534	3.80	0.28	1.07	0.45	0.45	1.07	1.07	0.45	D
PH	8	RG333			3.47	0.00							A
PH	7	CDO533			0.00	1.48							OD
PH	2	RZ987	10	RZ892	0.26	1.34	0.26	1.34	1.28	0.26	0.26	3.95	OD
PH	2	RG152	4	XNPB271	0.12	0.79	0.12	0.79	2.21	0.12	0.12	3.79	OD
PH	2	TW500	3	RG510	0.03	1.35	0.03	1.35	1.35	0.03	0.03	1.35	D
PH	1	RG173	2	TW500	1.15	1.02	1.15	1.02	1.02	1.15	1.15	1.02	D
PH	4	RG449	7	RG711	1.63	0.03	1.63	0.03	0.03	1.63	1.63	0.03	D
PH	5	RZ390	9	RZ12	0.90	0.31	0.90	0.31	0.31	0.90	0.90	0.31	D
PH	5	RZ70	8	RZ562	1.68	0.18	1.68	0.18	0.18	1.68	1.68	0.18	D
TP	1	RG541	7	RZ626	0.01	0.24	0.01	0.24	0.07	0.01	0.01	0.41	OD
TP	9	RZ422	11	XNPB179	0.04	0.25	0.04	0.25	0.05	0.04	0.04	0.45	OD
TP	2	RG555	7	CDO405	0.15	0.00	0.15	0.00	0.00	0.34	0.04	0.00	D
TP	6	RZ450	9	RZ12	0.03	0.05	0.03	0.05	0.06	0.12	0.18	0.06	D
TP	2	TW500	6	RZ828	0.04	0.29	0.04	0.29	0.29	0.04	0.04	0.29	D
TP	3	RG1356	6	RG213	0.12	0.00	0.12	0.00	0.00	0.12	0.12	0.00	D
TP	4	RG908	9	XNPB295	0.10	0.08	0.10	0.08	0.08	0.10	0.10	0.08	D
PL	5	RZ296	11	CDO534	0.05	0.32	0.05	0.32	0.10	0.05	0.05	0.55	OD
PL	4	RZ262	12	XNPB189	0.12	0.45	0.12	0.45	0.45	0.12	0.12	0.45	D
PL	4	RG449	11	CDO534	0.12	0.31	0.12	0.31	0.31	0.12	0.12	0.31	D
PL	9	RZ404	12	RG98	0.30	0.12	0.30	0.12	0.12	0.30	0.30	0.12	D
FGPP	3	CDO1081			6.12	0.00							A
FGPP	1	RG541	4	XNPB271	1.76	0.66	1.76	0.66	0.66	4.93	1.41	0.66	D
FGPP	1	RG469	5	RZ390	0.60	3.18	0.60	3.18	3.18	0.60	0.60	3.18	D
FGPP	1	RZ776	8	RG333	0.28	4.18	0.28	4.18	4.18	0.28	0.28	4.18	D
FGPP	1	RG375	10	RZ811	2.59	5.47	2.59	5.47	5.47	2.59	2.59	5.47	D
FGPP	2	RZ913	2	RG544	0.01	2.71	0.01	2.71	2.71	0.01	0.01	2.71	D
FGPP	2	CDO395	7	CDO405	0.22	4.98	0.22	4.98	4.98	0.22	0.23	4.98	D
FGPP	2	CDO395	8	RG333	0.22	2.14	0.22	2.14	2.14	0.22	0.22	2.14	D
FGPP	2	RZ987	6	WAXY	0.14	2.28	0.14	2.28	2.28	0.14	0.14	2.28	D
FGPP	2	RZ987	10	RZ892	2.27	4.02	2.27	4.02	4.02	2.27	2.27	4.02	D
FGPP	2	XNPB132	10	RZ811	1.97	3.42	1.97	3.42	3.42	1.97	1.97	3.42	D
FGPP	2	RG544	5	RG480	2.07	4.65	2.07	4.65	4.65	2.07	2.07	4.65	D
FGPP	2	TW500	8	RZ562	0.36	5.19	0.36	5.19	5.19	0.36	0.36	5.19	D
FGPP	3	XNPB249	8	RG136	0.19	2.33	0.19	2.33	2.33	0.19	0.19	2.33	D
FGPP	3	RZ16	10	RZ561	0.59	3.01	0.59	3.01	3.01	0.59	0.59	3.01	D
FGPP	4	CDO456	11	XNPB320	1.55	3.61	1.55	3.61	3.61	1.55	1.55	3.61	D
FGPP	5	RG573	9	RG667	0.39	3.70	0.39	3.70	3.70	0.39	0.39	3.70	D
FGPP	6	RG433	8	RZ28	0.10	2.10	0.10	2.10	2.10	0.10	0.10	2.10	D
FGPP	7	CDO405	10	RZ561	1.95	4.59	1.95	4.59	4.59	1.95	1.95	4.59	D
FGPP	1	RG375	12	XNPB316	2.62	1.52	2.62	1.52	1.52	2.62	2.62	1.52	D
FGPP	1	CDO962	2	RZ987	2.53	1.08	2.53	1.08	1.08	2.53	2.53	1.08	D
FGPP	1	RG173	5	RG480	3.60	0.14	3.60	0.14	0.14	3.60	3.60	0.14	D
FGPP	3	CDO1081	12	RZ816	4.27	0.07	1.85	0.07	0.07	1.85	1.85	0.07	D
FGPP	4	CDO456	6	XNPB317	1.96	0.34	1.96	0.34	0.34	1.96	1.96	0.34	D
FGPP	7	RG528	11	RZ638	3.67	2.60	3.67	2.60	2.60	3.67	3.68	2.60	D
FGPP	8	RG333	9	RZ927	4.54	2.46	4.54	2.46	2.46	4.54	4.54	2.46	D
FGPP	9	XNPB103	12	RG901	3.96	0.31	3.96	0.31	0.31	3.96	3.96	0.31	D
FGPP	9	RZ12	9	RG667	7.70	0.86	7.70	0.86	0.86	7.70	7.70	0.86	D
FGPP	12	RZ816	12	XNPB189	7.07	6.83	7.07	6.83	6.83	7.07	7.07	6.83	D
SS	1	XNPB302	10	RZ811	0.66	0.38	0.66	0.38	0.37	0.38	1.71	0.37	D
SS	2	RG544	7	RG711	0.35	3.04	0.35	3.04	3.04	0.35	0.35	3.04	D
SS	7	RG528	10	RZ400	2.05	0.91	2.05	0.91	0.91	2.05	2.05	0.91	D
GD	6	RG162			0.00	0.16							OD
GD	1	RG375	10	RZ811	0.10	0.29	0.10	0.29	0.11	0.10	0.10	0.48	OD
GD	2	CDO395	7	CDO405	0.03	0.33	0.03	0.33	0.33	0.03	0.03	0.33	D
GD	2	RZ599	6	RZ965	0.16	0.25	0.16	0.25	0.25	0.16	0.16	0.25	D
GD	4	RZ590	9	RZ12	0.23	0.03	0.23	0.03	0.03	0.23	0.23	0.03	D
KGW	5	RZ296			0.91	0.00							A
KGW	5	CDO202			0.00	0.41							OD
KGW	2	RG555	9	RG358	0.50	1.33	0.50	1.33	1.33	0.50	0.50	1.33	D
KGW	4	RG908	8	RG333	1.84	2.63	1.84	2.63	2.63	1.84	1.84	2.63	D
YD	8	RZ562			0.00	191.74							OD
YD	2	RZ825	5	RG480	177.88	378.17	177.88	378.17	378.17	177.88	177.88	378.17	D
YD	7	CDO405	10	RZ400	166.44	544.18	166.44	544.18	544.18	166.44	166.44	544.18	D
YD	4	RG908	8	RG333	481.55	686.50	481.55	686.50	686.50	481.55	481.55	686.50	D

a See footnote of Table 7.

b See footnote of Table 7.

HD: In the II hybrid, the only main effect QTL was classified as additive; in the other five epistatic QTL, most were classified as dominance, except one that showed overdominance. In the IJ hybrid, four main effect QTL were classified as additive, and only one main effect QTL was classified as partial dominance; four epistatic QTL were classified as dominance, and one epistatic QTL was classified as overdominance.

PH: In the II hybrid, two main effect QTL were classified as additive and overdominance, respectively; two epistatic QTL were classified as dominance. In the IJ hybrid, two main effect QTL were classified as additive and overdominance, respectively; in seven epistatic QTL, two were classified as overdominance and the other five were dominance.

Tillers per plant: In the II hybrid, the only main effect QTL was classified as overdominance; two of three epistatic QTL were classified as dominance and the remaining one was classified as overdominance. In the IJ hybrid, no main effect QTL was found; in seven epistatic QTL, two were classified as overdominance and the remaining five were dominance, which was similar to trait PH.

PL: In the II hybrid, the only main effect QTL was classified as additive; two of three epistatic QTL were classified as dominance and the remaining one showed overdominance. In the IJ hybrid, no main effect QTL was found; in four epistatic QTL, one was identified as overdominance and the rest were dominance.

FGPP: In the II hybrid, no main effect QTL was found; in four epistatic QTL, one was classified as overdominance and the rest were dominance. In the IJ hybrid, the only main effect QTL was classified as additive; a total of 28 epistatic QTL was dissected, all of which showed dominance.

Percentage of seed set: In the II hybrid, the only main effect QTL was classified as additive and the three epistatic QTL were classified as dominance. In the IJ hybrid, only three epistatic QTL were dissected and all of them were classified as dominance.

GD: In the II hybrid, no main effect QTL was found; among four epistatic QTL, one was classified as overdominance and the remaining three were dominance. In the IJ hybrid, the only main effect QTL was classified as overdominance; among four epistatic QTL, three were classified as dominance and the remaining one was classified as overdominance.

KGW: In the II hybrid, two main effect QTL were classified as overdominance; in four epistatic QTL, one showed overdominance and the remaining three showed dominance. In the IJ hybrid, two main effect QTL were classified as additive and overdominance, respectively; two epistatic QTL showed dominance.

YD: In the II hybrid, two main effect QTL were classified as additive and overdominance, respectively; all four epistatic QTL were classified as dominance. In the IJ hybrid, the only main effect QTL showed overdominance, and the three epistatic QTL were classified as dominance.

From Tables 7 and 8, we can see that little common loci were found. This phenomenon partially results from the mapping markers in IJ and II hybrid are different. But on the same chromosome, we found some nearby loci affected same trait in both II and IJ hybrids.

Table 9 summarizes the main and epistatic QTL revealed in the II and IJ hybrids. For main effect QTL, 10 QTL were identified in the II hybrid; five were classified as additive, and the rest were classified as overdominance. In the IJ hybrid, 12 QTL were found, more additive (58.33%) loci were identified than overdominance in number (33.34%). For epistatic QTL, dominance or overdominance are found in two hybrid combinations, and dominance played a leading role in epistatic QTL. Dominance accumulation and overdominance were the major genetic basis of heterosis.

Table 9

Summary of main and epistatic effects in II and IJ hybrids.

	II Main-effect QTL		IJ Main-effect QTL		II Epistatic-effect QTL		IJ Epistatic-effect QTL
	No.	Rate(%)	No.	Rate(%)	No.	Rate(%)	No.	Rate(%)
A	5	50	8	57.14
PD			1	7.14
D					26	81.25	56	86.15
OD	5	50	5	35.72	6	18.75	9	13.85
SUM	10		14		32		65

Discussion

Models comparison

Based on the aTTC design, this paper developed a QTL mapping method that fit for many base populations (RIL, F2, and DH); by employing multiple data set transformations (Z1i, Z2i, Z3i, Z4i, Z5i, and Z6i), many types of main and epistatic effects can be dissected. This paper took one combination (Z1, Z2, and Z4) of the aTTC design as an instance and proposed a two-step approach to dissect additive, dominance, and epistatic effects of QTL in the RIL-based aTTC design. A series of Monte Carlo simulation studies were carried out to confirm the proposed approach. Compared to previous studies on our methodologies, the proposed approach offered great advantages over previous methods.

aTTC design has many more transformations than do the TTC or NCIII designs, and with a series of transformation combinations (Z1, Z2, and Z4), (Z1, Z2, and Z5), or (Z1, Z2, and Z6), we can dissect main and epistatic effects of individual QTL or interactions by QTL mapping. It provides a new method for quantitative genetics research and especially for allowing scientists and breeders to understand the genetic basis for plant heterosis. In our study, we took the transformation combination Z1, Z2, and Z4 of RIL-based aTTC design as an instance to dissect genetic effects. There were some advantages when taking RIL as the base population. The genetic expectation mean of RIL was equivalent to L; therefore, there was no need to self-mate the base population. We simply used the RIL population data set substitute L, which saved labor and time. When using RIL-based TTC design for QTL mapping, we need generate four populations RIL, L1, L2 and L3. However, when using RIL-based aTTC design, breeders only need generate three populations RIL, L1 and L2. With combination (Z1 = L1+L2, Z2 = L1−L2 and Z4 = L1+L2−L4), we can dissect additive, dominance, and epistatic effects of QTL with high statistical powers and accuracies. In addition, many real mapping populations that derived from RIL-based NCIII design can be re-analyzed by the proposed method to develop main and epistatic effects to clearly decipher a genetic basis for heterosis.

In the present study, we used three different interaction patterns in one genetic model, which was much more complicated than that proposed by He et al. [37,39]. As shown in Tables A and B in S11 Supporting Information, with high detection power, all the augmented main effects in QTL (QTL1-6) and epistatic effects in digenic interactions (QTL1 and QTL2, QTL3 and QTL4, and QTL5 and QTL6) were estimated in an unbiased manner in Z1 and Z2. In Z4, two augmented epistatic effects and were further estimated precisely (shown in Table 3).

Actually, for the detection of small and linked QTL, low powers were observed. EBLASSO can handle the model that includes many effects [37, 39, 43]. In this sudy, we use a large number of effects, including main and epistatic QTL effects, simultaneously. EBLASSO shrinks weak effect into zero, which has little influence on large effect QTL. Simulation studies demonstrated that the fast EBLASSO greatly improved calculated speed and detected more true QTL effects without increasing the false-positive rate.

Comparison of QTL mapping results in II and IJ hybrid with previous mapping results

The QTL mapping results of this paper were compared with those of our previous study [24] in which the CIM was employed to mapping main effect QTL [44], and the mixed linear approach [45] was used to estimate epistatic QTL. QTL detected by both studies are listed in Tables 10 and 11 for II and IJ hybrids, respectively. As shown in Table 10, a total of nine main effects QTL and four epistatic QTL was found simultaneously in two studies; only one QTL revealed in trait PH showed opposite dominance degree. In an RIL mapping population, three main effects QTL were detected; two main effects QTL were detected in Z1, and both of them were identified as additive. Eight QTL were found in Z2, and half of them were main effects; dominance degree revealed by two methods was similar, except for marker R1925 in trait PH. For the IJ hybrid, shown in Table 11, no epistatic QTL was simultaneously detected. The number of main effects QTL detected by both studies was 17, 3, and 3 for RIL, Z1, and Z2, respectively. Except for marker CDO533 in trait PH, the detected main effects QTL showed the same dominance degree. If not taking threshold into consideration, the ratio of same main effect QTL detected by the fast EBLASSO algorithm to previous mapping results were 23.08%, 22.22%, and 50% for RIL, Z1, and Z2, respectively in the II hybrid, whereas in the IJ hybrid, they were 54.84%, 17.65%, and 21.430%.

Table 10

Comparison of the QTL mapping results by the proposed approach with previous results in Li et al. in II hybrid.

					RIL				Z1				Z2				Dominance degreeb
Trait	Chra	Marker	Chra	Marker	Method Ac		Method Bd		Method A		Method B		Method A		Method B		Method A	Method B
					beta	h2(%)e	beta	h2(%)e	beta	h2(%)e	beta	h2(%)e	beta	h2(%)e	beta	h2(%)e
HD	12	C996							-3.10	6.10	-2.09	2.86					A	A
HD	1	R2632	10	RM258									2.11	6.27	2.03	5.91	OD	D
PH	10	RG561			2.27	3.6	1.27	2.16									A	-
PH	3	R1925											-2.86	4.60	-1.86	1.99	A	OD
PL	5	R3166			0.43	4.2	-0.30	1.21									A	-
PL	6	G342							0.62	5.20	0.30	1.21					A	A
FGPP	8	RM223	9	RG570									-3.75	3.08	-3.70	3.30	OD	D
SS	6	RG424	13	C933									2.18	3.56	2.37	4.45	OD	D
GD	6	R1014			0.2	5.5	0.10	1.64									A	-
KGW	3	C1176											0.42	3.80	-0.33	2.48	D	OD
KGW	8	L363A											0.50	5.50	-0.22	1.13	OD	OD
YD	7	R1789											-1.99	5.20	1.67	3.67	OD	OD
YD	4	C2807	14	C477									-2.77	8.24	-1.94	5.33	OD	D

a See footnote of Table 7.

b See footnote of Table 7.

c Method A: Composite-interval mapping, with WinQTLcart (Zeng 1994) for main effect QTL mapping; Mixed linear approach, with QTLMAPPER ver.1.0 (Wang et al, 1999) for epistatic effect QTL mapping.

d Method B: Proposed method in this paper.

e Variation contributed by QTL or digenic interaction.

Table 11

Comparison of the QTL mapping results by the proposed approach with previous results in Li et al in IJ hybrid.

					RIL				Z1				Z2				Dominance degreeb
Trait	Chra	Marker	Chra	Marker	Method Ac		Method Bd		Method A		Method B		Method A		Method B		Method A	Method B
					beta	h2 (%)e	beta	h2 (%)e	beta	h2 (%)e	beta	h2 (%)e	beta	h2 (%)e	beta	h2 (%)e
HD	3	CDO1081			1.71	7.9	1.62	7.08	1.10	6.80	0.72	2.94					PD	A
HD	8	RG333			-5.1	58.3	-3.46	27.48	-2.72	34.80	-2.73	36.58	-1.22	17.40	-0.96	10.93	PD	PD
PH	8	RG333			-5.1	58.3	-2.22	55.37	-4.81	15.80	-3.47	8.47					A	A
PH	7	CDO533											-3.79	29.00	-1.48	4.29	A	OD
TP	4	RG214			0.27	8.7	0.21	5.12									A	-
TP	5	CDO1160			0.2	5.1	0.11	1.53									OD	-
PL	7	RG528			-1.07	34.6	-0.21	1.26									A	-
PL	9	RG358			0.44	5.2	0.21	1.17									A	-
FGPP	3	CDO1081			-7.02	12.5	-5.60	7.97									A	-
FGPP	4	RG214			-7.8	15.7	-7.28	13.06									A	-
FGPP	5	RG360			5.96	9	4.98	5.95									PD	-
SS	5	RG360			2.9	7.2	2.65	5.68									A	-
GD	3	RZ993			-0.3	11.5	-0.11	1.66									A	-
GD	4	RZ590			-0.38	17.8	-0.37	16.70									PD	-
GD	10	RZ561			0.2	5	0.14	2.17									A	-
KGW	8	RG333			3.07	15.5	2.86	14.55									OD	-
YD	3	CDO1081			-0.49	7.2	-0.45	6.03									A	-
YD	8	RZ562			0.66	11.3	0.56	8.09					0.17	5.90	0.19	4.17	OD	-

a See footnote of Table 7.

b See footnote of Table 7.

c See footnote of Table 10.

d See footnote of Table 10.

e See footnote of Table 10.

Among the identified QTL, some of them were pleiotropic. In the IJ hybrid, marker RG333 on chromosome 8 was simultaneously revealed in traits HD, PH, KGW, and YD; marker CDO1081 on chromosome 3 was simultaneously identified for traits HD, FGPP, and YD. These markers, especially for marker RG333 and marker CDO1081, were also found pleiotropic in the work of Xiao et al. [8] and Li et al. [24]. These regions deserve further attention, especially in marker-assisted breeding.

Genetic basis of heterosis with real data analysis

With our proposed approach, we dissected genetic effects of QTL and interactions for the II and IJ hybrids, respectively, and calculated the dominance degree of each QTL or digenic interaction (Tables 7 and 8). We summarized the classified dominance degree of real mapping populations (Table 9) and found that dominance degree in the Z2 data set that mainly characterized the heterosis showed overdominance and dominance for QTL and digenic interactions, and the ratio of dominance is greater than overdominance. Therefore, we conclude that dominance accumulation and overdominance are the major genetic basis of heterosis. This finding is consistent with Huang et al. [4], who pointed out that the accumulation of numerous rare superior alleles with positive dominance was an important contributor to heterotic phenomena after genomic analysis of hybrid rice varieties.

To explicitly elucidate the influence of single-locus (additive and dominance) and two-loci (aa, ad, da, and dd epistatic effect) genetic effects conditioning the heterosis of agronomic traits, models or genetic mating design (e.g., RIL-based TTC design) [30, 38], which can be used to study how interactions among multiple genes can lead to the phenotypic manifestations of heterosis, are probably the most relevant. Recent findings from genomic, proteomic, metabolic, epigenetic, and network studies in hybrids and polyploids also highlight some testable models for heterosis [46].

Expected genetic values of Z, Z, Z, Z, Z and Z under both the F∞ and F2 metric models in RIL-based aTTC design.

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Expected genetic values of Z, Z, Z, Z, Z and Z under both the F∞ and F2 metric models in F2-based aTTC design.

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Statistical genetic models for mapping heterotic QTL in the RIL-based aTTC design under the F2 metric model.

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Statistical genetic models for mapping heterotic QTL in the F2-based aTTC design under the F∞ metric model.

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Simulation data generate script.

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Simulation data of Z1.

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Simulation data of Z2.

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Simulation data of Z4.

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Simulation data of Z5.

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Simulation data of Z6.

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Simulation results of Z1,Z2,Z5,Z6.

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Real data augument effect results.

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The mean statistic power of augmented main and epistatic effect interactions in Z1.

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The mean statistic power of augmented main and epistatic effect interactions in Z2.

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The mean statistic power of augmented main and epistatic effect interactions in Z5.

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The mean statistic power of augmented main and epistatic effect interactions in Z6.

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