Introduction

The interaction among non-allelic genes, known as epistasis, is a crucial genetic component of quantitative traits with practical significance in determining the achievement of expected objectives in pyramiding breeding1,2. However, researches on epistasis have significantly lagged behind due to the limitations in experimental materials and statistical methods, making it challenging to quantitatively estimate epistatic effects at different levels and types3,4. Based on the single segment substitution lines, we effectively quantified the epistatic effects of various orders and types on lots of important agronomic traits in rice5,6,7. However, epistatic laws and mechanisms remained still mysterious due to the poverty in genetic theory and cognition. Eshed and Zamir8 observed that the combination effects between QTLs were always less than the additive of individual QTL effects, thus suggesting that the epistasis was accompanied by negative effects. Inspired by this, we carried out extensive researches on QTL epistasis to explore epistatic laws. Researches indicated that two positive QTLs always generated a negative epistatic effect, while two negative QTLs did a positive epistatic effect5,6,7,9,10,11.

In one of our recent articles, QTL epistasis on heading date were analyzed by four single segment substitution lines in rice12. We found that QTLs of three positive effects and one negative effect generated 62.5% negative dual QTL epistatic effects and 57.7% positive triple QTL epistatic effects, namely forming the relationship “positive QTLs—negative first-order epistasis—positive second-order epistasis”. Thus we suggested that the aggregation effect of QTLs was partially neutralized by the opposite epistatic effect sum, QTL epistasis playing a role of homeostasis on heading date in rice12. To provide additional evidences for this hypothesis, this paper analyzed QTL epistatic effects on two traits of plant height (ph, cm) and thousand kernel weight (tkw, g) based on five single segment substitution lines.

Results

Additive effects (a) and dominant effects (d)

A genotypic effect includes additive, dominance and epistasis, which can be estimated based on SSSLs and their pyramiding materials. Additive effects (a) and dominant effects (d) were estimated by the differences of genotypic values between the homozygotes and the heterozygotes of SSSLs and the receptor HJX74, respectively (Table 1).

Table 1 Additive effects (a) and dominant effects (d) of SSSLs on plant height (ph, cm) and thousand kernel weight (tkw, g).
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Except for S2 without ph QTL, all SSSLs carried with significant ph QTLs and tkw QTLs. The additive effects and dominant effects of all QTLs increased ph, while decreased tkw. They existed different estimations between the two season environments, and hadn’t significant tkw effects in the late season. These QTLs showed reverse effect directions between two measured traits, which was helpful to explore the regularity of QTL epistasis.

First-order epistatic effects

Based on SSSLs and their pyramiding materials, the epistatic effects between dual QTLs, known as first-order epistasis, were estimated by the residual effects between the pyramiding effects and the sum of two single QTL effects (Table 2). Here epistatic effects estimated included four components such as additive-additive (aa), additive-dominance (ad), dominance-additive (da) and dominance-dominance (dd) epistasis.

Table 2 The epistatic effects between dual SSSLs on plant height (ph, cm) and thousand kernel weight (tkw, g).
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All of nine pairs of SSSL combinations were detected with significant epistatic effects. Of 36 epistatic components, 20 and 21 estimations reached the significant level of p < 0.05 or 0.01 on ph and tkw in both seasons, respectively. One striking feature was that all significant epistatic components were negative on ph but positive on tkw. Additionally, despite the same effect directions, the effect magnitudes appeared differences between two season environments. Especially, no epistatic effects was detected significantly on tgw in the late season.

Second-order epistatic effects

The epistatic effects among three QTLs, known as second-order epistases, were estimated by the residual effects between the pyramiding effects and the sum of single QTL effects and the all dual QTL epistasis (Table 3). Here epistatic effects estimated included eight components such as additive–additive–additive (aaa), additive–additive–dominance (aad), additive–dominance–additive (ada), dominance–additive–additive (daa), additive–dominance–dominance (add), dominance–additive–dominance (dad), dominance–dominance–additive (dda), and dominance–dominance–dominance (ddd) epistases.

Table 3 The epistatic effects among three SSSLs on plant height (ph, cm) and thousand kernel weight (tkw, g).
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All of six SSSL combinations were detected with significant epistatic effects. Of 48 epistatic components, 35 and 18 estimations reached the significant level of p < 0.05 or 0.01 on ph and tkw in the both seasons, respectively. Oppositely with first-order epistasis, all significant second-order epistatic components were positive on ph but negative on tkw. They had the same effect directions but the different effect magnitudes in the two season environments.

Third-order epistatic effects

The epistatic effects among four QTLs, known as third-order epistases, were estimated by the residual effects between the pyramiding effects and the sum of single QTL effects and the all low-order epistases (Table 4). Here epistatic effects estimated included sixteen components such as additive–additive–additive–additive (aaaa), additive–additive–additive–dominance (aaad) and additive–additive–dominance–additive (aada) epistases etc., respectively.

Table 4 The epistatic effects among four SSSLs on plant height (ph, cm) and thousand kernel weight (tkw, g).
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Of sixteen epistatic effects, 13 and 11 estimations reached the significant level of p < 0.05 or 0.01 on ph and tkw in both seasons, respectively. All significant epistatic components were negative on ph but positive on tkw. They had the same effect directions but the different effect magnitudes appeared in the two season environments.

Discussions

The order and type of epistasis

Since genotype is composed of genes, genotypic effect can be divided into gene effects. Gene effects generally include additive, dominance and epistasis. Thus epistasis is one of important genetic components for a complex quantitative trait1,2. Epistasis was defined as the effect of one gene modified by another gene or several other genes (biological epistasis)1,2,3,4, and epistatic effect is estimated as the deviation from additivity in a linear statistical model (statistical epistasis)8. According to the genetic model \(G = a + d + e\), the additive (a), dominance (d) and epistasis (e) can be estimated on the genotypic effect G value8,13,14. This paper estimated effectively a and d by using the data from the homozygous and heterozygous of SSSLs, respectively. While e was done based the effects of pyramiding materials, a and d. When one locus is considered, there are a or d only. While involving in multiple loci, there are a, d and e. Epistasis between dual loci is known as first-order epistasis, including 22 types of components, aa, ad, da and dd. Epistasis among three and four loci are known as second-order and third-order epistasis, which includes 23 and 24 types of epistatic components, respectively. This paper estimated effectively various orders and types of epistases. High-order epistasis is even more common in the multiple gene system and more important to keep homeostasis of organism2,4,12. Various types of epistatic components play different roles in biological evolution and breeding practice12.

The estimations of epistasis

How to estimate epistasis? People went through a long process of exploration1,2. The additive-dominant genetic model has been around for more than half a century when epistasis was ignored3,4. Since traditional method to analysis of quantitative traits didn’t distinguish the effect of individual gene15, it could only estimate epistasis mixed from multi-gene system16. QTL mapping methods based on bi-parental populations couldn’t provide precise estimation of epistatic effects since the interference of genetic background17. Using near-isogenic lines or single segment substitution lines, some epistatic components between dual QTLs were estimated17,18. However, previous studies few estimated simultaneously various epistatic components8. Author ever constructed several secondary F2 populations derived from crossing of two SSSLs, each of which pyramided dual QTLs to allow simultaneously analysis of four epistatic components5,19. Subsequently, a half-diallel hybridization design was proposed to improve secondary F2 mapping population above6,7,9,10,11. A double QTL polymerization line was developed first, and then a half diallel crossing population from four parents (receptor, two SSSLs and their DSSL) was constituted to generate nine genotypes6,7. This method eased to get target genotypes, lowered the cost of molecular marker analysis, and could be constructed repeatedly9,10,11. Analyzing the genetic effects of the nine genotypes enabled to simultaneously estimate various epistatic components6,7,9,10,11. A mark advantage of this method is easy to extend to analyze epistasis among multiple QTLs. Using the extend half diallel crossing populations, we analyzed various epistatic components among three QTLs in a last paper12, and then among four QTLs in this paper.

The characteristics of epistasis

In our past series of studies, a several common features of epistasis were detected5,6,7,9,10,11. Epistasis was very prevalent, almost all QTL combinations appeared epistatic interactions5,6,7. One QTL always interacted with multiple QTLs9,10,11. In this paper, we detected that the first-order, second-order and third-order epistases were mostly with high frequency. Eshed and Zamir8 published the article of “Less-Than-Additive Epistatic Interactions of Quantitative Trait Loci in Tomato”, where dominance-dominance epistasis were detected to mostly be negative values. In fact, this is caused by positive QTL values. We found that, negative epistases were derived mainly from interactions between positive QTLs, while positive epistases from negative QTLs6,7,9,10,11,12. Further researches indicated that there was the regularity “positive QTLs–negative first-order epistasis–positive second-order epistasis, while negative QTLs–positive first-order epistasis–negative second-order epistasis”12. Hereby, we proposed the hypothesis that QTL epistasis plays a role homeostasis12. In this paper, the above regularity of epistasis was confirmed again on two target traits of ph and tkw. Since all QTL effects were positive on ph, then the first-order epistases were negative while the second-order positive. Furthermore, the third-order epistases were mostly negative. Inversely, all QTL effects were negative on tkw, then the first-order, second-order and third-order epistases were successively positive, negative and positive. Epistasis may be brought about by modification of gene function due to alterations in the signal-transducing pathway2,12. Epistatic genes are more deleterious in combination than separately, which are often accompanied by inverse epistatic interactions as homeostatic (that is, canalizing) mechanisms2,12. Further researches will provide additional evidences for this hypothesis.

Materials and methods

Materials

Huajingxian 74 (HJX74) and its five single-segment substitution lines (SSSLs) were applied in this trial. These experimental materials were described in our previous studies11,12. HJX74 is an elite indica variety with many excellent properties, which was cultured in our laboratory in South China20. Each SSSL possessed only a single substituted segment from a donor with the HJX74 genetic background, which was distributed in the related molecular marker regions on the corresponding chromosomes of given lengths21 (Fig. 1). With these markers, the foreground selections of the donors and the background selections of HJX74 were performed in order to ensure that the single fragment was unique.

Figure 1
figure 1

The approximate lengths and locations of substitution segments on the chromosomes for the five single segment substitution lines. Chr. and S were the abbreviation of chromosome and single segment substitution line, followed by serial numbers, respectively. The rectangular frames and the bold vertical lines represented chromosomes and substitution segments from donors of single segment substitution lines, respectively. The genetic distances (cM) between markers and the marker names were listed on either side of chromosomes, respectively.

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Some pyramiding materials of SSSLs (including homozygotes and heterozygotes) were configured to analyze epistasis among QTLs. The crossing between a SSSL and HJX74 would generate the heterozygote of SSSL. From the F2 populations derived from the F1 crossing combinations between two SSSLs, the homozygotes and the heterozygotes of dual segments could be obtained by marker assisted selection. Similarly, some pyramiding materials of triple- or tetra- segments could also be selected from the F2 generation of three or four SSSL combinations19.

Methods

Field trials. Phenotypic experiments were described in our previous studies11,12. The field trial site was the experimental farm of South China Agricultural University, Guangzhou (at ~ 113° east longitude and ~ 23°north latitude) and the field times were the early season (from March to July) and the late season (from July to November) in 2016, respectively. All experimental materials, including HJX74, homozygotes and heterozygotes of SSSLs, dual-, triple- and tetra-segment polymers, were grown in two season environments. In each experiment, the germinated seeds were sown in a seedling bed and seedlings were transplanted to a paddy field 20 days later, with one plant per hill spaced at 16.7 cm × 16.7 cm. A completely random design with three replications was adopted in the field trails, in which each plot was consisted of four rows with ten plants each row. The management of the field experiments was in accordance with local standard practices.

Traits measured

Plant height (ph, cm) per hill, the distance between the base and the top of the main panicle of the plant, was measured on 10 central plants in each plot when the plants approached maturity. After maturity, the seeds of single plant were harvested and put into a net bag to dry. The thousand kernel weight (tkw, g) was measured in each plot. The values of ph averages and tkw each plot were as inputting data for statistical analysis.

Analysis of variance. To estimate experimental error, analysis of variance was conducted on data of single season environment according to statistical model \(y_{ij} = \mu + G_{i} + e_{ij}\), where \(y,\mu ,G \, and \, e\) were plot average, population mean, genotypic effect and error on phenotypic values of target traits, respectively. Analysis of variance was performed by using the function aov() of R language (http://cran.r-project.org).

QTL analysis

According to the genetic model of additive(a)–dominance(d)–epistasis(e), a genotypic effect (G) can be expressed by \(G = a + d + e\). For instance, the effect of genotype \(AABBCc\) can be divided into \(G_{AABBCc} = a_{A} + a_{B} + d_{C} + e_{AB} + e_{AC}\) \(+ e_{BC} + e_{ABC}\). Using the homozygote and heterozygote of SSSLs and HJX74, a and d can be estimated by \(\hat{a}_{AA} = G_{AA} - G_{HJX74}\) and \(\hat{a}_{{A{\text{a}}}} = G_{Aa} - G_{HJX74}\), respectively. While based on the pyramiding materials, some of e are estimated by,

$$\hat{e}_{aa} = G_{AABB} - \hat{a}_{AA} - \hat{a}_{BB}$$
$$\hat{e}_{ad} = G_{AABb} - \hat{a}_{AA} - \hat{d}_{Bb}$$
$$\hat{e}_{aaa} = G_{AABBCC} - \hat{a}_{AA} - \hat{a}_{BB} - \hat{a}_{CC} - \hat{e}_{AABB} - \hat{e}_{AACC} - \hat{e}_{BBCC}$$
$$\hat{e}_{aad} = G_{AABBCc} - \hat{a}_{AA} - \hat{a}_{BB} - \hat{d}_{Cc} - \hat{e}_{AABB} - \hat{e}_{AACC} - \hat{e}_{BBCc}$$
$$\begin{aligned} \hat{e}_{{aaaa}} & = G_{{AABBCCDD}} - \hat{a}_{{AA}} - \hat{a}_{{BB}} - \hat{a}_{{CC}} - \hat{a}_{{DD}} - \hat{e}_{{AABB}} - \hat{e}_{{AACC}} - \hat{e}_{{AADD}} - \hat{e}_{{BBCC}} - \hat{e}_{{BBDD}} - \hat{e}_{{CCDD}} \\ & \quad - \hat{e}_{{AABBCC}} - \hat{e}_{{AABBDD}} - \hat{e}_{{AACCDD}} - \hat{e}_{{BBCCDD}} \\ \end{aligned}$$
$$\begin{aligned} \hat{e}_{{aaad}} & = G_{{AABBCCDd}} - \hat{a}_{{AA}} - \hat{a}_{{BB}} - \hat{a}_{{CC}} - \hat{d}_{{Dd}} - \hat{e}_{{AABB}} - \hat{e}_{{AACC}} - \hat{e}_{{AADd}} - \hat{e}_{{BBCC}} - \hat{e}_{{BBDd}} - \hat{e}_{{CCDd}} \\ & \quad - \hat{e}_{{AABBCC}} - \hat{e}_{{AABBDd}} - \hat{e}_{{AACCDd}} - \hat{e}_{{BBCCDd}} \\ \end{aligned}$$

where,

$$\hat{e}_{aa} = \hat{e}_{AABB} \, or \, \hat{e}_{AACC} \, or \, \hat{e}_{BBCC}$$
$$\hat{e}_{ad} = \hat{e}_{AABb} \, or \, \hat{e}_{AACc} \, or \, \hat{e}_{BBCc}$$
$$\hat{e}_{aaa} = \hat{e}_{AABBCC} \, or \, \hat{e}_{AABBDD} \, or \, \hat{e}_{AACCDD} \, or \, \hat{e}_{BBCCDD}$$
$$\hat{e}_{aad} = \hat{e}_{AABBCc} \, or \, \hat{e}_{AABBDd} \, or \, \hat{e}_{AABbCC} \, or \, \hat{e}_{AABbCC} \, or \, \hat{e}_{AABbDD} \, or \, \hat{e}_{BbCCDD} \, or \, \hat{e}_{AaBBCC} \, or \, \hat{e}_{AaBBDD} \, or \, \hat{e}_{AaCCDD}$$

The other epistatic components can also be estimated by the same manner. All QTL effects were performed by using the function lm() of R language (http://cran.r-project.org).