Introduction

Heterosis is an important way of increasing yield and improving quality in crops. Since 1914, when the term ‘heterosis’ was coined and first proposed by G. H. Shull (see Stuber et al., 1992), genetic research on heterosis has been an important issue. Since heterosis over environments is variable (Virmani et al., 1982; Young & Virmani, 1990) and environment-dependent (Knight, 1973), the genetic basis of heterosis per se is very complicated. Up to now a large number of methods addressing various aspects of heterosis have been developed based on biomathematical models (for a review see Schnell & Cockerham, 1992) and/or molecular techniques (Smith et al., 1990; Bernardo, 1992; Stuber et al., 1992; Zhang et al., 1994). Numerous authors have reported inconsistent relationships between yield heterosis and: (i) geographical distance (Moll et al., 1962); (ii) genealogical distance (Cowen & Fery, 1987); (iii) genetic divergence based on both quantitative traits and pedigree relationships (Cox & Murphy, 1990); and (iv) genetic distance based on molecular markers (Smith et al., 1990; Bernardo, 1992; Zhang et al., 1994). Because many traits of agronomic importance are quantitative in nature, controlled by polygenes and affected by environments, prediction of heterosis might not be very effective.

However, most of the approaches for predicting heterosis are based on information from the parents only. From a practical point of view, another key issue is whether we can predict the number of generations maintaining mid-parent or the better parent heterosis. An additional key issue is how to remove the influence of environmental effects and how to predict the performance of crosses (or maintenance of heterosis) in later generations from phenotypic observations on parents, the F1 and the F2 generations. Zhu (1993, 1997) and Zhu et al. (1993) proposed methods for directly predicting genotypic values and heterosis with an additive–dominance model. But an additive–dominance model may not fit all quantitatively inherited traits, because epistatic effects can also be an important component of genetic effects. Genetic models neglecting epistasis may result in biased information (Ketata et al., 1976). Schnell & Cockerham (1992) pointed out that heterosis without dominance arises from additive × additive epistasis alone. Zhu (1989) proposed an additive, dominance and additive × additive model (ADAA model) according to the principles of the general genetic model (Cockerham, 1980). By applying mixed-model approaches to analysing the ADAA model, variance components, heritability and genetic effects can be estimated (or predicted); then the genetic performance of parents and their crosses can be evaluated.

Based on the ADAA model with genotype × environment interaction, the objectives of present paper are: (i) to derive general formulae for predicting heterosis over mid-parent, heterosis over the better parent and deviation of heterosis caused by environment interaction of hybrids; (ii) to predict for how many generations heterosis of a cross could be maintained by using predicted genotypic values; and (iii) to show the importance of additive × additive effects in the utilization of heterosis for an F1 hybrid and its progeny. An example of yield traits in cotton is used to demonstrate the prediction of heterosis under different environments and its application in a breeding programme.

Genetic model

The assumptions for the ADAA model are: (i) normal diploid segregation; (ii) inbred parents in diallel mating are a random sample from a reference population; (iii) no additive × dominance epistasis and no dominance × dominance epistasis; (iv) linkage equilibrium. When genetic experiments are carried out with a randomized complete block design within multiple environments, a linear model can be written for the phenotypic mean value yhijkl of the kth mating type (k = 0 for parent, k = 1 for F1, k = 2 for F2) from lines i and j in the lth block within the hth environment:

where μ and Ehare the population mean and the environmental effect, respectively; Gijkis the total genetic main effect, and GEhijk is the total genotype × environment (GE) interaction; Bl(h) is the randomized complete block effect, Bl(h) ~ (0, σ2BΒ); ɛhijkl is the residual effect, ehijkl ~ (0, σ2ɛ).

If the experiment is conducted by a modified diallel mating with a set of parents and their F1s and F2s, Gijk and GEhijk can be partitioned into genetic components for different generations (Zhu, 1989):

for parent Pi (i = j, k = 0):

for F1ij from line i × line j (k = 1):

and for F2ij derived from selfing of F1ij (k = 2):

where Ai and Aj are cumulative additive effects from line i and line j, respectively, Ai and Aj ~ (0, σ2A); the cumulative dominance effect is Dij (or Dii, Djj) ~ (0, σ2D); the cumulative additive × additive effect is AAij (or AAii, AAjj) ~ (0, σ2AA); the cumulative additive × environment interaction is AEhi (or AEhj) ~ (0, σ2AE); the cumulative dominance × environment interaction is DEhij (or DEhii, DEhjj) ~ (0, σ2DE); the cumulative additive × additive × environment interaction is AAEhij (or AAEhii, AAEhjj) ~ (0, σ2AAE).

Prediction methods

Genetic model (1) can be expressed in the form of a mixed linear model:

where y is the vector of phenotype values with mean Xb and variance V; b is the vector of fixed effects; X is the known incidence matrix relating to the fixed effects; e u is the vector of the uth random factor, eu ~ (0, σ2uI); Uu is the known incidence matrix relating to the random vector eu.

Random effects in the genetic model are predictable without bias by the methods of linear unbiased prediction (LUP) (Zhu, 1992; Zhu & Weir, 1996a), and adjusted unbiased prediction (AUP) (Zhu, 1993; Zhu & Weir, 1996b). Because Monte Carlo simulation revealed that LUP could give prediction with unbiased mean but underestimated variance for random variables (Zhu & Weir, 1996b), the AUP method was suggested for predicting genetic effects:

αu is the prior value of the uth random factor. κu is an adjusted coefficient to ensure, êTu(α)êu(α)/(nu−1) = σ^2u, and set σ^2u=0 when σ^2u < 0.

When obtaining unbiased prediction of additive effects, dominance effects, additive × additive effects and their environment interaction effects, the genotypic values (μ + G) of parents and their offspring could be predicted without bias. Zhu (1993) proposed an approach to predict heterosis for an additive–dominance model and found that general heterosis over mid-parent for an F1 generation HM(F1) can be expressed as a function of dominance heterosis (ΔD) which is the difference between the heterozygote dominance effect and the average of the homozygote dominance effects (Dij – ½(Dii + Djj)). Expected heterosis over mid-parent for the Fn generation HM(Fn) can be expressed as a function of HM(F1) accordingly, i.e. HM(Fn) = (½)n–1 HM(F1) = (½)n–1ΔD. General heterosis over the better parent for the Fn generation HB(Fn) can also be expressed as a function of HM(F1) and parental genetic difference (ωG = |2(Ai– Aj) + (Dii– Djj)|), i.e. HB(Fn) = (½)n–1 HM(F1) – ½ ωG. When there exists GE interaction, formulae for predicting interaction heterosis are derived in a similar way (Zhu, 1997). Heterosis in a specific environment therefore consists of two components: general heterosis arising from genetic main effects, and interaction heterosis arising from GE interaction effects.

The total heterosis over mid-parent for Fn = HM(Fn) + HME(Fn) and the total heterosis over the better parent for Fn = HB(Fn) + HBE(Fn), where general heterosis is the performance of heterosis expected across different environments, whereas interaction heterosis is the deviation from general heterosis in a specific environment.

When the additive–dominance model is expanded by including an additive × additive effect, formulae for predicting general heterosis and interaction heterosis of each generation in different environments can be derived as follows.

General heterosis and interaction heterosis over mid-parent for the F1 are:

setting dominance heterosis ΔD = Dij − ½(Dii + Djj), epistasis heterosis ΔAA = AAij − ½(AAii + AAjj), dominance × environment heterosis ΔDE = DEhij − ½(DEhii + DEhjj) and epistasis × environment heterosis ΔAAE = AAEhij − ½(AAEhii + AAEhjj).

General heterosis and interaction heterosis over mid-parent for the F2 are:

Similarly, general heterosis and interaction heterosis over mid-parent for the Fn are:

The predicted results obtained from the above formulae are the real values of heterosis. Because different traits have different units of measurement, heterosis based on the population mean (μ) can be used for comparing among different traits:

It is thus clear that additive effects are not included in general heterosis over mid-parent of the different generations, but that ½ΔD is deducted for each succeeding generation. The amount of epistasis heterosis ΔAA, however, does not vary with generations. Similarly, interaction heterosis over mid-parent is not affected by additive × environment interaction effects. The Fn generation will be ½ΔDE less than the Fn–1 generation. 2ΔAAE remains constant for all generations.

If we set parent i as the better one, then the parental genetic difference ωG and parental interaction difference ωGE can be defined as:

which stand for the genetic difference and GE interaction difference between parent i and parent j, respectively.

General heterosis and interaction heterosis over the better parent for the F1 are:

General heterosis and interaction heterosis over the better parent for the F2 are:

Similarly, general heterosis and interaction heterosis over the better parent for the Fn are:

General heterosis and interaction heterosis over the better parent based on the population mean for Fn can be expressed as:

where δG = ωG/μ and δGE = ωGE/μ.

Thus it can be seen that the formulae for predicting heterosis over the better parent are different from those for predicting heterosis over mid-parent. The former is related not only to dominance effects and additive × additive effects, but also to additive effects (ωG including the difference of additive effects between the two parents). General heterosis over the better parent for each generation can be expressed as functions of ΔD, ΔAA and ωG, where ΔAA and ωG remain unchanged from F1 to Fn, and ΔD decreases by half with every selfing generation. In a similar way, interaction heterosis over the better parent is the functions of ΔDE, ΔAAE and ωGE, and the Fn generation will decrease by ½ΔDE from the Fn– 1 generation, but ΔAAE and ωGE will not change. It should be especially pointed out that a cross will show high heterosis in both F1 hybrids and their later generations when ΔAA is large, even if ΔD is small or less than zero.

If we set heterosis over the better parent based on the population mean HPB(Fn) equal to a prior value α (usually 0.10 or 0.05 for positive heterosis), the number of generations of a cross with HPB(Fn) = α is calculated by

when ΔD/(2μα – 4ΔAA +ωG) is larger than zero.

The jackknife resampling method (Miller, 1974) is appropriate for calculating the predictors of genetic merits and their standard errors for the t-test (Zhu, 1993; Zhu & Weir, 1996b). The null hypotheses of no difference between genotypic value and population mean (H0: G(Fn) = 0), and no heterosis (H0: H(Fn) = 0) can both be tested by t-tests.

A worked example

Yield data from an experiment on cotton (Gossypium hirsutum L.) conducted in two years is used as an example to illustrate the application of the methodology for predicting genotypic value and heterosis. The experiments with a randomized complete block design were carried out in the experimental station of Zhejiang Agricultural University in 1992 and 1993. There were three blocks in each year and 50 plots in each block. Five female parents, (1) A226, (2) A160, (3) A17, (4) Lumian 6 and (5) Zhongmian 12, and five male parents, (6) Zhongmian 13, (7) Xuzhou 184, (8) Shimian 2, (9) 4305 and (10) 4318, were used to produce 20 crosses of F1s and F2s. Cotton yield (kg/plot), boll number per plant, boll size (g), and lint percentage (%) were investigated. Genetic effects and genotype × environment interaction effects were predicted by the AUP method (Zhu, 1993; Zhu & Weir, 1996b).

The average genotypic values (μ + G) and general heterosis of the 20 crosses are listed in Table 1 for yield traits of F1s and F2s. The totals of genotypic values and general heterosis were less in the F2 than in the F1. The F1 genotypic value of each trait was not significantly different from the population mean. As far as cotton yield was concerned, general heterosis of the F1 and F2 over the mid-parent was 26.9% and 16.2%, respectively, and reached significance. On the other hand, general heterosis of the F1 over the better parent was 13.5% and significant at the 1% level, and decreased to as low as 2.8% in the F2. Boll number per plant showed significant positive HPM and nonsignificant negative HPB; lint percentage had highly significant negative HPB but nonsignificant HPM; both HPM and HPB for boll size were not significant at the 5% level.

Table 1 Predicted genotypic values and general heterosis for yield traits†
Full size table

Predicted interaction heterosis for the two years is presented in Table 2. Heterosis arising from GE interaction was found to varying degrees for different traits. Under two environments there was positive HPME and negative HPBE for both cotton yield and boll number per plant; HPME and HPBE were positive for boll size but negative for lint percentage with the exception of HPME (F2) in 1992.

Table 2 Predicted genotype × environment values and interaction heterosis for yield traits in 1992 and 1993
Full size table

It is important in practice to understand the performance of heterosis of each specific cross. Taking cotton yield as an example, the results of six crosses are given in Table 3. General heterosis over the mid-parent of these crosses, except for cross 4 × 10, was significant (P < 5%) or highly significant (P < 1%). Significant HPB of the F1 was 40.4% for cross 5 × 9 and 24.4% for cross 5 × 10, and decreased to 9.8% and 6.6% in the F2, respectively. This was because both crosses had large ΔD and small ΔAA (cross 5 × 9 having a negative ΔAA and cross 5 × 10 having only 1.9% of ΔAA). If the positive population heterosis over the better parent were set at 5%, the expected number of generations for 5 × 9 and 5 × 10 would be larger than two. However, there existed large negative heterosis deviations caused by GE interaction. From the sum of HPB and HPBE, the final heterosis over the better parent of 5 × 9 in the F2 was −3.1% in 1992 and 3.4% in 1993; similarly, that of 5 × 10 in the F2 was 1.9% in 1992 and –6.4% in 1993. Cross 1 × 7 also shared large ΔD and small ΔAA, and its expected number of generations was only one. HPBE (F1) and HPBE (F2) of cross 1 × 7 were negative in 1992 but positive in 1993. Because both ΔD and ΔAA were not large, cross 4 × 10 can not be used even for the F1 in practice. Heterosis over the better parent of cross 4 × 10 was negative so that zero was used as its expected number of generations.

Table 3 Genotypic values, heterosis and their interaction with environment for cotton yield of six crosses
Full size table

We did not list the expected number of generations for crosses 2 × 9 and 3 × 9 for the following reasons. Both crosses were rather special, and it was necessary to analyse them in detail. For cross 3 × 9, HPB of the nth generation was calculated as

. It is clear that as n becomes large enough,

approaches zero, and even under this circumstance the cross should still have heterosis of 15.7% over the better parent because of the large value for Δ^AA. Hence, when we predict the expected number of generations for this cross, the lower limit should be approximately larger than 16%. Meanwhile, it was indicated that additive × additive effects, which are the heritable genetic component, play an important role in hybrid breeding. In addition, this cross also maintained a large HPBE value in the F2, which was 4.6% in 1992 and −18.4% in 1993. Therefore this cross could make full use of yield potential in suitable environments.

Cross 2 × 9 showed the same trend as cross 3 × 9. HPB (F1) and HPB (F2) of cross 2 × 9 were 27.2% and 22.7%, respectively, and deviated from the population mean at P < 0.05. In further generations, ΔD would drop gradually and the smallest value (lower limit) of HPB would be

. Because cross 2 × 9 had similar ΔAA and smaller ΔD and ωG compared with cross 3 × 9, the lower limit of HPB of cross 2 × 9 was higher than that of cross 3 × 9. Moreover, the small interaction heterosis of cross 2 × 9 in both 1992 and 1993 indicated that this cross was hardly influenced by environments. From the above analysis we know that a cross might attain a high level of heterosis when ΔAA is large enough even though ΔD is small.

Discussion

F1 hybrids in maize, rice and sorghum have been successfully developed and cultivated. However, primarily because of the difficulty of producing F1 seed economically, utilization of F1 heterosis in many other crops has been limited. Significant heterosis of agronomic traits in the F2 generation has been reported in cotton (Meredith, 1990), winter wheat (Cox & Murphy, 1990), rapeseed (Engqvist & Becker, 1991), sesame (Ding et al., 1987), etc. Meredith (1990) indicated that F2 hybrids had the genetic potential for increasing cotton (Gossypium hirsutum) yields and fibre quality. When compared with the parents in spring oilseed rape (Brassica napus L.), the F2 was 11% higher in yield and earlier in flowering time; the descent lines of the F6 derived from random single seeds had an 8% lower yield and were later flowering (Engqvist & Becker, 1991).

The general formulae proposed in this paper might be useful for predicting heterosis in the offspring of crop hybrids. If we define the lowest positive value of heterosis over the better parent as α (e.g. 0.10 or 0.05), it is convenient to calculate the number of generations for which offspring of hybrids could maintain 10% or 5% heterosis in practical production. The predicted number of generations has close relationships with ΔD, ΔAA and ωG. The positive heterosis over the better parent will be maintained longer if ΔD and ΔAA are larger and ωG smaller. In order to use hybrids for longer, the difference between the two parents should not be too large (a small ωG might cause decreased heterosis of F1 hybrids). When ΔD/(2μα – 4ΔAA + ωG) reaches 1, 2, 4 and 8, respectively, the number of generations for positive heterosis over the better parent will be 2, 3, 4 and 5 accordingly.

Because GE interaction of heterosis is widely observed, it is reasonable to define the total heterosis in a specific environment as the sum of HPB (or HPM) and HPBE (or HPME). Here, heterosis arising from genetic main effects allows estimation of the stability of heterosis across different environments; heterosis arising from GE interaction reflects the deviation of heterosis in specific environments. In general, the genotypic value and heterosis of yield traits will show a declining trend with the increase of generations, but different characters behave differently. When both ΔD and ΔAA have large positive values, a cross (e.g. cross 3 × 9) could express high heterosis and maintain it for several generations. When ΔD is small but ΔAA is large enough, heterosis of a cross (e.g. 2 × 9) could remain relatively stable over generations; and particularly, when ΔD is less than zero, the upper limit of total heterosis could reach (1/μ)[2(ΔAA + ΔAAE) – ½(ωG + ωGE)]. When ΔD is large enough but ΔAA is small, these kinds of crosses (e.g. crosses 5 × 9 and 5 × 10) could be used mainly in the F1, and their offspring would have little genetic merit because heterosis would decline quickly in further generations. Hybrids with small values for both ΔD and ΔAA (e.g. cross 4 × 10) are not useful in practice. Furthermore, the above analysis indicates that maintainable heterosis cannot be caused by dominance, but rather results from epistatic effects like additive × additive effects. The additive × additive effects play an important role in the utilization of heterosis for F1 hybrids, and particularly for their later generations.

When the ADAA model is employed to predict heterosis for agronomic traits of crop hybrids, the formulae for predicting heterosis could be extended or reduced with flexibility depending on practical circumstances. If a genetic experiment is conducted only in one environment, and there is no GE interaction, the analysis of interaction heterosis is not necessary. If we do not consider epistatic effects, the prediction formulae for heterosis are the same as those proposed by Zhu (1993, 1997) based on an additive–dominance model. In an analysis of heterosis, if we do not know prior genetic information for some of the traits studied, the ADAA model can be used for an initial analysis. When the variance of additive × additive effects is not significant, the data can then be re-analysed by an AD model instead of the ADAA model. The ADAA model can give unbiased estimates of variance components even if these additive × additive effects do not exist (Zhu, 1992). In comparison with other genetic components, additive × dominance and dominance × dominance effects are very complicated, generally negligible and will decline quickly with increasing generations, so the ADAA model does not include additive × dominance epistasis and dominance × dominance epistasis. If we need to consider AD and DD interactions, the ADAA model should be expanded according to the principles of general genetic models (Cockherham, 1980), and the formulae for predicting heterosis should be derived accordingly. Because all kinds of gene effects are predictable by the AUP or LUP methods (Zhu, 1992, 1993; Zhu & Weir, 1996b), the predicted values of heterosis are also unbiased in theory. In addition, it is effective and feasible for the ADAA model to deal with balanced or unbalanced data from a diallel mating design.