Multivariate analyses to display interactions between environment and general or specific combining ability in hybrid crops.
Yan and Hunt (2002) used the site regression (SREG2) model to analyze a diallel mating dataset and identify an ideal tester. The biplot of the first two principal components (PC) of the SREG2 model displays the GCA of lines or testers (depending on the type of centering of the two-way lines x testers dataset) as well as the SCA of the line x tester interaction. In the SREG2 biplot, the ideal tester should be the one that has the longest PC1 vector (high power to discriminate lines) and zero projection on the average tester ordinate (more representative of all testers) (Yan and Hunt, 2002). Narro et al. (2003) pointed out that, since the SREG2 model contains in its multiplicative components both the main effects of the lines as well as the line x tester interaction, the visualization of the SCA of each line with each tester is often not clear and is imprecise. Hence, Narro et al. (2003) proposed the use of the additive main effects and multiplicative interaction (AMMI2) biplot, which uses only the line x tester interaction (SCA), to estimate the interaction parameters and to obtain a precise and clear ranking of the lines for their SCA with each tester.
Both the SREG2 (Yan and Hunt, 2002) and AMMI2 (Narro el al., 2003) biplots of lines x testers display the GCA and SCA of the lines averaged across environments. However, hybrid relative performance is not only a consequence of gamete segregation and recombination but also of the environmental effect where cultivars were evaluated (Narro et al., 2003). The G x E interactions result when there is a change in the relative performance of genotypes when they are tested in different environments and have the potential to influence the nature and magnitude of the selection response achieved by a breeding program (Cooper and DeLacy, 1994; Kang, 1998). Some understanding of the nature of the G x E interactions is needed to accommodate their effects through appropriate selection strategies aimed at exploiting broad and/or specific genotype-adaptation patterns (Basford and Cooper, 1998, and references therein). Similarly, consideration of the impact of G x E interactions on GCA and SCA is central to assess the relative merits of different inbred lines potentially used as testers in a breeding program targeting broad and/or specific adaptation.
The variance components estimated from multienvironment trials can be used to judge the relative magnitude of G and G x E interaction and to predict the response to selection (Cooper and DeLacy, 1994). In hybrid crops, the G component of variance can be partitioned into its components, namely female line (F), male line (M) and F x M interaction, allowing the determination of the relative size of the F x E, M x E and F x M x E interaction components of variance as well.
If F x E, M x E and/or F x M x E interactions explain a large portion of the variance in a half-sib mating design study (i.e., GC[A.sub.F] x E, GC[A.sub.M] x E and/or SCA x E interactions are important), then it would be useful to examine the three-way matrix of F x M x E means. Three-mode PCA (Tucker, 1966; Kroonenberg, 1983), an extension of the standard PCA to handle such three-way datasets, has been used for handling genotypes, environments, and attributes simultaneously (Kroonenberg and Basford, 1989; Basford et al., 1990; Chapman et al., 1997; de la Vega et al., 2002; Bertero et al., 2004), allowing an examination of the relationships between genotypes and attributes associated with specific patterns of environmental variability. In the same way, we propose that three-mode PCA can be used to display the relationships between female and male inbred lines associated with predictable or unpredictable environmental contrasts. If the F, M, F x M interaction and E effects are removed from the three-way dataset and F x E, M x E and F x M x E interactions are retained, then the graphical output of three-mode PCA will display the nature of the pattern-rich GCA x E and SCA x E interactions, complementing the analytical tools demonstrated by Yan and Hunt (2002) and Narro et al. (2003). In this analytical strategy (Strategy 1), SGREG2 (Yan and Hunt, 2002) could be used to display GC[A.sub.F] + GC[A.sub.M] + SCA (i.e., F, M and F x M interaction effects, respectively) and three-mode PCA to display GC[A.sub.F] x E + GC[A.sub.M] x E + SCA x E.
In Strategy 1, two- and three-mode multivariate analyses are used to analyze the sources of variation across environments and their interactions with the environments, respectively. Alternatively (Strategy 2), two-mode PCA on environment-centered and normalized (Fox and Rosielle, 1982; Cooper and DeLacy, 1994) two-way F (M) x E tables for hybrid yield can be used to display GC[A.sub.F] + GC[A.sub.F] x E + GC[A.sub.M] + GC[A.sub.M] x E effects, while three-mode PCA is used to display the SCA + SCA x E interaction effect. In this case, the F, M, E, F x E, and M x E interaction effects should be removed from the three-way dataset and F x M and F x M x E interactions should be retained. Together, the two strategies account for all components of the G and G x E interaction variances in a half-sib mating design analysis.
The main objective of this paper is to critically examine the use of two-mode PCA on environment-centered and normalized F (M) X E tables and three-mode PCA on F x M x E tables to graphically display the pattern rich variability associated with GCA, SCA and their interactions with the trial environments in a half-sib mating design study. The proposed methods (including Strategies 1 and 2) are evaluated together with the across-environments SREG2 (Yan and Hunt, 2002) and AMMI2 (Narro et al., 2003) biplots on the basis of their explanation of the total G + G x E variation of the system under study. The paper describes the methods completely and compares the methods for their suitability to detect elite tester lines for a breeding program that aims to target both broad and specific (i.e., regional) adaptation. Although the case study consists in a sunflower mating design, the methods described, as well as those proposed by Yan and Hunt (2002) and Narro et al. (2003), are applicable to any hybrid crop.
MATERIALS AND METHODS
A set of 16 sunflower single-cross hybrids (Table 1) was evaluated in 11 northern, central, and southern locations (environments, E) of Argentina (Table 2) during the 20042005 season. The hybrids were obtained from the North Carolina Design II (NCII) (Comstock and Robinson, 1952; Kearsey, 1965) between four female (one-headed, PET1 cytoplasmic male-sterile) inbred lines and four male (branched, fertility-restorer) inbred lines. All inbred lines used in this study were developed by the sunflower breeding program of Advanta Semillas SAIC at Venado Tuerto, Argentina, and constitute a representative sample of the elite core germplasm of this program, some of them being commercial and/or tester lines. They represent a wide range of genetic diversity according to their genetic origin and to molecular-marker fingerprints (A. Leon, Advanta Semillas SAIC, Balcarce, unpublished data) and were selected from the Advanta Semillas testing program on the basis of contrasting characteristics, such as maturity and patterns of relative performance across environments for oil yield. The 16 resultant sunflower hybrids exhibit differential adaptation to the different sunflower growing regions of Argentina. Since the Advanta Semillas line identifiers are protected under a confidentiality agreement, the lines were recoded for publication. However, seed of each of the 16 hybrids may be requested from the first author.
Most trials were located on-farm, except Venado Tuerto and Balcarce, which are Advanta Semillas research stations. The locations range from 28 to 38[degrees]S and are distributed between subtropical (northern) and temperate (central and south) sunflower growing environments.
Planting took place within the normal sowing window at each location (i.e., about October to mid-November in central and southern locations and about August to mid-September in the northern locations). In each environment, a 4 x 4 square lattice design with three replicates was used to test the 16 hybrids. The trials were over-planted and thinned to 47,600 plants [ha.sup.-1]. A plot size of four rows x 6 m and interrow spacing of 0.70 m was used. All trials were rain-fed and nutrient deficiencies were prevented with fertilization when necessary. Weeds and insect pests were controlled chemically. Plot data of grain yield were determined by hand harvesting of 3.99 [m.sup.2] (one central row, discarding the border plants). Grain oil concentration was determined on a 10 g oven-dried achene sample by nuclear magnetic resonance (Granlund and Zimmerman, 1975). Oil yield was calculated as the product of grain yield and grain oil concentration.
Analysis of Variance Components
A mixed model analysis was conducted to examine partitions of the G and G x E interaction components of variance for grain yield, oil content, and oil yield. The phenotypic observation [y.sub.ijkmn] on hybrid derived from the cross between female inbred line i and male inbred line j in incomplete block n of replicate m of environment k was modeled as:
 [y.sub.ijkmn] = [mu] + [e.sub.k] + [(r/e).sub.km] + [(b/r/e).sub.kmn] + [f.sub.i] + [m.sub.j] + [(fm).sub.ij] + [(fe).sub.ik] + [(me).sub.jk] + [(fme).sub.ijk] + [[epsilon].sub.ijkmn]
where [mu] is the grand mean; [e.sub.k] the fixed effect of the environment k; [(r/e).sub.km] the random effect of the replicate m nested within the environment k and is ~NID(0, [[sigma].sup.2.sub.r]); [(b/r/e).sub.kmn] the random effect of the incomplete block n nested within the replicate m of environment k and is ~NID(0, [[sigma].sup.2.sub.b]); [f.sub.i] the random effect of female inbred line i and is ~NID(0, [[sigma].sup.2.sub.b]); (m.sub.j) the random effect of the male inbred line j and is ~NID(0, [[sigma].sup.2.sub.m]); [(fm).sub.ij] the random effect of the interaction between female i and male j and is ~NID(0, [[sigma].sup.2.sub.fm]); [(fe).sub.ik] the random effect of the interaction between the female i and environment k and is ~NID(0, [[sigma].sup.2.sub.fe]); [(me).sub.jk] the random effect of the interaction between male j and environment k and is ~NID(0, [[sigma].sup.2.sub.me]); [(fme).sub.ijk] the random interaction effect for female i, male j, and environment k and is ~NID(0, [[sigma].sup.2.sub.fme]); and [[epsilon].sub.ijkm] is the random residual effect for the combination between female i and male j in replicate m of environment k (experimental error) and is ~NID(0, [[sigma].sup.2.sub.[epsilon]]). Restricted maximum likelihood (REML) methods (Patterson and Thompson, 1975) in GenStat v8.1 was used to estimate the variance components of the random terms in the model and their standard errors for grain yield, oil content, and oil yield.
Given that oil yield is the most important economic trait for sunflower, we used this attribute to compare different multivariate approaches aimed at understanding patterns of GCA, SCA, and their interactions with the trial environments. The best linear unbiased predictors (BLUPs) (Robinson, 1991) of the F x M, F x E, M x E, and F x M x E means for oil yield were calculated from REML by the mixed model of Eq.  and used further to construct the matrices for two- and three-mode multivariate analyses.
Two-Mode PCA-Derived Biplots of GCA and SCA across Environments
The 4 x 4 (F x M) two-way table of across-environments oil yield BLUPs was either female- and male-centered (i.e., centered by subtraction of female (male) mean and normalized by division of the remainder by the within-female (male) standard deviation (Fox and Rosielle, 1982; Cooper and DeLacy, 1994; Yan and Hunt, 2002) and double-centered [i.e., centered by subtracting the across-females male means and the across-males female means, and adding the overall mean, as in the AMMI model (residuals from additivity; Gauch, 1988; Narro et al., 2003)] to graphically display the across-environments GC[A.sub.F] + SCA, GC[A.sub.M] + SCA, and SCA, respectively (Yan and Hunt, 2002; Narro et al., 2003). The PCs of the squared Euclidean distance matrices of oil yield were estimated by a singular value decomposition procedure and biplots of the first two PCs were constructed from these analyses (Gabriel, 1971) by GenStat 8.1.
Two-Mode PCA-Derived Biplots of GCA + GCA x E Interaction
The two-way tables of F x E (4 x 11) and M x E (4 x 11) oil yield BLUPs were centered by subtraction of environment mean from columns and normalized by division of the remainder by the within-environment standard deviation (Fox and Rosielle, 1982; Cooper and DeLacy, 1994). The PCs of the squared Euclidean distance matrices of oil yield were estimated by a singular value decomposition procedure and biplots of the first two PCs were constructed from these analyses (Gabriel, 1971).
This procedure derives components, i.e., linear combinations of the levels of the modes, for each of the three modes (say, P, Q, and R components for female lines, male lines, and environments, respectively). It can be assumed that these components together contain the only relevant systematic variation of the three-way array dataset. The components of the three modes can be labeled on the basis of the patterns shown by the levels with high loadings on such components. In this model, each mode is allowed to have a different number of components. The number of components for each mode needs to be simultaneously determined for all modes. Therefore, several solutions have to be inspected to come to an adequate description of a dataset (Kroonenberg, 1983, chap. 2; Kroonenberg and Basford, 1989).
A three-way array of order P by Q by R (the core array) contains the weights assigned to each of the combinations of the components for the three modes. The model is written as (Kroonenberg, 1983, chap.2; Kroonenberg and Basford, 1989):
 [x.sub.ijk] = [P.summation over p=1][Q.summation over q=1][R.summation over r=1] [a.sub.ip][b.sub.jq][c.sub.kr][g.sub.pqr] + [e.sub.ijk]
where [a.sub.ip] represents the coefficient for female i in the female component p (p = 1, ..., P), [b.sub.jq] the coefficient for male j in the male component q (q = 1, ..., Q), and [c.sub.kr] the coefficient for environment k in the environmental component r (r = 1, ..., R). The term [g.sub.pqr] indicates the joint weight for the pth component of the male mode, the qth component of the female mode, and the rth component of the environment mode, and its squared value indicates the variation explained for that combination of components (Kroonenberg, 1983, chap. 2; Kroonenberg and Basford, 1989).
The component loadings of the female lines can be investigated jointly with the component loadings of the male lines, by projecting them together in one space, as it then becomes possible to display the interaction between female and male inbred lines. The plot of the common space is called the joint biplot, a variant of Gabriel's (1971) biplot (Kroonenberg, 1983; Basford et al., 1990) and is constructed from the core matrix as follows. For each component r of the environment mode, the female components and the male components are scaled by dividing the core slice associated with that component r between them (using singular value decomposition) and weighting the scaled female and male components by the relative number of elements in the modes to make the distances comparable. For the rationale behind this construction and more detailed discussion, see Kroonenberg and De Leeuw (1977).
The treatment of the raw data, i.e., centering and normalization of the F x M x E input data arrays, is a central decision to obtain the "best" analysis for a particular dataset (Kroonenberg, 1983, chap. 6). In three-way data, different types of centering and normalization lead to different solutions. There are also many more ways of centering and normalizing to choose from, compared with two-way data. Kroonenberg (1983, chap. 6) and Harshman and Lundy (1984, p. 225-253) give detailed discussions on this issue.
Centering and Normalization to Retain GCA x E + SCA x E Interactions in the Three-Way Table (3-Mode PCA1)
The two-way 16 x 11 (hybrids X environments) array of BLUPs for oil yield was double-centered by subtracting the across-hybrids environment means and the across-environment hybrid means, and adding the overall mean (residual from additivity, Gabriel, 1978), as in the AMMI model (Gauch, 1988) for two-way tables. The purpose of centering was to eliminate from the analysis those means that should not be modeled multiplicatively. To understand its implications, three-way data can be written as if they were generated by a three-way analysis of variance as that detailed in Eq. . The centering applied leads to
 [y".sub.ijk] = [(fe).sub.ik] + [(me).sub.jk] + [(fme).sub.ijk]
where F, M, and E main effects are removed from the data as well as the F x M interaction and the F x E, M x E, and F x M x E interactions remain.
Normalization, which is also named scaling or standardization, is the process of equalizing sums of squares. This process is necessary since environments showed different oil yield means. Normalization within-environments (i.e., [r.sub.ijk] = [y".sub.ijk/[S.sub...k) was used because it causes each environment to have a mean of zero and a standard deviation of one, this being the most appropriate treatment for reducing the influence of environmental main effects (Cooper and DeLacy, 1994). The standardized residuals from additivity were further arranged in a three-way 4 x 4 x 11 (F x M x E) matrix and three-mode PCA was applied by the program TUCKALS3 (Kroonenberg, 1994).
Centering and Normalization to Retain SCA + SCA x E Interaction in the Three-Way Table (3-Mode PCA2)
The three-way (F x M X E) table of oil yield BLUPs was arranged with the females as rows, the males as columns, and the environments as slices. The three-way dataset was centered within environment by subtracting both the across-females male means and the across-males female means and adding the overall environment mean (Kroonenberg, 1983, chap. 6), which leads to
 [y".sub.ijk] = [(fm).sub.ij] + [(fme).sub.ijk]
where F, M, and E main effects are removed from the data as well as the F x E and M x E interactions, and the F x M and F x M x E interactions remain.
With the aim of eliminating the environmental effects, normalization across slices was used (Kroonenberg, 1983, chap. 6), which in this type of analysis implies the division of the double-centered data by the within-environment standard deviation over all females and males (i.e., [r.sub.ijk] = [y".sub.ijk]/[S.sub...k]; Kroonenberg and Basford, 1989; Basford et al., 1990). Three-mode PCA was applied to the 4 x 4 x 11 (F x M X E)double-centered and normalized BLUPs residual array for oil yield using the program TUCKALS3 (Kroonenberg, 1994).
RESULTS AND DISCUSSION
Analysis of Variance
Large among-locations variation for oil yield and its components (grain yield and oil concentration) was observed (Table 2), with, in general terms, lower yields in the northern environments. The variance component analysis showed that, together, the F x E, M x E, and F X M x E interactions components of variance (GCA x E + SCA x E) explained 2.82 and 4.46 times of the variation for oil yield and grain yield, respectively, when compared with variation explained by the F, M, and F x M variance components (GCA + SCA) (Table 3). This observation strongly suggests that GCA x E and SCA x E interactions for these traits should not be ignored when selecting new testers for a breeding program aimed at releasing hybrids to this complex target population of environments. The ratio of total G x E interaction to G components of variance for oil concentration was 0.35 (Table 3), confirming the relatively high heritability previously reported for this trait (Chapman and de la Vega, 2002). This also suggests that GCA and SCA for oil concentration could be studied averaged across environments. However, in this study, oil yield is the focus to compare different multivariate approaches aimed at understanding patterns of GCA, SCA, and their interactions with the trial environments
GCA and SCA Averaged across Environments SREG2 Model
The male-centered and normalized SREG2 biplot (Fig. 1a) explains 98% of the GC[A.sub.F] (since the input matrix was male-centered) and the SCA between males and females across environments and can be interpreted as follows (Kroonenberg, 1997; Yan and Hunt, 2002). Females that are close together show similar patterns of combining ability across males, while adjacent males are similar in the way they discriminate among females. For any particular male, females can be compared by projecting a perpendicular from the female symbols to the male vector, i.e., females that are further along in the positive direction of the male vector show a higher yield with this male and vice versa. The cosine of the angle between any two male vectors approximates their correlation with equality if the overall fit is perfect. Thus, acute angles between any two vectors indicate positive associations, i.e., they influence the female relative performance in a similar manner; 90[degrees] angles indicate no association, and angles greater than 90[degrees] indicate negative associations. In these biplots, if the objective was to select a tester showing a high GCA across all environments, the ideal tester line should be that which its perpendicular projection on the average environment vector (Yan, 2001) intercepts it at greater distance from the origin. If subregion ("specific") adaptation is likely to be exploited in selection, the environment vector to be considered will be the average across all environments belonging to the same subregion. The female-centered and normalized SREG2 biplot of the first two components for oil yield explains 97% of the GCAM (since the input matrix was female-centered) and the SCA between males and females across environments (Fig. 1b) and can be interpreted in analogous way to the biplot of Fig. 1a.
[FIGURE 1 OMITTED]
According to Yan and Hunt (2002), the ideal tester vector should have a large PC1 component, to discriminate among lines, and zero ordinate on PC2, to be a good representative of all testers. However, in Fig. 1a, which retains the GCAF and the SCA, the average male tester vector is almost parallel to the PC2 and discriminates the highest yielding females (A2 and A3) on the top half of the biplot and the lowest yielding females (A4 and A1) on the bottom half of the biplot. The PC1 (73% of the variation) is dominated by the SCA among males and females, showing that there is no single female that combines positively with all males and vice versa, which complicates the selection of an appropriate male tester to test new females across environments. This pattern is consistent with the finding that the female component of variance was 0 (Table 3).
In Fig. 1b, which retains the GCAM and the SCA, the ideal female tester is apparently A3, which is parallel to the PC1 and shows an almost zero ordinate on PC1 (Yan and Hunt, 2002). The PC1 is related to the GCAM across environments, with the highest yielding males at the right hand side of the diagram. The PC2 is dominated by the strong SCA between A2 and R3.
The AMMI2 biplot of the first two components for oil yield explains 99% of the SCA between males and females across environments (Fig. 1c) and can be interpreted as follows (Narro et al., 2003). A positive perpendicular projection of a male point on an individual female vector indicates that both lines combine well (positive SCA). The greater the distance from the origin to the intersection of a male projection on a female vector, the better this male combines with this female line. A male projection around the origin of a female vector indicates nearly null SCA, while a male negative projection on an individual female vector indicates that these lines do not combine well in relative terms (negative SCA).
The patterns of SCA across environments revealed by Fig. 1c indicate for female A2: a strong positive SCA with the male R3, a positive SCA with R2, a negative SCA with R4, and a strong negative SCA with R1. Since the main effects of the lines (GCA) have been removed, this pattern does not directly reflect yield rankings. In this particular case, the hybrid A2/R4 (negative SCA) still had a greater average oil yield across environments than did hybrid A2/R2 (positive SCA) (Table 1). However, this analysis allows the detection of SCA across environments, which is highly useful information for a breeder, especially when selecting parents for new breeding populations. For example, male lines of similar SCA pattern for oil yield with an individual elite female (e.g., R1 and R4 show positive SCA with A3, Fig. 1c), but that are genetically distant, could be combined in experimental crosses to deliver inbred lines that bring better opportunities for yield improvement in hybrid combinations with this or similar female. In the genotype--environment system under study, the females A1 and A2 showed a positive SCA with the males R3 and R2, while the females A3 and A4 showed a positive SCA with the males R1 and R4 across environments (Fig. 1c).
GCA x E and SCA x E Interactions (Three-Mode PCA1)
The three-mode model with 2 x 2 x 3 components for F x M x E, respectively, was considered adequate for fitting the data ([r.sup.2] = 0.51, Table 4), on the basis of informal judgments of the small increases in r2 when more dimensions were used and on the increased difficulty of interpretation (Kroonenberg, 1983). In this model, the two components of the female mode accounted for 40 and 11% of the variation; the two components of the male mode accounted for 31 and 20%; and the three components for the environments accounted for 35, 10, and 5% of the variation (Table 4). Not all females, males, and environments were fitted equally well by the model. For example, the model only explained 13% of the interactions of the female line A4 compared to the overall fit of 51% (Table 4). While environments TA and VM were less well fitted by the model, the selected model accounted for more than 50% of the contribution of BA, GE QQ, SB, VO, and VT to the total GCA x E and SCA x E interactions (Table 4).
The components of the female and male modes do not have obvious interpretations, and the lower-dimensional representations primarily serve the purpose of data reduction (Kroonenberg and Basford, 1989). The environment components reveal some contrasts between regions in terms of their genotype discrimination effects, together with some unpredictable G x E interactions. The first environmental component contrasts two northern environments (SB and VO), one central environment (VT) and three southern environments (BA, CM, and TA), with positive scores, versus the central environments QQ, GE and DX, with negative scores (Table 4). Although this environmental component reflects some unpredictable (i.e., within-subregion) G x E interactions, it could be labeled for the purposes of this example as "Central versus North and South." The second environmental component reflects the portion of the G x E interaction that separates CM, QQ, VM, and SB, with negative scores, from the northern environments MA and VO, with positive scores. This environment component could be labeled as "North versus Central and South."
Northern and central regions of Argentina are different subregions for sunflower, i.e., their differences in terms of genotypic discrimination for oil yield are large and repeatable over years (de la Vega et al., 2001). The northern region shows a higher variation across years than the central region in terms of the environmental factors affecting sunflower growth and development (Chapman and de la Vega, 2002). Consistent with this observation, the ratio of the genotype x year and genotype x year x location interaction effects to the genotypic effect within-region is higher in the northern region (de la Vega and Chapman, 2006). In the central region, 1 yr of trials can adequately represent the target environment if sufficient locations are sampled (de la Vega and Chapman, 2006). Conversely, in the northern region different years can show opposite discrimination effects on northern-adapted hybrids differing in maturity. According to the results of the sunflower private trials of Advanta Semillas SAIC (unpublished data), the 2004-2005 season could be considered representative of the northern environments that promote a relative improvement of early-maturity hybrids.
The joint biplots of female and male inbred lines for the first and the second environment components will be used to investigate the relationships between females and males (GCA and SCA) associated with the two specific patterns of genotype discrimination, namely "Central versus North and South" and "North versus Central and South." The third environmental component reflects unpredictable G x E interactions (Table 4). The proportion of variability explained by this component (5%) is too small to justify discussion.
First Environment Component: Interpreting the Contrasting Effects of Central versus Northern and Southern Regions on GCA and SCA
The component weights for the first and second axes of the joint biplot of female and male inbred lines in the first environmental component were 0.29 and 0.07, respectively (Fig. 2a). This bidimensional joint biplot displays those aspects of the relationships between female and male lines that are influenced by the differences between northern and southern versus central environments, after the genotype and environment effects have been removed. These effects can be selected in breeding for specific adaptation to the central sub-region or discarded in breeding for broad adaptation to the undivided target region.
In this bidimensional joint biplot (Fig. 2a), males were represented by points and females by vectors from the origin. The F x M x E interactions associated with this environmental contrast can be interpreted as a product of any combination of the scores of the three modes. For example, male R2 will have a positive product with female A2, since its perpendicular projection on the female vector intercepts it in positive direction. The product of the hybrid combination and the environment score will indicate the strength and the sign of the G x E interaction effect of each hybrid-environment combination. This means that hybrid A2/R2 (positive score) will show a strong positive G x E interaction effect in VT (score 0.99), a weak positive effect in TA (score 0.32), a weak negative effect in DX (score -0.21), and a strong negative G x E interaction effect in QQ (score -0.75). Hybrid A2/R4 (negative score, since the R4 perpendicular projection on A2 vector intercepts it in negative direction) will show an opposite pattern of G x E interaction effects.
In this study (Fig. 2a), the male inbred lines showed a wider pattern of interactions than the female lines, since they occupy a wider range of the Euclidean space. While all females can interact positively (negatively) with some males (e.g., all females have a positive G x E interaction effect with R2 in VO and a negative effect with R4 in the same environment), all males together cannot interact in the same direction with any female.
A practical application of the above observation is related to the definition of selection strategies for broad or specific adaptation. If division of the breeding program's target population of environments in subregions is appropriate and sources of specific adaptation to those subregions have to be identified, they are more likely to be found within the male germplasm base of the program than within the female one. This is assuming that such germplasm bases are adequately represented by the sampled lines. Regarding the selection of testers, for example, a male interacting positively with all females in the southern region (e.g., R2) could be used as a tester for this subregion, while a male interacting positively with all females in the central region (e.g., R4) could be used as a tester for this contrasting subregion. The male R3 showed positive and negative G x E interaction effects in the southern region with A3 and Al, respectively, while R1 is located around the origin of the joint biplot, showing a weak pattern of responses in this environmental component.
Second Environment Component: Interpreting the Contrasting Effects of Northern versus Central and Southern Regions on GCA and SCA
The component weights for the first and the second axes of the joint biplot of female and male inbred lines in the second environment component were 0.10 and 0.01, respectively. Therefore, major effects of this particular environmental contrast can be described in a single dimension, corresponding to the first component of the joint biplot (Fig. 2b). When one dimension effectively explains most of variability, the joint biplots collapse into a single line, in which it is possible to include the component loadings of the environment mode as well. In such a case, a product term to compare scores may be calculated as a product of any combination of the scores of the three modes, e.g., Basford et al. (1990) and Chapman et al. (1997), for genotypes, environment, and attributes. From Fig. 2b, female A2 (score -1.37), for example, will show a negative G x E interaction effect with male R1 (score -0.15) in CM (score -0.44) (i.e., -1.37 x -0.15 x -0.44 = -0.09) and a positive G x E interaction effect in VO (score 0.74).
All females have a negative (or near zero) score in this joint biplot, while the males clearly split into two groups, showing, as in the case of the first environment component, a higher variability in terms of their G x E interaction effects (Fig. 2b). This confirms the notion that a search for specific adaptation traits in this particular breeding program should focus on the male germplasm base. Male R1 (negative score) showed a positive G x E interaction effect with all females (negative score) in the northern environments (positive score). Males R2, R3, and R4 (positive score), on the other hand, showed a positive G x E interaction effect with the same females in the southern and central environments (negative score).
In summary, a joint analysis of the two joint biplots (Fig. 2), which together account for the pattern-rich G x E interaction variability of the system under study, demonstrates that male R4 showed a positive G x E interaction effect with all females in the central environments; males R2 and R3 showed a positive G x E interaction effect with most females in the southern environments (except Al for R3 in environment component 1); and R1 showed a positive G x E interaction effect with all females in the northern environments associated with environment component 2 and a weak response in environment component 1. This information is highly relevant for the definition of the use of tester lines in the breeding program. Regarding the female lines, they showed similar patterns of combining ability by environment interactions. However, line A4 was poorly explained by the model when comparing with the other females, suggesting that it does not constitute an appropriate representation of the interaction patterns of the female germplasm base of the program and should not be used to test the combining ability of new male inbred lines.
GCA and GCA x E Interaction Using 2-Mode PCA
The environment-centered and normalized biplot of the first two components for female oil yield explains 88% of the GC[A.sub.F] + GC[A.sub.F] x E interaction (Fig. 3a). The environment-centered and normalized biplot of the first two components for male oil yield explains 86% of the GC[A.sub.M] + GC[A.sub.M] x E interaction (Fig. 3b) and can be interpreted in analogous way to the biplot of Fig. 2a.
The environment vectors covered a wide range of Euclidean space of the two biplots (Fig. 3), indicating the existence of strong G x E interactions among the 11 environments evaluated. The PC1 appears to be associated with line average oil yield in the two biplots (Fig. 3a: r = 0.78 P = 0.22; Fig. 3b: r = 0.97 P = 0.02), with most of environments having positive scores for this component. Female (Fig. 3a) or male (Fig. 3b) projections on the first principal component reflect the rankings of GCA, with the lines showing the highest GCA at the right hand side of each biplot and vice versa. Consistently with SREG2 analysis (Fig. 1a and 1b), the lines showing the highest GCA were A3 and A2 on the female side and R4 and R1 on the male side.
However, in the two biplots (Fig. 3), the PC2 adds the environmental dimension to the analysis, reflecting the main pattern of GCA x E interaction for females and males and discriminating between the lines that showed high GCA in both the SREG2 model (Fig. 1a and 1b) and in the direction of the PC1 of Fig. 3a and 3b. In the GC[A.sub.F] + GC[A.sub.F] x E interaction biplot (Fig. 3a), the three environments of the northern region (i.e., VO, MA and SB) had negative scores for the PC2, while most central environments had positive scores for that PC. The female A3 had, on average, a higher GCA than A2 in the northern environments and a lower GCA in the central environments. Thus, a program aimed at exploiting specific adaptations to the subregions should select A3 and A2 as female testers for the northern and central subregions, respectively. Similar analysis could be done for the GC[A.sub.M] + GC[S.sub.M] x E interaction biplot (Fig. 3b), where, on average, the male R1 had a higher GCA than R4 in the northern environments and a lower GCA in the central environments.
SCA and SCA x E Interactions (Three-Mode PCA2)
The three-mode model with 2 x 2 x 2 components for F x M x E, respectively, was considered adequate ([r.sup.2] = 0.76, Table 4). The first environmental component displays the overall variability across environments, with positive scores for all environments. The second environment component reflects the portion of the G x E interaction that separates CM, SB, and VM, with negative scores, from GP, MA, TA, and VO, with positive scores. This environmental contrast is considered as unpredictable as there is no obvious regional association. The proportion of variability explained by this component (5%) is too small to justify discussion.
The joint biplot of female and male inbred lines for the first environment component can be described in one dimension, which retains 63% of the SCA + SCA x E interaction variability (Fig. 4). The interpretation rules of this one-dimensional joint biplot are the same as those for the second environment component of three-mode PCA1 analysis (Fig. 2b).
The female and male dimensions of the joint biplot of the first environmental component (Fig. 4) are consistent with the AMMI2 biplot (Fig. 1c) in showing that the females A1 and A2 have a positive SCA with the males R3 and R2, while the females A3 and A4 have a positive SCA with the males R1 and R4. However, when the environmental dimension is added through three-mode PCA (Fig. 4), this general pattern of SCA shown in Fig. 1c actually appears to be strong in BA, VT, and VM and weak in TA, QQ, GP, VO, and MA, the last two being northern environments. This suggests that caution must be exercised when applying the patterns of SCA revealed by the AMMI2 biplot (Fig. 1c) for the northern subregion. As a practical consequence, if patterns of SCA are going to be used in the design of breeding populations to develop subregion-specifically adapted lines, it has to be considered that the subregions could influence the SCA of the lines in different ways.
Comparison of the Models to Display GCA, SCA, and Their Interactions with Environments
Table 5 summarizes the main features of all methods utilized in this study to partially explain GCA, SCA, and their interactions in a half-sib mating (NCII) multi-environment trial. Two combinations of methods were explored to account for all sources of variation. In Strategy 1, two-mode PCA was used to display GCA + SCA across environments and three-mode PCA to display GCA x E + SCA x E interactions. In Strategy 2, two-mode PCA was used to analyze GCA + GCA x E interaction and three-mode PCA to display SCA + SCA x E interaction. In this example, Strategy 2 explained almost 18% more of the total G + G x E interaction variation of the system than did Strategy 1 (Table 3), proving to be an adequate analytical approach to understand the combining ability of the core germplasm of a hybrid crop breeding program within its target population of environments. However, in another dataset, Strategy 1 may be superior to Strategy 2, so both strategies need to be considered when dealing with a F x M x E dataset.
In half-sib mating designs, G x E interactions have the potential of confounding breeding decisions made on the basis of genetic information such as GCA and SCA determined across environments. When the G x E interactions explain a high portion of variability of the genotype-environment system under study relative to the G effect, an understanding of the GCA x E and SCA x E interactions will help accommodate these effects through appropriate breeding strategies aimed at exploiting broad and/or specific adaptation.
The SREG2 and AMMI2 models are able to identify the lines showing the higher GCA and the main patterns of SCA across environments. However, the compression of the environmental dimension implicit in these analytical approaches do not permit display of the variation of those patterns across the population of environments targeted by a breeding program. Two-mode PCA of environment-centered and normalized F (M) x E tables and three-mode (F x M x E) PCA provide effective tools to complement these methods in visualizing and studying GCA x E and SCA x E interactions, allowing the selection of the best tester for each selection strategy (broad or specific adaptation) and displaying the variability of the tested lines for adaptation and combining ability.
We would like to thank Advanta Semillas (Steven Quast, Alan Scott) for supporting this research. We also thank Aldo Martinez, Sergio Solian, Hugo Baravalle, Carlos Ghanem, Ney Flores, and Cesar Sanchez for collaborating in the field experiments and Drs. Ian DeLacy and David Bonnett for critically reading the manuscript.
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Abbreviations: AMMI, additive main effects and multiplicative interaction; BLUE, best linear unbiased estimate; E, environment; F, female; G, genotype; GCA, general combining ability; M, male; NCII, North Carolina Experiment II; PC, principal component; PCA, principal component analysis; SCA, specific combining ability; REML, restricted maximum likelihood; SREG, site regression.
Abelardo J. de la Vega * and Scott C. Chapman
A.J. de la Vega, Advanta Semillas S.A.I.C., Ruta Nac. 33 Km 636, CC 559, (2600) Venado Tuerto, Argentina; S.C. Chapman, CSIRO Plant Industry, Queensland Bioscience Precinct, 306 Carmody Rd., St. Lucia, Qld 4067, Australia. Received 31 Aug. 2005. * Corresponding author (firstname.lastname@example.org).
Table 1. Sunflower hybrids obtained from the NCH mating design between four female inbred lines and four male inbred lines, which were evaluated in 11 environments of Argentina during the 2004-2005 season. Hybrid code Time to Grain (female/male) anthesis ([dagger]) yield ([dagger]) d kg [ha.sup.-1] A1/R1 82 2964 A2/R1 81 3012 A3/R1 77 3194 A4/R1 79 3207 A1/R2 81 2975 A2/R2 81 2962 A3/R2 77 2813 A4/R2 80 2939 A1/R3 77 3036 A2/R3 74 3225 A3/R3 73 2977 A4/R3 76 3013 A1/R4 77 2976 A2/R4 75 3063 A3/R4 74 3135 A4/R4 77 3153 s.e.d. ([double dagger]) 73 Hybrid code Oil Oil (female/male) content ([dagger]) yield ([dagger]) % kg [ha.sup.-1] A1/R1 50.0 1477 A2/R1 50.3 1505 A3/R1 49.8 1586 A4/R1 48.5 1558 A1/R2 49.1 1456 A2/R2 49.9 1470 A3/R2 48.9 1361 A4/R2 47.8 1401 A1/R3 48.6 1469 A2/R3 49.5 1592 A3/R3 48.4 1424 A4/R3 46.6 1393 A1/R4 50.4 1496 A2/R4 50.4 1544 A3/R4 51.3 1601 A4/R4 47.9 1512 s.e.d. ([double dagger]) 0.3 37 ([dagger]) Agronomic traits reported are best linear unbiased predictors (BLUPs) of the hybrid means derived from a linear mixed model (Eg. ) applied to the multienvironment sunflower hybrid trial dataset. ([double dagger]) Standard error of the difference. Table 2. Trial environments of Argentina in which the sunflower hybrids obtained from the NCII mating design between four female inbred lines and four male inbred lines were tested during the 2004-2005 season. Trial Sowing date Environment code Latitude Region (d/m/y) Balcarce CA 37.8 S S 23/10/2004 Cristiano Muerto CM 38.5 S S 21/10/2004 Daireaux DX 36.6 S C 07/10/2004 General Pico GP 35.7 S C 29/10/2004 Malabrigo MA 29.2 S N 14/09/2004 Quemu Quemu QQ 36.2 S C 27/10/2004 San Bernardo SB 27.4 S N 22/08/2004 Des Arroyos TA 38.4 S S 08/11/2004 Villa Maza VM 36.8 S C 28/10/2004 Villa Ocampo VO 28.5 S N 03/08/2004 Venado Tuerto VT 33.7 S C 14/10/2004 s.e.d. ([double dagger]) Time to anthesis Grain yield Environment ([dagger]) ([dagger]) d kg [ha.sup.-1] Balcarce 73 2889 Cristiano Muerto 3250 Daireaux 3335 General Pico 4072 Malabrigo 77 2006 Quemu Quemu 3762 San Bernardo 81 2403 Des Arroyos 76 3134 Villa Maza 4102 Villa Ocampo 83 1515 Venado Tuerto 79 2975 s.e.d. ([double dagger]) 105 Oil yield Environment Oil content ([dagger]) ([dagger]) % kg [ha.sup.-1] Balcarce 47.0 1355 Cristiano Muerto 47.7 1551 Daireaux 50.4 1684 General Pico 47.9 1950 Malabrigo 52.7 1060 Quemu Quemu 48.8 1840 San Bernardo 48.6 1165 Des Arroyos 47.3 1482 Villa Maza 50.0 2050 Villa Ocampo 51.4 779 Venado Tuerto 49.7 1478 s.e.d. ([double dagger]) 0.3 52 ([dagger]) Agronomic traits reported are best linear unbiased estimates (BLUES) of the environment means derived from a linear mixed model (Eq. [11) applied to the multienvironment sunflower hybrid trial dataset. ([double dagger]) Standard error of the difference. Table 3. Estimated variance components ([+ or -] standard errors) for grain yield, oil content and oil yield, derived from a mixed model analysis (all random terms except environments, fitted as fixed) applied to the 4 x 4 x 11 (females X males x environments) sunflower hybrid trial dataset. Source of variationt Grain yield kg [ha.sup.-1] [[sigma].sup.2.sub.f] 0 [[sigma].sup.2.sub.m] 2700 [+ or -] 6666 [[sigma].sup.2.sub.fm] 9644 [+ or -] 6381 [[sigma].sup.2.sub.fe] 17353 [+ or -] 8229 [[sigma].sup.2.sub.me] 16311 [+ or -] 8133 [[sigma].sup.2.sub.fme] 21332 [+ or -] 8731 [[sigma].sup.2.sub.r/e] 9277 [+ or -] 5171 [[sigma].sup.2.sub.b/r/e] 4212 [+ or -] 5672 [[sigma].sup.2.sub.[epsilon]] 100231 [+ or -] 8847 Source of variationt Oil content % [[sigma].sup.2.sub.f] 0.980 [+ or -] 0.903 [[sigma].sup.2.sub.m] 0.524 [+ or -] 0.517 [[sigma].sup.2.sub.fm] 0.299 [+ or -] 0.172 [[sigma].sup.2.sub.fe] 0.351 [+ or -] 0.141 [[sigma].sup.2.sub.me] 0.168 [+ or -] 0.095 [[sigma].sup.2.sub.fme] 0.106 [+ or -] 0.119 [[sigma].sup.2.sub.r/e] 0.001 [+ or -] 0.048 [[sigma].sup.2.sub.b/r/e] 0.100 [+ or -] 0.102 [[sigma].sup.2.sub.[epsilon]] 1.799 [+ or -] 0.160 Source of variationt Oil yield kg [ha.sup.-1] [[sigma].sup.2.sub.f] 0 [[sigma].sup.2.sub.m] 1699 [+ or -] 2863 [[sigma].sup.2.sub.fm] 3685 [+ or -] 2145 [[sigma].sup.2.sub.fe] 4074 [+ or -] 2049 [[sigma].sup.2.sub.me] 5272 [+ or -] 2376 [[sigma].sup.2.sub.fme] 5819 [+ or -] 2259 [[sigma].sup.2.sub.r/e] 2158 [+ or -] 1276 [[sigma].sup.2.sub.b/r/e] 1399 [+ or -] 1480 [[sigma].sup.2.sub.[epsilon]] 25106 [+ or -] 2233 Source of variationt % of oil yield G + G x E [[sigma].sup.2.sub.f] 0.0 [[sigma].sup.2.sub.m] 8.3 [[sigma].sup.2.sub.fm] 17.9 [[sigma].sup.2.sub.fe] 19.8 [[sigma].sup.2.sub.me] 25.7 [[sigma].sup.2.sub.fme] 28.3 [[sigma].sup.2.sub.r/e] [[sigma].sup.2.sub.b/r/e] [[sigma].sup.2.sub.[epsilon]] Source of variationt Figure ([double dagger]) [[sigma].sup.2.sub.f] 1a, 3a [[sigma].sup.2.sub.m] 1b, 3b [[sigma].sup.2.sub.fm] 1a, 1b, 1c, 4 [[sigma].sup.2.sub.fe] 2a, 2b, 3a [[sigma].sup.2.sub.me] 2a, 2b, 3b [[sigma].sup.2.sub.fme] 2a, 2b, 4 [[sigma].sup.2.sub.r/e] [[sigma].sup.2.sub.b/r/e] [[sigma].sup.2.sub.[epsilon]] ([dagger]) [[sigma].sup.2]: variance component, f: female inbred lines, m: male inbred lines, e: environments, r: replicates, b: incomplete blocks, [epsilon]: experimental error. ([double dagger]) The figures of this paper in which each source of variation is summarized are indicated. Table 4. Mode component scores (with adequacy of fit) for 16 sunflower hybrids (four inbred females x four inbred males) grown over 11 trial environments of Argentina during the 2004-2005 season. See Table 2 for environment codes. GCA x E + SCA X E model(3-mode PCA1) ([dagger]) Component scores Code Name or region (E) 1 2 3 Females (F) 1 A1 0.37 -0.38 2 A2 0.91 -0.21 3 A3 0.72 0.50 4 A4 0.32 -0.09 Proportion of 0.40 0.11 SS explained Males (M) 1 R1 -0.27 -0.01 2 R2 0.88 -0.03 3 R3 0.45 0.64 4 R4 -0.42 0.63 Proportion of 0.31 0.20 SS explained Environments (E) BA S 0.69 -0.09 0.00 CM S 0.24 -0.44 -0.08 DX C -0.21 0.07 0.07 GP C -0.69 0.10 0.46 MA N -0.10 0.48 -0.30 QQ C -0.75 -0.19 -0.14 SB N 0.99 -0.26 -0.12 TA S 0.32 -0.08 -0.13 VM C -0.01 -0.12 -0.16 VO N 0.43 0.74 -0.11 VT C 0.99 0.13 0.42 Proportion of 0.35 0.10 0.05 SS explained GCA x E + SCA X E model(3-mode PCA1) SCA + SCA x E model(3-mode ([dagger]) PCA2) ([double dagger]) Component scores Code Proportion of SS explained 1 2 Females (F) 1 0.34 0.46 0.16 2 0.68 1.07 -0.17 3 0.71 -0.89 -0.47 4 0.13 -0.63 0.48 Proportion of 0.51 0.63 0.13 SS explained Males (M) 1 0.10 -1.01 0.19 2 0.59 0.60 0.38 3 0.60 0.99 -0.24 4 0.59 -0.58 -0.34 Proportion of 0.51 0.67 0.09 SS explained Environments (E) BA 0.55 1.23 -0.09 CM 0.31 0.88 -0.26 DX 0.09 0.85 0.17 GP 0.59 0.42 0.27 MA 0.44 0.36 0.30 QQ 0.78 0.52 0.03 SB 0.64 0.90 -0.27 TA 0.14 0.57 0.16 VM 0.10 1.12 -0.20 VO 0.51 0.58 0.40 VT 0.75 1.22 0.09 Proportion of 0.51 0.71 0.05 SS explained SCA + SCA x E model(3-mode PCA2) ([double dagger]) Code Proportion of SS explained Females (F) 1 0.47 2 0.87 3 0.83 4 0.66 Proportion of 0.76 SS explained Males (M) 1 0.85 2 0.57 3 0.85 4 0.69 Proportion of 0.76 SS explained Environments (E) BA 0.91 CM 0.77 DX 0.80 GP 0.55 MA 0.70 QQ 0.73 SB 0.59 TA 0.45 VM 0.94 VO 0.71 VT 0.81 Proportion of 0.76 SS explained ([dagger]) Retained the GCA x E + SCA R E. ([double dagger]) Retained the SCA + SCA x E. Table 5. Comparison of the analytical models for graphical displaying of general combining ability (GCA), specific combining ability (SCA) and their interactions with testing environments explored in this paper. Treatment of the F x M x E raw data Averaged Method ([dagger]) Author across SREG[2.sub.F] Yan and Hunt E (2002) SREG[2.sub.M] Yan and Hunt E (2002) AMM12 Narro et al. E (2003) 3-mode PCAI This paper None 2-mode PC[A.sub.F+FxE] This paper M 2-mode PC[A.sub.M+MxE] This paper F 3-mode PCA2 This paper None Combined-method strategies Strategy 1: SREG[2.sub.F] + SREG[2.sub.M] + 3-mode PCA1 Strategy 2: 2-mode PC[A.sub.F+FxE] + 2-mode PC[A.sub.M+Mxe] + 3-mode PCA2 Treatment of the F x M x E raw data Centered Normalized by within Method ([dagger]) M M SREG[2.sub.F] F F SREG[2.sub.M] M and F None AMM12 F X M and E None 3-mode PCAI E E E E 2-mode PC[A.sub.F+FxE] M and F E 2-mode PC[A.sub.M+MxE] 3-mode PCA2 Combined-method strategies Strategy 1: SREG[2.sub.F] + SREG[2.sub.M] + 3-mode PCA1 Strategy 2: 2-mode PC[A.sub.F+FxE] + 2-mode PC[A.sub.M+Mxe] + 3-mode PCA2 Source of G and G x E variation retained Method ([dagger]) GC[A.sub.F] + SCA SREG[2.sub.F] GC[A.sub.M] + SCA SREG[2.sub.M] SCA AMM12 GC[A.sub.F] x E + GC[A.sub.M] x E + SCA x E 3-mode PCAI GC[A.sub.F] + GC[A.sub.F] x E GC[A.sub.M] + GC[A.sub.M] x E 2-mode PC[A.sub.F+FxE] SCA + SCA x E 2-mode PC[A.sub.M+MxE] 3-mode PCA2 All Combined-method strategies Strategy 1: SREG[2.sub.F] + SREG[2.sub.M] + 3-mode All PCA1 Strategy 2: 2-mode PC[A.sub.F+FxE] + 2-mode PC[A.sub.M+Mxe] + 3-mode PCA2 % of total oil yield Figure G + G x E explained ([dagger]) Method ([dagger]) 17.5 1a SREG[2.sub.F] 25.3 1b SREG[2.sub.M] 17.6 1c AMM12 33.5 2a, 2b 3-mode PCAI 17.4 3a 30.1 3b 2-mode PC[A.sub.F+FxE] 29.0 4 2-mode PC[A.sub.M+MxE] 3-mode PCA2 58.8 1a, 1b, 2a, 2b Combined-method strategies Strategy 1: SREG[2.sub.F] + SREG[2.sub.M] + 3-mode 76.5 3a, 3b, 4 PCA1 Strategy 2: 2-mode PC[A.sub.F+FxE] + 2-mode PC[A.sub.M+Mxe] + 3-mode PCA2 ([dagger]) SREG2: site regression, AMM12: additive main effects and multiplicative interaction, PCA: principal component analysis, F: female, M: male, E: environment. ([double dagger]) The figures of this paper in which each source of variation is summarized are indicated.
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|Author:||de la Vega, Abelardo J.; Chapman, Scott C.|
|Date:||Mar 1, 2006|
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