Genotype-by-environment interaction (GEI) is an important phenomenon in plant breeding. This paper presents a series of models for describing, exploring, understanding, and predicting GEI. All models depart from a two-way table of genotype by environment means. First, a series of descriptive and explorative models/approaches are presented: Finlay–Wilkinson model, AMMI model, GGE biplot. All of these approaches have in common that they merely try to group genotypes and environments and do not use other information than the two-way table of means. Next, factorial regression is introduced as an approach to explicitly introduce genotypic and environmental covariates for describing and explaining GEI. Finally, QTL modeling is presented as a natural extension of factorial regression, where marker information is translated into genetic predictors. Tests for regression coefficients corresponding to these genetic predictors are tests for main effect QTL expression and QTL by environment interaction (QEI). QTL models for which QEI depends on environmental covariables form an interesting model class for predicting GEI for new genotypes and new environments. For realistic modeling of genotypic differences across multiple environments, sophisticated mixed models are necessary to allow for heterogeneity of genetic variances and correlations across environments. The use and interpretation of all models is illustrated by an example data set from the CIMMYT maize breeding program, containing environments differing in drought and nitrogen stress. To help readers to carry out the statistical analyses, GenStat® programs, 15th Edition and Discovery® version, are presented as “Appendix.”

To feed a world population growing by up to 160 people per minute, with >90% of them in developing countries, will require an astonishing increase in food production. Forecasts call for wheat to become the most important cereal in the world, with maize close behind; together, these crops will account for Ϸ80% of developing countries' cereal import requirements. Access to a range of genetic diversity is critical to the success of breeding programs. The global effort to assemble, document, and utilize these resources is enormous, and the genetic diversity in the collections is critical to the world's fight against hunger. The introgression of genes that reduced plant height and increased disease and viral resistance in wheat provided the foundation for the ''Green Revolution'' and demonstrated the tremendous impact that genetic resources can have on production. Wheat hybrids and synthetics may provide the yield increases needed in the future. A wild relative of maize, Tripsacum, represents an untapped genetic resource for abiotic and biotic stress resistance and for apomixis, a trait that could provide developing world farmers access to hybrid technology. Ownership of genetic resources and genes must be resolved to ensure global access to these critical resources. The application of molecular and genetic engineering technologies enhances the use of genetic resources. The effective and complementary use of all of our technological tools and resources will be required for meeting the challenge posed by the world's expanding demand for food.

In most maize-growing areas yield reductions due to drought have been observed. Drought at flowering time is, in some cases, the most damaging. In the experiment reported here, trials with F families, derived from a segregating F population, were conducted in the field under well-watered conditions (WW) and two other water-stress regimes affecting flowering (intermediate stress, IS, and severe stress, SS). Several yield components were measured on equal numbers of plants per family: grain yield (GY), ear number (ENO), kernel number (KNO), and 100-kernel weight (HKWT). Correlation analysis of these traits showed that they were not independent of each other. Drought resulted in a 60% decrease of GY under SS conditions. By comparing yield under WW and SS conditions, the families that performed best under WW conditions were found to be proportionately more affected by stress, and the yield reductions due to SS conditions were inversely proportional to the performance under drought. Moreover, no positive correlation was observed between a drought-tolerance index (DTI) and yield under WW conditions. The correlation between GY under WW and SS conditions was 0.31. Therefore, in this experiment, selection for yield improvement under WW conditions only, would not be very effective for yield improvement under drought. Quantitative trait loci (QTLs) were identified for GY, ENO and KNO using composite interval mapping (CIM). No major QTLs, expressing more then 13% of the phenotypic variance, were detected for any of these traits, and there were inconsistencies in their genomic positions across water regimes. The use of CIM allowed the evaluation of QTL-by-environment interactions (Q;E) and could thus identify ''stable'' QTLs Communicated by G. E. Hart

Drought is an important climatic phenomenon which, after soil infertility, ranks as the second most severe limitation to maize production in developing countries. When drought stress occurs just before or during the flowering period, a delay in silking is observed, resulting in an increase in the length of the anthesis-silking interval (ASI) and in a decrease in grain yield. Selection for reduced ASI in tropical open-pollinated varieties has been shown to be correlated with improved yields under drought stress. Since efficient selection for drought tolerance requires carefully managed experimental conditions, molecular markers were used to identify the genomic segments responsible for the expression of ASI, with the final aim of developing marker-assisted selection (MAS) strategies. An F2population of 234 individuals was genotyped at 142 loci and F3 families were evaluated in the field under several water regimes for male flowering (MFLW), male sterility (STER), female flowering (FFLW) and ASI. The genetic variance of ASI increased as a function of the stress intensity, and the broad-sense heritabilites of MFLW, FFLW and ASI were high under stress conditions, being 86%, 82% and 78%, respectively. Putative quantitative trait loci (QTLs) involved in the expression of MFLW and/or FFLW under drought were detected on chromosomes 1, 2, 4, 5, 8, 9 and 10, accounting for around 48% of the phenotypic variance for both traits. For ASI, six putative QTLs were identified under drought on chromosomes 1, 2, 5, 6, 8 and 10, and together accounted for approximately 47% of the phenotypic variance. Under water stress conditions, four QTLs were common for the expression of MFLW and FFLW, one for the expression of ASI and MFLW, and four for the expression of ASI and FFLW. The number of common QTLs for two traits was related to the level of linear correlation between these two traits. Segregation for ASI was found to be transgressive with the drought-susceptible parent contributing alleles for reduced ASI (4 days) at two QTL positions. Alleles contributed by the resistant line at the other four QTLs were responsible for a 7-day reduction of ASI. These four QTLs represented around 9% of the linkage map, and were stable over years and stress levels. It is argued that MAS based on ASI QTLs should be a powerful tool for improving drought tolerance of tropical maize inbred lines.

A recombinant inbred line (RIL) population was evaluated in seven field experiments representing four environments: water stress at flowering (WS) and well-watered (WW) conditions in Mexico and Zimbabwe. The QTLs were identified for each trait in each individual experiment (single-experiment analysis) as well as per environment, per water regime across locations and across all experiments (joint analyses). For the six target traits (male flowering, anthesis-to-silking interval, grain yield, kernel number, 100-kernel fresh weight and plant height) 81, 57, 51 and 34 QTLs were identified in the four step-wise analyses, respectively. Despite high values of heritability, the phenotypic variance explained by QTLs was reduced, indicating epistatic interactions. About 80, 60 and 6% of the QTLs did not present significant QTL-by-environment interactions (QTL x E) in the joint analyses per environment, per water regime and across all experiments. The expression of QTLs was quite stable across years at a given location and across locations under the same water regime. However, the stability of QTLs decreased drastically when data were combined across water regimes, reflecting a different genetic basis of the target traits in the drought and well-watered trials. Several clusters of QTLs for different traits were identified by the joint analyses of the WW (chromosomes 1 and 8) and WS (chromosomes 1, 3 and 5) treatments and across water regimes (chromosome 1). Those regions are clear targets for future marker-assisted breeding, and our results confirm that the best approach to breeding for drought tolerance includes selection under water stress.

It has long been recognized that epistasis or interactions between non-allelic genes plays an important role in the genetic control and evolution of quantitative traits. However, the detection of epistasis and estimation of epistatic effects are difficult due to the complexity of epistatic patterns, insufficient sample size of mapping populations and lack of efficient statistical methods. Under the assumption of additivity of QTL effects on the phenotype of a trait in interest, the additive effect of a QTL can be completely absorbed by the flanking marker variables, and the epistatic effect between two QTL can be completely absorbed by the four marker-pair multiplication variables between the two pairs of flanking markers. Based on this property, we proposed an inclusive composite interval mapping (ICIM) by simultaneously considering marker variables and marker-pair multiplications in a linear model. Stepwise regression was applied to identify the most significant markers and marker-pair multiplications. Then a two-dimensional scanning (or interval mapping) was conducted to identify QTL with significant digenic epistasis using adjusted phenotypic values based on the best multiple regression model. The adjusted values retain the information of QTL on the two current mapping intervals but exclude the influence of QTL on other intervals and chromosomes. Epistatic QTL can be identified by ICIM, no matter whether the two interacting QTL have any additive effects. Simulated populations and one barley doubled haploids (DH) population were used to demonstrate the efficiency of ICIM in mapping both additive QTL and digenic interactions.

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