Introduction

The urgent need to increase grain production presents a serious challenge to agricultural systems globally and locally; the increase must come from raising grain yield per unit area, given that the degree to which the area of cultivated land can be expanded is very limited. The Green Revolution enhanced overall agricultural productivity in many areas of the world by generating improved wheat varieties with high yield potential under optimal high-input environments. However, low-input and less favourable environments have poor agroclimatic potential and are highly affected by biotic and abiotic stresses that show marked climatic fluctuations from year to year. In these less favourable environments, the plant breeding approach should be different from those used in more favourable high-input environments. Furthermore, in the near future many favourable environments may become less favourable (in terms of soil fertility and general climatic conditions) and be plagued by biotic and abiotic stresses due to extreme climate change. Climate change is due to many factors such as rising global mean temperatures, increased intensity and frequency of storms, drought and flooding, weather extremes, and altered hydrological cycles and precipitation patterns. Annual crop production will be greatly affected by increases in mean temperature throughout

© CAB International 2010. Climate Change and Crop Production (ed. M.P. Reynolds)

this century, and climate change will be likely to reduce agricultural production and decrease food availability (Lobell et al., 2005).

Plant breeding will play a paramount role in developing more sustainable farming systems in less favourable environments subject to extreme biotic and abiotic stresses (see Chapters 4-8, this volume). Developing cultivars with enhanced tolerance to heat and moisture stress and salinity is essential to a long-term adaptation response to climate change. In developing crops for the 21st century, breeders must keep in mind that production environments will be more variable and more stressful, yearly climate variation will be greater, and field sites and test environments will essentially be rapidly moving targets. Appropriate breeding strategies will ensure the development: (i) in the long term, of improved varieties, lines and hybrids with adaptation to less favourable environments and high yield stability; and (ii) in the short term, of appropriate varieties, lines and hybrids to meet local farmers' needs. Breeding strategies are based mainly on breeders' in-depth knowledge of their germplasm in general and of how genotypes will respond under different environmental conditions.

Regardless of the breeding strategy used, in any breeding programme multienvironment trials (METs) are essential for assessing varietal adaptation and stability, and for studying and understanding genotype x environment (GE) interaction. For example, significant progress has been made in maize grain yield under drought stress by selecting for component traits such as kernel set, rapid silking and reduced barrenness in METs. GE interaction refers to the differential response of a set of plant materials (such as lines, open-pollinated varieties or populations, etc., referred to as 'genotypes') when evaluated in a set of environments characterized by certain soil, climatic, pest, disease and management conditions (referred to as 'environments') in a given location and year. In general, GE may be due to heterogeneity of within-environment variance (HV) or scale changes, or to crossover interaction (COI) or changes in rank of genotypes in different environments. In agricultural production, the most important GE is that due to COI.

Conventional breeding in conjunction with marker assisted selection (MAS) may bring about significant and predictable incremental improvements in the drought tolerance of new maize lines and hybrids (Banziger and Araus, 2007). Likewise, the genetic dissection of maize performance in drought-prone environments has greatly benefited from the use of DNA markers (Ribaut and Ragot, 2007). The use of MAS in plant breeding has increased consistently since 1980, and molecular markers are now considered a valuable breeding tool. Advances in high-throughput genotyping have reduced the cost of using molecular markers, and their abundance and low cost have led to selection based only on molecular markers (called marker-assisted recurrent selection, or MARS). Applying MARS for one cycle based on phenotypic and marker scores followed by two or three cycles of selection based solely on marker score information has increased genetic gains. Genome-wide dense marker maps are now available for many plant and animal species, and genome-wide selection has become an interesting option for increasing genetic gains in different crops and animals (Bernardo and Yu, 2007).

Important challenges are how: (i) marker information should be incorporated into statistical models that could be useful for predicting genetic values in animal and plant breeding programmes, or for predicting diseases; (ii) the large number of candidate genes known to have specific trait effects could be used in a practical breeding programme; (iii) the large number of environmental variables and pests affecting genotypes in METs could be used to better predict genotypic and phenotypic performance so that the best genotypes are selected as parents for the next generation; and (iv) the powerful computer algorithms used in crop modelling and simulation methods could help breeders to better achieve their goals.

The massive accumulation of genetic and environmental data confirms the urgent need for suitable and efficient bioinforma-tics, biometrical and statistical methods to assess and incorporate GE studies into conventional as well as MARS and genome-wide selection breeding schemes for less favourable environments. The objective of this chapter is to describe the theory and practical applications of statistical models and methods normally used for studying and understanding GE and how they can be applied in combination with molecular markers in plant breeding.

Phenotypic Values, Genotypic Values and Environments

Before describing statistical models for studying the response of genotypes under different environmental conditions, we should explain that phenotypic (observed) values are a function of genes that produce genotypic (unobserved) values under certain environmental conditions. This is clearly explained by Bernardo (2002) for modelling the phenotypic value of the kth individual having a genotype AjAm (locus A has two alleles, lth and mth), which is in turn affected by a non-genetic component elml. Then, the phenotypic value will be Pmk = genetic

+ elmt (for the deviation from the population mean, |, of gm = Glm ) assuming the genotypic value glm and the non-genetic elml values are uncorrelated. In general, genotypic values include additive, and dominance within locus and all types of epistatic effects between loci. The expected value of Plmk for all individuals with genotype AlAm is equal to the genotypic value glm plus the expectation of the non-genetic effects emk. Under the assumption that the expectation of elml is equal to zero, then the expected value of Plmk = | + glm and the expected value of Plmk across all genotypes (and not only genotype AAm) is | (implying that the expected value of glm is, in fact, zero). This two-allele locus model can be extended to any number of loci.

Although genotypic values cannot be measured directly, they are estimated based on phenotypic values and environmental effects. This is the main reason why breeding programmes need to have not only a clear set of genotypes to be tested but also a clear set of target environments where those genotypes should be tested. Therefore, in METs, just as genotypic values (estimated based on phenotypic values) depend on the environments in which the genotypes are grown and the trait measured, so environmental values depend on the genotypes grown in those environments. In most METs, the genotypic values gm for different genotypes are different in different environments; this constitutes GE.

The Basic Two-way Fixed-effect Linear Model

Early approaches to GE analyses included the conventional fixed-effect two-way model with sum to zero constraints running over indices. The empirical response yijr of the ith genotype (i = 1, 2, ..., I) in the jth environment (j = 1, 2, ..., J) with r replications in each of the I x J cells is expressed as:

where | is the grand mean (over all genotypes and environments), Ti is the additive effect of the ith genotype, 8j is the additive effect of the jth environment, ( t 8 )lj is the non-additivity interaction (GE) of the ith genotype in the jth environment (forming matrix Z), and eijr is the within-environment error associated with the ith genotype in the jth environment and the rth replicate.

The phenotypic value averaged across replicates in each environment is y..( and the least squares estimates of the genotypic effect and the environmental effects are Ti = yi - y (which satisfies the constraint

the constraint ^8j = 0 ) (Table 14.1), where y is the least squares estimate of the overall mean | and y. is the mean of the ith genotype averaged across environments and replicates, and yjis the mean of the jth environment across all genotypes and replicates. Therefore, the least squares estimate of the GE term in Eqn 14.1 is ( t 8 ).. = z.. = y.. - y. - yj + y... (which satisfies the constraints XX (t d)lj =

X ( t 8)j = £ ( t 8)j = 0) (Table l4.2). i ' Note that the notation in Eqn 14.1 can be used for models with fixed, mixed, or random effects. For a complete random model, it is assumed that t, 8i and (x8)ij are normally and independently distributed, with variances ctT , ct8 and ct^ , respectively.

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