The basic form of the “breeder’s equation” provides a framework to predict the response to selection (ΔG ) from one cycle (L) of a breeding program, following application of a selection strategy (Moose and Mumm, 2008; Cobb et al., 2019). Here we consider selection strategies that incorporate applications of genomic prediction (Meuwissen et al., 2001; Bernardo and Yu, 2007; Heffner et al., 2009; Cooper et al., 2014a; Voss-Fels et al., 2019). Selection pressure is implemented by applying truncation selection to the distributions of observed or predicted values for one or more traits within the reference population of genotypes (RPG) of a breeding program; for example, selection to increase crop yield, improve grain quality and improve abiotic and biotic stress tolerances to reduce the extent of yield losses due to the occurrence of the frequent stresses in the TPE (Chenu et al., 2011; Kholová et al., 2013; Hajjarpoor et al., 2021; Messina et al., 2022a). The structure of the breeder’s equation has a long history in animal and plant breeding (Lush, 1937; Hallauer and Miranda, 1988; Nyquist and Baker, 1991; Comstock, 1996; Moose and Mumm, 2008) and is frequently used as a quantitative framework for the design and optimization of crop breeding programs (Araus and Cairns, 2014; Araus et al., 2018; Cobb et al., 2019; Voss-Fels et al., 2019; Kholová et al., 2021; Cooper and Messina, 2023). For applications of genomic prediction, a common form of the breeder’s equation is given as:

Where i represents the selection differential applied to the selection units, based on the trait variation within the RPG, r _a represents the prediction accuracy for breeding values for the selection units within the RPG, and σ _a represents the additive genetic variation among the selection units within the RPG for the traits that are targeted for improvement by selection. For genomic breeding, the quantification of prediction accuracy r _a is based on G2P models for traits that are constructed using suitable training data sets. These G2P models are created algorithmically using the genetic marker fingerprints and trait phenotypes for the genotypes included in breeding multi-environment trials (METs) used as training data sets (Meuwissen et al., 2001; Crossa et al., 2017; Messina et al., 2018; Diepenbrock et al., 2021). The foundation of the MET training data sets is typically based on data collected from the relevant stages of the breeding program (Cooper et al., 2014a; Voss-Fels et al., 2019; Smith et al., 2021a). Environmental covariates and model-based characterizations of the sample of environments present in the MET can be used to create environmental predictors to be included in the G2P model. These environmental predictors provide a basis to adjust genomic predictions of genotype breeding value and performance for different environments to account for effects of GxE interactions (Jarquín et al., 2014; Crossa et al., 2017; Messina et al., 2018; de los Campos et al., 2020; Diepenbrock et al., 2021). Importantly, the samples of environments included in the METs are considered to represent the environmental composition of the TPE (Comstock and Moll, 1963; Nyquist and Baker, 1991; Cooper and DeLacy, 1994; Chenu et al., 2011). The environmental composition of the METs can be augmented in many ways using specifically designed field-based and controlled-environment experiments (Cooper et al., 1995; Cooper et al., 1997, Campos et al., 2004; Cooper et al., 2014a; Cooper et al., 2014b; Rebetzke et al., 2013; van Eeuwijk et al., 2019; Langstroff et al., 2022; Cooper and Messina, 2023). Many assumptions are made when applying the breeder’s equation, as represented by equation (1). We consider some of these assumptions in more detail as they relate to the prediction of response to selection for improved on-farm performance within the TPE. We focus on the influence of the MET-TPE relationship in the presence of GxE interactions within the TPE of the breeding program and use this as the basis for deriving the extended breeder’s equation introduced below.

The breeder’s equation, as represented in equation (1), quantifies the per cycle rate of change of the trait mean value for the RPG (Nyquist and Baker, 1991; Moose and Mumm, 2008; Araus et al., 2018; Cobb et al., 2019; Voss-Fels et al., 2019). However, this form of the breeder’s equation does not explicitly quantify the directionality of the changes in trait values, that are based on the results and predictions from METs, relative to their requirements for improved performance in the TPE. Instead, it relies on the assumption that the environmental composition of the MET is a good representation of the environmental composition of the TPE, i.e., that there is good MET-TPE alignment (Comstock and Moll, 1963; Nyquist and Baker, 1991). To enable efficient design of a breeding program, targeted on creation of new products to close on-farm yield gaps within the TPE, it is desirable to have a form of the breeder’s equation that includes both the rate and the directionality components of genetic gain for the TPE. One approach is to explicitly include a term in the breeder’s equation that quantifies the influence of the MET-TPE alignment on the predicted rate of change within the TPE. Applying correlated response selection theory (Falconer, 1952; Cooper and DeLacy, 1994; Rogers et al., 2021; Cooper and Messina, 2023), we provide an extended form of the breeder’s equation that combines both the rate and directionality components of trait change under the influence of selection, explicitly accounting for the influence of the MET-TPE alignment on the directionality of the change relative to the requirements for the TPE. Considering the environmental composition of the MET to be a sample of the environmental composition of the TPE (MET∈TPE ), an equation for trait genetic gain within the TPE, based on selection decisions made using predictions from G2P trait information obtained from METs (ΔG _(MET,TPE) ), can be given as:

Two of the terms in equation (2) are equivalent to terms in equation (1): i _MET is the selection differential applied to phenotypic and G2P prediction information obtained from analyses of the MET training data sets, as for i in equation (1), r _a(MET) is the prediction accuracy for the selection units based on applications of the training data available from the MET, as for r _a in equation (1). In equation (2) the σ _a term of equation (1) is replaced by the product of two terms r _a(MET,TPE) and σ _a(TPE) . The term r _a(MET,TPE) is the genetic correlation between the additive genetic effects estimated by applying G2P models developed using the MET training data sets, and the additive genetic effects for the trait targets required for realized trait performance in the TPE. The term σ _a(TPE) represents the relevant target additive genetic variation for the traits within the TPE. Thus, the r _a(MET,TPE) term of the extended breeder’s equation provides a quantitative measure of the impact of the MET-TPE alignment for the prediction of additive genetic variation for traits in the TPE, and thus for predicting their contributions to genotype performance in the TPE. The r _a(MET,TPE) can range from +1, with good MET-TPE alignment, to -1, with poor MET-TPE alignment. Additional forms of equation (2) can be given, for example for prediction at the level of the total genotypic trait performance level. Equally equation (2) can be further extended to examine the contributions of quantitative trait loci (QTL) and combinations of haplotypes and specific QTL to the additive or total genotypic variance for multiple traits in the RPG for the TPE.

Applying the extended form of the breeder’s equation given in equation (2), statements can be made regarding the design of genomic prediction strategies based on applications of equation (1).

To demonstrate the implications of GxE interactions on realized genetic gain in the on-farm TPE we consider two examples of the application of the extended form of the breeder’s equation to investigate the MET-TPE alignment and its potential impact on the r _a(MET,TPE) component of equation (2). The first considers a familiar theoretical example from the study of crossover GxE interactions (Haldane, 1947; Ceccarelli, 1989; Ceccarelli, 1994; Cooper and DeLacy, 1994; van Eeuwijk et al., 2001). The second considers an empirical example based on a previously published MET-TPE data set for wheat in Australia (Cooper et al., 1995; Cooper et al., 1997; Cooper et al., 2001). The wheat example was previously used to investigate the implications of GxE interactions for grain yield in the TPE, and also the MET-TPE relationship for the design of METs to accelerate genetic gain for yield from wheat breeding in a TPE where complex GxE interactions for grain yield are ubiquitous (Brennan et al., 1981; Cooper and DeLacy, 1994; Cooper et al., 1995; Cooper et al., 1997; Basford and Cooper, 1998; Cooper et al., 2001; Chenu et al., 2011; Lobell et al., 2015; Bustos-Korts et al., 2021).

Theoretical and empirical considerations of the influences of GxE interactions for breeding have consistently emphasized the importance of crossover GxE interactions ( Figure 1A ; Haldane, 1947; Ceccarelli, 1989; Ceccarelli, 1994; Cooper and DeLacy, 1994; Cooper et al., 2021; Rogers et al., 2021; Smith et al., 2021a; Smith et al., 2021b). Examples of such crossover interactions in breeding METs have been demonstrated at the genotypic (Cooper et al., 1995; Cooper et al., 1997; van Eeuwijk et al., 2001; Xiong et al., 2021; Smith et al., 2021b) and QTL levels (Boer et al., 2007, Millet et al., 2019). For the theoretical example of crossover GxE interactions shown in Figure 1A , the yield performance responses for two genotypes (G2 and G8) in two environments (Env_1 and Env_2) are considered. The potential impact of the crossover interactions depicted in Figure 1A on selection decisions can be examined using equation (2) by considering the influence of changes in the frequency of occurrence of the two environments within both the MET and TPE on the genetic correlation r _a(MET,TPE) term from equation (2). Here we consider the genotypic correlation r _g(MET,TPE) between weighted average yield of the two genotypes between the MET and the TPE, where the weights are based on the frequencies of occurrence of the two environments in the MET and the TPE (Podlich et al., 1999). This provides a simulated scan of the range of possible MET-TPE alignment scenarios based on the potential range in frequency of occurrence of the two environments within the MET and the TPE.

In Figure 1B the genotypic covariance σ _g(MET,TPE) of the average performance of the two genotypes in the MET and the TPE is plotted against the frequency of Env_1 in the MET and the TPE. The genotypic covariance is the numerator of the genetic correlation r _g(MET,TPE) term of equation (2) and is used here in place of r _g(MET,TPE) to smooth out the response surface for illustration purposes. The shape of the response surface for the genotypic covariance ( Figure 1B ) fluctuates between negative and positive values depending on the frequency of occurrence of both environments in the MET and the TPE. Two aspects are noted.

Building on the theoretical example ( Figures 1A, B ), we apply the extended breeder’s equation to quantify the impact of the MET-TPE alignment for an empirical example by estimating the genotypic correlation r _g(MET,TPE) term of equation (2) for a range of wheat MET-TPE alignment scenarios for north-eastern Australia ( Figures 1C, D ). We utilize grain yield data available from a previously published wheat data set (Cooper et al., 1995; Cooper et al., 1997; Cooper et al., 2001). The example provides grain yield data for 15 genotypes and 53 environments. Importantly, for current considerations, the 53 environments were previously organized to represent a breeding MET (27 environments) and the TPE (26 environments) for the north-eastern region of the Australian wheat belt (Brennan et al., 1981; Cooper et al., 1995; Cooper et al., 1997; Chenu et al., 2011). The MET was specifically designed to represent the current understanding of GxE interactions and MET-TPE alignment scenarios for the wheat breeding program at that time. The set of 15 genotypes was chosen to represent groupings of key germplasm from the reference population of genotypes for the wheat breeding program (Cooper and DeLacy, 1994; Cooper et al., 1995; Cooper et al., 1997; Cooper et al., 2001). Further, we identify that the data for the two genotypes (G2 and G8), used to illustrate crossover GxE interactions in the theoretical example ( Figure 1A ), were chosen from the larger set of 15 genotypes included in the empirical example ( Figure 1C ). Also, the two environments (Env_1 and Env_2) used in the theoretical example were taken from the empirical example. Thus, the numerical values for the example of crossover GxE interaction for grain yield ( Figure 1A ) used for the theoretical investigations of MET-TPE alignment ( Figure 1B ) were representative of important crossover GxE interactions under consideration within the target breeding program, as considered in the empirical example ( Figures 1C, D ; Brennan et al., 1981; Cooper and DeLacy, 1994; Basford and Cooper, 1998; Cooper et al., 2001).

Improving grain yield stability for the TPE of the north-eastern region of the Australian wheat-belt was a primary objective of the wheat breeding program at that time (Brennan et al., 1981). A weighted selection strategy, combined with field-based managed-environments, was developed to account for GxE interactions in the TPE (Cooper et al., 1995; Cooper et al., 1997; Cooper et al., 2001; Podlich et al., 1999). Spatial and temporal variability for water availability was identified as primary driver of grain yield variation within the TPE, and drought was a major source of crossover GxE interactions for grain yield. Thus, the environments included in the MET were managed to sample a gradient of water availability scenarios, ranging from severe drought to water-sufficient environments, by managing combinations of irrigation and nitrogen inputs at a restricted number of locations. The TPE set of environments was designed by sampling a range of water availability scenarios from a wider range of locations and years within the north-eastern region of Australia. The objective was to design a MET for the stages of the wheat breeding program that could be consistently managed at a few locations to provide a stratified sample of the range of water availability environments expected within the TPE (Brennan et al., 1981; Cooper et al., 1995; Cooper et al., 1997; Cooper et al., 2001).

Grain yield GxE interactions were previously identified for both the MET and TPE data sets (Cooper et al., 1995; Cooper et al., 1997; Cooper et al., 2001). Crossover GxE interactions were frequent ( Figure 1A ; Cooper and DeLacy, 1994). For the purposes of demonstrating an application of equation (2) to the empirical wheat example, the prior envirotyping was used to identify two groups of environment-types for both the MET and TPE sets; environment-type 1 (E1) characterized by mild water-deficits, and environment-type 2 (E2) characterized by severe water-deficits. There were GxE interactions between the two environment-types within the MET and TPE sets ( Figure 2 ; Cooper et al., 1995; Cooper et al., 1997; Cooper et al., 2001). There was a moderate to weak positive genotypic correlation for grain yield variation among the 15 genotypes between both environment-types E1 and E2 for the MET ( Figure 2A ) and TPE ( Figure 2B ). Importantly, for interpretation of the genotypic correlation r _g(MET,TPE) between the MET and TPE ( Figure 1D ), the genotypic correlation for grain yield variation between the mild stress environment-type E1 was positive and strong between the MET and the TPE ( Figure 2C ). However, there was no relationship for grain yield variation between the severe drought stress environment-type E2 between the MET and the TPE ( Figure 2D ). The details of the lack of relationship for environment-type E2 are discussed in detail elsewhere (Cooper et al., 1995; Cooper et al., 1997). In summary the MET was designed to focus on the expected water availability gradient in the absence of other abiotic and biotic stresses that could also occur within the TPE. Occurrences of these other abiotic and biotic stresses within the TPE set were interpreted to be contributing factors to the low relationship observed for severe drought stress environment-type E2 between the MET and TPE ( Figure 2D ). In the absence of the drought stress for environment-type E1 these other abiotic and biotic stresses were less influential on the genotypic correlation for grain yield ( Figure 2C ).

For purposes of demonstrating an application of the extended breeder’s equation to the wheat MET-TPE data set ( Figures 1C, D ) it is sufficient to note that there was GxE interaction for grain yield between Environment-types E1 and E2 in both the MET ( Figure 2A ) and the TPE ( Figure 2B ) data sets and that there was positive predictability between the MET and TPE sets for environment-type E1 ( Figure 2C ), but no predictability for environment-type E2 ( Figure 2D ). Using this level of envirotyping we can simulate the influence of changes in the MET-TPE alignment on r _g(MET,TPE) and prediction of average grain yield in the TPE based on average grain yield estimated from the MET ( Figure 1D ). Following the same procedures applied to the theoretical example ( Figures 1A, B ), the potential range of MET-TPE alignment scenarios was simulated by changing the frequencies of environment-types E1 and E2 within the MET and the TPE in steps of 0.1 from 0.0 to 1.0, calculating the weighted average grain yield of the 15 genotypes for both the MET and TPE, taking into consideration the frequencies of both environment-types, and calculating the genotypic correlation r _g(MET,TPE) between the estimates of weighted average grain yield for the 15 genotypes between the MET and TPE for all MET-TPE alignment combinations. We then plotted the r _g(MET,TPE) against the frequency of environment-type E1 in the MET and TPE to generate a simulated r _g(MET,TPE) genotypic correlation response surface for all MET-TPE alignment configurations ( Figure 1D ). The genotypic correlation r _g(MET,TPE) between the simulated MET and TPE alignments ranged from a high value of 0.90 to a low value of -0.07 ( Figure 1D ). The r _g(MET,TPE) response surface for the wheat example has interesting features. Firstly, there is a relatively broad plateau of high r _g(MET,TPE) values for many of the MET-TPE alignment scenarios. This plateau of high r _g(MET,TPE) values occurred for scenarios where the frequency of the water-sufficient environment-type E1 was higher than 0.5 in both the MET and TPE ( Figure 1D ), taking advantage of the high predictability between environment-type E1 in the MET and TPE ( Figure 2C ). Secondly, when the frequency of environment-type E1 falls below 0.5 in the MET or TPE, and therefore the frequency of the water-limited environment-types E2 increases above 0.5, the r _g(MET,TPE) is degraded from the high levels of the plateau ( Figure 1D ), reflecting the increased influence of the poor predictability between the MET and TPE for the water-limited environment-type E2 ( Figure 2D ). This impact of the MET-TPE alignment on predictability for performance in the TPE using MET results will apply to all levels of prediction, including genomic prediction, phenotypic prediction, and combined prediction approaches.

For the specific environment-type configuration realized for the empirical example ( Figure 2 ), the estimate of r _g(MET,TPE) for prediction of average grain yield for the TPE based on average gain yield obtained for the MET was intermediate ( Figure 1C );r _g(MET,TPE) = 0.70 for MET f(E1) = 0.41, f(E2) = 0.59 and for TPE f(E1) = 0.31, f(E2) = 0.69. Thus, the MET-TPE alignment for the empirical example was located on the r _g(MET,TPE) response surface ( Figure 1D ) slightly off of the plateau of higher r _g(MET,TPE) levels, but still above the precipice where the r _g(MET,TPE) value is severely degraded. This empirical realization of MET-TPE alignment is just one of the many possible scenarios that can occur as the frequencies of environment-types change between the MET and the TPE ( Figure 1D ).

The empirical wheat example ( Figures 1 , 2 ) was used to demonstrate the utility of the extended form of the breeder’s equation for applications in prediction-based breeding. Here we have emphasized the use of the extended breeder’s equation as a useful framework to guide the design MET data sets for training G2P models for applications of genomic prediction and genomic selection at different stages of a breeding program to take aim at the TPE (Cooper et al., 2014a; Cooper et al., 2014b; Gaffney et al., 2015; Messina et al., 2022a). Many other possible prediction scenarios can also be investigated, and these will be the subject of future research.