Mapping Environment-Specific Quantitative Trait Loci

Environment-specific quantitative trait loci (QTL) refer to QTL that express differently in different environments, a phenomenon called QTL-by-environment (Q × E) interaction. Q × E interaction is a difficult problem extended from traditional QTL mapping. The mixture model maximum-likelihood method is commonly adopted for interval mapping of QTL, but the method is not optimal in handling QTL interacting with environments. We partitioned QTL effects into main and interaction effects. The main effects are represented by the means of QTL effects in all environments and the interaction effects are represented by the variances of the QTL effects across environments. We used the Markov chain Monte Carlo (MCMC) implemented Bayesian method to estimate both the main and the interaction effects. The residual error covariance matrix was modeled using the factor analytic covariance structure. A simulation study showed that the factor analytic structure is robust and can handle other structures as special cases. The method was also applied to Q × E interaction mapping for the yield trait of barley. Eight markers showed significant main effects and 18 markers showed significant Q × E interaction. The 18 interacting markers were distributed across all seven chromosomes of the entire genome. Only 1 marker had both the main and the Q × E interaction effects. Each of the other markers had either a main effect or a Q × E interaction effect but not both.GENOTYPE-BY-ENVIRONMENT (G × E) interaction is a very important phenomenon in quantitative genetics. With the advanced molecular technology and statistical methods for quantitative trait loci (QTL) mapping (Lander and Botstein 1989; Jansen 1993; Zeng 1994), G × E interaction analysis has shifted to QTL-by-environment (Q × E) interaction. In the early stage of QTL mapping, almost all statistical methods were developed in a single environment (Paterson et al. 1991; Stuber et al. 1992). Data from different environments were analyzed separately and the conclusions were drawn from the separate analyses of QTL across environments. These methods do not consider the correlation of data under different environments and thus may not extract maximum information from the data. Composite interval mapping for multiple traits can be used for Q × E interaction if different traits are treated as the same trait measured in different environments (Jiang and Zeng 1995). This multivariate composite interval mapping approach makes good use of all data simultaneously and increases statistical power of QTL detection and accuracy of the estimated QTL positions. However, the number of parameters of this method increases dramatically as the number of environments increases. Therefore, the method may not be applied when the number of environments is large. Several other models have been proposed to solve the problem of a large number of environments (Jansen et al. 1995; Beavis and Keim 1996; Romagosa et al. 1996). These methods were based on some special situations and assumptions. One typical assumption was independent errors or constant variances across environments. These assumptions are often violated in real QTL mapping experiments.Earlier investigators realized the problem and adopted the mixed-model methodology to solve the problem (Piepho 2000; Boer et al. 2007). Under the mixed-model framework, people can choose which model effects are random and which are fixed. The mixed-model methodology is very flexible, leading to an easy way to model genetic and environmental correlation between environments using a suitable error structure. Piepho (2000) proposed a mixed model to detect QTL main effect across environments. Similar to the composite interval mapping analysis, his model incorporated one putative QTL and a few cofactors. The Q × E effects in the model were assumed to be random, which greatly reduced the number of estimated parameters. However, the fact that only one QTL is included in the model means that Piepho''s (2000) model remains a single-QTL model rather than a multivariate model. Boer et al. (2007) proposed a step-by-step mixed-model approach to detecting QTL main effects, Q × E interaction effects, and QTL responses to specific environmental covariates. In the final step, Boer et al. (2007) rewrote the model to include all QTL in a multiple-QTL model and reestimated their effects.In this study, we extended the Bayesian shrinkage method (Xu 2003) to map Q × E interaction effects of QTL. In the original study (Xu 2003), we treated each marker as a putative QTL and used the shrinkage method to simultaneously estimate marker effects of the entire genome. In the multiple-environment case, we can still use this approach to simultaneously evaluate marker effects under multiple environments but we can further partition the marker effects into main and Q × E interaction effects. For any particular marker, the mean of the marker effects represents the main effect and the variance of the marker effects represents the Q × E interaction effect for that marker. Under the Bayesian framework, we assigned a normal prior with zero mean and an unknown variance to each marker main effect and a multivariate normal prior with zero vector mean and homogeneous diagonal variance–covariance matrix to the Q × E interaction effects of each maker. In multiple environments, the structure of the error terms might be very complicated since we need to consider the correlation of the same genotype under different environments. In our analysis, we used different variance–covariance structures to model the error terms. The simplest case was the homogeneous diagonal matrix, and the most complex choice was an unstructured matrix. We also used a heterogeneous diagonal matrix whose parameters are somewhere between the two models. Finally, we considered several factor analytic models. The reason to use the factor analytic structure is that it can separate genetic effects into common effects and environment-specific effects. In addition, the factor analytic structure is parsimonious and thus can substantially reduce the computational burden of the mixed-model analyses.