Inference about quantitative traits under selection: a Bayesian revisitation for the post-genomic era.

BACKGROUND: Selection schemes distort inference when estimating differences between treatments or genetic associations between traits, and may degrade prediction of outcomes, e.g., the expected performance of the progeny of an individual with a certain genotype. If input and output measurements are not collected on random samples, inferences and predictions must be biased to some degree. Our paper revisits inference in quantitative genetics when using samples stemming from some selection process. The approach used integrates the classical notion of fitness with that of missing data. Treatment is fully Bayesian, with inference and prediction dealt with, in an unified manner. While focus is on animal and plant breeding, concepts apply to natural selection as well. Examples based on real data and stylized models illustrate how selection can be accounted for in four different situations, and sometimes without success. RESULTS: Our flexible "soft selection" setting helps to diagnose the extent to which selection can be ignored. The clear connection between probability of missingness and the concept of fitness in stylized selection scenarios is highlighted. It is not realistic to assume that a fixed selection threshold t holds in conceptual replication, as the chance of selection depends on observed and unobserved data, and on unequal amounts of information over individuals, aspects that a "soft" selection representation addresses explicitly. There does not seem to be a general prescription to accommodate potential distortions due to selection. In structures that combine cross-sectional, longitudinal and multi-trait data such as in animal breeding, balance is the exception rather than the rule. The Bayesian approach provides an integrated answer to inference, prediction and model choice under selection that goes beyond the likelihood-based approach, where breeding values are inferred indirectly. CONCLUSIONS: The approach used here for inference and prediction under selection may or may not yield the best possible answers. One may believe that selection has been accounted for diligently, but the central problem of whether statistical inferences are good or bad does not have an unambiguous solution. On the other hand, the quality of predictions can be gauged empirically via appropriate training-testing of competing methods.

Omics-based hybrid prediction in maize.

KEY MESSAGE: Complementing genomic data with other "omics" predictors can increase the probability of success for predicting the best hybrid combinations using complex agronomic traits. Accurate prediction of traits with complex genetic architecture is crucial for selecting superior candidates in animal and plant breeding and for guiding decisions in personalized medicine. Whole-genome prediction has revolutionized these areas but has inherent limitations in incorporating intricate epistatic interactions. Downstream "omics" data are expected to integrate interactions within and between different biological strata and provide the opportunity to improve trait prediction. Yet, predicting traits from parents to progeny has not been addressed by a combination of "omics" data. Here, we evaluate several "omics" predictors-genomic, transcriptomic and metabolic data-measured on parent lines at early developmental stages and demonstrate that the integration of transcriptomic with genomic data leads to higher success rates in the correct prediction of untested hybrid combinations in maize. Despite the high predictive ability of genomic data, transcriptomic data alone outperformed them and other predictors for the most complex heterotic trait, dry matter yield. An eQTL analysis revealed that transcriptomic data integrate genomic information from both, adjacent and distant sites relative to the expressed genes. Together, these findings suggest that downstream predictors capture physiological epistasis that is transmitted from parents to their hybrid offspring. We conclude that the use of downstream "omics" data in prediction can exploit important information beyond structural genomics for leveraging the efficiency of hybrid breeding.

QTL mapping under truncation selection in homozygous lines derived from biparental crosses.

In plant breeding, a large number of progenies that will be discarded later in the breeding process must be phenotyped and marker genotyped for conducting QTL analysis. In many cases, phenotypic preselection of lines could be useful. However, in QTL analyses even moderate preselection can have a significant effect on the power of QTL detection and estimation of effects of the target traits. In this study, we provide exact formulas for quantifying the change of allele frequencies within marker classes, expectations of marker contrasts and the variance of the marker contrasts under truncation selection, for the general case of two QTL affecting the target trait and a correlated trait. We focused on homozygous lines derived at random from biparental crosses. The effects of linkage between the marker and the QTL under selection as well as the effect of selection on a correlated trait can be quantified with the given formulas. Theoretical results clearly show that depending on the magnitude of QTL effects, high selection intensities can lead to a dramatic reduction in power of QTL detection and that approximations based on the infinitesimal model deviate substantially from exact solutions. The presented formulas are valuable for choosing appropriate selection intensity when performing QTL mapping experiments on the data on phenotypically preselected traits and enable the calculation and bias correction of the effects of QTL under selection. Application of our theory to experimental data revealed that selection-induced bias of QTL effects can be successfully corrected.

Unraveling epistasis with triple testcross progenies of near-isogenic lines.

Libraries of near-isogenic lines (NILs) are a powerful plant genetic resource to map quantitative trait loci (QTL). Nevertheless, QTL mapping with NILs is mostly restricted to genetic main effects. Here we propose a two-step procedure to map additive-by-additive digenic epistasis with NILs. In the first step, a generation means analysis of parents, their F(1) hybrid, and one-segment NILs and their triple testcross (TTC) progenies is used to identify in a one-dimensional scan loci exhibiting QTL-by-background interactions. In a second step, one-segment NILs with significant additive-by-additive background interactions are used to produce particular two-segment NILs to test for digenic epistatic interactions between these segments. We evaluated our approach by analyzing a random subset of a genomewide Arabidopsis thaliana NIL library for growth-related traits. The results of our experimental study illustrated the potential of the presented two-step procedure to map additive-by-additive digenic epistasis with NILs. Furthermore, our findings suggested that additive main effects as well as additive-by-additive digenic epistasis strongly influence the genetic architecture underlying growth-related traits of A. thaliana.

High congruency of QTL positions for heterosis of grain yield in three crosses of maize.

The genetic basis of heterosis in maize has been investigated in a number of studies but results have not been conclusive. Here, we compare quantitative trait loci (QTL) mapping results for grain yield, grain moisture, and plant height from three populations derived from crosses of the heterotic pattern Iowa Stiff Stalk Synthetic x Lancaster Sure Crop, investigated with the Design III, and analyzed with advanced statistical methods specifically developed to examine the genetic basis of mid-parent heterosis (MPH). In two populations, QTL analyses were conducted with a joint fit of linear transformations Z (1) (trait mean across pairs of backcross progenies) and Z (2) (half the trait difference between pairs of backcross progenies) to estimate augmented additive and augmented dominance effects of each QTL, as well as their ratio. QTL results for the third population were obtained from the literature. For Z (2) of grain yield, congruency of QTL positions was high across populations, and a large proportion of the genetic variance (~70%) was accounted for by QTL. This was not the case for Z (1) or the other two traits. Further, almost all congruent grain yield QTL were located in the same or an adjacent bin encompassing the centromere. We conclude that different alleles have been fixed in each heterotic pool, which in combination with allele(s) from the opposite heterotic pool lead to high MPH for grain yield. Their positive interactions very likely form the base line for the superior performance of the heterotic pattern under study.

Heterosis for biomass-related traits in Arabidopsis investigated by quantitative trait loci analysis of the triple testcross design with recombinant inbred lines.

Arabidopsis thaliana has emerged as a leading model species in plant genetics and functional genomics including research on the genetic causes of heterosis. We applied a triple testcross (TTC) design and a novel biometrical approach to identify and characterize quantitative trait loci (QTL) for heterosis of five biomass-related traits by (i) estimating the number, genomic positions, and genetic effects of heterotic QTL, (ii) characterizing their mode of gene action, and (iii) testing for presence of epistatic effects by a genomewide scan and marker x marker interactions. In total, 234 recombinant inbred lines (RILs) of Arabidopsis hybrid C24 x Col-0 were crossed to both parental lines and their F1 and analyzed with 110 single-nucleotide polymorphism (SNP) markers. QTL analyses were conducted using linear transformations Z1, Z2, and Z3 calculated from the adjusted entry means of TTC progenies. With Z1, we detected 12 QTL displaying augmented additive effects. With Z2, we mapped six QTL for augmented dominance effects. A one-dimensional genome scan with Z3 revealed two genomic regions with significantly negative dominance x additive epistatic effects. Two-way analyses of variance between marker pairs revealed nine digenic epistatic interactions: six reflecting dominance x dominance effects with variable sign and three reflecting additive x additive effects with positive sign. We conclude that heterosis for biomass-related traits in Arabidopsis has a polygenic basis with overdominance and/or epistasis being presumably the main types of gene action.

Quantitative trait locus mapping based on resampling in a vast maize testcross experiment and its relevance to quantitative genetics for complex traits.

From simulation studies it is known that the allocation of experimental resources has a crucial effect on power of QTL detection as well as on accuracy and precision of QTL estimates. In this study, we used a very large experimental data set composed of 976 F(5) maize testcross progenies evaluated in 19 environments and cross-validation to assess the effect of sample size (N), number of test environments (E), and significance threshold on the number of detected QTL, the proportion of the genotypic variance explained by them, and the corresponding bias of estimates for grain yield, grain moisture, and plant height. In addition, we used computer simulations to compare the usefulness of two cross-validation schemes for obtaining unbiased estimates of QTL effects. The maximum, validated genotypic variance explained by QTL in this study was 52.3% for grain moisture despite the large number of detected QTL, thus confirming the infinitesimal model of quantitative genetics. In both simulated and experimental data, the effect of sample size on power of QTL detection as well as on accuracy and precision of QTL estimates was large. The number of detected QTL and the proportion of genotypic variance explained by QTL generally increased more with increasing N than with increasing E. The average bias of QTL estimates and its range were reduced by increasing N and E. Cross-validation performed well with respect to yielding asymptotically unbiased estimates of the genotypic variance explained by QTL. On the basis of our findings, recommendations for planning of QTL mapping experiments and allocation of experimental resources are given.