Martin P. Boer

Bayesian analysis of complex traits in pedigreed plant populations

A Bayesian approach to analyze complex traits is presented that can help plant eneticists and breeders in exploiting the marker and phenotypic data on pedigreed populations as available from ongoing breeding programs. The statistical model for the quantitative trait may include non-genetic and genetic components. The latter component can be divided into QTL on known marker linkage groups, major genes and a polygenic component. The full probability model, prior assumptions on model variables are presented and criterion for model selection and posterior inferences are given. Simulated data on a known pedigreed population structure of the EU project HiDRAS was used to illustrate the use of the Bayesian approach to analyze complex traits. It was shown that estimates for QTL parameters were more accurate when non-genetic factors were included in the model and when a polygenic component was included when not all linkage groups were analyzed simultaneously. The Bayesian approach has been implemented into the software package FlexQTL and allows plant breeders explore their pedigreed populations for segregating QTL alleles that are relevant in their breeding program.

Marker-Based Estimation of Heritability in Immortal Populations

Heritability is a central parameter in quantitative genetics, from both an evolutionary and a breeding perspective. For plant traits heritability is traditionally estimated by comparing within- and between-genotype variability. This approach estimates broad-sense heritability and does not account for different genetic relatedness. With the availability of high-density markers there is growing interest in marker-based estimates of narrow-sense heritability, using mixed models in which genetic relatedness is estimated from genetic markers. Such estimates have received much attention in human genetics but are rarely reported for plant traits. A major obstacle is that current methodology and software assume a single phenotypic value per genotype, hence requiring genotypic means. An alternative that we propose here is to use mixed models at the individual plant or plot level. Using statistical arguments, simulations, and real data we investigate the feasibility of both approaches and how these affect genomic prediction with the best linear unbiased predictor and genome-wide association studies. Heritability estimates obtained from genotypic means had very large standard errors and were sometimes biologically unrealistic. Mixed models at the individual plant or plot level produced more realistic estimates, and for simulated traits standard errors were up to 13 times smaller. Genomic prediction was also improved by using these mixed models, with up to a 49% increase in accuracy. For genome-wide association studies on simulated traits, the use of individual plant data gave almost no increase in power. The new methodology is applicable to any complex trait where multiple replicates of individual genotypes can be scored. This includes important agronomic crops, as well as bacteria and fungi.

Studying the genetic basis of drought tolerance in sorghum by managed stress trials and adjustments for phenological and plant height differences

Managed environments in the form of well watered and water stressed trials were performed to study the genetic basis of grain yield and stay green in sorghum with the objective of validating previously detected QTL. As variations in phenology and plant height may influence QTL detection for the target traits, QTL for flowering time and plant height were introduced as cofactors in QTL analyses for yield and stay green. All but one of the flowering time QTL were detected near yield and stay green QTL. Similar co-localization was observed for two plant height QTL. QTL analysis for yield, using flowering time/plant height cofactors, led to yield QTL on chromosomes 2, 3, 6, 8 and 10. For stay green, QTL on chromosomes 3, 4, 8 and 10 were not related to differences in flowering time/plant height. The physical positions for markers in QTL regions projected on the sorghum genome suggest that the previously detected plant height QTL, Sb-HT9-1, and Dw2, in addition to the maturity gene, Ma5, had a major confounding impact on the expression of yield and stay green QTL. Co-localization between an apparently novel stay green QTL and a yield QTL on chromosome 3 suggests there is potential for indirect selection based on stay green to improve drought tolerance in sorghum. Our QTL study was carried out with a moderately sized population and spanned a limited geographic range, but still the results strongly emphasize the necessity of corrections for phenology in QTL mapping for drought tolerance traits in sorghum.

Gene and QTL detection in a three-way barley cross under selection by a mixed model with kinship information using SNPs

Quantitative trait locus (QTL) detection is commonly performed by analysis of designed segregating populations derived from two inbred parental lines, where absence of selection, mutation and genetic drift is assumed. Even for designed populations, selection cannot always be avoided, with as consequence varying correlation between genotypes instead of uniform correlation. Akin to linkage disequilibrium mapping, ignoring this type of genetic relatedness will increase the rate of false-positives. In this paper, we advocate using mixed models including genetic relatedness, or ‘kinship’ information for QTL detection in populations where selection forces operated. We demonstrate our case with a three-way barley cross, designed to segregate for dwarfing, vernalization and spike morphology genes, in which selection occurred. The population of 161 inbred lines was screened with 1,536 single nucleotide polymorphisms (SNPs), and used for gene and QTL detection. The coefficient of coancestry matrix was estimated based on the SNPs and imposed to structure the distribution of random genotypic effects. The model incorporating kinship, coancestry, information was consistently superior to the one without kinship (according to the Akaike information criterion). We show, for three traits, that ignoring the coancestry information results in an unrealistically high number of marker–trait associations, without providing clear conclusions about QTL locations. We used a number of widely recognized dwarfing and vernalization genes known to segregate in the studied population as landmarks or references to assess the agreement of the mapping results with a priori candidate gene expectations. Additional QTLs to the major genes were detected for all traits as well.

Mixed model approaches for the identification of QTLs within a maize hybrid breeding program

Two outlines for mixed model based approaches to quantitative trait locus (QTL) mapping in existing maize hybrid selection programs are presented: a restricted maximum likelihood (REML) and a Bayesian Markov Chain Monte Carlo (MCMC) approach. The methods use the in-silico-mapping procedure developed by Parisseaux and Bernardo (2004) as a starting point. The original single-point approach is extended to a multi-point approach that facilitates interval mapping procedures. For computational and conceptual reasons, we partition the full set of relationships from founders to parents of hybrids into two types of relations by defining so-called intermediate founders. QTL effects are defined in terms of those intermediate founders. Marker based identity by descent relationships between intermediate founders define structuring matrices for the QTL effects that change along the genome. The dimension of the vector of QTL effects is reduced by the fact that there are fewer intermediate founders than parents. Furthermore, additional reduction in the number of QTL effects follows from the identification of founder groups by various algorithms. As a result, we obtain a powerful mixed model based statistical framework to identify QTLs in genetic backgrounds relevant to the elite germplasm of a commercial breeding program. The identification of such QTLs will provide the foundation for effective marker assisted and genome wide selection strategies. Analyses of an example data set show that QTLs are primarily identified in different heterotic groups and point to complementation of additive QTL effects as an important factor in hybrid performance.

Numerical bifurcation analysis of a tri-trophic food web with omnivory.

We study the consequences of omnivory on the dynamic behaviour of a three species food web under chemostat conditions. The food web consists of a prey consuming a nutrient, a predator consuming a prey and an omnivore which preys on the predator and the prey. For each trophic level an ordinary differential equation describes the biomass density in the reactor. The hyperbolic functional response for single and multi prey species figures in the description of the trophic interactions. There are two limiting cases where the omnivore is a specialist; a food chain where the omnivore does not consume the prey and competition where the omnivore does not prey on the predator. We use bifurcation analysis to study the long-term dynamic behaviour for various degrees of omnivory. Attractors can be equilibria, limit cycles or chaotic behaviour depending on the control parameters of the chemostat. Often multiple attractor occur. In this paper we will discuss community assembly. That is, we analyze how the trophic structure of the food web evolves following invasion where a new invader is introduced one at the time. Generally, with an invasion, the invader settles itself and persists with all other species, however, the invader may also replace another species. We will show that the food web model has a global bifurcation, being a heteroclinic connection from a saddle equilibrium to a limit cycle of saddle type. This global bifurcation separates regions in the bifurcation diagram with different attractors to which the system evolves after invasion. To investigate the consequences of omnivory we will focus on invasion of the omnivore. This simplifies the analysis considerably, for the end-point of the assembly sequence is then unique. A weak interaction of the omnivore with the prey combined with a stronger interaction with the predator seems advantageous.

QTL linkage analysis of connected populations using ancestral marker and pedigree information

The common assumption in quantitative trait locus (QTL) linkage mapping studies that parents of multiple connected populations are unrelated is unrealistic for many plant breeding programs. We remove this assumption and propose a Bayesian approach that clusters the alleles of the parents of the current mapping populations from locus-specific identity by descent (IBD) matrices that capture ancestral marker and pedigree information. Moreover, we demonstrate how the parental IBD data can be incorporated into a QTL linkage analysis framework by using two approaches: a Threshold IBD model (TIBD) and a Latent Ancestral Allele Model (LAAM). The TIBD and LAAM models are empirically tested via numerical simulation based on the structure of a commercial maize breeding program. The simulations included a pilot dataset with closely linked QTL on a single linkage group and 100 replicated datasets with five linkage groups harboring four unlinked QTL. The simulation results show that including parental IBD data (similarly for TIBD and LAAM) significantly improves the power and particularly accuracy of QTL mapping, e.g., position, effect size and individuals’ genotype probability without significantly increasing computational demand.

A mixed model QTL analysis for a complex cross population consisting of a half diallel of two-way hybrids in Arabidopsis thaliana: analysis of simulated data

To improve QTL detection power for QTL main effects and interactions and QTL mapping resolution, new types of multi-founder crossing populations are created in plants and animals. Some recent examples are complex intercrossed populations in mice and Arabidopsis thaliana. For the latter, a set of eight accessions was intercrossed to produce four two-way hybrids that were subsequently intercrossed again in a half diallel fashion leading to six subpopulations of four-way hybrids, each subpopulation containing 100 individuals. Within each subpopulation, individuals were inbred for four generations via single seed descent. QTL mapping in the complex crosses requires new statistical tools. We present a first sketch of a QTL mapping methodology for the complex cross in Arabidopsis based on mixed model analyses. As experimental data were not yet available, we illustrate our methodology on simulated but realistic data.

Genotype–environment interactions affecting preflowering physiological and morphological traits of Brassica rapa grown in two watering regimes

Plant growth and productivity are greatly affected by drought, which is likely to become more threatening with the predicted global temperature increase. Understanding the genetic architecture of complex quantitative traits and their interaction with water availability may lead to improved crop adaptation to a wide range of environments. Here, the genetic basis of 20 physiological and morphological traits is explored by describing plant performance and growth in a Brassica rapa recombinant inbred line (RIL) population grown on a sandy substrate supplemented with nutrient solution, under control and drought conditions. Altogether, 54 quantitative trait loci (QTL) were identified, of which many colocated in 11 QTL clusters. Seventeen QTL showed significant QTL–environment interaction (Q×E), indicating genetic variation for phenotypic plasticity. Of the measured traits, only hypocotyl length did not show significant genotype–environment interaction (G×E) in both environments in all experiments. Correlation analysis showed that, in the control environment, stomatal conductance was positively correlated with total leaf dry weight (DW) and aboveground DW, whereas in the drought environment, stomatal conductance showed a significant negative correlation with total leaf DW and aboveground DW. This correlation was explained by antagonistic fitness effects in the drought environment, controlled by a QTL cluster on chromosome A7. These results demonstrate that Q×E is an important component of the genetic variance and can play a great role in improving drought tolerance in future breeding programmes.

Identity-by-Descent Matrix Decomposition Using Latent Ancestral Allele Models

Genetic linkage and association studies are empowered by proper modeling of relatedness among individuals. Such relatedness can be inferred from marker and/or pedigree information. In this study, the genetic relatedness among n inbred individuals at a particular locus is expressed as an n × n square matrix Q. The elements of Q are identity-by-descent probabilities, that is, probabilities that two individuals share an allele descended from a common ancestor. In this representation the definition of the ancestral alleles and their number remains implicit. For human inspection and further analysis, an explicit representation in terms of the ancestral allele origin and the number of alleles is desirable. To this purpose, we decompose the matrix Q by a latent class model with K classes (latent ancestral alleles). Let P be an n × K matrix with assignment probabilities of n individuals to K classes constrained such that every element is nonnegative and each row sums to 1. The problem then amounts to approximating Q by PPT, while disregarding the diagonal elements. This is not an eigenvalue problem because of the constraints on P. An efficient algorithm for calculating P is provided. We indicate the potential utility of the latent ancestral allele model. For representative locus-specific Q matrices constructed for a set of maize inbreds, the proposed model recovered the known ancestry.