On Flexible finite polygenic models for multiple-trait evaluation

Finite polygenic models (FPM) might be an alternative to the infinitesimal model (TIM) for the genetic evaluation of pedigreed multiple-generation populations for multiple quantitative traits. I present a general flexible Bayesian method that includes the number of genes in the FPM as an additional random variable. Markov-chain Monte Carlo techniques such as Gibbs sampling and the reversible jump sampler are used for implementation. Sampling of genotypes of all genes in the FPM is done via the use of segregation indicators. A broad range of FPM models, some combined with TIM, are empirically tested for the estimation of variance components and the number of genes in the FPM. Four simulation scenarios were studied, including genetic models with 5 or 50 additive independent diallelic genes affecting the traits, and random selection or selection on one of the traits was performed. The results in this study were based on ten replicates per simulation scenario. In the case of random selection, uniform priors on additive gene effects led to posterior mean estimates of genetic variance that were positively correlated with the number of genes fitted in the FPM. In the case of trait selection, assuming normal priors on gene effects also led to genetic variance estimates for the selected trait that were negatively correlated with the number of genes in the FPM. This negative correlation was not observed for the unselected trait. Treating the number of genes in the FPM as random revealed a positive correlation between prior and posterior mean estimates of this number, but the prior hardly affected the posterior estimates of genetic variance. Posterior inferences about the number of genes should be considered to be indicative where trait selection seems to improve the power of distinguishing between TIM and FPM. Based on the results of this study, I suggest not replacing TIM by the FPM, but combining TIM and FPM with the number of genes treated as random, to facilitate a highly flexible and thereby robust method for variance component estimation in pedigreed populations. Further study is required to explore the full potential of these models under different genetic model assumption

Approximating a similarity matrix by a latent class model: A reappraisal of additive fuzzy clustering

Let Q be a given n×n square symmetric matrix of nonnegative elements between 0 and 1, similarities. Fuzzy clustering results in fuzzy assignment of individuals to K clusters. In additive fuzzy clustering, the n×K fuzzy memberships matrix P is found by least-squares approximation of the off-diagonal elements of Q by inner products of rows of P. By contrast, kernelized fuzzy c-means is not least-squares and requires an additional fuzziness parameter. The aim is to popularize additive fuzzy clustering by interpreting it as a latent class model, whereby the elements of Q are modeled as the probability that two individuals share the same class on the basis of the assignment probability matrix P. Two new algorithms are provided, a brute force genetic algorithm (differential evolution) and an iterative row-wise quadratic programming algorithm of which the latter is the more effective. Simulations showed that (1) the method usually has a unique solution, except in special cases, (2) both algorithms reached this solution from random restarts and (3) the number of clusters can be well estimated by AIC. Additive fuzzy clustering is computationally efficient and combines attractive features of both the vector model and the cluster mode

Searching for interacting QTL in related populations of an outbreeding species

Many important crop species are outbreeding. In outbreeding species the search for genes affecting traits is complicated by the fact that in a single cross up to four alleles may be present at each locus. This paper is concerned with the search for interacting quantitative trait loci (QTL) in populations which have been obtained by crossing a number of parents. It will be assumed that the parents are unrelated, but the methods can be extended easily to allow a pedigree structure. The approach has two goals: (1) finding QTL that are interacting with other loci and also loci which behave additively; (2) finding parents which segregate at two or more interacting QTL. Large populations obtained by crossing these parents can be used to study interactions in detail. QTL analysis is carried out by means of regression on predictions of QTL genotypes

Comparison of analyses of the QTLMAS XIII common dataset. II: QTL analysis

Background - Five participants of the QTL-MAS 2009 workshop applied QTL analyses to the workshop common data set which contained a time-related trait: cumulative yield. Underlying the trait were 18 QTLs for three parameters of a logistic growth curve that was used for simulating the trait. Methods - Different statistical models and methods were employed to detect QTLs and estimate position and effect sizes of QTLs. Here we compare the results with respect to the numbers of QTLs detected, estimated positions and percentage explained variance. Furthermore, limiting factors in the QTL detection are evaluated. Results - All QTLs for the asymptote and the scaling factor of the logistic curve were detected by at least one of the participants. Only one out of six of the QTLs for the inflection point was detected. None of the QTLs were detected by all participants. Dominant, epistatic and imprinted QTLs were reported while only additive QTLs were simulated. The power to map QTLs for the inflection point increased when more time points were added. Conclusions - For the detection of QTLs related to the asymptote and the scaling factor, there were no strong differences between the methods used here. Also, it did not matter much whether the time course data were analyzed per single time point or whether parameters of a growth curve were first estimated and then analyzed. In contrast, the power for detection of QTLs for the inflection point was very low and the frequency of time points appeared to be a limiting factor. This can be explained by a low accuracy in estimating the inflection point from a limited time range and a limited number of time points, and by the low correlation between the simulated values for this parameter and the phenotypic data available for the individual time point

Markov chain Monte Carlo for mapping a quantitative trait locus in outbred populations

A Bayesian approach is presented for mapping a quantitative trait locus (QTL) using the 'Fernando and Grossman' multivariate Normal approximation to QTL inheritance. For this model, a Bayesian implementation that includes QTL position is problematic because standard Markov chain Monte Carlo (MCMC) algorithms do not mix, i.e. the QTL position gets stuck in one marker interval. This is because of the dependence of the covariance structure for the QTL effects on the adjacent markers and may be typical of the 'Fernando and Grossman' model. A relatively new MCMC technique, simulated tempering, allows mixing and so makes possible inferences about QTL position based on marginal posterior probabilities. The model was implemented for estimating variance ratios and QTL position using a continuous grid of allowed positions and was applied to simulated data of a standard granddaughter design. The results showed a smooth mixing of QTL position after implementation of the simulated tempering sampler. In this implementation, map distance between QTL and its flanking markers was artificially stretched to reduce the dependence of markers and covariance. The method generalizes easily to more complicated applications and can ultimately contribute to QTL mapping in complex, heterogeneous, human, animal or plant populations

Exploratory QTL analyses of some pepper physiological traits in two environments

behind phenotypic differences and led to selection of genotypes having favourable traits. Continuous monitoring of environmental conditions has also become an accessible option. Rather than single trait evaluation, we would prefer smarter approaches capable of evaluating multiple, often correlated and time dependent traits simultaneously as a function of genes (QTLs) and environmental inputs, where we would The use of molecular breeding techniques has increased insight into the genetics like to include intermediate genomic information as well. In this paper, an exploratory QTL analysis over two environments was undertaken using available genetic and phenotypic data from segregating recombinant inbred lines (RIL) of pepper (Capsicum annuum). We focused on vegetative traits, e.g. stem length, speed of stem development, number of internodes etc. We seek to improve the estimation of allelic values of these traits under the two environments and determine possible QTL x E interaction. Almost identical QTLs are detected for each trait under the two environments but with varying LOD scores. No clear evidence was found for presence of QTL by environment interactions, despite differences in phenotypes and in magnitude of QTLs expression. Within the EU project SPICY (Voorrips et al., 2010 this issue), a larger number of environments will be studied and more advanced statistical analysis tools will be considered. The correlation between the traits will also be modelled. The identification of markers for the important QTL (Nicolaï et al., 2010 this issue) will improve the speed and accuracy of genomic prediction of these complex phenotype

Crop growth models for the -omics era: the EU-SPICY project

The prediction of phenotypic responses from genetic and environmental information is an area of active research in genetics, physiology and statistics. Rapidly increasing amounts of phenotypic information become available as a consequence of high throughput phenotyping techniques, while more and cheaper genotypic data follow from the development of new genotyping platforms. , A wide array of -omics data can be generated linking genotype and phenotype. Continuous monitoring of environmental conditions has become an accessible option. This wealth of data requires a drastic rethinking of the traditional quantitative genetic approach to modeling phenotypic variation in terms of genetic and environmental differences. Where in the past a single phenotypic trait was partitioned in a genetic and environmental component by analysis of variance techniques, nowadays we desire to model multiple, interrelated and often time dependent, phenotypic traits as a function of genes (QTLs) and environmental inputs, while we would like to include transcription information as well. The EU project 'Smart tools for Prediction and Improvement of Crop Yield' (KBBE-2008-211347), or SPICY, aims at the development of genotype-to-phenotype models that fully integrate genetic, genomic, physiological and environmental information to achieve accurate phenotypic predictions across a wide variety of genetic and environmental configurations. Pepper (Capsicum annuum) is chosen as the model crop, because of the availability of genetically characterized populations and of generic models for continuous crop growth and greenhouse production. In the presentation the objectives and structure of SPICY as well as its philosophy will be discussed

Consumer-Oriented New Product Development in Fruit Flavour Breeding : A Bayesian Approach

Taking consumer quality perceptions into account is very important for new-fruit product development in todays competitive food market. To this end, consumer-oriented quality improvement models like the Quality Guidance Model (QGM) have been proposed. Implementing such mod- els in the agro industry is challenging. We propose the use of Bayesian Structure Equation Modelling (SEM) for parameterizing the Quality Guid- ance Model, allowing for the integration of elicited expert knowledge. Such casual modelling would furnish important insights for determining the opti- mal fruit product in terms of consumer avour-quality perceptions. In the context of tomato breeding, where we have data about metabolites, sensory- panel judgments, and consumer avour-quality perceptions, we estimated a benchmark Bayesian SEM using non-informative priors, starting from an initial causal model derived from the data with a score-based Bayesian Network (BN) learning algorithm. The results so far have given some in- dication of the importance of accounting for consumer heterogeneity in the modeling process

Genomic Prediction with 12.5 Million SNPs for 5503 Holstein Friesian Bulls

This study reports the first preliminary results of genomic prediction with whole-genome sequence data (12,590,056 SNPs) for 5503 bulls with accurate phenotypes. Two methods were compared: genome-enabled best linear unbiased prediction (GBLUP) and a Bayesian approach (BSSVS). Results were compared with results using BovineHD genotypes (631,428 SNPs). Results were reported for somatic cell score, interval between first and last insemination, and protein yield. For all traits, and both methods genomic prediction with sequence data showed similar results compared to BovineHD and GBLUP showed similar results compared to BSSVS. However, it remains to be seen if reliability of BSSVS with sequence data will improve after more sampling cycles have been finished

Alternative Models in Genetic Analyses of Carcass Traits Measured by Ultrasonography in Guzerá cattle: A Bayesian Approach

The objective was to study alternative models for genetic analyses of carcass traits assessed by ultrasonography in Guzerá cattle. Data from 947 measurements (655 animals) of Rib-eye area (REA), rump fat thickness (RFT) and backfat thickness (BFT) were used. Finite polygenic models (FPM), infinitesimal polygenic models (IPM) and FPM combined with IPM (IPM + FPM) were empirically tested, adjusting for the effects of permanent environment, age and weight at measurement and the contemporary group. A Bayesian analysis using the computer package FlexQTLTM was adopted. The combined model adjusted to the data, allowing reliable genetic analyses of REA and BFT. For the RFT, the IPM model was the only one to have convergence and, in this case, the trait should be analyzed by a polygenic model. The presence of up to three major genes (MGs) controlling the expression of REA and two MGs for BFT was detected. The additive genetic action was over dominance to REA, and for BFT the dominance genetic action was greater. Heritability estimates, and respective standard error, adjusted for the combined model to REA were 0.15 (0.00025) for the polygenic fraction and 0.10 (0.00019) for the oligogenic fraction; for BFT was 0.19 (0.00027) and 0.13 (0.00025), respectively. Heritability of 0.17 (0.00028) was estimated for RFT when the model was adjusted to IPM. There are major genes segregating within the population studied for REA and BFT traits, thus, their genetic analyses must be studied considering oligogenic effects. The major gene effects detected for a small number of genes, may possibly help to increase the reliability in detecting chromosomal regions that explain and control the phenotypic expression of these traits, facilitating research on detection and validation of molecular markers in this populatio