Patrik Waldmann

BAPS 2: enhanced possibilities for the analysis of genetic population structure

UNLABELLED Bayesian statistical methods based on simulation techniques have recently been shown to provide powerful tools for the analysis of genetic population structure. We have previously developed a Markov chain Monte Carlo (MCMC) algorithm for characterizing genetically divergent groups based on molecular markers and geographical sampling design of the dataset. However, for large-scale datasets such algorithms may get stuck to local maxima in the parameter space. Therefore, we have modified our earlier algorithm to support multiple parallel MCMC chains, with enhanced features that enable considerably faster and more reliable estimation compared to the earlier version of the algorithm. We consider also a hierarchical tree representation, from which a Bayesian model-averaged structure estimate can be extracted. The algorithm is implemented in a computer program that features a user-friendly interface and built-in graphics. The enhanced features are illustrated by analyses of simulated data and an extensive human molecular dataset. AVAILABILITY Freely available at http://www.rni.helsinki.fi/~jic/bapspage.html.

Hierarchical Spatial Process Models for Multiple Traits in Large Genetic Trials

This article expands upon recent interest in Bayesian hierarchical models in quantitative genetics by developing spatial process models for inference on additive and dominance genetic variance within the context of large spatially referenced trial datasets of multiple traits of interest. Direct application of such multivariate models to large spatial datasets is often computationally infeasible because of cubic order matrix algorithms involved in estimation. The situation is even worse in Markov chain Monte Carlo (MCMC) contexts where such computations are performed for several thousand iterations. Here, we discuss approaches that help obviate these hurdles without sacrificing the richness in modeling. For genetic effects, we demonstrate how an initial spectral decomposition of the relationship matrices negates the expensive matrix inversions required in previously proposed MCMC methods. For spatial effects we discuss a multivariate predictive process that reduces the computational burden by projecting the original process onto a subspace generated by realizations of the original process at a specified set of locations (or knots). We illustrate the proposed methods using a synthetic dataset with multivariate additive and dominant genetic effects and anisotropic spatial residuals, and a large dataset from a scots pine (Pinus sylvestris L.) progeny study conducted in northern Sweden. Our approaches enable us to provide a comprehensive analysis of this large trial which amply demonstrates that, in addition to violating basic assumptions of the linear model, ignoring spatial effects can result in downwardly biased measures of heritability.

Comparing Bayesian estimates of genetic differentiation of molecular markers and quantitative traits: an application to Pinus sylvestris

Comparison of the level of differentiation at neutral molecular markers (estimated as FST or GST) with the level of differentiation at quantitative traits (estimated as QST) has become a standard tool for inferring that there is differential selection between populations. We estimated QST of timing of bud set from a latitudinal cline of Pinus sylvestris with a Bayesian hierarchical variance component method utilizing the information on the pre-estimated population structure from neutral molecular markers. Unfortunately, the between-family variances differed substantially between populations that resulted in a bimodal posterior of QST that could not be compared in any sensible way with the unimodal posterior of the microsatellite FST. In order to avoid publishing studies with flawed QST estimates, we recommend that future studies should present heritability estimates for each trait and population. Moreover, to detect variance heterogeneity in frequentist methods (ANOVA and REML), it is of essential importance to check also that the residuals are normally distributed and do not follow any systematically deviating trends.

Efficient Markov Chain Monte Carlo Implementation of Bayesian Analysis of Additive and Dominance Genetic Variances in Noninbred Pedigrees

Accurate and fast computation of quantitative genetic variance parameters is of great importance in both natural and breeding populations. For experimental designs with complex relationship structures it can be important to include both additive and dominance variance components in the statistical model. In this study, we introduce a Bayesian Gibbs sampling approach for estimation of additive and dominance genetic variances in the traditional infinitesimal model. The method can handle general pedigrees without inbreeding. To optimize between computational time and good mixing of the Markov chain Monte Carlo (MCMC) chains, we used a hybrid Gibbs sampler that combines a single site and a blocked Gibbs sampler. The speed of the hybrid sampler and the mixing of the single-site sampler were further improved by the use of pretransformed variables. Two traits (height and trunk diameter) from a previously published diallel progeny test of Scots pine (Pinus sylvestris L.) and two large simulated data sets with different levels of dominance variance were analyzed. We also performed Bayesian model comparison on the basis of the posterior predictive loss approach. Results showed that models with both additive and dominance components had the best fit for both height and diameter and for the simulated data with high dominance. For the simulated data with low dominance, we needed an informative prior to avoid the dominance variance component becoming overestimated. The narrow-sense heritability estimates in the Scots pine data were lower compared to the earlier results, which is not surprising because the level of dominance variance was rather high, especially for diameter. In general, the hybrid sampler was considerably faster than the blocked sampler and displayed better mixing properties than the single-site sampler.

Simultaneous estimation of multiple quantitative trait loci and growth curve parameters through hierarchical Bayesian modeling

A novel hierarchical quantitative trait locus (QTL) mapping method using a polynomial growth function and a multiple-QTL model (with no dependence in time) in a multitrait framework is presented. The method considers a population-based sample where individuals have been phenotyped (over time) with respect to some dynamic trait and genotyped at a given set of loci. A specific feature of the proposed approach is that, instead of an average functional curve, each individual has its own functional curve. Moreover, each QTL can modify the dynamic characteristics of the trait value of an individual through its influence on one or more growth curve parameters. Apparent advantages of the approach include: (1) assumption of time-independent QTL and environmental effects, (2) alleviating the necessity for an autoregressive covariance structure for residuals and (3) the flexibility to use variable selection methods. As a by-product of the method, heritabilities and genetic correlations can also be estimated for individual growth curve parameters, which are considered as latent traits. For selecting trait-associated loci in the model, we use a modified version of the well-known Bayesian adaptive shrinkage technique. We illustrate our approach by analysing a sub sample of 500 individuals from the simulated QTLMAS 2009 data set, as well as simulation replicates and a real Scots pine (Pinus sylvestris) data set, using temporal measurements of height as dynamic trait of interest.

EASY AND FLEXIBLE BAYESIAN INFERENCE OF QUANTITATIVE GENETIC PARAMETERS

There has been a tremendous advancement of Bayesian methodology in quantitative genetics and evolutionary biology. Still, there are relatively few publications that apply this methodology, probably because the availability of multipurpose and user-friendly software is somewhat limited. It is here described how only a few rows of code of the well-developed and very flexible Bayesian software WinBUGS (Lunn et al. 2000) can be used for inference of the additive polygenic variance and heritabilty in pedigrees of general design. The presented code is illustrated by application to an earlier published dataset of Scots pine.

Genetic changes in flowering and morphology in response to adaptation to a high-latitude environment in Arabidopsis lyrata

BACKGROUND AND AIMS The adaptive plastic reactions of plant populations to changing climatic factors, such as winter temperatures and photoperiod, have changed during range shifts after the last glaciation. Timing of flowering is an adaptive trait regulated by environmental cues. Its genetics has been intensively studied in annual plants, but in perennials it is currently not well characterized. This study examined the genetic basis of differentiation in flowering time, morphology, and their plastic responses to vernalization in two locally adapted populations of the perennial Arabidopsis lyrata: (1) to determine whether the two populations differ in their vernalization responses for flowering phenology and morphology; and (2) to determine the genomic areas governing differentiation and vernalization responses. METHODS Two A. lyrata populations, from central Europe and Scandinavia, were grown in growth-chamber conditions with and without cold treatment. A QTL analysis was performed to find genomic regions that interact with vernalization. KEY RESULTS The population from central Europe flowered more rapidly and invested more in inflorescence growth than the population from alpine Scandinavia, especially after vernalization. The alpine population had consistently a low number of inflorescences and few flowers, suggesting strong constraints due to a short growing season, but instead had longer leaves and higher leaf rosettes. QTL mapping in the F2 population revealed genomic regions governing differentiation in flowering time and morphology and, in some cases, the allelic effects from the two populations on a trait were influenced by vernalization (QTL × vernalization interactions). CONCLUSIONS The results indicate that many potentially adaptive genetic changes have occurred during colonization; the two populations have diverged in their plastic responses to vernalization in traits closely connected to fitness through changes in many genomic areas.

The effect of inbreeding on fluctuating asymmetry in Scabiosa canescens (Dipsacaceae)

Developmental instability and fluctuating asymmetry (FA) describe the inability of organisms to correct for random accidents under development and has become a major but controversial topic in evolutionary biology. Theoretical models predict that the level of FA should increase as a result of inbreeding, but empirical results are ambiguous. Moreover, the relationship between fitness and FA is still debated. In the current study, plants from a population of Scabiosa canescens, a locally rare species in southern Sweden, were raised under uniform growth conditions to examine the effects of one-generation of selfing and outcrossing on FA in flower morphology. The level of flower FA was significantly higher (p = 0.038) for inbred progeny than for offspring derived from outcross pollinations. Given that earlier studies of this species have found no negative relation between heterozygosity and FA, the results support the conclusion that expression of deleterious recessive alleles are responsible for the increase of FA. There was no correlation between FA and estimates of five fitness-related traits when estimated at the individual level. However, a companion study found significant inbreeding depression for all fitness traits, and a negative association between FA and fitness could therefore be asserted at the treatment level (inbred/outbred progeny). Hence, FA seems to be useful to predict inbreeding depression in S. canescens, but specific individuals with high fitness cannot be identified based on their FA levels.

Bayesian inference of genetic parameters based on conditional decompositions of multivariate normal distributions.

It is widely recognized that the mixed linear model is an important tool for parameter estimation in the analysis of complex pedigrees, which includes both pedigree and genomic information, and where mutually dependent genetic factors are often assumed to follow multivariate normal distributions of high dimension. We have developed a Bayesian statistical method based on the decomposition of the multivariate normal prior distribution into products of conditional univariate distributions. This procedure permits computationally demanding genetic evaluations of complex pedigrees, within the user-friendly computer package WinBUGS. To demonstrate and evaluate the flexibility of the method, we analyzed two example pedigrees: a large noninbred pedigree of Scots pine (Pinus sylvestris L.) that includes additive and dominance polygenic relationships and a simulated pedigree where genomic relationships have been calculated on the basis of a dense marker map. The analysis showed that our method was fast and provided accurate estimates and that it should therefore be a helpful tool for estimating genetic parameters of complex pedigrees quickly and reliably.

Optimization of selection contribution and mate allocations in monoecious tree breeding populations

BackgroundThe combination of optimized contribution dynamic selection and various mating schemes was investigated over seven generations for a typical tree breeding scenario. The allocation of mates was optimized using a simulated annealing algorithm for various object functions including random mating (RM), positive assortative mating (PAM) and minimization of pair-wise coancestry between mates (MCM) all combined with minimization of variance in family size and coancestry. The present study considered two levels of heritability (0.05 and 0.25), two restrictions on relatedness (group coancestry; 1 and 2%) and two maximum permissible numbers of crosses in each generation (100 and 400). The infinitesimal genetic model was used to simulate the genetic architecture of the trait that was the subject of selection. A framework of the long term genetic contribution of ancestors was used to examine the impacts of the mating schemes on population parameters.ResultsMCM schemes produced on average, an increased rate of genetic gain in the breeding population, although the difference between schemes was small but significant after seven generations (up to 7.1% more than obtained with RM). In addition, MCM reduced the level of inbreeding by as much as 37% compared with RM, although the rate of inbreeding was similar after three generations of selection. PAM schemes yielded levels of genetic gain similar to those produced by RM, but the increase in the level of inbreeding was substantial (up to 43%).ConclusionThe main reason why MCM schemes yielded higher genetic gains was the improvement in managing the long term genetic contribution of founders in the population; this was achieved by connecting unrelated families. In addition, the accumulation of inbreeding was reduced by MCM schemes since the variance in long term genetic contributions of founders was smaller than in the other schemes. Consequently, by combining an MCM scheme with an algorithm that optimizes contributions of the selected individuals, a higher long term response is obtained while reducing the risk within the breeding program.