The genomics of linkage drag in inbred lines of sunflower.

Crop wild relatives represent valuable sources of alleles for crop improvement, including adaptation to climate change and emerging diseases. However, introgressions from wild relatives might have deleterious effects on desirable traits, including yield, due to linkage drag. Here, we analyzed the genomic and phenotypic impacts of wild introgressions in inbred lines of cultivated sunflower to estimate the impacts of linkage drag. First, we generated reference sequences for seven cultivated and one wild sunflower genotype, as well as improved assemblies for two additional cultivars. Next, relying on previously generated sequences from wild donor species, we identified introgressions in the cultivated reference sequences, as well as the sequence and structural variants they contain. We then used a ridge-regression best linear unbiased prediction (BLUP) model to test the effects of the introgressions on phenotypic traits in the cultivated sunflower association mapping population. We found that introgression has introduced substantial sequence and structural variation into the cultivated sunflower gene pool, including >3,000 new genes. While introgressions reduced genetic load at protein-coding sequences, they mostly had negative impacts on yield and quality traits. Introgressions found at high frequency in the cultivated gene pool had larger effects than low-frequency introgressions, suggesting that the former likely were targeted by artificial selection. Also, introgressions from more distantly related species were more likely to be maladaptive than those from the wild progenitor of cultivated sunflower. Thus, breeding efforts should focus, as far as possible, on closely related and fully compatible wild relatives.

Gene regulatory network inference methodology for genomic and transcriptomic data acquired in genetically related heterozygote individuals.

MOTIVATION: Inferring gene regulatory networks in non-independent genetically related panels is a methodological challenge. This hampers evolutionary and biological studies using heterozygote individuals such as in wild sunflower populations or cultivated hybrids. RESULTS: First, we simulated 100 datasets of gene expressions and polymorphisms, displaying the same gene expression distributions, heterozygosities and heritabilities as in our dataset including 173 genes and 353 genotypes measured in sunflower hybrids. Secondly, we performed a meta-analysis based on six inference methods [least absolute shrinkage and selection operator (Lasso), Random Forests, Bayesian Networks, Markov Random Fields, Ordinary Least Square and fast inference of networks from directed regulation (Findr)] and selected the minimal density networks for better accuracy with 64 edges connecting 79 genes and 0.35 area under precision and recall (AUPR) score on average. We identified that triangles and mutual edges are prone to errors in the inferred networks. Applied on classical datasets without heterozygotes, our strategy produced a 0.65 AUPR score for one dataset of the DREAM5 Systems Genetics Challenge. Finally, we applied our method to an experimental dataset from sunflower hybrids. We successfully inferred a network composed of 105 genes connected by 106 putative regulations with a major connected component. AVAILABILITY AND IMPLEMENTATION: Our inference methodology dedicated to genomic and transcriptomic data is available at https://forgemia.inra.fr/sunrise/inference_methods. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Discovery of interesting new polymorphisms in a sugar beet (elite [Formula: see text] exotic) progeny by comparison with an elite panel.

KEY MESSAGE: The comparison of QTL detection performed on an elite panel and an (elite [Formula: see text] exotic) progeny shows that introducing exotic germplasm into breeding programs can bring new interesting allelic diversity. Selection of stable varieties producing the highest amount of extractable sugar per hectare (ha), resistant to diseases, and respecting environmental criteria is undoubtedly the main target for sugar beet breeding. As sodium, potassium, and [Formula: see text]-amino nitrogen in sugar beets are the impurities that have the biggest negative impact on white sugar extraction, it is interesting to reduce their concentration in further varieties. However, domestication history and strong selection pressures have affected the genetic diversity needed to achieve this goal. In this study, quantitative trait locus (QTL) detection was performed on two populations, an (elite [Formula: see text] exotic) sugar beet progeny and an elite panel, to find potentially new interesting regions brought by the exotic accession. The three traits linked with impurities content were studied. Some QTLs were detected in both populations, the majority in the elite panel because of most statistical power. Some of the QTLs were colocated and had favorable effect in the progeny since the exotic allele was linked with a decrease in the impurity content. A few number of favorable QTLs were detected in the progeny, only. Consequently, introgressing exotic genetic material into sugar beet breeding programs can allow the incorporation of new interesting alleles.

Comparison of GWAS models to identify non-additive genetic control of flowering time in sunflower hybrids.

KEY MESSAGE: This study compares five models of GWAS, to show the added value of non-additive modeling of allelic effects to identify genomic regions controlling flowering time of sunflower hybrids. Genome-wide association studies are a powerful and widely used tool to decipher the genetic control of complex traits. One of the main challenges for hybrid crops, such as maize or sunflower, is to model the hybrid vigor in the linear mixed models, considering the relatedness between individuals. Here, we compared two additive and three non-additive association models for their ability to identify genomic regions associated with flowering time in sunflower hybrids. A panel of 452 sunflower hybrids, corresponding to incomplete crossing between 36 male lines and 36 female lines, was phenotyped in five environments and genotyped for 2,204,423 SNPs. Intra-locus effects were estimated in multi-locus models to detect genomic regions associated with flowering time using the different models. Thirteen quantitative trait loci were identified in total, two with both model categories and one with only non-additive models. A quantitative trait loci on LG09, detected by both the additive and non-additive models, is located near a GAI homolog and is presented in detail. Overall, this study shows the added value of non-additive modeling of allelic effects for identifying genomic regions that control traits of interest and that could participate in the heterosis observed in hybrids.

Genetic control of plasticity of oil yield for combined abiotic stresses using a joint approach of crop modelling and genome-wide association.

Understanding the genetic basis of phenotypic plasticity is crucial for predicting and managing climate change effects on wild plants and crops. Here, we combined crop modelling and quantitative genetics to study the genetic control of oil yield plasticity for multiple abiotic stresses in sunflower. First, we developed stress indicators to characterize 14 environments for three abiotic stresses (cold, drought and nitrogen) using the SUNFLO crop model and phenotypic variations of three commercial varieties. The computed plant stress indicators better explain yield variation than descriptors at the climatic or crop levels. In those environments, we observed oil yield of 317 sunflower hybrids and regressed it with three selected stress indicators. The slopes of cold stress norm reaction were used as plasticity phenotypes in the following genome-wide association study. Among the 65 534 tested Single Nucleotide Polymorphisms (SNPs), we identified nine quantitative trait loci controlling oil yield plasticity to cold stress. Associated single nucleotide polymorphisms are localized in genes previously shown to be involved in cold stress responses: oligopeptide transporters, lipid transfer protein, cystatin, alternative oxidase or root development. This novel approach opens new perspectives to identify genomic regions involved in genotype-by-environment interaction of a complex traits to multiple stresses in realistic natural or agronomical conditions.

Molecular diversity of sunflower populations maintained as genetic resources is affected by multiplication processes and breeding for major traits.

KEY MESSAGE: SNP genotyping of 114 cultivated sunflower populations showed that the multiplication process and the main traits selected during breeding of sunflower cultivars drove molecular diversity of the populations. The molecular diversity in a set of 114 cultivated sunflower populations was studied by single-nucleotide polymorphism genotyping. These populations were chosen as representative of the 400 entries in the INRA collection received or developed between 1962 and 2011 and made up of land races, open-pollinated varieties, and breeding pools. Mean allele number varied from 1.07 to 1.90. Intra-population variability was slightly reduced according to the number of multiplications since entry but some entries were probably largely homozygous when received. A principal component analysis was used to study inter-population variability. The first 3 axes accounted for 17% of total intra-population variability. The first axis was significantly correlated with seed oil content, more closely than just the distinction between oil and confectionary types. The second axis was related to the presence or absence of restorer genes and the third axis to flowering date and possibly to adaptation to different climates. Our results provide arguments highlighting the effect of the maintenance process on the within population genetic variability as well as on the impact of breeding for major agronomic traits on the between population variability of the collection. Propositions are made to improve sunflower population maintenance procedures to keep maximum genetic variability for future breeding.

Genetic diversity, linkage disequilibrium and power of a large grapevine (Vitis vinifera L) diversity panel newly designed for association studies.

BACKGROUND: As for many crops, new high-quality grapevine varieties requiring less pesticide and adapted to climate change are needed. In perennial species, breeding is a long process which can be speeded up by gaining knowledge about quantitative trait loci linked to agronomic traits variation. However, due to the long juvenile period of these species, establishing numerous highly recombinant populations for high resolution mapping is both costly and time-consuming. Genome wide association studies in germplasm panels is an alternative method of choice, since it allows identifying the main quantitative trait loci with high resolution by exploiting past recombination events between cultivars. Such studies require adequate panel design to represent most of the available genetic and phenotypic diversity. Assessing linkage disequilibrium extent and panel power is also needed to determine the marker density required for association studies. RESULTS: Starting from the largest grapevine collection worldwide maintained in Vassal (France), we designed a diversity panel of 279 cultivars with limited relatedness, reflecting the low structuration in three genetic pools resulting from different uses (table vs wine) and geographical origin (East vs West), and including the major founders of modern cultivars. With 20 simple sequence repeat markers and five quantitative traits, we showed that our panel adequately captured most of the genetic and phenotypic diversity existing within the entire Vassal collection. To assess linkage disequilibrium extent and panel power, we genotyped single nucleotide polymorphisms: 372 over four genomic regions and 129 distributed over the whole genome. Linkage disequilibrium, measured by correlation corrected for kinship, reached 0.2 for a physical distance between 9 and 458 Kb depending on genetic pool and genomic region, with varying size of linkage disequilibrium blocks. This panel achieved reasonable power to detect associations between traits with high broad-sense heritability (> 0.7) and causal loci with intermediate allelic frequency and strong effect (explaining > 10 % of total variance). CONCLUSIONS: Our association panel constitutes a new, highly valuable resource for genetic association studies in grapevine, and deserves dissemination to diverse field and greenhouse trials to gain more insight into the genetic control of many agronomic traits and their interaction with the environment.

Breeding patterns and cultivated beets origins by genetic diversity and linkage disequilibrium analyses.

KEY MESSAGE: Genetic diversity in worldwide population of beets is strongly affected by the domestication history, and the comparison of linkage disequilibrium in worldwide and elite populations highlights strong selection pressure. Genetic relationships and linkage disequilibrium (LD) were evaluated in a set of 2035 worldwide beet accessions and in another of 1338 elite sugar beet lines, using 320 and 769 single nucleotide polymorphisms, respectively. The structures of the populations were analyzed using four different approaches. Within the worldwide population, three of the methods gave a very coherent picture of the population structure. Fodder beet and sugar beet accessions were grouped together, separated from garden beets and sea beets, reflecting well the origins of beet domestication. The structure of the elite panel, however, was less stable between clustering methods, which was probably because of the high level of genetic mixing in breeding programs. For the linkage disequilibrium analysis, the usual measure (r (2)) was used, and compared with others that correct for population structure and relatedness (r S (2) , r V (2) , r VS (2)). The LD as measured by r (2) persisted beyond 10 cM within the elite panel and fell below 0.1 after less than 2 cM in the worldwide population, for almost all chromosomes. With correction for relatedness, LD decreased under 0.1 by 1 cM for almost all chromosomes in both populations, except for chromosomes 3 and 9 within the elite panel. In these regions, the larger extent of LD could be explained by strong selection pressure.

Clusthaplo: a plug-in for MCQTL to enhance QTL detection using ancestral alleles in multi-cross design.

KEY MESSAGE: We enhance power and accuracy of QTL mapping in multiple related families, by clustering the founders of the families on their local genomic similarity. MCQTL is a linkage mapping software application that allows the joint QTL mapping of multiple related families. In its current implementation, QTLs are modeled with one or two parameters for each parent that is a founder of the multi-cross design. The higher the number of parents, the higher the number of model parameters which can impact the power and the accuracy of the mapping. We propose to make use of the availability of denser and denser genotyping information on the founders to lessen the number of MCQTL parameters and thus boost the QTL discovery. We developed clusthaplo, an R package ( http://cran.r-project.org/web/packages/clusthaplo/index.html ), which aims to cluster haplotypes using a genomic similarity that reflects the probability of sharing the same ancestral allele. Computed in a sliding window along the genome and followed by a clustering method, the genomic similarity allows the local clustering of the parent haplotypes. Our assumption is that the haplotypes belonging to the same class transmit the same ancestral allele. So their putative QTL allelic effects can be modeled with the same parameter, leading to a parsimonious model, that is plugged in MCQTL. Intensive simulations using three maize data sets showed the significant gain in power and in accuracy of the QTL mapping with the ancestral allele model compared to the classical MCQTL model. MCQTL_LD (clusthaplo outputs plug in MCQTL) is a versatile and powerful tool for QTL mapping in multiple related families that makes use of linkage and linkage disequilibrium (web site http://carlit.toulouse.inra.fr/MCQTL/ ).

Combined linkage and association mapping of flowering time in Sunflower (Helianthus annuus L.).

Association mapping and linkage mapping were used to identify quantitative trait loci (QTL) and/or causative mutations involved in the control of flowering time in cultivated sunflower Helianthus annuus. A panel of 384 inbred lines was phenotyped through testcrosses with two tester inbred lines across 15 location × year combinations. A recombinant inbred line (RIL) population comprising 273 lines was phenotyped both per se and through testcrosses with one or two testers in 16 location × year combinations. In the association mapping approach, kinship estimation using 5,923 single nucleotide polymorphisms was found to be the best covariate to correct for effects of panel structure. Linkage disequilibrium decay ranged from 0.08 to 0.26 cM for a threshold of 0.20, after correcting for structure effects, depending on the linkage group (LG) and the ancestry of inbred lines. A possible hitchhiking effect is hypothesized for LG10 and LG08. A total of 11 regions across 10 LGs were found to be associated with flowering time, and QTLs were mapped on 11 LGs in the RIL population. Whereas eight regions were demonstrated to be common between the two approaches, the linkage disequilibrium approach did not detect a documented QTL that was confirmed using the linkage mapping approach.

Sunflower Hybrids and Inbred Lines Adopt Different Physiological Strategies and Proteome Responses to Cope with Water Deficit.

Sunflower is a hybrid crop that is considered moderately drought-tolerant and adapted to new cropping systems required for the agro-ecological transition. Here, we studied the impact of hybridity status (hybrids vs. inbred lines) on the responses to drought at the molecular and eco-physiological level exploiting publicly available datasets. Eco-physiological traits and leaf proteomes were measured in eight inbred lines and their sixteen hybrids grown in the high-throughput phenotyping platform Phenotoul-Heliaphen. Hybrids and parental lines showed different growth strategies: hybrids grew faster in the absence of water constraint and arrested their growth more abruptly than inbred lines when subjected to water deficit. We identified 471 differentially accumulated proteins, of which 256 were regulated by drought. The amplitude of up- and downregulations was greater in hybrids than in inbred lines. Our results show that hybrids respond more strongly to water deficit at the molecular and eco-physiological levels. Because of presence/absence polymorphism, hybrids potentially contain more genes than their parental inbred lines. We propose that detrimental homozygous mutations and the lower number of genes in inbred lines lead to a constitutive defense mechanism that may explain the lower growth of inbred lines under well-watered conditions and their lower reactivity to water deficit.

Effect of population structure corrections on the results of association mapping tests in complex maize diversity panels.

Association mapping of sequence polymorphisms underlying the phenotypic variability of quantitative agronomical traits is now a widely used method in plant genetics. However, due to the common presence of a complex genetic structure within the plant diversity panels, spurious associations are expected to be highly frequent. Several methods have thus been suggested to control for panel structure. They mainly rely on ad hoc criteria for selecting the number of ancestral groups; which is often not evident for the complex panels that are commonly used in maize. It was thus necessary to evaluate the effect of the selected structure models on the association mapping results. A real maize data set (342 maize inbred lines and 12,000 SNPs) was used for this study. The panel structure was estimated using both Bayesian and dimensional reduction methods, considering an increasing number of ancestral groups. Effect on association tests depends in particular on the number of ancestral groups and on the trait analyzed. The results also show that using a high number of ancestral groups leads to an over-corrected model in which all causal loci vanish. Finally the results of all models tested were combined in a meta-analysis approach. In this way, robust associations were highlighted for each analyzed trait.

Asymptotic distribution of the "orthogonal" quantitative transmission disequilibrium test in a structured population: exact formula.

Population structure is a recurrent problem for the detection of associations between a marker and a trait, because it can lead to an excess of false positives of the association tests. One popular way of circumventing this problem is the use of family based tests, which consider the transmission of the genotype from the parents to the offspring. Here we focus on quantitative traits and study the Abecasis "orthogonal" quantitative transmission disequilibrium test, which is commonly used in family based association studies. We derive the probability distribution of this test under a general model of structured population. Our derivations show that this test leads to a small excess of false positives due to population structure. They also illustrate and quantify how the heterogeneity in genotypes and phenotypes between populations affect the power of the test. We finally show that the excess of false positives observed for the Abecasis "orthogonal" test may also be found for the Allison "linear" test, though at a lower extent.

Does probabilistic modelling of linkage disequilibrium evolution improve the accuracy of QTL location in animal pedigree?

BACKGROUND: Since 2001, the use of more and more dense maps has made researchers aware that combining linkage and linkage disequilibrium enhances the feasibility of fine-mapping genes of interest. So, various method types have been derived to include concepts of population genetics in the analyses. One major drawback of many of these methods is their computational cost, which is very significant when many markers are considered. Recent advances in technology, such as SNP genotyping, have made it possible to deal with huge amount of data. Thus the challenge that remains is to find accurate and efficient methods that are not too time consuming. The study reported here specifically focuses on the half-sib family animal design. Our objective was to determine whether modelling of linkage disequilibrium evolution improved the mapping accuracy of a quantitative trait locus of agricultural interest in these populations. We compared two methods of fine-mapping. The first one was an association analysis. In this method, we did not model linkage disequilibrium evolution. Therefore, the modelling of the evolution of linkage disequilibrium was a deterministic process; it was complete at time 0 and remained complete during the following generations. In the second method, the modelling of the evolution of population allele frequencies was derived from a Wright-Fisher model. We simulated a wide range of scenarios adapted to animal populations and compared these two methods for each scenario. RESULTS: Our results indicated that the improvement produced by probabilistic modelling of linkage disequilibrium evolution was not significant. Both methods led to similar results concerning the location accuracy of quantitative trait loci which appeared to be mainly improved by using four flanking markers instead of two. CONCLUSIONS: Therefore, in animal half-sib designs, modelling linkage disequilibrium evolution using a Wright-Fisher model does not significantly improve the accuracy of the QTL location when compared to a simpler method assuming complete and constant linkage between the QTL and the marker alleles. Finally, given the high marker density available nowadays, the simpler method should be preferred as it gives accurate results in a reasonable computing time.

The estimator of the optimal measure of allelic association: mean, variance and probability distribution when the sample size tends to infinity.

The allelic association or linkage disequilibrium between two loci is a parameter of fundamental interest in modern population genetics for evolutionary inference and association mapping studies. Among the many measures available, the optimal measure of allelic association rho presents a strong evolutionary theory basis and is modeled on the physical distance along the chromosome with the Malécot equation for isolation by distance. Moreover, rho is equal to the absolute value of D', the standardized measure of gametic disequilibrium. We studied here the statistical properties of the rho sample estimator. We derived its asymptotic probability distribution and showed that it is neither asymptotically normal nor unbiased when rho=0 or when allelic frequencies are equal at both loci, in contrast to previous claims. This asymptotic study leads to propose a new test for absence of linkage disequilibrium. We compared it to Pearson's Chi2 test for independence in a contingency table and showed by simulations that the range in power of these two tests depends on the sign of D'. The new test outperformed slightly the Chi2 test, when D', polarized with respect to major alleles, is negative. Finally, we derived the asymptotic bias and information of the rho estimator that are due to the experimental sampling and showed by simulation that its bias is large in small samples. The consequences of these findings on applications using the rho measure are then discussed in particular for constructing LD unit maps, and call for a revised statistical treatment.

Training set optimization of genomic prediction by means of EthAcc.

Genomic prediction is a useful tool for plant and animal breeding programs and is starting to be used to predict human diseases as well. A shortcoming that slows down the genomic selection deployment is that the accuracy of the prediction is not known a priori. We propose EthAcc (Estimated THeoretical ACCuracy) as a method for estimating the accuracy given a training set that is genotyped and phenotyped. EthAcc is based on a causal quantitative trait loci model estimated by a genome-wide association study. This estimated causal model is crucial; therefore, we compared different methods to find the one yielding the best EthAcc. The multilocus mixed model was found to perform the best. We compared EthAcc to accuracy estimators that can be derived via a mixed marker model. We showed that EthAcc is the only approach to correctly estimate the accuracy. Moreover, in case of a structured population, in accordance with the achieved accuracy, EthAcc showed that the biggest training set is not always better than a smaller and closer training set. We then performed training set optimization with EthAcc and compared it to CDmean. EthAcc outperformed CDmean on real datasets from sugar beet, maize, and wheat. Nonetheless, its performance was mainly due to the use of an optimal but inaccessible set as a start of the optimization algorithm. EthAcc's precision and algorithm issues prevent it from reaching a good training set with a random start. Despite this drawback, we demonstrated that a substantial gain in accuracy can be obtained by performing training set optimization.

Heliaphen, an Outdoor High-Throughput Phenotyping Platform for Genetic Studies and Crop Modeling.

Heliaphen is an outdoor platform designed for high-throughput phenotyping. It allows the automated management of drought scenarios and monitoring of plants throughout their lifecycles. A robot moving between plants growing in 15-L pots monitors the plant water status and phenotypes the leaf or whole-plant morphology. From these measurements, we can compute more complex traits, such as leaf expansion (LE) or transpiration rate (TR) in response to water deficit. Here, we illustrate the capabilities of the platform with two practical cases in sunflower (Helianthus annuus): a genetic and genomic study of the response of yield-related traits to drought, and a modeling study using measured parameters as inputs for a crop simulation. For the genetic study, classical measurements of thousand-kernel weight (TKW) were performed on a biparental population under automatically managed drought stress and control conditions. These data were used for an association study, which identified five genetic markers of the TKW drought response. A complementary transcriptomic analysis identified candidate genes associated with these markers that were differentially expressed in the parental backgrounds in drought conditions. For the simulation study, we used a crop simulation model to predict the impact on crop yield of two traits measured on the platform (LE and TR) for a large number of environments. We conducted simulations in 42 contrasting locations across Europe using 21 years of climate data. We defined the pattern of abiotic stresses occurring at the continental scale and identified ideotypes (i.e., genotypes with specific trait values) that are more adapted to specific environment types. This study exemplifies how phenotyping platforms can assist the identification of the genetic architecture controlling complex response traits and facilitate the estimation of ecophysiological model parameters to define ideotypes adapted to different environmental conditions.

Genomic Prediction of Sunflower Hybrids Oil Content.

Prediction of hybrid performance using incomplete factorial mating designs is widely used in breeding programs including different heterotic groups. Based on the general combining ability (GCA) of the parents, predictions are accurate only if the genetic variance resulting from the specific combining ability is small and both parents have phenotyped descendants. Genomic selection (GS) can predict performance using a model trained on both phenotyped and genotyped hybrids that do not necessarily include all hybrid parents. Therefore, GS could overcome the issue of unknown parent GCA. Here, we compared the accuracy of classical GCA-based and genomic predictions for oil content of sunflower seeds using several GS models. Our study involved 452 sunflower hybrids from an incomplete factorial design of 36 female and 36 male lines. Re-sequencing of parental lines allowed to identify 468,194 non-redundant SNPs and to infer the hybrid genotypes. Oil content was observed in a multi-environment trial (MET) over 3 years, leading to nine different environments. We compared GCA-based model to different GS models including female and male genomic kinships with the addition of the female-by-male interaction genomic kinship, the use of functional knowledge as SNPs in genes of oil metabolic pathways, and with epistasis modeling. When both parents have descendants in the training set, the predictive ability was high even for GCA-based prediction, with an average MET value of 0.782. GS performed slightly better (+0.2%). Neither the inclusion of the female-by-male interaction, nor functional knowledge of oil metabolism, nor epistasis modeling improved the GS accuracy. GS greatly improved predictive ability when one or both parents were untested in the training set, increasing GCA-based predictive ability by 10.4% from 0.575 to 0.635 in the MET. In this scenario, performing GS only considering SNPs in oil metabolic pathways did not improve whole genome GS prediction but increased GCA-based prediction ability by 6.4%. Our results show that GS is a major improvement to breeding efficiency compared to the classical GCA modeling when either one or both parents are not well-characterized. This finding could therefore accelerate breeding through reducing phenotyping efforts and more effectively targeting for the most promising crosses.

Linkage disequilibrium interval mapping of quantitative trait loci.

BACKGROUND: For many years gene mapping studies have been performed through linkage analyses based on pedigree data. Recently, linkage disequilibrium methods based on unrelated individuals have been advocated as powerful tools to refine estimates of gene location. Many strategies have been proposed to deal with simply inherited disease traits. However, locating quantitative trait loci is statistically more challenging and considerable research is needed to provide robust and computationally efficient methods. RESULTS: Under a three-locus Wright-Fisher model, we derived approximate expressions for the expected haplotype frequencies in a population. We considered haplotypes comprising one trait locus and two flanking markers. Using these theoretical expressions, we built a likelihood-maximization method, called HAPim, for estimating the location of a quantitative trait locus. For each postulated position, the method only requires information from the two flanking markers. Over a wide range of simulation scenarios it was found to be more accurate than a two-marker composite likelihood method. It also performed as well as identity by descent methods, whilst being valuable in a wider range of populations. CONCLUSION: Our method makes efficient use of marker information, and can be valuable for fine mapping purposes. Its performance is increased if multiallelic markers are available. Several improvements can be developed to account for more complex evolution scenarios or provide robust confidence intervals for the location estimates.

On the Accuracy of Genomic Selection.

Genomic selection is focused on prediction of breeding values of selection candidates by means of high density of markers. It relies on the assumption that all quantitative trait loci (QTLs) tend to be in strong linkage disequilibrium (LD) with at least one marker. In this context, we present theoretical results regarding the accuracy of genomic selection, i.e., the correlation between predicted and true breeding values. Typically, for individuals (so-called test individuals), breeding values are predicted by means of markers, using marker effects estimated by fitting a ridge regression model to a set of training individuals. We present a theoretical expression for the accuracy; this expression is suitable for any configurations of LD between QTLs and markers. We also introduce a new accuracy proxy that is free of the QTL parameters and easily computable; it outperforms the proxies suggested in the literature, in particular, those based on an estimated effective number of independent loci (Me). The theoretical formula, the new proxy, and existing proxies were compared for simulated data, and the results point to the validity of our approach. The calculations were also illustrated on a new perennial ryegrass set (367 individuals) genotyped for 24,957 single nucleotide polymorphisms (SNPs). In this case, most of the proxies studied yielded similar results because of the lack of markers for coverage of the entire genome (2.7 Gb).