Comparison of genomic-enabled cross selection criteria for the improvement of inbred line breeding populations.

A crucial step in inbred plant breeding is the choice of mating design to derive high-performing inbred varieties while also maintaining a competitive breeding population to secure sufficient genetic gain in future generations. In practice, the mating design usually relies on crosses involving the best parental inbred lines to ensure high mean progeny performance. This excludes crosses involving lower performing but more complementary parents in terms of favorable alleles. We predicted the ability of crosses to produce putative outstanding progenies (high mean and high variance progeny distribution) using genomic prediction models. This study compared the benefits and drawbacks of 7 genomic cross selection criteria (CSC) in terms of genetic gain for 1 trait and genetic diversity in the next generation. Six CSC were already published, and we propose an improved CSC that can estimate the proportion of progeny above a threshold defined for the whole mating plan. We simulated mating designs optimized using different CSC. The 835 elite parents came from a real breeding program and were evaluated between 2000 and 2016. We applied constraints on parental contributions and genetic similarities between selected parents according to usual breeder practices. Our results showed that CSC based on progeny variance estimation increased the genetic value of superior progenies by up to 5% in the next generation compared to CSC based on the progeny mean estimation (i.e. parental genetic values) alone. It also increased the genetic gain (up to 4%) and/or maintained more genetic diversity at QTLs (up to 4% more genic variance when the marker effects were perfectly estimated).

Genomic Prediction of Complex Traits, Principles, Overview of Factors Affecting the Reliability of Genomic Prediction, and Algebra of the Reliability.

The quality of the predictions of genetic values based on the genotyping of neutral markers (GEBVs) is a key information to decide whether or not to implement genomic selection. This quality depends on the part of the genetic variability captured by the markers and on the precision of the estimate of their effects. Selection index theory provided the framework for evaluating the accuracy of GEBVs once the information had been gathered, with the genomic relationship matrix (GRM) playing a central role. When this accuracy must be known a priori, the theory of quantitative genetics gives clues to calculate the expectation of this GRM. This chapter makes a critical inventory of the methods developed to calculate these accuracies a posteriori and a priori. The most significant factors affecting this accuracy are described (size of the reference population, number of markers, linkage disequilibrium, heritability).

An analytical framework to derive the expected precision of genomic selection.

BACKGROUND: Formulae to predict the precision or accuracy of genomic estimated breeding values (GEBV) are important when modelling selection schemes. Simple versions of such formulae have been proposed in the past, based on a number of simplifying hypotheses, including absence of linkage disequilibrium and linkage between loci, a population made up of unrelated individuals, and that all genetic variability of the trait is explained by the genotyped loci. These formulae were based on approximations that were not always clear. The objective of this paper is to offer a unique framework to derive equations that predict the precision of GEBV from the size of the reference population and the heritability of and number of QTL controlling the quantitative trait. RESULTS: The exact formulation of the precision of GEBV involves the expectation of the inverse of a linear function of the genomic matrix, which cannot be calculated from simple algebra but can be approximated using a Taylor polynomial expansion. First order approximations performed better than the initial prediction equations published in the literature. Second order approximations produced almost perfect estimates of precision when compared to results obtained when simulating situations that agreed with the assumptions that were required to derive the precision equations. Using this proposed framework, we present several generalizations, including multi-trait genomic evaluation. CONCLUSIONS: Although further improvements are needed to account for the complexity of practical situations, the equations proposed here can be used to derive the precision of GEBV when comparing breeding schemes a priori.

Using a very low-density SNP panel for genomic selection in a breeding program for sheep.

BACKGROUND: Building an efficient reference population for genomic selection is an issue when the recorded population is small and phenotypes are poorly informed, which is often the case in sheep breeding programs. Using stochastic simulation, we evaluated a genomic design based on a reference population with medium-density genotypes [around 45 K single nucleotide polymorphisms (SNPs)] of dams that were imputed from very low-density genotypes (≤ 1000 SNPs). METHODS: A population under selection for a maternal trait was simulated using real genotypes. Genetic gains realized from classical selection and genomic selection designs were compared. Genomic selection scenarios that differed in reference population structure (whether or not dams were included in the reference) and genotype quality (medium-density or imputed to medium-density from very low-density) were evaluated. RESULTS: The genomic design increased genetic gain by 26% when the reference population was based on sire medium-density genotypes and by 54% when the reference population included both sire and dam medium-density genotypes. When medium-density genotypes of male candidates and dams were replaced by imputed genotypes from very low-density SNP genotypes (1000 SNPs), the increase in gain was 22% for the sire reference population and 42% for the sire and dam reference population. The rate of increase in inbreeding was lower (from - 20 to - 34%) for the genomic design than for the classical design regardless of the genomic scenario. CONCLUSIONS: We show that very low-density genotypes of male candidates and dams combined with an imputation process result in a substantial increase in genetic gain for small sheep breeding programs.

Approximated prediction of genomic selection accuracy when reference and candidate populations are related.

BACKGROUND: Genomic selection is still to be evaluated and optimized in many species. Mathematical modeling of selection schemes prior to their implementation is a classical and useful tool for that purpose. These models include formalization of a number of entities including the precision of the estimated breeding value. To model genomic selection schemes, equations that predict this reliability as a function of factors such as the size of the reference population, its diversity, its genetic distance from the group of selection candidates genotyped, number of markers and strength of linkage disequilibrium are needed. The present paper aims at exploring new approximations of this reliability. RESULTS: Two alternative approximations are proposed for the estimation of the reliability of genomic estimated breeding values (GEBV) in the case of non-independence between candidate and reference populations. Both were derived from the Taylor series heuristic approach suggested by Goddard in 2009. A numerical exploration of their properties showed that the series were not equivalent in terms of convergence to the exact reliability, that the approximations may overestimate the precision of GEBV and that they converged towards their theoretical expectations. Formulae derived for these approximations were simple to handle in the case of independent markers. A few parameters that describe the markers' genotypic variability (allele frequencies, linkage disequilibrium) can be estimated from genomic data corresponding to the population of interest or after making assumptions about their distribution. When markers are not in linkage equilibrium, replacing the real number of markers and QTL by the "effective number of independent loci", as proposed earlier is a practical solution. In this paper, we considered an alternative, i.e. an "equivalent number of independent loci" which would give a GEBV reliability for unrelated individuals by considering a sub-set of independent markers that is identical to the reliability obtained by considering the full set of markers. CONCLUSIONS: This paper is a further step towards the development of deterministic models that describe breeding plans based on the use of genomic information. Such deterministic models carry low computational burden, which allows design optimization through intensive numerical exploration.

Genomic BLUP including additive and dominant variation in purebreds and F1 crossbreds, with an application in pigs.

BACKGROUND: Most developments in quantitative genetics theory focus on the study of intra-breed/line concepts. With the availability of massive genomic information, it becomes necessary to revisit the theory for crossbred populations. We propose methods to construct genomic covariances with additive and non-additive (dominance) inheritance in the case of pure lines and crossbred populations. RESULTS: We describe substitution effects and dominant deviations across two pure parental populations and the crossbred population. Gene effects are assumed to be independent of the origin of alleles and allelic frequencies can differ between parental populations. Based on these assumptions, the theoretical variance components (additive and dominant) are obtained as a function of marker effects and allelic frequencies. The additive genetic variance in the crossbred population includes the biological additive and dominant effects of a gene and a covariance term. Dominance variance in the crossbred population is proportional to the product of the heterozygosity coefficients of both parental populations. A genomic BLUP (best linear unbiased prediction) equivalent model is presented. We illustrate this approach by using pig data (two pure lines and their cross, including 8265 phenotyped and genotyped sows). For the total number of piglets born, the dominance variance in the crossbred population represented about 13 % of the total genetic variance. Dominance variation is only marginally important for litter size in the crossbred population. CONCLUSIONS: We present a coherent marker-based model that includes purebred and crossbred data and additive and dominant actions. Using this model, it is possible to estimate breeding values, dominant deviations and variance components in a dataset that comprises data on purebred and crossbred individuals. These methods can be exploited to plan assortative mating in pig, maize or other species, in order to generate superior crossbred individuals in terms of performance.

Genome-wide association study and biological pathway analysis of the Eimeria maxima response in broilers.

BACKGROUND: Coccidiosis is the most common and costly disease in the poultry industry and is caused by protozoans of the Eimeria genus. The current control of coccidiosis, based on the use of anticoccidial drugs and vaccination, faces serious obstacles such as drug resistance and the high costs for the development of efficient vaccines, respectively. Therefore, the current control programs must be expanded with complementary approaches such as the use of genetics to improve the host response to Eimeria infections. Recently, we have performed a large-scale challenge study on Cobb500 broilers using E. maxima for which we investigated variability among animals in response to the challenge. As a follow-up to this challenge study, we performed a genome-wide association study (GWAS) to identify genomic regions underlying variability of the measured traits in the response to Eimeria maxima in broilers. Furthermore, we conducted a post-GWAS functional analysis to increase our biological understanding of the underlying response to Eimeria maxima challenge. RESULTS: In total, we identified 22 single nucleotide polymorphisms (SNPs) with q value <0.1 distributed across five chromosomes. The highly significant SNPs were associated with body weight gain (three SNPs on GGA5, one SNP on GGA1 and one SNP on GGA3), plasma coloration measured as optical density at wavelengths in the range 465-510 nm (10 SNPs and all on GGA10) and the percentage of ß2-globulin in blood plasma (15 SNPs on GGA1 and one SNP on GGA2). Biological pathways related to metabolic processes, cell proliferation, and primary innate immune processes were among the most frequent significantly enriched biological pathways. Furthermore, the network-based analysis produced two networks of high confidence, with one centered on large tumor suppressor kinase 1 (LATS1) and 2 (LATS2) and the second involving the myosin heavy chain 6 (MYH6). CONCLUSIONS: We identified several strong candidate genes and genomic regions associated with traits measured in response to Eimeria maxima in broilers. Furthermore, the post-GWAS functional analysis indicates that biological pathways and networks involved in tissue proliferation and repair along with the primary innate immune response may play the most important role during the early stage of Eimeria maxima infection in broilers.

A comparison of methods for whole-genome QTL mapping using dense markers in four livestock species.

BACKGROUND: With dense genotyping, many choices exist for methods to detect quantitative trait loci (QTL) in livestock populations. However, no across-species study has been conducted on the performance of different methods using real data. We compared three methods that correct for relatedness either implicitly or explicitly: linkage and linkage disequilibrium haplotype-based analysis (LDLA), efficient mixed-model association (EMMA) analysis, and Bayesian whole-genome regression (BayesC). We analyzed one chromosome in each of five datasets (dairy cattle, beef cattle, sheep, horses, and pigs) using real genotypes based on dense single nucleotide polymorphisms and phenotypes. The P values corrected for multiple testing or Bayes factors greater than 150 were considered to be significant. To complete the real data study, we also simulated quantitative trait loci (QTL) for the same datasets based on the real genotypes. Several scenarios were chosen, with different QTL effects and linkage disequilibrium patterns. A pseudo-null statistical distribution was chosen to make the significance thresholds comparable across methods. RESULTS: For the real data, the three methods generally agreed within 1 or 2 cM for the locations of QTL regions and disagreed when no signals were significant (e.g. in pigs). For certain datasets, LDLA had more significant signals than EMMA or BayesC, but they were concentrated around the same peaks. Therefore, the three methods detected approximately the same number of QTL regions. For the simulated data, LDLA was slightly less powerful and accurate than either EMMA or BayesC but this depended strongly on how thresholds were set in the simulations. CONCLUSIONS: All three methods performed similarly for real and simulated data. No method was clearly superior across all datasets or for any particular dataset. For computational efficiency and ease of interpretation, EMMA is recommended, but using more than one method is suggested.

Using haplotypes for the prediction of allelic identity to fine-map QTL: characterization and properties.

BACKGROUND: Numerous methods have been developed over the last decade to predict allelic identity at unobserved loci between pairs of chromosome segments along the genome. These loci are often unobserved positions tested for the presence of quantitative trait loci (QTL). The main objective of this study was to understand from a theoretical standpoint the relation between linkage disequilibrium (LD) and allelic identity prediction when using haplotypes for fine mapping of QTL. In addition, six allelic identity predictors (AIP) were also compared in this study to determine which one performed best in theory and application. RESULTS: A criterion based on a simple measure of matrix distance was used to study the relation between LD and allelic identity prediction when using haplotypes. The consistency of this criterion with the accuracy of QTL localization, another criterion commonly used to compare AIP, was evaluated on a set of real chromosomes. For this set of chromosomes, the criterion was consistent with the mapping accuracy of a simulated QTL with either low or high effect. As measured by the matrix distance, the best AIP for QTL mapping were those that best captured LD between a tested position and a QTL. Moreover the matrix distance between a tested position and a QTL was shown to decrease for some AIP when LD increased. However, the matrix distance for AIP with continuous predictions in the [0,1] interval was algebraically proven to decrease less rapidly up to a lower bound with increasing LD in the simplest situations, than the discrete predictor based on identity by state between haplotypes (IBS hap), for which there was no lower bound. The expected LD between haplotypes at a tested position and alleles at a QTL is a quantity that increases naturally when the tested position gets closer to the QTL. This behavior was demonstrated with pig and unrelated human chromosomes. CONCLUSIONS: When the density of markers is high, and therefore LD between adjacent loci can be assumed to be high, the discrete predictor IBS hap is recommended since it predicts allele identity correctly when taking LD into account.

Genetic parameters for growth and faecal worm egg count following Haemonchus contortus experimental infestations using pedigree and molecular information.

BACKGROUND: Haemonchosis is a parasitic disease that causes severe economic losses in sheep industry. In recent years, the increasing resistance of the parasite to anthelmintics has raised the need for alternative control strategies. Genetic selection is a promising alternative but its efficacy depends on the availability of genetic variation and on the occurrence of favourable genetic correlations between the traits included in the breeding goal. The objective of this study was twofold. First, to estimate both the heritability of and the genetic correlations between growth traits and parasite resistance traits, using bivariate linear mixed animal models, from the phenotypes and genotypes of 1004 backcross lambs (considered as a single population), which underwent two subsequent experimental infestations protocols with Haemonchus contortus. Second, to compare the precision of the estimates when using two different relationship matrices: including pedigree information only or including also SNP (single nucleotide polymorphism) information. RESULTS: Heritabilities were low for average daily gain before infestation (0.10 to 0.15) and average daily gain during the first infestation (0.11 to 0.16), moderate for faecal egg counts during the first infestation (0.21 to 0.38) and faecal egg counts during the second infestation (0.48 to 0.55). Genetic correlations between both growth traits and faecal egg count during the naïve infestation were equal to zero but the genetic correlation between faecal egg count during the second infestation and growth was positive in a Haemonchus contortus free environment and negative in a contaminated environment. The standard errors of the estimates obtained by including SNP information were smaller than those obtained by including pedigree information only. CONCLUSIONS: The genetic parameters estimates suggest that growth performance can be selected for independently of selection on resistance to naïve infestation. Selection for increased growth in a non-contaminated environment could lead to more susceptible animals with long-term exposure to the infestation but it could be possible to select for increased growth in a contaminated environment while also increasing resistance to the long-term exposure to the parasite. The use of molecular information increases the precision of the estimates.

Graphics processing unit-accelerated quantitative trait Loci detection.

Mapping quantitative trait loci (QTL) using genetic marker information is a time-consuming analysis that has interested the mapping community in recent decades. The increasing amount of genetic marker data allows one to consider ever more precise QTL analyses while increasing the demand for computation. Part of the difficulty of detecting QTLs resides in finding appropriate critical values or threshold values, above which a QTL effect is considered significant. Different approaches exist to determine these thresholds, using either empirical methods or algebraic approximations. In this article, we present a new implementation of existing software, QTLMap, which takes advantage of the data parallel nature of the problem by offsetting heavy computations to a graphics processing unit (GPU). Developments on the GPU were implemented using Cuda technology. This new implementation performs up to 75 times faster than the previous multicore implementation, while maintaining the same results and level of precision (Double Precision) and computing both QTL values and thresholds. This speedup allows one to perform more complex analyses, such as linkage disequilibrium linkage analyses (LDLA) and multiQTL analyses, in a reasonable time frame.

Statistical distributions of test statistics used for quantitative trait association mapping in structured populations.

BACKGROUND: Spurious associations between single nucleotide polymorphisms and phenotypes are a major issue in genome-wide association studies and have led to underestimation of type 1 error rate and overestimation of the number of quantitative trait loci found. Many authors have investigated the influence of population structure on the robustness of methods by simulation. This paper is aimed at developing further the algebraic formalization of power and type 1 error rate for some of the classical statistical methods used: simple regression, two approximate methods of mixed models involving the effect of a single nucleotide polymorphism (SNP) and a random polygenic effect (GRAMMAR and FASTA) and the transmission/disequilibrium test for quantitative traits and nuclear families. Analytical formulae were derived using matrix algebra for the first and second moments of the statistical tests, assuming a true mixed model with a polygenic effect and SNP effects. RESULTS: The expectation and variance of the test statistics and their marginal expectations and variances according to the distribution of genotypes and estimators of variance components are given as a function of the relationship matrix and of the heritability of the polygenic effect. These formulae were used to compute type 1 error rate and power for any kind of relationship matrix between phenotyped and genotyped individuals for any level of heritability. For the regression method, type 1 error rate increased with the variability of relationships and with heritability, but decreased with the GRAMMAR method and was not affected with the FASTA and quantitative transmission/disequilibrium test methods. CONCLUSIONS: The formulae can be easily used to provide the correct threshold of type 1 error rate and to calculate the power when designing experiments or data collection protocols. The results concerning the efficacy of each method agree with simulation results in the literature but were generalized in this work. The power of the GRAMMAR method was equal to the power of the FASTA method at the same type 1 error rate. The power of the quantitative transmission/disequilibrium test was low. In conclusion, the FASTA method, which is very close to the full mixed model, is recommended in association mapping studies.

Fine-mapping quantitative trait loci with a medium density marker panel: efficiency of population structures and comparison of linkage disequilibrium linkage analysis models.

Recently, a Haley-Knott-type regression method using combined linkage disequilibrium and linkage analyses (LDLA) was proposed to map quantitative trait loci (QTLs). Chromosome of 5 and 25 cM with 0·25 and 0·05 cM, respectively, between markers were simulated. The differences between the LDLA approaches with regard to QTL position accuracy were very limited, with a significantly better mean square error (MSE) with the LDLA regression (LDLA_reg) in sparse map cases; the contrary was observed, but not significantly, in dense map situations. The computing time required for the LDLA variance components (LDLA_vc) model was much higher than the LDLA_reg model. The precision of QTL position estimation was compared for four numbers of half-sib families, four different family sizes and two experimental designs (half-sibs, and full- and half-sibs). Regarding the number of families, MSE values were lowest for 15 or 50 half-sib families, differences not being significant. We observed that the greater the number of progenies per sire, the more accurate the QTL position. However, for a fixed population size, reducing the number of families (e.g. using a small number of large full-sib families) could lead to less accuracy of estimated QTL position.

XVth QTLMAS: simulated dataset.

BACKGROUND: Our aim was to simulate the data for the QTLMAS2011 workshop following a pig-type family structure under an oligogenic model, each QTL being specific. RESULTS: The population comprised 3000 individuals issued from 20 sires and 200 dams. Within each family, 10 progenies belonged to the experimental population and were assigned phenotypes and marker genotypes and 5 belonged to the selection population, only known on their marker genotypes. A total of 10,000 SNPs carried by 5 chromosomes of 1 Morgan each were simulated. Eight QTL were created (1 quadri-allelic, 2 linked in phase, 2 linked in repulsion, 1 imprinted and 2 epistatic). Random noise was added giving an heritability of 0.30. The marker density, LD and MAF were similar to real life parameters.

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

BACKGROUND: The QTLMAS XVth dataset consisted of the pedigrees, marker genotypes and quantitative trait performances of 2,000 phenotyped animals with a half-sib family structure. The trait was regulated by 8 QTL which display additive, imprinting or epistatic effects. This paper aims at comparing the QTL mapping results obtained by six participants of the workshop. METHODS: Different regression, GBLUP, LASSO and Bayesian methods were applied for QTL detection. The results of these methods are compared based on the number of correctly mapped QTL, the number of false positives, the accuracy of the QTL location and the estimation of the QTL effect. RESULTS: All the simulated QTL, except the interacting QTL on Chr5, were identified by the participants. Depending on the method, 3 to 7 out of the 8 QTL were identified. The distance to the real location and the accuracy of the QTL effect varied to a large extent depending on the methods and complexity of the simulated QTL. CONCLUSIONS: While all methods were fairly efficient in detecting QTL with additive effects, it was clear that for non-additive situations, such as parent-of-origin effects or interactions, the BayesC method gave the best results by detecting 7 out of the 8 simulated QTL, with only two false positives and a good precision (less than 1 cM away on average). Indeed, if LASSO could detect QTL even in complex situations, it was associated with too many false positive results to allow for efficient GWAS. GENMIX, a method based on the phylogenies of local haplotypes, also appeared as a promising approach, which however showed a few more false positives when compared with the BayesC method.

Comparison of analyses of the XVth QTLMAS common dataset III: Genomic Estimations of Breeding Values.

BACKGROUND: The QTLMAS XVth dataset consisted of pedigree, marker genotypes and quantitative trait performances of animals with a sib family structure. Pedigree and genotypes concerned 3,000 progenies among those 2,000 were phenotyped. The trait was regulated by 8 QTLs which displayed additive, imprinting or epistatic effects. The 1,000 unphenotyped progenies were considered as candidates to selection and their Genomic Estimated Breeding Values (GEBV) were evaluated by participants of the XVth QTLMAS workshop. This paper aims at comparing the GEBV estimation results obtained by seven participants to the workshop. METHODS: From the known QTL genotypes of each candidate, two "true" genomic values (TV) were estimated by organizers: the genotypic value of the candidate (TGV) and the expectation of its progeny genotypic values (TBV). GEBV were computed by the participants following different statistical methods: random linear models (including BLUP and Ridge Regression), selection variable techniques (LASSO, Elastic Net) and Bayesian methods. Accuracy was evaluated by the correlation between TV (TGV or TBV) and GEBV presented by participants. Rank correlation of the best 10% of individuals and error in predictions were also evaluated. Bias was tested by regression of TV on GEBV. RESULTS: Large differences between methods were found for all criteria and type of genetic values (TGV, TBV). In general, the criteria ranked consistently methods belonging to the same family. CONCLUSIONS: Bayesian methods - A

Statistical properties of interval mapping methods on quantitative trait loci location: impact on QTL/eQTL analyses.

BACKGROUND: Quantitative trait loci (QTL) detection on a huge amount of phenotypes, like eQTL detection on transcriptomic data, can be dramatically impaired by the statistical properties of interval mapping methods. One of these major outcomes is the high number of QTL detected at marker locations. The present study aims at identifying and specifying the sources of this bias, in particular in the case of analysis of data issued from outbred populations. Analytical developments were carried out in a backcross situation in order to specify the bias and to propose an algorithm to control it. The outbred population context was studied through simulated data sets in a wide range of situations.The likelihood ratio test was firstly analyzed under the "one QTL" hypothesis in a backcross population. Designs of sib families were then simulated and analyzed using the QTL Map software. On the basis of the theoretical results in backcross, parameters such as the population size, the density of the genetic map, the QTL effect and the true location of the QTL, were taken into account under the "no QTL" and the "one QTL" hypotheses. A combination of two non parametric tests - the Kolmogorov-Smirnov test and the Mann-Whitney-Wilcoxon test - was used in order to identify the parameters that affected the bias and to specify how much they influenced the estimation of QTL location. RESULTS: A theoretical expression of the bias of the estimated QTL location was obtained for a backcross type population. We demonstrated a common source of bias under the "no QTL" and the "one QTL" hypotheses and qualified the possible influence of several parameters. Simulation studies confirmed that the bias exists in outbred populations under both the hypotheses of "no QTL" and "one QTL" on a linkage group. The QTL location was systematically closer to marker locations than expected, particularly in the case of low QTL effect, small population size or low density of markers, i.e. designs with low power. Practical recommendations for experimental designs for QTL detection in outbred populations are given on the basis of this bias quantification. Furthermore, an original algorithm is proposed to adjust the location of a QTL, obtained with interval mapping, which co located with a marker. CONCLUSIONS: Therefore, one should be attentive when one QTL is mapped at the location of one marker, especially under low power conditions.

Optimization of gene-assisted selection in small-sized populations: comparison of deterministic and stochastic approaches.

Many of the models used to optimize selection processes in livestock make the assumption that the population is of infinite size and are built on deterministic equations. The finite size case should however be considered explicitly when selection involves one identified gene. Indeed, drift can cause the loss of a favorable allele if its initial frequency is low. In this paper, a stochastic approach was developed to simultaneously optimize selection on two traits in a limited size population: a quantitative trait with underlying polygenic variation and a monogenic trait. We outline the interests of considering the limited size of the population in stochastic modeling with a simple example. Such stochastic models raise some technical problems (uncertain convergence to the maximum, computational burden) which could obliterate their usefulness as compared to simpler but approximate deterministic models which can be used when the population size is large. By way of this simple example, we show the feasibility of the optimization of this type of model using a genetic algorithm and demonstrate its interest compared with the corresponding deterministic model which assumes that the population is of infinite size.

A fast algorithm for estimating transmission probabilities in QTL detection designs with dense maps.

BACKGROUND: In the case of an autosomal locus, four transmission events from the parents to progeny are possible, specified by the grand parental origin of the alleles inherited by this individual. Computing the probabilities of these transmission events is essential to perform QTL detection methods. RESULTS: A fast algorithm for the estimation of these probabilities conditional to parental phases has been developed. It is adapted to classical QTL detection designs applied to outbred populations, in particular to designs composed of half and/or full sib families. It assumes the absence of interference. CONCLUSION: The theory is fully developed and an example is given.

Performance of genomic selection in mice.

Selection plans in plant and animal breeding are driven by genetic evaluation. Recent developments suggest using massive genetic marker information, known as "genomic selection." There is little evidence of its performance, though. We empirically compared three strategies for selection: (1) use of pedigree and phenotypic information, (2) use of genomewide markers and phenotypic information, and (3) the combination of both. We analyzed four traits from a heterogeneous mouse population (http://gscan.well.ox.ac.uk/), including 1884 individuals and 10,946 SNP markers. We used linear mixed models, using extensions of association analysis. Cross-validation techniques were used, providing assumption-free estimates of predictive ability. Sampling of validation and training data sets was carried out across and within families, which allows comparing across- and within-family information. Use of genomewide genetic markers increased predictive ability up to 0.22 across families and up to 0.03 within families. The latter is not statistically significant. These values are roughly comparable to increases of up to 0.57 (across family) and 0.14 (within family) in accuracy of prediction of genetic value. In this data set, within-family information was more accurate than across-family information, and populational linkage disequilibrium was not a completely accurate source of information for genetic evaluation. This fact questions some applications of genomic selection.