The fluctuation experiment is the preferred method for estimating microbial mutation rates. A difficult task facing the data analyst is to infer the mean number of mutations from the number of mutant cells that only indirectly reflects the number of mutations. Partial plating, commonly practised in the laboratory, renders this task even more challenging by allowing only a portion of the mutant cells to be counted. In this paper, we propose a Bayesian approach to correcting for partial plating in the analysis of fluctuation experiments.

The covariance of heterozygosity serves as a measure of linkage disequilibrium (LD) between genes at two loci, although one that does not have as much information as a parameter such as r2. However, it may be extended to blocks of loci (single nucleotide polymorphisms, SNPs) along a chromosome. This has two advantages when searching for significant associations between different chromosomal regions. Calculations for a data set such as Hapmap are complicated by the large number of pairs of loci (SNPs) that need to be considered. For example, a search for significant associations between SNPs on different chromosomes involves around 1012 calculations for a single population. Furthermore, this may not be an efficient way of detecting associations since r2 values calculated from neighbouring pairs will not be independent of each other. The covariance of heterozygosity provides an average measure of association between blocks of any size, and reduces the number of calculations by a factor of b2, where b is the block size. Unlike the calculation of r2, the covariance of heterozygosity uses just diploid data and is not biased by sample size. Calculations using a block size of 50 have been used to look for associations in the Hapmap data set between regions within and between chromosomes. Within chromosomes, a signal is detected up to around 10 cM. No obviously significant associations have been detected between regions on different chromosomes, although there is a low level of association consistent with departures from random mating.

For genomic selection methods, the statistical challenge is to estimate the effect of each of the available single-nucleotide polymorphism (SNP). In a context where the number of SNPs (p) is much higher than the number of bulls (n), this task may lead to a poor estimation of these SNP effects if, as for genomic BLUP (gBLUP), all SNPs have a non-null effect. An alternative is to use approaches that have been developed specifically to solve the ‘p>>n’ problem. This is the case of variable selection methods and among them, we focus on the Elastic-Net (EN) algorithm that is a penalized regression approach. Performances of EN, gBLUP and pedigree-based BLUP were compared with data from three French dairy cattle breeds, giving very encouraging results for EN. We tried to push further the idea of improving SNP effect estimates by considering fewer of them. This variable selection strategy was considered both in the case of gBLUP and EN by adding an SNP pre-selection step based on quantitative trait locus (QTL) detection. Similar results were observed with or without a pre-selection step, in terms of correlations between direct genomic value (DGV) and observed daughter yield deviation in a validation data set. However, when applied to the EN algorithm, this strategy led to a substantial reduction of the number of SNPs included in the prediction equation. In a context where the number of genotyped animals and the number of SNPs gets larger and larger, SNP pre-selection strongly alleviates computing requirements and ensures that national evaluations can be completed within a reasonable time frame.

Genetic resistance to gastrointestinal worms is a complex trait of great importance in both livestock and humans. In order to gain insights into the genetic architecture of this trait, a mixed breed population of sheep was artificially infected with Trichostrongylus colubriformis (n=3326) and then Haemonchus contortus (n=2669) to measure faecal worm egg count (WEC). The population was genotyped with the Illumina OvineSNP50 BeadChip and 48 640 single nucleotide polymorphism (SNP) markers passed the quality controls. An independent population of 316 sires of mixed breeds with accurate estimated breeding values for WEC were genotyped for the same SNP to assess the results obtained from the first population. We used principal components from the genomic relationship matrix among genotyped individuals to account for population stratification, and a novel approach to directly account for the sampling error associated with each SNP marker regression. The largest marker effects were estimated to explain an average of 0·48% (T. colubriformis) or 0·08% (H. contortus) of the phenotypic variance in WEC. These effects are small but consistent with results from other complex traits. We also demonstrated that methods which use all markers simultaneously can successfully predict genetic merit for resistance to worms, despite the small effects of individual markers. Correlations of genomic predictions with breeding values of the industry sires reached a maximum of 0·32. We estimate that effective across-breed predictions of genetic merit with multi-breed populations will require an average marker spacing of approximately 10 kbp.

Although the expected relationship or proportion of genome shared by pairs of relatives can be obtained from their pedigrees, the actual quantities deviate as a consequence of Mendelian sampling and depend on the number of chromosomes and map length. Formulae have been published previously for the variance of actual relationship for a number of specific types of relatives but no general formula for non-inbred individuals is available. We provide here a unified framework that enables the variances for distant relatives to be easily computed, showing, for example, how the variance of sharing for great grandparent–great grandchild, great uncle–great nephew, half uncle–nephew and first cousins differ, even though they have the same expected relationship. Results are extended in order to include differences in map length between sexes, no recombination in males and sex linkage. We derive the magnitude of skew in the proportion shared, showing the skew becomes increasingly large the more distant the relationship. The results obtained for variation in actual relationship apply directly to the variation in actual inbreeding as both are functions of genomic coancestry, and we show how to partition the variation in actual inbreeding between and within families. Although the variance of actual relationship falls as individuals become more distant, its coefficient of variation rises, and so, exacerbated by the skewness, it becomes increasingly difficult to distinguish different pedigree relationships from the actual fraction of the genome shared.

Prediction of genetic merit or disease risk using genetic marker information is becoming a common practice for selection of livestock and plant species. For the successful application of genome-wide marker-assisted selection (GWMAS), genomic predictions should be accurate and unbiased. The effect of selection on bias and accuracy of genomic predictions was studied in two simulated animal populations under weak or strong selection and with several heritabilities. Prediction of genetic values was by best-linear unbiased prediction (BLUP) using data either from relatives summarized in pseudodata for genotyped individuals (multiple-step method) or using all available data jointly (single-step method). The single-step method combined genomic- and pedigree-based relationship matrices. Predictions by the multiple-step method were biased. Predictions by a single-step method were less biased and more accurate but under strong selection were less accurate. When genomic relationships were shifted by a constant, the single-step method was unbiased and the most accurate. The value of that constant, which adjusts for non-random selection of genotyped individuals, can be derived analytically.

Knowledge of the genetic architecture of a quantitative trait is useful to adjust methods for the prediction of genomic breeding values and to discover the extent to which common assumptions in quantitative trait locus (QTL) mapping experiments and breeding value estimation are violated. It also affects our ability to predict the long-term response of selection. In this paper, we focus on additive and dominance effects of QTL. We derive formulae that can be used to estimate the number of QTLs that affect a quantitative trait and parameters of the distribution of their additive and dominance effects from variance components, inbreeding depression and results from QTL mapping experiments. It is shown that a lower bound for the number of QTLs depends on the ratio of squared inbreeding depression to dominance variance. That is, high inbreeding depression must be due to a sufficient number of QTLs because otherwise the dominance variance would exceed the true value. Moreover, the second moment of the dominance coefficient depends only on the ratio of dominance variance to additive variance and on the dependency between additive effects and dominance coefficients. This has implications on the relative frequency of overdominant alleles. It is also demonstrated how the expected number of large QTLs determines the shape of the distribution of additive effects. The formulae are applied to milk yield and productive life in Holstein cattle. Possible sources for a potential bias of the results are discussed.

Linkage between markers and genes that affect a phenotype of interest may be determined by examining differences in marker allele frequency in the extreme progeny of a cross between two inbred lines. This strategy is usually employed when pooling is used to reduce genotyping costs. When the cross progeny are asexual, the extreme progeny may be selected by multiple generations of asexual reproduction and selection. We analyse this method of measuring phenotype in asexual progeny and examine the changes in marker allele frequency due to selection over many generations. Stochasticity in marker frequency in the selected population arises due to the finite initial population size. We derive the distribution of marker frequency as a result of selection at a single major locus, and show that in order to avoid spurious changes in marker allele frequency in the selected population, the initial population size should be in the low to mid hundreds.

Tolerance to infections is the ability of a host to limit the impact of a given pathogen burden on host performance. This simulation study demonstrated the merit of using random regressions to estimate unbiased genetic variances for tolerance slope and its genetic correlations with other traits, which could not be obtained using the previously implemented statistical methods. Genetic variance in tolerance was estimated as genetic variance in regression slopes of host performance along an increasing pathogen burden level. Random regressions combined with covariance functions allowed genetic variance for host performance to be estimated at any point along the pathogen burden trajectory, providing a novel means to analyse infection-induced changes in genetic variation of host performance. Yet, the results implied that decreasing family size as well as a non-zero environmental or genetic correlation between initial host performance before infection and pathogen burden led to biased estimates for tolerance genetic variance. In both cases, genetic correlation between tolerance slope and host performance in a pathogen-free environment became artificially negative, implying a genetic trade-off when it did not exist. Moreover, recording a normally distributed pathogen burden as a threshold trait is not a realistic way of obtaining unbiased estimates for tolerance genetic variance. The results show that random regressions are suitable for the genetic analysis of tolerance, given suitable data structure collected either under field or experimental conditions.

The fat mass and obesity associated (FTO) gene has been implicated with obesity and dietary intake predominantly in European populations. We assessed the association between the FTO rs9939609 variant with body fat distribution and dietary intake in a multi-ethnic population. Aboriginal, Chinese, European and South Asian participants living in Canada (n=706) were assessed for body fat and inner-abdominal fat using imaging techniques, dietary intake and genotyped for the FTO rs9939609 variant. Linear regression was used to study the associations between the minor allele of the variant and measures of adiposity and dietary intake. Minor allele frequencies were: Aboriginals (17%), Chinese (17%), Europeans (39%) and South Asians (31%). The rs9939609 variant was associated with intake of dietary macronutrients in Aboriginals and Europeans only. In the total population, there were positive associations between the rs9939609 minor allele and greater fat mass (0·94±0·56 kg, P=0·045), per cent body fat (0·7±0·4%, P=0·031), relative greater subcutaneous abdominal adipose tissue (4·9±2·8%, P=0·039) and percent daily calories from fat (0·4±0·2%, P=0·064). Our findings suggest that the FTO rs9939609 minor allele may be associated with dietary intake in adults and is positively associated with regional fat deposition.

Crow et al. (1990) and Barton (1992) have examined the critical migration rate for swamping selection in the nuclear system. Here, I use the same methodology to examine the critical migration rate in the cytonuclear system for hermaphrodite plants with a mixed mating system. Two selection schemes for a nuclear gene (heterozygote disadvantage and directional selection) and the directional selection scheme for organelle genes are considered. Results show that under random mating, the previous results are applicable to plant species by appropriate re-parameterization of the migration rate for nuclear and paternal organelle genes. A simple complementary relationship exists between seed and pollen flow in contributing to the critical migration rate. Under the mixed mating system, the critical migration rate of seeds and pollen for nuclear and paternal organelle genes can be changed due to the effects of selection and the cytonuclear linkage disequilibrium generated by migration and inbreeding. A negative but not complementary relationship exists between seed and pollen flow in contributing to the critical migration rate, varying with the mating system. Partial selfing can also adjust the critical seed flow for the maternal organelle gene, with a small critical migration rate for species of a high selfing rate. Both concordance and discordance among cytonuclear genes can occur under certain conditions during the process of swamping selection. This theory predicts the presence of various contributions of seed versus pollen flow to genetic swamping for plants with diverse mating systems.

Artificial-selection experiments constitute an important source of empirical information for breeders, geneticists and evolutionary biologists. Selected characters can generally be shifted far from their initial state, sometimes beyond what is usually considered as typical inter-specific divergence. A careful analysis of the data collected during such experiments may thus reveal the dynamical properties of the genetic architecture that underlies the trait under selection. Here, we propose a statistical framework describing the dynamics of selection-response time series. We highlight how both phenomenological models (which do not make assumptions on the nature of genetic phenomena) and mechanistic models (explaining the temporal trends in terms of e.g. mutations, epistasis or canalization) can be used to understand and interpret artificial-selection data. The practical use of the models and their implementation in a software package are demonstrated through the analysis of a selection experiment on the shape of the wing in Drosophila melanogaster.

Understanding the relative influence of genetic drift and selection is fundamental in evolutionary biology. The theory of neutrality predicts that the genetic differentiation of a quantitative trait (QST) equals the genetic differentiation at neutral molecular markers (FST) if the quantitative trait has not been under selection. Thus, the relative magnitude of observed QST and expected QST under neutral expectations suggests the importance of selection and genetic drift for any observed phenotypic divergence. Because QST is based on additive genetic variance, estimating QST based on phenotypic measurements is problematic due to unknown environmental effects. To account for this, we used a model where the environmental component was allowed to vary when estimating QST. The model was used on data from 14 house sparrow (Passer domesticus) populations in Norway. In accordance with the significant phenotypic inter-population differences our analyses suggested that directional selection may have favoured different optimal phenotypes for some morphological traits across populations. In particular, different body mass and male ornamental phenotypes seemed to have been favoured. The conclusions are, however, dependent on assumptions regarding the proportion of the observed inter-population variation that is due to additive genetic differences, showing the importance of collecting such information in natural populations. By estimating QST, allowing the additive genetic proportion of phenotypic inter-population variation to vary, and by making use of recent statistical methods to compare observed QST with neutral expectations, we can use data that are relatively easy to collect to identify adaptive variation in natural populations.

In Caenorhabditis elegans, the Bone Morphogenetic Protein (BMP)-related ligand Dpp- and BMP-like-1 (DBL-1) regulates body size by promoting the larval and adult growth of the large epidermal syncytium hyp7 without affecting cell division. This system provides an excellent model for dissecting the growth-promoting activities of BMP ligands, since in this context the growth and differentiation functions of DBL-1 are naturally uncoupled. dbl-1 is expressed primarily in neurons and the DBL-1 ligand signals to its receptors and Smad signal transducers in the target tissue of the epidermis. The requirements constraining the source(s) of DBL-1, however, have not previously been investigated. We show here that dbl-1 expression requirements are strikingly relaxed. Expression in non-overlapping subsets of the endogenous expression pattern, as well as ectopic expression, can provide sufficient levels of activity for rescue of the small body size of dbl-1 mutants. By analysing dbl-1 expression levels in transgenic strains with different degrees of rescue, we corroborate the model that DBL-1 is a dose-dependent regulator of growth. We conclude that, for body size regulation, the site of expression of dbl-1 is less important than the level of expression.

MADS-box genes encode transcription factors that are key regulators of plant inflorescence and flower development. We examined DNA sequence variation in 32 maize MADS-box genes and 32 randomly chosen maize loci and investigated their involvement in maize domestication and improvement. Using neutrality tests and a test based on coalescent simulation of a bottleneck model, we identified eight MADS-box genes as putative targets of the artificial selection associated with domestication. According to neutrality tests, one additional MADS-box gene appears to have been under selection during modern agricultural improvement of maize. For random loci, two genes were indicated as targets of selection during domestication and four additional genes were indicated to be candidate-selected loci for maize improvement. These results suggest that MADS-box genes were more frequent targets of selection during domestication than genes chosen at random from the genome.

A new estimation-based Bayesian variable selection approach is presented for genetic analysis of complex traits based on linear or logistic regression. By assigning a mixture of uniform priors (MU) to genetic effects, the approach provides an intuitive way of specifying hyperparameters controlling the selection of multiple influential loci. It aims at avoiding the difficulty of interpreting assumptions made in the specifications of priors. The method is compared in two real datasets with two other approaches, stochastic search variable selection (SSVS) and a re-formulation of Bayes B utilizing indicator variables and adaptive Student's t-distributions (IAt). The Markov Chain Monte Carlo (MCMC) sampling performance of the three methods is evaluated using the publicly available software OpenBUGS (model scripts are provided in the Supplementary material). The sensitivity of MU to the specification of hyperparameters is assessed in one of the data examples.