The identification of causes of genetic diseases has been carried out by several approaches with increasing complexity. Innovation of genetic methodologies leads to the production of large amounts of data that needs the support of statistical and computational methods to be correctly processed. The aim of the paper is to provide an overview of statistical and computational methods paying attention to methods for the sequence analysis and complex diseases. As a result, the need for computer/statistical/machine learning techniques is today stronger rather than weaker”. Today, after fourteen years, the need has become even stronger as the human knowledge of genetic mechanisms still increases, making the research on genetic diseases an amazing adventure as well as difficult and demanding. In case of diseases with a complex etiopathogenesis, for example, those caused by several variants in different genes, more advanced investigations are required. Some examples of methods for association studies are here reported together with methods for meta-analysis of different studies. The study of quantitative traits associated with specific variants is a hot topic in the field of complex diseases, as well as gene expression studies. The presence of a genetic mutation/variant is not the only dysfunction cause of the encoded protein; in fact also alterations in its levels could be responsible for a pathological phenotype. Here we report the example of the combination of both studies, the analysis of expression quantitative trait loci that investigates the association of the quantitative data about gene expression with the presence of specific variants across the genome. Thus the aim of this paper is to provide the reader with an overview of the statistical and computational methodologies, focusing on sequence analysis and complex diseases. Further hot topics, such as methods for next-generation sequencing, gene expression studies, miRNA regulation, and epigenetics, are not discussed merely for sake of space.The concept that some disease could be inherited by parents was always present, but only after the discovery of DNA as the genetic material, the research about molecular causes of diseases started. Since the first associations of a disease to a defect in a specific gene, the genetic diagnosis becomes an aim of medical scientists in order to early identify the affected patients and to improve their treatments. For simple monogenic diseases, the conventional way to search for mutations in a gene is the sequencing of amplified fragments corresponding to the gene regions. Innovation in molecular methods together with innovation in computational methods allowed developing new analytical techniques useful to unravel most complicated cases. When the gene responsible for the disease is unknown, in order to identify the genetic defects, the next-generation sequencing could be applied to sequence the whole genome/exome of affected patients, producing then a huge amount of data. In early 2001, during the first assemblies of the human genome, Baldi and Brunak, in their seminal book p-value fχ2 test , 40. Froks (NNs) , Supportne (SVM) , and Rants (RFs)  may morets (RFs) . A numbets (RFs) , Parkinsts (RFs) , Alzheimts (RFs) , and mults (RFs) . SVM hasts (RFs)  and typets (RFs) , while Rts (RFs) , familiats (RFs) , and colts (RFs) . Meta-analyses can be defined as the statistical integration of information from multiple independent studies with the aim of obtaining an overall estimator  of the investigated association [To date, a huge number of association studies identified many genetic variants associated with complex diseases. However, these studies often explain only a small proportion of the disease trait's variability , 54. Genociation . Most geociation , 59. Mosociation . These eociation . GWAS meociation , 59 for ociation  for a moociation .p-values. Here the null hypothesis that the true effect is null in each of the combined data sets is checked against the alternative hypothesis that there is nonnull association in at least one data set. A closely related approach to p-value combination is based on the average of Z-values [Z-score approach, over the Fisher method, is that it takes into account the direction of the effect, and it is rather straightforward to introduce the weights for each study. An alternative and popular approach is fixed effects meta-analysis, used for synthesizing GWAS data and resulting to be very effective for prioritizing and discovering phenotype-associated SNPs [One possible approach, that is, the Fisher's approach , is baseZ-values . Althougted SNPs . Fixed eted SNPs  is the mted SNPs . Cochranted SNPs  approachted SNPs . A well ted SNPs  and refeted SNPs . Althougted SNPs . In Han ted SNPs , a novelted SNPs  has beented SNPs . Moreoveted SNPs  and the ted SNPs . Neverthted SNPs .Quantitative trait locus (QTL) is a DNA region associated with a quantitative phenomenon. In most genetic diseases, quantitative traits are often a measure of the disease severity, such as the lipid levels in a dyslipidemia. Genetic variants could be studied for its capacity to affect these quantitative traits and then to influence the disease severity. Differences in gene expression levels between patients and controls are now recognized as an additional mechanism influencing the development of a complex disease. We are here reporting an example of QTL study based on gene expression levels, the expression Quantitative Trait Locus (eQTL), for example, the study of the effect of a DNA variant on the gene expression. Experimental data from eQTL mapping are mainly formed by a genetic map, marker genotypes, and microarray data extracted by a set of individuals. After the removal of systematic effects, it can obtain measures of gene expression levels. This section does not deal with statistical issues related to a correct eQTL experimental design. To this purpose the reader can refer to  and refe hot spots [ logarithms of odd (LOD) score curve is computed for each transcript. LOD score allows comparing the probability of measuring the observed values if two loci are linked with respect to the probability of observing the same values at random. LOD score curve is obtained computing LOD score for all genomic positions. Several approaches have been proposed to control the FDR based on p-values and q-values [eQTL data were used for the identification of the so-calledot spots , construot spots  and the ot spots , and shoot spots . All theq-values .p-values (corresponding to the peaks of LOD score curves from each transcript) are used to yield and to control the FDR for a list of transcripts mapping to one location. Since this approach takes into account only LOD score peaks, it cannot be used for transcript mapping to multiple loci [α) percent of all probabilities for the trait. A typical value for α is 5.Having said that, in eQTL studies ple loci . In ordeple loci , 78. In  hot spots is usually the next task. Hot spots are genomic regions where there is plenty of transcript maps. The simpler method for identifying the hot spots is the following. For each genomic region, the overall number of mapping transcript is computed. Hot spot candidates are the region whose overall number is ranked among highest ones. Although very simple, the method above can fail if there are several loci with effects whose intensity is not adequately large to be considered statistically significant. A strategy for coping with the problem above has been proposed by Kendziorski et al. [ ghost hot spots, that is, candidate spots that have been considered erroneously hot. This problem has been partially addressed by a Poisson-based test [k transcripts linked to it if there were not any hot spots. Unfortunately, this test cannot be applied when the candidate hot spots are identified by summing the evidence of linkage across all transcripts.After having generated the list of transcripts, the identification of thei et al. . The stri et al.  and profi et al. . After hsed test  that can graphs. A graph is a couple of a set of vertices and a set of edges, connecting couples of vertices. In this case, a vertex represents either a gene or a transcript. An edge connects two vertices when there is some relationship between them; besides, a weight, measured by correlation coefficient, is generally associated to the edge. Pairwise correlations among all transcripts are used to identify cliques [The detection of hot spots yields list of comapping transcripts and involves the inspection of further candidates controlling the whole collection. This is motivated by the observation that comapping is the result of comembership in a biological pathway where functional information is deduced by means of temporally correlated transcripts. Jansen and Nap  showed f cliques , namely  cliques . This im bayesian networks [ best model, namely, the model that optimally describes the data  in some given model space. Finding the best model usually requires the computation of penalized likelihood that manages the trade-off between the goodness of the fit of the model and the number of model parameters. In order to guarantee that the problem is computationally feasible, the model space has to be moderate. Narrowing down the model space for eQTL mapping is usually performed considering only the transcripts that maps to at least one location [Other approaches that can permit the identification of potentially causal relationships among transcripts are the ones based onnetworks . Bayesialocation , 85.We conclude the section quoting that several software tools for eQTL analysis are currently available –88.In the paper an overview of statistical and computational methods focused on sequence analysis and complex diseases has been presented. Among the different techniques discussed in this overview, bayesian techniques seem to be promising in terms of performance in some fields, for example, complex diseases . Since t


Genealogy and productive information of 48621 dual-purpose buffaloes born in Colombia between years 1996 and 2014 was used. The following traits were assessed using one-trait models: milk yield at 270 days (MY270), age at first calving (AFC), weaning weight (WW), and weights at the following ages: first year (W12), 18 months (W18), and 2 years (W24). Direct additive genetic and residual random effects were included in all the traits. Maternal permanent environmental and maternal additive genetic effects were included for WW and W12. The fixed effects were: contemporary group , sex , parity . Age was included as covariate for WW, W12, W18 and W24. Principal component analysis (PCA) was conducted using the genetic values of 133 breeding males whose breeding-value reliability was higher than 50% for all the traits in order to define the number of principal components (PC) which would explain most of the variation. The highest heritabilities were for W18 and MY270, and the lowest for AFC; with 0.53, 0.23, and 0.17, respectively. The first three PCs represented 66% of the total variance. Correlation of the first PC with meat production traits was higher than 0.73, and it was -0.38 with AFC. Correlations of the second PC with maternal genetic component traits for WW and W12 were above 0.75. The third PC had 0.84 correlation with MY270. PCA is an alternative approach for analyzing traits in dual-purpose buffaloes and reduces the dimension of the traits. Buffalo herds are managed under dual-purpose production systems in Colombia so farmers are interested in improving traits related with breeding, milk and meat production. A strategy to improve herd productivity is to select animals according to their breeding values (BV), which allow programming mating according to specific objectives. However, when BV are available for various traits it can be difficult to select the animals, especially when the traits have negative genetic correlation.The principal components analysis (PCA) is a multivariate technique that reduces the amount of originally-correlated variables into a smaller set of non-correlated variables, keeping most of the original variability, and reducing the dimensionality to a new set of variables named principal components (PC), under the assumption of losing the least possible amount of information. This technique creates orthogonal axes which are linear combinations of the original variables, based on the matrix eigenvalues of the variables considered. The eigenvalues are generated in order from highest to lowest and each eigenvalue is assigned a principal component allowing each PC to retain more variability than the following PC . AccordiQuantitative genetics has developed three uses for principal components (PCs): as a tool to visualize genetic variation patterns, to define the genetic parameters to be estimated, and to separate the original number of variables into a smaller set of principal components to estimate the genetic parameters of these PCs .The PCA technique has been incorporated into genetic evaluations in beef cattle –6, dairyThe aim of this study was to explore the relationship between BV for growth, milk yield, and age at first calving in dual-purpose buffaloes by using PCA.This study was approved by the Ethics Committee for Animal Experimentation of Universidad de Antioquia .The Colombian Association of Buffalo Breeders (ACB) provided the database used in this study. The traits evaluated were: weaning weight (WW), yearling weight W12), weight at 18 months of age  .a), maternal additive genetic (m), maternal permanent environmental (pe), and residual effect (ε). The fixed effects were: sex , number of calving (1 to 14) and contemporary group . Age at weighing was used as a covariate (linear effect). The matrix representation of the model is:For WW and W12 the random effects were: direct additive genetic  and the residual effect (ε). The fixed effects were sex , number of calving (1 to14), and contemporary group (defined as for WW and W12). The age at weighing was used as a covariate (linear effect).For W18 and W24 random effects were the additive genetic random .The matrix representation of the model was:a), permanent environmental (pe), and residual effect (ε). The fixed effects were parity (1 to 14) and contemporary group .For MY270 the random effects were: additive genetic , and maternal genetic effect for yearling weight (MGW12), data are also available in The estimated heritability of the studied traits is presented in PCA was performed using BV of WW, W12, W18, W24, MY270, AFC, MGWW and MGW12 from 133 breeding males chosen from 961 males. The first three PC had eigenvalues greater than one, and explained 65.78% of the original variance of the breeding values for the aforementioned traits, view Distribution of traits in each of the first three components  is shown in The values found in this study for WW and W12 were higher, and W18 and W24 were lower than those reported in Colombia for those traits: 182, 201, 278 and 363 kg, respectively , and 159The performance parameters of buffaloes for WW, W12, W18, W24, MY270, and AFC were better than the data reported for the dual purpose cattle in Colombia [The estimated WW, W12 and W24 heritability was lower than the figures reported in Colombia by Bolivar et al. : 0.42, 0In this study, the estimated heritability for milk yield was lower than previously reported for buffaloes in Brazil: 0.30, 0.25, and 0.28 ,25,26, rThe estimated heritability for AFC was higher than that reported in Nellore heifers: between 0.08 and 0.16 ; but lesThe estimated maternal heritability for WW and W12 coincide with values reported by Albuquerque and Meyer  for NellThe PCA results in this study are consistent with other reports, evidencing the usefulness of PCA to reduce dimensionality. According to the report by Val and Ferraudo , the firAccording to the heritability figures obtained, W18 and MY270 would be the most responsive traits to the selection process, while AFC would be less responsive. PCA facilitates and improves efficiency of the animal selection process by using correlations between traits and components, hence reducing the range of the analysis. It is concluded that the traits studied in this work can be analyzed with the first three PCs.S1 Dataset(XLSX)Click here for additional data file.S2 Dataset(XLS)Click here for additional data file.S3 DatasetInformation of each column corresponds to BV for: milk yield at 270 days (MY270), weaning weight (WW), weight at one year (W12), weight at 18 months (W18), weight at two years (W24), age at first calving (AFC), maternal genetic effect for weaning weight (MGWW), and maternal genetic effect for yearling weight (MGW12).(TXT)Click here for additional data file.S1 File(DOCX)Click here for additional data file.


The small contribution sometimes corresponds to incomplete penetrance, which may result from probabilistic effects on molecular regulations. In such cases, genetic mapping may benefit from the wealth of data produced by single-cell technologies. We present here the development of a novel genetic mapping method that allows to scan genomes for single-cell Probabilistic Trait Loci that modify the statistical properties of cellular-level quantitative traits. Phenotypic values are acquired on thousands of individual cells, and genetic association is obtained from a multivariate analysis of a matrix of Kantorovich distances. No prior assumption is required on the mode of action of the genetic loci involved and, by exploiting all single-cell values, the method can reveal non-deterministic effects. Using both simulations and yeast experimental datasets, we show that it can detect linkages that are missed by classical genetic mapping. A probabilistic effect of a single SNP on cell shape was detected and validated. The method also detected a novel locus associated with elevated gene expression noise of the yeast galactose regulon. Our results illustrate how single-cell technologies can be exploited to improve the genetic dissection of certain common traits. The method is available as an open source R package called  Genetic association studies are usually conducted on phenotypes measured at the scale of whole tissues or individuals, and not at the scale of individual cells. However, some common traits, such as cancer, can result from a minority of cells that adopted a special behavior. From one individual to another, DNA variants can modify the frequency of such cellular behaviors. The body of one of the individuals then harbours more misbehaving cells and is therefore predisposed to a macroscopic phenotypic change, such as disease. Such genetic effects are probabilistic, they contribute little to trait variation at the macroscopic level and therefore largely escape detection in classical studies. We have developed a novel statistical method that uses single-cell measurements to detect variants of the genome that have non-deterministic effects on cellular traits. The approach is based on a comparison of distributions of single-cell traits. We applied it to colonies of yeast cells and showed that it can detect mutations that change cellular morphology or molecular regulations in a probabilistic manner. This opens the way to study multicellular organisms from a novel angle, by exploiting single-cell technologies to detect genetic variants that predispose to certain diseases or common traits. Modern genetics aims to identify DNA variants contributing to common trait variation between individuals. A high motivation to map such variants is shared worldwide because many heritable traits relate to social and economical preoccupations, such as human health or agronomical and industrial yields. In addition to the molecular knowledge they provide, these variants fuel the development of personalized and predictive medicine as well as the improvement of economically-relevant plants, animal breeds or biotechnology materials. However, this high ambition is accompanied by a major challenge: common traits are under the control of numerous variants that each contribute little to phenotypic variation , and thide novo mutations in the cell's daughter. Other loci may modulate the heterogeneity between isogenic cancer cells that underlies tumour progression [Small-effect variants are typically associated with predisposition (or incomplete penetrance): carriers of a mutation display a phenotype at increased frequency, but not all of them do. In this probabilistic context, the statistical properties of cellular traits may sometimes become informative: a tissue may break because cells have an increased probability to detach, a tumor may emerge because a cell type has an increased probability of somatic mutations, a chemotherapy may fail if cancer cells have an increased probability to be in a persistent state. In other words, molecular events in one or few cells can have devastating consequences at the multicellular level. As discussed previously , cellulagression ,8 and regression –11. TheyFortunately, the experimental throughput of single-cell measurements has recently exploded. Technological developments in high-throughput flow cytometry , multiplSaccharomyces cerevisiae offers a powerful experimental framework for developing such methods. Using this model organism, scPTL were discovered by treating one statistical property of the single-cell trait, such as its variance in the population of cells, as a quantitative trait and by applying Quantitative Trait Locus (QTL) mapping to it [a priori which summary statistics must be used.Methods are needed to detect scPTL. With its fast generation time, high recombination rate and reduced genome size, the unicellular yeast ng to it ,20. HoweWe present here the development of a genome-scan method that exploits all single-cell values with no prior simplification of the cell population phenotype. Using simulations and existing single-cell data from yeast, we show that it can detect genetic effects that were missed by conventional linkage analysis. When applied to a novel experimental dataset, the method detected a locus of the yeast genome where natural polymorphism modifies cell-to-cell variability of the activation of the GAL regulon. This work shows how single-cell quantitative data can be exploited to detect probabilistic effects of DNA variants. Our approach is conceptually and methodologically novel in quantitative genetics. Although we validated it using a unicellular organism, it opens alternative ways to apprehend the genetic predisposition of multicellular organisms to certain complex traits.X be a quantitative trait that can be measured at the level of individual cells. X is affected by the genotype of the cells and by their environmental context. However, even for isogenic cells sharing a common, supposedly homogeneous environment, X may differ between the cells. To describe the values of X among cells sharing a common genotype and environment, we define a single-cell quantitative trait density function f [X at a given level  constitutes the 'phenotype' of the individual from whom the cells are studied. As for any macroscopic phenotype, it can depend on the environmental context of the individual  as well as on its genotype. Single-cell trait density functions also obviously depend on the properties of the cells that are studied, such as their differentiation state or proliferation rate.We specify here the concepts and definitions that are used in the present study. Let nction f  as the fen level . StatistX that is different from the one followed by cells of another individual, because of genotypic differences between the two individuals .We focus here on the effect of the genotype. Conceptually, cells from one individual may follow a density function of ividuals . The imp disease . The terX is a location on a chromosome where a genetic variant changes the mean or the median of X in the cell population. Similarly, a varQTL is a genetic locus changing the variance of X and a cvQTL is a genetic locus changing the coefficient of variation  of X in the cell population. All three types of loci  assume a change in f and they are therefore special cases of scPTL. However, not all scPTL are QTL: many properties of f may change while preserving its mean, median, variance or CV. The purpose of the present study was to develop an approach that could identify scPTL without knowing a priori how it might change f.A quantitative trait locus (QTL) linked to f without affecting its expected value (the mean of X). If not, then QTL mapping will capture the genetic modifiers of f and searching for more complex scPTL is not justified. In contrast, if other-than-mean genotypic changes of f are frequent, then scPTL can considerably complement QTL to control single-cell traits. In this case, scPTL mapping becomes important.An important question before investing efforts in scPTL mapping is whether genotypes can modify S. cerevisiae provides the opportunity to study individual cells that all belong to a single cell type, in the context of a powerful genetic experimental system. By analysing specific gene expression traits in this organism, we and others identified loci that meet the definition of scPTL but not of QTL [In multicellular organisms, cell types and intermediate differentiation states constitute the predominant source of cellular trait variation. Studying their single-cell statistical characteristics requires accounting for the developmental status of the cells. This constitutes a major challenge that can be avoided by studying unicellular organisms. The yeast t of QTL ,21. ThisS. cerevisiae, individual segregants had been amplified by mitotic  divisions and traits of cellular morphology were acquired by semi-automated fluorescent microscopy and image analysis [f should coincide with a high genetic heritability of the expected value of f. In particular, the coefficient of variation (CV) of a single-cell trait should then display high heritability only if the mean value of the trait also does. To see if this was the case, we computed for each trait the broad-sense genetic heritabilities of both the mean and CV of the trait. Note that the genetic heritability computed here is not the same as the mitotic heritability of cellular traits transmitted from mother to daughter cells. Here, a value (mean or CV) is computed on a population of cells, and its heritability corresponds to the proportion of its variation that can be attributed to genetic differences between the cell populations , a cellular trait has been quantified in many individual cells of the same type. This way, the observed distribution of the trait constitutes the phenotypic measure of individuals. We also assume that a genetic map is available and the individuals have been genotyped at marker positions on the map. The method we propose is based on three steps. First, a distance is computed for all pairs of individuals in order to quantify how much their phenotype differs. We chose the Kantorovich metric  to measure this distance because, unlike the Kullback-Leibler divergence, it satisfies the conditions of non-negativity, symmetry and triangle inequality and, unlike the Hellinger distance, it does not converge to a finite upper limit when the overlap between distributions diminishes [f-function) into another heap (a second f-function). It has enabled developments in various fields, ranging from mathematics [p-values produced by permutations where the identities of individuals are re-sampled. The full procedure is described in details in the methods section.Scanning for scPTL considers the entire distribution of single-cell trait values as the phenotype of interest and searches the genome for a statistical association with minishes . The Kanhematics  to econohematics ,30 to thhematics . The nexhematics ,32. In shematics . The thiη was added to introduce intra-genotype inter-individual variation which, in real datasets, could originate from limited precision of measurements or from non-genetic biological differences between individuals. For each of five increasing values of η, about 130 individuals were simulated.We first evaluated if our method could detect scPTL from simulated datasets. To do this, we considered a probabilistic single-cell trait governed by a positive feedback of molecular regulations. This is representative of the expression level of a gene with positive autoregulation. As depicted in We first scanned the generated dataset by QTL mapping, treating either the mean trait or its variance as the phenotype of interest. This way, the central scPTL locus was detected only when intra-genotype noise was null or very low . This waThe results described above using a simulated dataset suggest that the method can complement usual QTL mapping strategies. To explore if this was also the case when using real experimental data, we applied scPTL scans to the dataset of Nogami et al.  mentioneExamples of scPTL of yeast cellular morphology are shown in α protein Gpa1p. It targets a domain that is essential for physical interaction with pheromone receptors Ste2p and Ste3p [I469S to the receptor causes residual activation of the pathway in the BY strain, as seen by transcriptomic profiling [I469S mutation, we examined the data from a BY strain where this mutation was cured [I469S SNP. Another trait, corresponding to the distance between the bud tip and the short axis of the mother cell, also mapped to this locus, with the RM allele associated to greater cell-cell variability, and data from the GPA1I469S allele-replaced strain validated this SNP as the causal polymorphism  at 30 minutes post induction, the time at which heterogeneity markedly differed between the BY and RM strains . Importantly, the scPTL was not a QTL: the locus genotype did not correlate with the mean level of expression of the population of cells (LOD score < 2.8).Visualizing the distributions of single-cell expression levels at 30 minutes revealed that the RM and BY genotypes at this locus corresponded to high and low cell-cell heterogeneity, respectively . Thus, tDOT6RM  displayed very similar transcriptional responses at intermediate times of induction  scale can be investigated.When considering macroscopic phenotypes, it is important to distinguish the situations where scPTL mapping is biologically relevant from those where it is not. The determinants of human height, for example, act via countless cells, of multiple types, and over a very long period of time (~ 16 years). In such cases, the macroscopic trait results from multiple effects that are cumulated and considering the probabilistic individual contribution of specific cells is inappropriate. Similarly, many tissular traits heavily rely on communications between cells and probabilistic changes in a few may not affect the collective output of the cell population. In contrast, a number of macroscopic traits can be affected by particular events happening in rare cells or at a very precise time (see below). In these cases, studying the probabilities of a biological outcome in the relevant cells or of a molecular event within the critical time interval can provide invaluable information on the emergence of the macroscopic phenotype, and scPTL mapping then becomes relevant.de novo mutations in the cell's daughter. These variants have classically been identified by genetic linkage of the macroscopic trait (disease frequency in families and cohorts), and their role on the maintenance of DNA integrity was deduced afterwards by molecular characterizations. For a review on the genetics of cancer syndrome predisposition, see [A striking example of such traits is cancer. Genetic predisposition is conferred by variants affecting somatic mutation rates and these loci are special cases of scPTL: the cellular trait they modify is the amount of ion, see ,43. scPTion, see ,8 and trion, see –11. Genep-value lower than 0.01  in D. santomea but not in D. yakuba. These and other examples [The interest of scPTL mapping is not restricted to cancer biology. Developmental processes and cellular differentiation are also vulnerable to mis-regulations happening in few cells or during short time intervals. Their macroscopic outcome can therefore be affected by probabilistic events at the cellular scale. For example, stochastic variation in the expression of the stem cell marker Sca-1 is associated with different cellular fates in mouse hematopoietic lineages , suggestelection . The secelection . Embryosexamples  illustraOur new method based on the Kantorovich distance is not the only one by which scPTL can be identified. Applying classical QTL mapping to summary statistics of the cellular traits can also be efficient. We emphasize that the two approaches are complementary. For example, our method missed to detect linkage for 9 yeast morphological traits for which cvQTL scans were successful, but it detected several significant scPTL that were missed by the QTL-based approach Figs  and 6. SWhile the principle of genotype-phenotype genetic linkage dates back to several decades ago, the statistical methods that test for linkage are still being improved, especially regarding multi-loci interactions or population structure corrections ,56. The 'captured variance' as quantified by the eigenvalues of the LDA is not the same as the 'explained variance' which must be re-computed by regression; and if linearity of the data is questionable, the method remains a useful tool if it detects scPTL but interpreting variance proportions need justifications.Estimating the proportion of variance explained by scPTL is not straightforward: the A phenotypic space can be constructed by alternative ways that do not require the Kantorovich metric. For example, we considered representing individuals in a "space of moments", where the coordinate of every individual on the i-th axis was the i-th moment of the cellular trait distribution associated to this individual. We applied this to the yeast morphological data and we searched for genetic linkages by linear discriminant analysis as described above. This approach detected many significant scPTL but we encountered a difficulty that was avoided by our Kantorovich-metric based method. When searching for significant linear discrimination, the dimensionality of the phenotypic space is important. At high dimensionality, discriminant axes are more likely to be found. This improves detection in the actual data but at the expense of increasing the degrees of freedom and therefore the false positive hits estimated from the permuted data. In a "space of moments", the properties of the single-cell trait distributions are very important because they define which axis (moments) are relevant to separate individuals. Keeping the 4-th axis may be crucial even if all individuals have very similar first, second or third moments. Choosing the appropriate dimension for LDA is then arbitrary and it becomes difficult to keep a good detection power while still controlling the FDR. In fact, applying QTL mapping on the 3rd and 4th moments of all traits was fruitless because the FDR could not be controlled at the genome-by-phenome scale. This issue is avoided in the case of Kantorovich distances because multi-dimensional scaling can be applied without normalization and the axes of the phenotypic space are ranked by descending order of their contribution to the inter-individual differences. The 4-th axis, for example, contributes less than the first three axes to the separation of individuals in the space. If keeping the 4-th dimension prior to LDA is beneficial for linkage, then keeping the first three axes is also highly relevant, and this is true regardless of the properties of the single-cell trait distributions. We found this very useful: our algorithm adds dimensions one by one and evaluates the benefit of each increase see .There are at least three lines along which our method may be further improved. First, LDA is only appropriate if genotypic categories can be distinguished along linear axis. If individuals in the phenotypic space are separated in non-linear patterns, other methods such as those based on kernel functions  may be mFurthermore, although our study was focused on probability density functions, steps (ii), constructing the phenotypic space, and (iii), testing for genetic linkage, could in principle be applied to other types of functions, provided that a relevant metric estimating the dissimilarity between such functions exists. This could be interesting in the case of function-valued traits, such as speech sound or other time-series functions. The evolution of these functions is being studied using phylogenetic methods that present challenging statistical issues ,63. ExteFinally, we can anticipate that gene-gene and gene-environment interactions also shape the probability density function of cellular traits. Our results on the activation of the yeast galactose network remarkably illustrate this: the effect of the scPTL on chromosome V is apparent only transiently, and in response to a change of environmental conditions. It is tempting to extrapolate that signaling pathways in plants and animals may be affected by scPTL that act at various times and steps along molecular cascades.In conclusion, our study provides a novel method that can detect genetic loci with probabilistic effects on single-cell phenotypes, with no prior assumption on their mode of action. By exploiting the power of single-cell technologies, this approach has the potential to detect small-effect genetic variants that may underlie incomplete trait penetrance at the multicellular scale.X that represented the number of proteins in one cell at a given time. Under the model, the dynamics of X is controlled by two processes: (1) protein production with rate α and (2) protein degradation with rate β. We assume that the gene is positively auto-regulated by a 4-mer complex, meaning that α is an increasing function of X with a typical Hill-like shapeα0 the leaky production rate in absence of X 4-mers at the promoter, α1 the production rate in presence of 4-mers, and K the dissociation constant of the 4-mer. We set β = 1, which corresponds to scaling time units. The dynamics of the mean value of X in a population of isogenic cells follows the equation shown in X, we performed exact stochastic simulations of the chemical system defined by the two reactions rate α and β, using the Stochastic Simulation Algorithm [Single-cell gene activity was modeled by a stochastic variable lgorithm .α0, α1, K) was controlled by one locus that could exist in two alleles (A and B) with mean values  and, for simplicity, that the individuals were haploids. To account for sources of inter-individual variability within genotypic groups, the values of the parameters for one individual were drawn from normal distributions of mean values μ0A/B, μ1A/B and μKA/B and of standard deviations ημ0A/B, ημ1A/B and ημKA/B where η represented the strength of inter-individual variability. η was assumed to be the same for A and B alleles. Values were: μ0A = 6.3, μ0B = 0.1, μ1A = 12, μ1B = 10, μKA= 10 and μKB = 1.6.To generate two groups of individuals, we assumed that the set of parameters (H2= (varT − varE) / varT, with varT and varE being the total and environmental variance, respectively. For varT by randomly choosing one of the three replicate sample of each segregant and computing the variance across these 59 values. This was repeated 100 times and the estimates were averaged. Our estimate of varE was the pooled variance of varBY, varRM, varSeg1, varSeg2, …, varSeg59 which were the between-replicates variance of each strain. Confidence intervals on H2 values were computed by bootstrapping the strains. For the filtering step prior to linkage, H2 was computed slightly differently in order to be consistent across mapping methods (see below).All statistical analysis were done using R (version 3.1.2) . The datf1 and f2 as the area under the absolute value of the cumulative sum of the difference between the two distributions:We first normalized the distributions as densities . Following , we thencmdscale from the stats package. The number of dimensions retained (ndim) was the number of eigenvalues exceeding the expected value under the hypothesis of no structure in the data . We computed the heritability of each yeast morphological trait in this multidimensional space. This was done as above for one dimension, by computing the total variance of the data, and estimating the environmental variance from the replicated experiments made on the parental strains. For 147 traits, heritability was greater than 0.5 and scPTL were searched. Details on how these steps were implemented in R are described in ptlmapper R package (https://github.com/fchuffar/ptlmapper).Multi-dimensional scaling of the resulting distance matrix was then performed using the R function d ranging from 2 to ndim. For each value of d, we applied LDA at every marker position and we recorded the Wilks' lambda statistics:j was the j-th eigenvalue of the discriminant analysis. Low values of this statistics allow to reject the null hypothesis of no discrimination by the factor of interest [P is the p-value of the Wilk's test (deviation of Λ from the F-statistics with relevant degrees of freedom). Note that P is not interpreted directly as a significance value for linkage (see the permutation test below).The yeast genotypes we used were from Smith and Kruglyak . For theinterest  which, iWbest, <W> and σW were the highest, the mean and the standard deviation of all W scores found on the genome, respectively. Finally, we chose the dimension that maximized this Z-score (i.e. dimension where the linkage peak had highest contrast). Very importantly, the same degrees of freedom  were allowed when applying the permutation test of significance (see below). The distribution of the dimensionalities retained for the morphological traits is shown in ptlmapper R package (https://github.com/fchuffar/ptlmapper).We then quantified how much the best marker position was distinguished from the rest of the genome by computing a Z-score:H2 > 0.5 prior to linkage. To do so, we re-computed H2 values in a way that was consistent with the heritability calculation of the phenotypic space prior to scPTL mapping, where replicates of segregants were pooled together before analysis of inter-strain variation (see above). In this case, only 3 replicates of BY and 3 replicates of RM are then available to estimate the environmental variance. Therefore, we estimated varT as the variance of the 59 macro-trait values of the segregants and varE as (varBY+ varRM) / 2, with varBY (resp. varRM) being the variance of the three macro-trait values of the BY (resp. RM) strain. We then scanned the genome using the scanone function from r/qtl [r/qtl is available in the open source ptlmapper R package (https://github.com/fchuffar/ptlmapper).QTL-based mapping was performed as follows. A quantitative trait was considered at the cell-population level. This macro-trait was either the mean (for QTL), the coefficient of variation  or the variance (for varQTL) of the cellular trait in the population of cells. For the yeast morphology data, we selected the traits with om r/qtl  with a som r/qtl . Our codndim, the dimension showing the best contrast (high Z score) was retained, and the highest W score obtained at this dimension was recorded. The empirical threshold corresponding to genome-wide error rates of 0.1%, 1% and 5% were the 99.9th, 99th and 95th percentiles of the 1,000 values produced by the permutations, respectively. These thresholds are typically those employed in whole-genome scans for a single trait.We first explain the case where a single trait is studied. When the trait was mapped using R/qtl, significance was assessed by the permutation test implemented in function scanone of the package . For scPNtraits traits only. For each of these traits, LOD scores were computed on the genome by interval mapping using the macro-trait value as the quantitative phenotype of interest. Significance was assessed by random re-assignment of the macro-trait values to the individuals (yeast segregants). We did 1,000 such permutations. For each one, the genome was scanned as above and the highest LOD score on the genome was retained. This generated a 1,000 x Ntraits matrix Mperm of the hits expected by chance. At a LOD threshold L, the FDR was computed as:NActualL was the number of linkages obtained from the actual dataset at LOD > L, and NFalseL was the expected number of false positives at LOD > L, which was estimated by the fraction of elements of Mperm exceeding L.We now explain the case of the morphological study, where multiple traits (220) were considered. This case is similar to system genetics studies, where the FDR must be controlled. Keeping it below 10% ensures that 9 out of 10 results are true positives, which is often considered as acceptable. Four different methods were used. For three of them, single-cell trait values were resumed to a scalar macro-trait and QTL was searched. The three methods differed by the choice of this macro-trait, which was either the mean or the coefficient of variation of single-cell traits, or the coordinate of individuals on the first axis of the phenotypic space. For each of the three methods, morphological traits with less than 50% genetic heritability (see above) were not considered further, and QTL was searched for the remaining ndim dimensions had been built as explained above by computing Kantorovich distances and applying multi-dimensional scaling. Let P1, P2 and PS be the phenotypic matrices of parent 1 (strain BY), parent 2 (strain RM) and segregants, respectively, with rows being the samples  and columns being the ndim coordinates of each sample in the phenotypic space. These matrices had dimensions 3 x ndim for P1 and P2 and 59 x ndim for PS. Genetic heritability was computed as H2= (varT—varE) / varT, where the total variance varT was the variance of the samples in PS, and where the environmental variance varE was estimated as (varP1+ varP2) / 2, with varP1 (resp. varP2) being the variance of the samples in P1 (resp. P2). Morphological traits showing H2 < 0.5 were discarded, and scPTL mapping was applied to the remaining Ntraits traits as described above . Significance of W scores was assessed as described above for the LOD scores, by performing 1,000 permutations and determining the FDR associated to various thresholds of W scores.The fourth method considered all coordinates of the individuals in the phenotypic space. At this step, for each morphological trait, a phenotypic space of β0 = 0.2, ϑ = 2.5, β∞ = 0.05 and n = 2.5.For each cell, the probability to divide depended on the concentration of gene product X according to the following Hill-like function :P(X) = η = 0.16. For each individual, parameters alpha0, alpha1 and K were drawn from the same normal distributions as above, where mean and variance depended on the genotype (A or B). At age 0, a population of 1,000 cells was initiated with X = 5. This population was then evolved by Stochastic Simulation Algorithm [The regulation of X was governed by the same model as above, with lgorithm , with a The yeast strains and oligonucleotides used in this study are listed in PGAL1-yEGFP-NatMX cassette that could be integrated at the HIS3 genomic locus.To construct the Gal-GFP reporter, we first removed the MET17 promoter of plasmid pGY8  by digesHIS3 locus of strain BY4716 (isogenic to S288c), YEF1946 (a non- clumpy derivative of RM11-1a) and in 61 F1 non-clumpy segregants from BY471xRM11-1a described in [HIS3:PGAL1-yEGFP-NatMX:HIS3 set, respectively.Plasmid pGY37 was linearized at NheI and integrated at the ribed in  to generPEST reporter coding for a destabilized fluorescent protein [PEST was under the control of the PGK promoter. pGY334 was constructed in several steps. The PGK promoter was PCR-amplified from pJL49 (gift from Jean-Luc Parrou) using primers 1A23 and 1A24, digested by BamHI and cloned into the BamHI site of pGY10. The resulting plasmid was digested with EcoRV and XbaI, subjected to Klenow fill-in end blunting and religated, generating plasmid pGY13 carrying a HIS3:PPGK-yEGFP-NatMX:HIS3 cassette. The lox-CEN/ARS-lox sequence from pALREP [PEST sequence was PCR-amplified from pSVA18 [in vivo into pGY252 (digested by MfeI and DraIII), leading to pGY334. The GAL1 promoter fragment was amplified by PCR from pGY37 using primers 1J33 and 1I42 and cloned into plasmid pGY334 by recombination at homologous sequences flanking the BamHI site of the plasmid. The CEN/ARS cassette of the resulting plasmid was excised by transient expression of the Cre recombinase in bacteria [HIS3:PGAL1-GFPPEST-NatMX:HIS3 cassette.In parallel, we also constructed a GAL1-GFP protein . We derim pALREP  was amplm pSVA18  using prbacteria , generatHIS3 locus of BY4724 (isogenic to S288c) and GY1561 to create GY1566 and GY1567 strains, respectively. Strain GY1561 is a non-clumpy derivative of RM11-1a where the KanMX4 cassette was removed. It was obtained by first transforming RM11-1a with an amplicon from plasmid pUG73 [pGY338 was linearized by NheI and integrated at the id pUG73  to generLiquid cultures in synthetic medium with 2% raffinose were inoculated with a single colony and incubated overnight, then diluted to OD600 = 0.1  and grown for 3 to 6 hours. Cells were then resuspended in synthetic medium with 2% raffinose and 0.5% galactose and grown for the desired time . Cells were then washed with PBS1X, incubated for 8 min in 2% paraformaldehyde (PFA) at room temperature, followed by 12 min of incubation in PBS supplemented with Glycine 0.1M at room temperature and finally resuspended in PBS. They were then analyzed on a FACSCalibur (BD Biosciences) flow cytometer to record 10,000 cells per sample.flowCore package from Bioconductor [Flow cytometry data was analysed using the onductor . Cells oDOT6RM allele was amplified by PCR from genomic DNA of the RM strain using primers 1K87 and 1K88. It was then cloned into plasmid pALREP [HIS3:PGAL1-GFPPEST:HIS3 cassette). The pop-in pop-out strategy was applied as previously described [The d pALREP  by homold pALREP , generatescribed  and fourThe yeast morphological data corresponds to the experiments described in . For thehttp://flowrepository.org under accession number FR-FCM-ZZPA.The flow cytometry data corresponding to yeast galactose response is made available from https://github.com/fchuffar/ptldata.The simulated data of https://github.com/fchuffar/ptlmapper. A tutorial of this package explains how to run the analysis on the simulated dataset.The scPTL mapping method is made available as an open source R package (ptlmapper) which can be downloaded from S1 FigDOT6RM) were cultivated in raffinose 2% and were shifted to a medium containing Raffinose 2% and Galactose 0.5%. After the indicated time, cultures were fixed with paraformaldehyde and analysed by flow cytometry. Histograms correspond to the fluorescent values obtained on cells gated for cell-size (see Strains GY1566 (BY), GY1567 (RM) and GY1604, GY1605, GY1606, GY1607 (BY-size see .(PDF)Click here for additional data file.S2 FigA) Confidence interval of scPTL location. An example of confidence intervals obtained by bootstrap is shown. In this case, we used the data from 90 individuals shown in B) Dimensionalities retained for the yeast morphological traits. Let n1 be the Kaiser’s based number of dimensions retained after MDS, and n2 the Z-score based number of dimensions retained for linkage. The figure shows the distribution of these values for yeast morphological traits where multi-dimensional scPTL outperformed the one-dimension only. On this plot, the number printed at position x = n2, y = n1 is the number of morphological traits for which the corresponding  values were chosen.(PDF)Click here for additional data file.S1 Table(XLS)Click here for additional data file.S2 Table(DOCX)Click here for additional data file.S1 Methods(DOCX)Click here for additional data file.S2 Methods(TGZ)Click here for additional data file.S1 Dataset(ZIP)Click here for additional data file.


Our simulation showed that, compared to the existing methods, the GWSS method had better performance in most of the scenarios. The GWSS (in particular VDWSS-t) method is particularly robust for opposite association directions, association strength, and varying distributions of minor-allele frequencies. It is therefore promising for detecting disease-associated loci. For empirical data application, we also applied our GWSS method to the Genetic Analysis Workshop 17 data, and the results were consistent with the simulation, suggesting good performance of our method. As re-sequencing studies become more popular to identify putative disease loci, we recommend the use of this newly developed GWSS to detect associations with both common and rare variants.The rapid advances in sequencing technologies and the resulting next-generation sequencing data provide the opportunity to detect disease-associated variants with a better solution, in particular for low-frequency variants. Although both common and rare variants might exert their independent effects on the risk for the trait of interest, previous methods to detect the association effects rarely consider them simultaneously. We proposed a class of test statistics, the generalized weighted-sum statistic (GWSS), to detect disease associations in the presence of common and rare variants with a case-control study design. Information of rare variants was aggregated using a weighted sum method, while signal directions and strength of the variants were considered at the same time. Permutations were performed to obtain the empirical  CLEC16A gene for type-1 diabetes and ANGPTL gene for metabolism traits wSSU [23]uggested , the opt samples . We noti samples  to detec samples  and sum  samples , respect samples , often s samples  clusterst-statistic, to account for association directions, the magnitude of association, and the presence of noise variants. In addition, we utilized a “filtering algorithm” with or without a threshold to collapse information of rare variants collectively in detecting disease associated variants. In the cases where rare and common variants altogether contribute to the etiology of complex traits, our new methods had higher power in detecting signal variants. We conducted a series of simulations to evaluate detection power in different scenarios, including linkage disequilibrium (LD) structure among variants, association directions, MAF distribution of RVs, and the ratio of signal and noise variants. In addition, we applied our method to the Genetic Analysis Workshop 17 (GAW17) dataset to evaluate the performance of our method.In this study, we reviewed recently proposed methods for the detection of disease associations for both CVs and RVs for comparisons. To overcome aforementioned methodological limitations, we also proposed a class of new methods based on p-values. From these two considerations, we propose a generalized weighted sum statistic (GWSS), which includes various existing methods as special cases. Note that most of existing methods need a predetermined value for MAF to define “rare variants”. We believe that a data-driven approach for defining rare variants is more suitable, especially in the presence of signal CVs. We thus also incorporate the idea of VT [k variants with n subjects (n1 cases and n0 controls). For subject i, let Gij ∊ {0,1,2} be the number of minor allele in variant j and Di ∊ {1,0} be the disease status (yes/no). For variant j, let qj be its adjusted MAF [ORj be its estimated odds ratio. The algorithm of GWSS is described below.One major aim of this research is to propose a robust test method to overcome the difficulties encountered by existing methods. In view of the robustness, we aim to extend the idea of WSS without specifying any distributional assumption, which includes two core components. First, a weighted sum of RVs is used to aggregate the rare information. Commonly used weights include MAF and OR, but they will suffer the problem of low power as mentioned in the previous section. Second, with the aggregated variants, a summary measure is adopted to obtain the final genetic scores and the corresponding ea of VT  into GWSsted MAF  in the cθ of MAF, obtaining the aggregated genotype for each subject i aswj is one of the following four possible weights:Given a threshold Sθ using one of the following two summary measures:rank-sum:t-sum: Based onθ is known a priori, define S = Sθ. Otherwise, repeat Steps 1–2 for all θ ∊ Θ and calculate S = supθ∊ΘSθ as the summary statistic, where Θ is a predetermined set of possible values of θ.If p-value of S via randomly permuting the disease status, while keeping the case/control ratio constant.Obtain the empirical The robustness nature of GWSS is threefold as described below:et al. [wj.The weightting WSS , implicig et al.  proposedoff as 1 ,28, that of SKAT  as wj.Gi(θ) for each subject i, two summary measures oft-sum”, are considered. The “rank-sum” statistic, which borrows the idea of the Wilcoxon rank-sum statistic, is robust against model assumptions, but at the cost of being less efficient when the distributional assumption is valid. Thus, when the sample size is moderate , we suggest to use the more efficient “t-sum” to summarize information.In Step 2, based on the aggregated genotype θ for defining RVs should be determined. The same idea is also used in GWSS, which involves the indicator function I(qj ≤ θ) in Step 1 of GWSS algorithm. This indicator function, however, ignores all variants with MAF >θ, which will lose detection power in the presence of signal CVs. Needless to say, it is also hardly the case that researchers have prior knowledge about the value of such θ. It turns out that a data-driven approach for determining θ is preferable. We thus follow the idea of VT to use the maximum value of Sθ among possible values of θ (see Step 3) as the summary statistic, which is able to adapt to various situations of signal/noise RVs and CVs .Before implementing WSS, a threshold t” denotes the method using t-sum in Step 2, and “V” denotes the method that taking maximum over θ in Step 3.We close this section by noting that GWSS includes many existing test methods as special cases, such as WSS  and ORWSk SNPs with 500 cases and 500 controls. Two settings for MAF (denoted by MAFj) of RV were considered. For the case of identical distribution, signal and noise RVs were both generated from Uniform. For the case of different distributions, signal and noise RVs were generated from Uniform and Uniform, respectively, i.e. the MAFs of signal variants are lower than noise variants [MAFj, let the latent vector Di is generated from the logistic regression modelk − 8 noise variants. The effects of SNP 1–4 are additive, and the effects of SNP 5–8 are non-additive. As to SNP 5–8, the effect is 3 if any of the SNP 5–8 has mutation. We note that the non-additive effect of multi-markers may be jointly incorporated in haplotypes [ρ = 0 or 0.7 for LD between markers; (2) ORj for  is  or ; (3) p-values are calculated based on 500 permutations. The detection power is calculated as the proportion of test statistic attained significance level of 0.05 over 1000 simulation datasets.We conducted simulation studies to compare the performance of the proposed GWSS with existing methods. Assume that a study examines variants . CVs arere index  and Φ(⋅)plotypes  to captup-value for CMC statistic instead in the following simulation. We only present methods with better performance in the Tables, including five existing methods  and five tests of our GWSS method . The results of comparisons with other methods are listed in the Supporting Tables. t have the best performance. When there are noise variants, MAF-based methods such as wSSU, VT, WSS-t, and VWSS-t have better performances than others, since the noise CVs are down-weighted by the value of MAF in these methods. In contrast, the detection powers of SSU, Sum Test, KMR, C-alpha, SKAT1  and DSS-t decrease dramatically, suggesting that some underlying assumptions in these methods are inappropriate in the presence of noise CVs. Comparing WSS with WSS-t for which one uses “rank-sum” and the other uses “t-sum”, WSS-t has better performance than WSS, especially when noise CVs are present. The same pattern exists between ORWSS and ORWSS-t. This observation suggests that, in the presence of noise CVs and with moderate sample size, using “t-sum" to obtain the summary score is able to extract more information and, hence, is more efficient than using “rank-sum”.In t . Taking two special cases of GWSS, ORWSS-1, SKATb, SKAT-C and SKAT-A) if there is no noise CV. In the presence of noise CVs, SSUw , wSSU , SKATb  density function evaluated at qj as the weight) and SKAT-C/A  perform better as expected. As to the class of GWSS methods, ORWSS-t, DWSS-t, VORWSS-t, VDWSS-t and VDSS-t perform well uniformly in all situations. These results reveal that using OR-based weighting  still have improved performances. In contrast, we observe power losses in Sum Test, CMC-p (cutoff-point is fixed at 0.01), WSS-t and VWSS-t when opposite association directions are present. The failure to detect associations using Sum Test and CMC-p is due to the offset of harmful and protective effects during the summation process. The power losses in WSS-t and VWSS-t are because the weight t, WSS using rank-sum is less affected by the presence of RVs with opposite directions and has a higher detection power than using “t-sum”, which is a benefit of using the more robust “rank-sum” summary measure.Simulation results are provided in ρ = 0: (1) SSU, KMR, C-alpha, ORWSS and SKAT1 performed worse in the presence of CVs; (2) for GWSS, using t-sum is better than using rank-sum in most of the situations; (3) methods using OR-based weighting scheme can identify protective association while methods using MAF-based weighting scheme cannot; (4) the VT-version of GWSS methods are superior to non-VT-version ones whether the CVs and protective association exists or not. The impact of LD among variants depends heavily on the association directions among variants. When the effects of signal variants are in the same direction, the joint signals become stronger than the case of uncorrelated variants. In this situation, it is easier to detect association and the detection power is much higher than the case of ρ = 0 . The GAW17 dataset consists of 697 unrelated individuals, each of whom provided genotypes  that were sequenced from the 1000 Genomes Project (http://www.1000genomes.org). The simulated phenotypes used binary disease status that was generated for each individual . We calculated association p-values for disease affected status based on 1000 permutations. We compared our GWSS method to a few commonly implemented methods, including SSUw, WSS, ORWSS, VT, and SKAT-type methods  consisted of at least one RV and CV were considered. We listed fourteen genes that showed significantly aggregated signals to the disease affected status in ARL6IP2, TRIM42, SHC3, FRMPD2, NFKBIA, MBD1, GDF15 and SUSD2) had the smallest p-value detected by one of our GWSS method . The p-values of the other six genes  were also ranked in the top three among all methods. We found that SKAT-type methods performed well  in seven genes, but completely failed to detect signals for three genes . Notably, three genes  can only be detected by our WSS-t and VWSS-t. These results suggested that our GWSS method had good performance among all methods. Taken together, we concluded that our GWSS method robustly showed good performance under different scenarios.To conduct the association tests of the GAW17 data, only genes that (1) had significant Various statistical testing strategies have been developed for identifying associations between RVs and complex disease traits ,21,23,24Many existing methods suffer from low to moderate power loss when noise variants (rare or common) are present. The presence of noise variants can dramatically decrease the detection powers and increase the false positive results, which further complicate the interpretations of the analysis results. Importantly, by incorporating the idea of VT, our GWSS method can separate signal variants from noise ones in a data-driven approach. In practice, noise may also be introduced from calling rare variants and using imputed genotypes . Methodst, and VWSS-t ignore associations with protective effects (i.e. OR<1), also encounter power loss. Although the protective associations can be detected by using OR as the weighting scheme, the unstable estimation of OR arises another difficulty. The newly proposed VDWSS-t exhibits a more robust estimation in both association strength and direction, which are more applicable in many situations.Power loss may also occur when using methods assuming the same effect direction of variants, since in reality both risk and protective variants are present . It is eThe power of association testing also depends on the pattern of LD among markers , since nt and ORWSS-t performed better than WSS and ORWSS, suggesting that the use of “t-sum” is beneficial especially when CVs are present. One reason is that “t-sum” utilizes more information than “rank-sum”, through considering both the mean shift and variation. In particular, when only RVs are involved and all variants have harmful effects, most of the genotypes are common homozygotes (i.e. Gij = 0) and the aggregated genotypes (i.e. Gi(θ)) close to zero in most of the subjects, forming a right-skewed distribution. In this situation, “rank-sum” is more appropriate due to its robustness against the normality assumption. Similarly, when the assumption of the same association direction among variants was violated, both WSS and WSS-t had poor performance, while the rank-sum slightly improved the detection power. In contrast, when CVs are involved  with moderate sample size, the distribution of aggregated genotypes would approximately follow the normal distribution, so using “t-sum” is suitable.It is found that WSS-t perform nearly equally well under identical MAF distribution Click here for additional data file.S1 Table(DOC)Click here for additional data file.S2 Table(DOC)Click here for additional data file.S3 Table(DOC)Click here for additional data file.S4 Table(DOC)Click here for additional data file.S5 Table(DOC)Click here for additional data file.


Different genome wide association methods (GWAS) including multivariate analysis techniques were applied to identify quantitative trait loci (QTL) and pleiotropy in the simulated data set provided by the QTL-MAS workshop 2012 held in Alghero .Genetic correlations and heritabilities for all three quantitative traits were obtained by a multivariate animal model. In a second step the data were corrected for a polygenic component containing the genomic-based kinship matrix. Residuals from this model were later used for QTL detection in a regression analysis, to achieve genome-wide rapid association (GRAMMAR). In order to take pleiotropic effects into account, all three traits were condensed via principle component techniques to two principal components (PC) which reflect the phenotypic variance covariance structure of all traits. The PCs were analyzed by single trait analysis by GRAMMAR. As an alternative to GRAMMAR, the data set was analyzed by Bayesian methods implemented in the package snptest. The program allows the analysis of the data in a univariate and a multivariate way, where all three traits are investigated simultaneously.g = 0.84). All applied GWAS revealed 10 QTL on four different chromosomes. No QTL was detected on chromosome 5. The Bayesian multivariate analysis revealed significant pleiotropic SNPs.According to the polygenic model, analyses the three traits revealed high heritability . Traits 1 and 2 were highly correlated (rPrincipal component and multivariate analyses seem to be promising in order to characterize the genetic basis of trait relationships. Recently, the high-density single nucleotide polymorphism (SNP) arrays have been developed for almost all domestic animals. These tools offer the prerequisite of genome-wide association studies (GWAS), a powerful approach for high-resolution mapping of loci controlling phenotypic traits . In agriThe aim of this study was to apply different genome wide association methods to identify quantitative trait loci (QTL) in the simulated data set provided by the QTL-MAS workshop 2012 Alghero  and to investigate pleiotropic effects among the three simulated quantitative traits.In a first step genetic correlations and heritabilities for all three quantitative traits were obtained by a multivariate animal model analyzed by VCE6 . In ordeA quality control was performed for 10000 SNPs equally spaced on five chromosomes using a minor allele frequency < 0.01 and a significant deviation from Hardy Weinberg equilibrium (p < 0.01). Markers, which deviated from these criteria, were removed from the data set, so that 9596 were used for the GWAS.The GWAS was performed with the genome-wide rapid association using mixed model and regression (GRAMMAR) . Residuai yis the phenotype  of the ith individual, i aare the random additive polygenic effects with where ei are the random residual effects. The kinship coefficients G from genomic data were estimated using the formula [and  formula :ik gis the genotype of the th iperson at the th kSNP, k pis the frequency of the major allele and n is the number of SNPs used for kinship estimation.where The residuals were estimated asThe test for association was performed with these residuals using a linear model:* yrepresents residuals from model (3) of th iindividuals , * µthe intercept, k is the regression on the genotype (ig), where g contains a dose effect of a target allele for each SNP and * eis the random residual [2 test-statistic is used to determine whether a SNP is significantly associated with the trait.where residual . A Χ2 teIn addition permutation resampling techniques, as implemented in GenABEL , were usAs an alternative to GRAMMAR, the data set was analyzed with Bayesian methods implemented in the package snptest . The prowith ith individual. The residuals of the three traits (q) were scaled to a mean of zero and a unit variance. Ci is the coded version of the genotype of the ith individual. For this model a conjugated prior was used that based on an inverse Wishart prior IW on the error covariance matrix ∑ and a matrix normal (N) prior on the vector of parameters:is the vector of the residuals from model (3) measured on the M is a mean vector and V is a constant. Further information of the matrix nomal distribution can be found in Dawid [M = 0, V = 0.02) were used as recommended by the authors Marchini and Howie [where in Dawid . For thend Howie .1) and a null model (M0) of no association:The Bayes Factor (BF) is the ratio of marginal likelihoods between a model of association (Mg = 0.84) . As a result of step 1, particular traits 1 and 2 were highly correlated  SNPs for trait 1, 2 and 3, respectively  were treated as independent phenotypes analyzed with GRAMMAR and allowed to investigate pleiotropy between the traits. For PC1 three and for PC2 seven QTL regions were identified Table . PrincipThe Bayesian multivariate analysis revealed significant SNPs only involved in pleiotropy for all traits Table . In totaThe effects of the SNPs on the genetic correlations were evaluated including the SNPs identified for the particular trait or PC Table  Table 4 The investigation of the QTL-MAS 2012 data set using different multivariate approaches allowed identifying most of the simulated QTL with large effects. Smaller effects might not be detected due to the chosen threshold correction. The analysis of the PCs and multivariate approaches seem to be promising in order to detect QTLs mainly involved in pleiotropic effects.The authors declare that they have no competing interests.CG compiled the data set and performed the GRAMMAR and multivariate analyses. SB estimated the genetic parameters. CG and SB wrote the manuscript. ET advised analyses, data interpretation and revised the manuscript.


Combining samples across multiple cohorts in large-scale scientific research programs is often required to achieve the necessary power for genome-wide association studies. Controlling for genomic ancestry through principal component analysis (PCA) to address the effect of population stratification is a common practice. In addition to local genomic variation, such as copy number variation and inversions, other factors directly related to combining multiple studies, such as platform and site recruitment bias, can drive the correlation patterns in PCA. In this report, we describe the combination and analysis of multi-ethnic cohort with biobanks linked to electronic health records for large-scale genomic association discovery analyses. First, we outline the observed site and platform bias, in addition to ancestry differences. Second, we outline a general protocol for selecting variants for input into the subject variance-covariance matrix, the conventional PCA approach. Finally, we introduce an alternative approach to PCA by deriving components from subject loadings calculated from a reference sample. This alternative approach of generating principal components controlled for site and platform bias, in addition to ancestry differences, has the advantage of fewer covariates and degrees of freedom. To reach the statistical power needed for genome-wide association studies, large numbers of participants are needed. This can be achieved through large research networks such as the Electronic Medical Records and Genomics (eMERGE) Network, which comprises a multi-ethnic cohort of ~57,000 participants linked to electronic health records (EHRs) for phenotype mining from nine participating sites  in the United States (U.S.) .Through the course of the eMERGE project, multiple genotyping platforms from both Illumina and Affymetrix were utilized  approach in human genetics research  in the United States  at Pennsylvania State University (PSU). Detailed quality assurance/quality control (QA/QC) measures are outlined in the imputation guide provided on the PSU eMERGE CC web site (see Web Resources). Before imputation, study site data were converted to the same build (Build 37) as the imputation reference data set. Next, strand flipping was employed to account for different strand alignments including Illumina TOP/BOT strand, plus(+) / minus(−), and forward/reverse  after imputation using the BEAGLE software package (Version 3.3.1)  sliding window, and a minor allele frequency (MAF) > 0.05. In addition, a variant missingness filter of 0.02 was applied. For both PCA on the combined imputed and the combined preimputed, which is basically the singular value decomposition on the sample covariance matrix as outlined in Patterson et al. , we applied the following thresholds. The autosomal variants were selected after LD pruning at n et al. .snpgdsPCASampLoading and snpgdsPCASNPLoading functions (see Web Resources).We also assessed PCA using the Zhu et al.  method ba = 1,2,…, B) unrelated. Because of this, the ijg marker genotype value of the jth individual in the ith family as utilized by Zhu et al. T of l = 1,2,…, M biallelic markers, and was coded as an additive model of inheritance.We treated the entire eMERGE cohort as one “related” family, and the imputation reference sample as  of the entire cohort or “related” family by tjl = (Xj − X)Tle.The variance-covariance matrix for the marker data from the reference sample (unrelated), took on the form u et al. , we let The venous thromboembolism (VTE) phenotype was extracted using an EHR-driven algorithm from African ancestry participants , excluding patients with cancer. A total of 400 VTE cases and 5,065 controls were selected from 4 sites and 4 different genotype platforms . We performed two logistic regressions for association using the software PLINK v1.07  and standard deviations (SD) calculated for each genetic ancestry group, as illustrated in Figures QA2 ± 2SD), European (Q2E ± 4SD) and Hispanic (QH2 ± 1SD) groups, respectively.Eigenvectors 1 and 2 for the 38,288 adult eMERGE participants are illustrated in Figure To assess the variance explained from each PCA, we plotted the first ten PCs against the variance explained as illustrated in Figure We evaluated the population structure by plotting eigenvectors 1 and 2 for the joint data set Figure  as well Figures We also illustrate component-genotype absolute correlation plots generated using the SNPRelate R package for the imputed and pre-imputed data sets. Ideally, a component will be driven by genome-wide correlation patterns, as illustrated by eigenvector 3 of the pre-imputed data in Figure We applied our approach using the eMERGE VTE African ancestry cohort that consists of four adult sites and four genotyping platforms that had previously been analyzed controlling for site, platform and genomic ancestry  are tighter for chromosome 22  capture the true underlying LD. The more completely LD is represented, the more accurately the imputation will extend the LD to non-genotyped markers. There is always an inherent risk that the imputed genotypes will not represent the true state of nature accurately; this risk increases as the genotyped density decreases and the genotypes do not capture the underlying LD. We detected effects from the genotyping platform when performing the PCA (here we use platform to indicate the design as well as the method). The effect was most evident when a low-density platform such as the MetaboChip (data not shown) were combined with high-density platforms: the MetaboChip data set was an outlier even at overview scale. Platform differences re-appear when PCA is performed on apparently homogeneous subsets, e.g., African and European genetic ancestry subsets. These platform differences in homogeneous racial groups are amplified as the overall variance in the data set diminishes. Some of the differences might actually reflect subtle differences in LD in the populations due to ethnic stratification correlated with platform, because the populations were not randomly represented in the Biobank and therefore not randomized to platform.In addition to difference of LD capture by platform, genotype encoding remains problematic when combining large data sets genotyped at different sites and on different platforms. A number of tools, e.g., liftOver , probably are reflecting rare chromosomes in the reference panel.r = 0.50 − 0.84), in a sliding window of 500 kbp; (4) Filter variants on MAF >0.05, and for missingness <0.02; and (5) Examine plots of absolute correlation between PC and genotype as illustrated in Figure We have outlined a general checklist for filtering variants to be utilized with PCA: (1) Ensure uniformity of strand representation among different platforms to avoid the bias induced by site; (2) Select variants on autosomal chromosomes only, no sex chromosomes; (3) Filter variants with LD pruning , a component of the National Institutes of Health (NIH), Bethesda, MD, USA: (1) U01HG006375 ; (2) U01HG006382 ; (3) U01HG006379 (Mayo Clinic); (4) U01HG006389 ; (5) U01HG006388 (Northwestern University); (6) HG004438 ; (7) HG004424 (Broad Institute of Harvard and MIT); (8) U01HG006378, U01HG006385, U01HG006385 (Vanderbilt University and Pennsylvania State University); (9) U01HG006380 ; (10) U01HG006828 ; (11) U01HG006830 . Additional support was provided by a State of Washington Life Sciences Discovery Fund award to the Northwest Institute of Genetic Medicine (Gail P. Jarvik).The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


In this study, we performed genomic prediction in Chinese Simmental beef cattle and treated degrees of freedom and scale parameters as unknown with inappropriate priors. Furthermore, we compared the modified methods  with their corresponding counterparts using simulation datasets. We found that the modified methods with distribution assumed to the two hyperparameters were beneficial for improving the predictive accuracy. Our results showed that the predictive accuracies of the modified methods were slightly higher than those of their counterparts especially for traits with low heritability and a small number of QTLs. Moreover, cross-validation analysis for three traits, namely carcass weight, live weight and tenderloin weight, in 1136 Simmental beef cattle suggested that predictive accuracy of BayesFCπ noticeably outperformed BayesCπ with the highest increase (3.8%) for live weight using the cohort masking cross-validation.Three conventional Bayesian approaches  have been demonstrated to be powerful in predicting genomic merit for complex traits in livestock. A priori, these Bayesian models assume that the non-zero SNP effects  follow a  Genomic prediction uses all available molecular markers as covariates in a linear regression model to estimate genomic breeding values for quantitative traits. This has aroused a strong interest on the explorations of genomic selection in animals and plants –5.et al. coined the term “Bayesian alphabet” using the numbers of letters of the alphabet /dx.Where θ = 1/x obtained from the Eq (h(x) using the substitution of θ in Eq 11) equal θ in Eq .x / 2) + 1 − ψ(x / 2) was a decreasing function, we could find h(x) was a decreasing function. The range of function (12) was provided below.Given that ln ≤ 0 was obtained through Eq (h(x) equal to zero with the range less than zero to positive infinity should be existed. This value was assumed to be x* that was used to obtain the maximum value M, hence the function Q obtained the maximum value M with x being equal to x*, and θ was equal to 1 / x*. To calculate x*, we used the algorithm of exponential dichotomy [In Eq , η≥12q+λrough Eq . A valueichotomy .Full conditional posterior distribution of Simulation population structure: All starting with a base population of 100 animals , 2000 non-overlapping historical generations with the same population size denoted as generation -1999 to generation 0 were simulated. In the base population and each historical generation, 50 males randomly mated with 50 females, and each mating produced two offspring . Six additional generations were simulated after the 2000 historical generations, which formed generation 1 to generation 6. The population size was expanded from 100 to 2000 in generation 1. The process involved 50 males and 50 females from generation 0 randomly mating, and 40 progenies  were produced by each mating pair. Then 50 males were randomly selected from the 500 males as the sires of the next generation, and different population sizes of females  were used as dams in generations 1–5. Finally, three population sizes  from generations 2 to 6 were obtained by each male randomly mating 10 females, and each female produced two offspring. The simulation program was same to that of previous study .Genomic structure: The frequently used mutation-drift equilibrium (MDE) model was applied to simulate whole-genome data. Five chromosomes were simulated, and each contained 1 Morgan with 2000 SNPs that were randomly located in the genome. There was a potential quantitative trait locus (QTL) between two SNPs, so 10000 SNPs produced 9995 potential QTL in total. The true QTL effects, just as the allele substitution effects, were drawn from standard normal distribution as typically done ,22. To eIn simulation, genotypes and true breeding values (TBVs) were assigned to each animal from generation 1 to 6. Phenotypic records were assigned only to animals in generation 1 , and the animals in generation 2–6 were validation groups. Finally, forty-eight combinations were obtained by different genetic architecture considering reference population size, number of QTL and heritability. Among them, 10 replicates were simulated for each scenario.The resource population was established in Ulgai, Xilingole League, Inner Mongolia of China. All the young Simmental beef cattle were born between 2008 and 2013. After weaning, Simmental cattle were moved to Beijing Jinweifuren cattle farm for fattening and put under the same feeding and management conditions. Each animal was measured for growth and developmental traits on time until its slaughter when they were 18 to 24 months of age. Live weight was measured after 24 hours of fasting. During the period of slaughter, carcass traits and meat quality traits were measured according to the Institutional Meat Purchase Specification for fresh beef guidelines. Three response variables  were analyzed in this study. Systematic environment factors including farm, year of measurement and age at slaughter (seasons) effects were adjusted in the mixed linear model. Genetic parameters were calculated using residual maximum likelihood method in animal model using G matrix .−6). After quality control, any missing genotypes were replaced with the average value  for each SNP, and the final data consisted of 1136 Simmental cattle and 697,769 SNPs in the autosomes.In total, 1173 Simmental cattle were genotyped with Illumina Bovine 770K SNP BeadChip. Before statistical analysis, SNP quality control was pre-processed as following: PLINK v1.07  softwareTo assess the predictive accuracies of modified methods in our study, we explored cross-validation methods using real data. Overall, 1136 Simmental cattle were divided into a reference and validation population. Phenotypes of the animals in the validation set were masked to be unknown. Two cross-validation methods were used to select the reference population . The firWe treated two hyperparameters as unknown and estimated them in both simulation and real data. Markov chains were run for 50,000 cycles of iterations. The first 10,000 were discarded as burn-in. In BayesB and BayesFB, 100 additional cycles of Metropolis Hastings sampling were performed for the variances of marker effects. In the modified methods, every MCMC cycle consisted of Gibbs sampling and accept-reject sampling steps. Scale parameter was sampled by the Gibbs sampling process, and degrees of freedom were acquired by efficient accept-reject scheme. A stable estimation was obtained in these sufficient iterations and little improvement could be made in higher settings. All the samples of SNP effects and the two variable hyperparameters were averaged after burn-in to obtain the estimates. In simulation, π was defined as the number of QTL divided by the total number of loci in BayesB, BayesFB, BayesCπ and BayesFCπ. In the real analysis, π was set as 0.996 for BayesB and BayesFB, and 0.8 for BayesCπ and BayesFCπ.GEBVs for animals were calculated as the sum of all SNP effects, according to their marker genotypes. The GEBV of animals was calculated asZij was a genotype for SNP j of animal i, and αj was the posterior mean of the j-th SNP effect.To compare the difference of SNP effects estimated by the six Bayesian models, we calculated the absolute SNP effects . We founTo estimate the predictive accuracies of the modified methods comparing with their counterparts, we analyzed the simulation datasets with different genetic architectures. Generally, the predictive accuracies of BayesFA and BayesFCπ performed better than their counterparts in all generations. However, the predictive accuracies between BayesFB and BayesB were very similar. In generation 1 with population size of 1000, different number of QTL and heritability, the mean accuracies of GEBVs estimated by GBLUP generated the lowest accuracy . In conth2 = 0.1), the predictive accuracies improved with the highest (2.5% and 1.5%) for BayesFA and BayesFCπ compared to their counterparts, respectively. Additionally, the predictive accuracies for high (h2 = 0.8) and medium (h2 = 0.5) heritability were not robust than those for low heritability.To evaluate the effects of heritability and QTL on predictive accuracy, we simulated four different heritabilities  and numbers of QTL . In v or v increased with the number of QTLs.For simulated data, QTL effects were drawn from the standard normal distribution. These were produced by the process of recombination over thousands of generations in terms of the relationships between the number of QTL and BayesFCπ . Under pHeritability estimates of three traits were 0.38, 0.43 and 0.47 for CW, LW and TW, respectively . To assev and Genomic selection has revolutionized cattle breeding by greatly increasing the predictive accuracies of genetic merit for young animals and shortening the generation interval. In our study, we performed genomic prediction in Chinese Simmental beef cattle with addressing the impact of  effects . To addr effects ,17. Baseet al. used antedependence model for whole genomic prediction to explore the relationship between et al. discussed the impact of h2 = 0.5, the value decreased with the increasing number of QTL in BayesFB and BayesFCπ as shown in et al. developed a full hierarchical approach to estimate v [v in BayesFA and BayesFCπ with exponential distribution, and BayesFB with gamma distribution. The larger degrees of freedom sampled by efficient accept-reject can result in greater impact on the posterior SNP variances inference. The posterior median degrees of freedom estimated by BayesFA, BayesFB and BayesFCπ were 2.6, 4.3, 118.54, respectively, but different from the fixed value 4.2 as suggested by Meuwissen et al. [For simulation study, Yang ensities , and fouanalysis . In our timate v  and founn et al. . Therefoet al. [et al. [For real dataset, Simmental beef cattle was partitioned into training and validation data sets, cross-validation analyses were carried out using CW, LW and TW traits to assess the accuracies of genomic prediction. Previous studies have conducted genomic selection of CW and LW traits in other Simmental cattle –31. We fet al. . Therefo [et al. .v and et al. [The proposal distribution of locus-specific variance is an inverse chi-square distribution with parameters g et al. . Their ev and i.e., with population size of 1000, QTL = 200, h2 = 0.5 using 3.3GHz, Intel core(TM) i3-2120 processor, computing time for 50000 iterations was 2.39, 2.56, 5.08, 5.36, 2.12 and 2.29 hr for BayesA, BayesFA, BayesB, BayesFB, BayesCπ and BayesFCπ, respectively. However, these modified methods produced higher predictive accuracies than their conventional counterparts. For the posterior distribution in a complicated non-standard form, the degrees of freedom sampled by the efficient accept-reject algorithm may produce anticipated results.Due to the additional sampling the variable In conclusion, we performed genomic prediction for live weight, carcass weight and tenderloin weight in Chinese Simmental beef cattle. Our study indicated the degrees of freedom and scale parameters could slightly impact the predictive accuracies in both simulation and real dataset.