Current genome-wide association studies (GWAS) have high power to detect intermediate frequency SNPs making modest contributions to complex disease, but they are underpowered to detect rare alleles of large effect (RALE). This has led to speculation that the bulk of variation for most complex diseases is due to RALE. One concern with existing models of RALE is that they do not make explicit assumptions about the evolution of a phenotype and its molecular basis. Rather, much of the existing literature relies on arbitrary mapping of phenotypes onto genotypes obtained either from standard population-genetic simulation tools or from non-genetic models. We introduce a novel simulation of a 100-kilobase gene region, based on the standard definition of a gene, in which mutations are unconditionally deleterious, are continuously arising, have partially recessive and non-complementing effects on phenotype , and are interspersed with neutral markers that can be genotyped. Genes evolving according to this model exhibit a characteristic GWAS signature consisting of an excess of marginally significant markers. Existing tests for an excess burden of rare alleles in cases have low power while a simple new statistic has high power to identify disease genes evolving under our model. The structure of linkage disequilibrium between causative mutations and significantly associated markers under our model differs fundamentally from that seen when rare causative markers are assumed to be neutral. Rather than tagging single haplotypes bearing a large number of rare causative alleles, we find that significant SNPs in a GWAS tend to tag single causative mutations of small effect relative to other mutations in the same gene. Our results emphasize the importance of evaluating the power to detect associations under models that are genetically and evolutionarily motivated. Current GWA studies typically only explain a small fraction of heritable variation in complex traits, resulting in speculation that a large fraction of variation in such traits may be due to rare alleles of large effect (RALE). The most parsimonious evolutionary mechanism that results in an inverse relationship between the frequency and effect size of causative alleles is an equilibrium between newly arising deleterious mutations and selection eliminating those mutations, resulting in an inverse relation between effect size and average frequency. This assumption is not built into many current models of RALE and, as a result, power calculations may be misleading. We use forward population genetic simulations to explore the ability of GWAS to detect genes in which unconditionally deleterious, partially recessive mutations arise each generation. Our model is based on the standard definition of a gene as a region within which loss-of-function mutations fail to complement, consistent with the multi-allelic basis for Mendelian disorders. Our model predicts that it may not be uncommon for single genes evolving under our model to contribute upwards of 5% to variation in a complex trait, and that such genes could be routinely detected via modified GWAS approaches. However, these associations explain only a small fraction of the known heritability of risk for the diseases examined Genome-wide association studies (GWAS) genotype upwards of 500,000 common SNPs and test for allele frequency differences in case/control panels consisting of several thousand individuals. Such studies have identified highly significant and replicable associations, and as a result have uncovered entirely new pathways contributing to complex disease risk  between causative alleles and linked neutral markers. Ideally, the predictions of various RALE models would come from explicit population-genetic models of disease, with concrete assumptions about the fitness effects of causative mutations and the relationship between phenotype and fitness determining the frequency of causative mutations in the population. To date, Prichard's Pritchard's e.g. heterozygous for at least two different causative mutations at different positions in the gene region), as is commonly-observed for Mendelian disorders sensucis-regulatory regions controlling expression of a gene contributing to risk of a complex disease. We use our simulation machinery to explore the population-genetic signals of selection against causative sites, and to explore the power of GWAS to test the hypothesis that variation within a gene region contributes to complex disease.Here, we propose an explicit model of a quantitative trait subject to natural selection, with case/control status treated as a liability trait. Our model is similar to that of Pritchard Our model results in gene regions evolving under the “allelic heterogeneity” model involving many non-complementing risk mutations segregating within a gene region. Since the 1990s, many human geneticists believed that this model was likely to explain complex variation Under our gene-based model, the effect size of a causative mutation is exponentially-distributed with mean λ (λ = 0 implies a mutation that does not contribute to a complex disease phenotype), and the effect of a maternal or paternal haplotype is additive over causative mutations. The phenotype of a diploid is the geometric mean effect of the maternal and paternal haplotype plus a random Gaussian environmental effect scaled so that the gene-region being modeled accounts for some fraction of the total disease burden. In our simulated populations, liabilities are close to normally distributed, except in the extreme “diseased” tail, where there is a slight excess of extremely affected individuals . FurtherGiven the computational demands of forward simulation, we focus our attention on a set of parameters (see Methods) that results in the proportion of total phenotypic variation in the population attributable to the focal gene region reaching a plateau at ∼4% as λ increases to ∼0.075–0.10 . Althougd  an λ , our model is able to generate a single gene of small to large effect contributing to disease risk for plausible parameters (d and λ) is quite large. This implies that even for parameter combinations that predict equilibrium heritability values of ∼2% . Therefore, despite our focus on parameter values that result in a heritability plateau of ∼4%, parameter values predicting lower plateaus are clearly of interest.The value of heritability at the plateau depends on the model parameters. Plateau height is approximately linear as a function of the deleterious mutation rate , holding balance . Thus, irameters . It is a∼2% e.g., 2–9% ofWe examined the frequencies of mutations in a sample of 100 diploids drawn from each of the simulated regions. On average, causative mutations are more rare than expected in the absence of natural selection –S3d. Thei.e., an evolutionary outlier). This observed relationship between common SNPs significant in a GWAS and causative alleles is inconsistent with the claim of recently published work that introduced the idea of a synthetic association resulting from a common marker tagging a haplotype(s) harboring an excess of rare causative mutations We also observe examples of significant, common, non-causative markers e.g. , consist−8. Further, the cumulative distribution of p-values for λ = 0 is a line with a slope less than one, indicating that the logistic regression test is conservative when applied to our simulated data (data not shown). For small values of λ>0, broad-sense heritability is also lower  for these gene-based tests, as they integrate over markers, and thus fewer tests are carried out when doing a genome-wide scan. The Madsen and Browning test results in an excess of small p-values (compared to the same test in λ = 0 controls) across a wide range of λ, with the excess being greater in resequencing studies than chip-based GWAS  of the M most significant markers in a genomic region (see Methods for details). Control simulations with no causative mutations show that a permutation procedure (see Methods) results in the correct distribution of p-values . Our model indeed suggests that tests focusing on detecting such an excess of rare mutations in cases have low power when there are no a priori weights applied to different sites within a region and when causative mutations are a small fraction of the total number of variants in a region  that we assume to be detectable via a chip or resequencing assay.We implemented a forward-time simulation of a Wright-Fisher population with mutation following the infinitely-many sites model We note that there are a variety of forward-time simulation programs in the literature. However, the majority of these either simulate non-gene-based models 1 and c2 causative mutations on each haplotype. The effect size of the ith mutation on the jth haplotype is e, which we fix at 0.075 in the simulations. In words, the phenotypic effect of a single haplotype is additive over causal mutations, and the phenotype of an individual is the geometric mean of the effects of each haplotype plus Gaussian noise. Since phenotypes are continuous they represent the underlying liability of developing a disease s = 1, and w, the fitness of a diploid, is proportional to An individual carries cIn our simulations, the effect sizes of causative mutations are exponentially distributed with means of λ = 0, 0.01, 0.025, 0.05, 0.075, 0.1, 0.125, 0.175, 0.25, 0.35, or 0.5. For each λ>0, we performed 250 independent simulations. For an effect size of 0, representing “control” simulations where there is no genetic contribution to risk, we simulated 1000 independent replicates. All simulations were run for 8N generations prior to sampling.d = 0.1 µ, σs = 1, σe = 0.075, and r = 0.00125/diploid or 0, we simulated both neutral and causative markers, allowing us to examine the properties of GWAS in detail. In order to reduce computational time, for all other parameter values explored, we set μ = 0  and only simulated the causative sites. By not simulating the neutral mutations, simulations run orders of magnitude faster, allowing us to look at heritability across a broader parameter space.For the parameters μ = 0.00125/gamete, μFor each simulated population, 3000 cases and 3000 controls were sampled. A case was defined as being in the upper 15% of the phenotypic distribution, and controls were within 1 standard deviation of the population mean. For each case-control panel, we define a GWAS to include all markers present in the panel with a minor allele frequency ≥5%, and a resequencing study to include all markers. For both types of study, we performed a logistic regression of case/control status onto genotype under an additive model.−8, representing a typical cutoff used in current GWAS The significance threshold used was 10http://www.molpopgen.org/ThorntonForanLongPLoSGenetics.html.The forward simulations required approximately six weeks on a cluster of 96 computing cores . To facilitate the further development of tests for detecting associations in gene regions, we have made all source code, forward simulation output, and case/control files available online at −8 used for the single-marker test. Our simulated data are 100 kilobase regions, from a genome of approximately 3×109 base pairs, giving 3×109/105 = 3×104 non-overlapping windows. A conservative significance threshold would thus be 0.05/(3×104) = 1.67×10−6. Here, we take p< = 10−6 as the significance threshold for all region-based tests  comparing allele counts in cases and controls for M unique markers  from a gene region, we define 1Y to be the largest value of negative base ten logarithm of p, 2Y the second largest, etc. Then our test statistic is:p-values  of the M most significant markers in a region.We developed a statistical test that attempts to integrate significance over marginally significant variants in a single gene. Under the gene-based model, genes harboring causative mutations tend to display such a genetic signature, and the ESM statistic is larger when there are more marginally significant mutations in a genomic region. Given a vector of Fisher's exact test p-values . For GWAS in non-recombining regions, we considered values of M = 25, as too few simulations had more than 25 unique markers with the requisite MAF to consider larger M. For both GWA and resequencing studies, the minimum count of a minor allele had to be 4 in order for a marker to be included in the analysis.The test statistic was calculated for two different conditions. First, for GWA studies where, as above, only minor allele frequencies (MAF) ≥0.05 were included. The second condition assumed complete resequencing of individuals and included all markers. For the latter case, and for GWAS assuming a recombining region, we considered values of We have also applied several other “region-based” tests designed to detect a contribution of rare alleles to disease risk within a defined genomic region. The first test is Madsen and Browning's The second test is Li and Leal's e.g., present only once in the entire case/control panel), then those two mutations would count more towards case/control differences in the permutation test than they would in the ESM test statistic. Thus, any differences in the power between ESM and the other two statistics should be viewed as conservative.For Madsen and Browning's, and for Li and Leal's, test statistics, we did not first collapse redundant markers. The rationale for not collapsing is that if a “case” contains, for example, two singleton mutations (http://www.hsph.harvard.edu/xlin/software.html) to all of our simulated data. We applied the software in two different ways. First, we used default weights on individual markers and the optimal p-value approach described in not equivalent to Madsen and Browning's rank-sum test Finally, we applied the SKAT software For the ESM, Madsen-Browning, and Li and Leal tests, we assessed statistical significance following the permutation procedure outlined in In the analyses described above, we assume that a GWA study is conducted using perfect genotyping technology able to assay 100% of markers with minor allele frequencies >5%. However, the majority of GWAS to date have used genotyping chips that assay a subset of ascertained markers whose minor allele frequencies are uniform in the range of 0.05 to 0.5 We use these imperfect chips to look at the MAF distribution of the most significant marker (defined by a logistic regression test described above) in a gene region  The model of gene action results in partial recessivity of haplotypes. The panel shows the empirical cumulative distribution of phenotypes that result from our simulations with mean effect size λ = 0.10 per causative mutation (black line), based on 250 independent simulations. Using the output of each simulation, we calculated each individual's phenotype under the standard models used in quantitative genetics–the additive model (red line), recessive model (blue line) and dominant model (purple line) of gene action. These other models were not explicitly simulated. Rather, the haplotype effect sizes output from our gene-based simulation were used to generate phenotypes under these alternative genotype-to-phenotype models. The gene-based model results in a distribution of phenotypes in between that of the additive and recessive models. (b) Average fitness of individuals in the simulations. Red dots show the mean of the population mean fitness. Blue triangles are the average fitness of individuals in the upper 15% of the phenotypic distribution of the population, who were treated as cases in the case-control analyses. The black diamonds are the mean fitness of the least fit individual observed in each simulated population. (c) Mean ±1 standard deviation of broad-sense heritability, as a function of λ. The points with solid lines are the same parameters as in (PDF)Click here for additional data file.Figure S2d) are the same as in e) is changed (open triangles and solid diamonds), heritability plateaus at different values. However, if d, plateauing at approximately 0.02 when the deleterious mutation rate is halved. (c) Estimated broad-sense heritability as a function of predicted broad-sense heritability . Thus, the heritability under the gene-based model is roughly one-half of that predicted under house-of-cards, likely a result of our assumed weak selection Broad-sense heritability in different parts of the parameter space. (a) The deleterious mutation rate has an approximately linear effect on broad-sense heritability at large mean effect sizes of causative mutations (λ). All model parameters except the deleterious mutation rate (μ(PDF)Click here for additional data file.Figure S32N refers to the mean number of mutations present in the entire population, and 2N is plotted on a log10 scale. In the absence of selection, the theoretical expectation of S2N is 2N for neutral markers for all λ shows that the total strength of selection against causative mutations does not result in a loss of variability in the region (because selection is weak on a per-marker basis). For all λ, there is at least a 1 order of magnitude difference in the number of causative and neutral mutations, and Population-genetic properties of a locus. (a–d) The mean, normalized site frequency spectrum (SFS) of derived mutations is shown for three different mean effect sizes (λ), calculated from a sample of 100 randomly-chosen diploids from each simulated population. Shown are the first ten entries of the SFS for neutral sites (red), causative variants (black), all polymorphisms (dashed blue), and the expected values for a Wright-Fisher population experiencing no natural selection (black circles). (e) Mean (± ¼ standard deviation) of the number of causative mutations per diploid in a case/control panel. For both cases and controls, the mean total number of causative mutations (open circles) and rare causative mutations  are shown. (f) Summaries of the amount of variation in the entire population. Here, S(PDF)Click here for additional data file.Figure S4−8, we measured LD using the r2 statistic between the significant marker and all causal markers in the case-control panel, and recorded the top two r2 values. The distribution of r2 for the top marker is summarized in white boxplots, and the distribution for r2 for the second-strongest association is summarized in red. (c) White boxes summarize the distribution of the number of significant, common, neutral markers, conditional on there being at least one such marker. The red boxes summarize the distribution of the number of unique causal markers amongst the top r2 values for each significant marker. Taken together, panels a and b suggest that significant common markers tend to tag a single causative site. (d) For each of the most strongly-tagged causal mutations making up the red boxes in panel b, the frequency and effect size of each mutant was recorded. The frequencies are summarized in the white boxes, and effect sizes are in red.Statistical properties of association studies. (a) Average proportion of rare variants which are causative in either the general population, in a case-control panel, or amongst significant markers in a GWAS where individuals are completely sequenced (b) For every neutral, common marker in a GWAS that was significant in a logistic regression test at p≤10(PDF)Click here for additional data file.Figure S510 p-values for the Madsen and Browning −6 is shown. (a) The empirical cumulative distribution function (ECDF) of p-values for control simulations with no deleterious alleles. As expected, the ECDF of p-values is a straight line with a slope of approximately 1. (b–k) ECDF of p-values for simulations with non-zero mean effect sizes of causative mutations (λ>0).Distributions of −log(PDF)Click here for additional data file.Figure S610 p-values for the Hotelling T statistic −6 is shown. (a) The empirical cumulative distribution function (ECDF) of p-values for control simulations with no deleterious alleles. The ECDF of p-values is a straight line with a slope of approximately one when the number of markers is ≥200. (b–k) ECDF of p-values for simulations with non-zero mean effect sizes of causative mutations (λ>0).Distributions of −log(PDF)Click here for additional data file.Figure S710 p-values for the Madsen and Browning −6 is shown. (a) The empirical cumulative distribution function (ECDF) of p-values for control simulations with no deleterious alleles. As expected, the ECDF of p-values is a straight line with a slope of approximately 1 (b–k) ECDF of p-values for simulations with non-zero mean effect sizes of causative mutations (λ>0).Distributions of −log(PDF)Click here for additional data file.Figure S810 p-values for the Hotelling T statistic −6 is shown. (a) The empirical cumulative distribution function (ECDF) of p-values for control simulations with no deleterious alleles. The ECDF of p-values is a straight line with a slope of approximately 1 when the number of markers is ≥200. (b–k) ECDF of p-values for simulations with non-zero mean effect sizes of causative mutations (λ>0).Distributions of −log(PDF)Click here for additional data file.Figure S910 p-values for the ESM statistic (see Methods). For all panels, the significance threshold of 10−6 is shown. (a) The empirical cumulative distribution function (ECDF) of p-values for control simulations with no deleterious alleles. As expected, the ECDF of p-values is a straight line with a slope of approximately 1 (b–k) ECDF of p-values for simulations with non-zero mean effect sizes of causative mutations (λ>0).Distributions of −log(PDF)Click here for additional data file.Figure S1010 p-values for the ESM statistic (see Methods) with no recombination. For all panels, the significance threshold of 10−6 is shown. (a) The empirical cumulative distribution function (ECDF) of p-values for control simulations with no deleterious alleles. As expected, the ECDF of p-values is a straight line with a slope of approximately 1. (b–k) ECDF of p-values for simulations with non-zero mean effect sizes of causative mutations (λ>0).Distributions of −log(PDF)Click here for additional data file.Figure S11−4) and frequency-matched to the most-associated marker in the same replicate. Each panel of the figure is labeled by the mean effect size of a causative mutation (λ).A comparison of the burden of risk mutations between significant and non-significant markers in a GWAS. Shown are mean and standard errors of the number of causative singletons in individuals with genotypes defined as having either zero, one, or two copies of the derived allele at the marker most significantly associated  with case/control status in a GWAS analyzed by a single-marker test (red). Pale blue lines show the mean and standard errors of the number of causative singletons associated with zero, one, or two copies of the derived mutation of a marker that is both not associated with case/control status in a logistic regression analysis (p>10(PDF)Click here for additional data file.


Non-additive genetic variation is usually ignored when genome-wide markers are used to study the genetic architecture and genomic prediction of complex traits in human, wild life, model organisms or farm animals. However, non-additive genetic effects may have an important contribution to total genetic variation of complex traits. This study presented a genomic BLUP model including additive and non-additive genetic effects, in which additive and non-additive genetic relation matrices were constructed from information of genome-wide dense single nucleotide polymorphism (SNP) markers. In addition, this study for the first time proposed a method to construct dominance relationship matrix using SNP markers and demonstrated it in detail. The proposed model was implemented to investigate the amounts of additive genetic, dominance and epistatic variations, and assessed the accuracy and unbiasedness of genomic predictions for daily gain in pigs. In the analysis of daily gain, four linear models were used: 1) a simple additive genetic model (MA), 2) a model including both additive and additive by additive epistatic genetic effects (MAE), 3) a model including both additive and dominance genetic effects (MAD), and 4) a full model including all three genetic components (MAED). Estimates of narrow-sense heritability were 0.397, 0.373, 0.379 and 0.357 for models MA, MAE, MAD and MAED, respectively. Estimated dominance variance and additive by additive epistatic variance accounted for 5.6% and 9.5% of the total phenotypic variance, respectively. Based on model MAED, the estimate of broad-sense heritability was 0.506. Reliabilities of genomic predicted breeding values for the animals without performance records were 28.5%, 28.8%, 29.2% and 29.5% for models MA, MAE, MAD and MAED, respectively. In addition, models including non-additive genetic effects improved unbiasedness of genomic predictions. Non-additive genetic variation results from interactions between genes. Interactions between genes at the same locus are called dominance, and interactions between genes at different loci are called epistasis. Although many studies have shown that non-additive effects have a substantial contribution to variation of complex traits Genome-wide dense single nucleotide polymorphism (SNP) markers have been widely used for association analysis Many statistical models and algorithms have been proposed to predict breeding values using genome-wide dense markers, which differ in the assumption of distributions of SNP effects. Best linear unbiased prediction (BLUP) models There are two BLUP models which have been widely used for genomic predictions. One estimates marker effects using random regression on marker genotypes and genomic breeding value of an individual is calculated as the sum of estimated marker effects . The other estimates genomic breeding value directly using a marker-based relationship matrix (hereafter denoted as GBLUP). It has been shown that the GBLUP model is equivalent to the SNP-BLUP model Similar to traditional genetic evaluations, genomic predictions are usually carried out using a model that ignores non-additive effects. The hypothesis of this study is that statistical models which include additive and non-additive genetic effects will predict genetic merit more accurately and with less bias, when non-additive genetic effects have a substantial contribution to the genetic variation.The objectives of this study were twofold. The first was to describe an approach to estimate additive and non-additive genetic variations and predict genetic values for complex traits using models integrating additive and non-additive genomic relationship matrices. The second was to estimate additive, epistatic and dominance genetic variances and access the accuracy of genomic predictions for daily gain in Danish Duroc pigs using models with or without including non-additive genetic effects.y is the vector of observations, b is the vector of non-genetic effects, a is the vector of additive genetic effects, i is the vector of epistatic effects (second order epistasis in this study), d is the vector of dominance effects, and e is the vector of random residuals.A linear mixed model including additive and non-additive genetic effects can be written as:It is assumed thatI is an identity matrix, G, Gaa and D are the additive, epistatic and dominance genetic relationship matrices, respectively. These matrices can be constructed from either the information of pedigree or the information of genome-wide markers. The marker-based relationship matrices have the advantage to capture both the Mendelian segregation and the genetic links through unknown common ancestors, which are not available in the known pedigree. The present study will demonstrate the calculation of additive, dominance and epistatic genetic relationship matrices based on genome-wide markers. In the context, the three marker-based relationship matrices will be denoted as additive, epistatic and dominance genomic relationship matrices.where G can be constructed using SNP marker information according to the previous studies M is a n×m matrix  which specifies SNP genotype coefficients at each locus. The coefficients of the ith column in the M matrix are (0 - 2pi) for genotype A1A1, (1-2pi) for A1A2, and (2-2pi) for A2A2, where qi and pi are the frequencies of allele 1 (A1) and allele 2 (A2) at locus i, respectively.The additive genomic relationship matrix According to the previous study D can be derived as follows. Denote u as the dominance value at a single locus and h0 as the heterozygosity of an individual at the locus. The dominance effect of an individual at a locus can be described as 1 and A2), 1A1 and A2A2, and 1A2. Since h0 is either 0 or 1, The dominance genomic relationship matrix To simplify the algorithm in the above linear mixed models, it is assumed that dominance effects (d) are normally distributed with mean equal to zero. In order to meet this assumption, dominance effects are expressed as deviations from the population mean, h is the m-dimensional vector of heterozygosity coefficients, and u is the m-dimensional vector of dominance values. Assuming that dominance values at different loci are identically and independently distributed normal variables, the variance of genome-wide dominance effects in a Hardy-Weinberg equilibrium population isExpanding to m loci, the genome-wide dominance effect of an individual is Expanding to n individuals and m loci,d is the n-dimensional vector of dominance effects, H is the n×m matrix of heterozygosity coefficients with element d isD is,Consequently, the dominance genomic relationship matrix ki with 2piqi and scaling HH' with D matrix has the properties that the expectation of an off-diagonal element is zero for two unrelated individuals, and the expectation of a diagonal element is one for a non-inbred individual. The D matrix is a realized dominance relationship matrix and is analogous to the pedigree-based numerator dominance relationship matrix.By centering hSNP marker data were obtained from 1911 Danish Duroc pigs (most were boars) which were genotyped using Illumina PorcineSNP60 BeadChip . The SNP data were edited using the following criteria: 1) Minor allele frequency higher than 5%; 2) locus call rate larger than 0.95; and 3) individual animal call rate larger than 0.95. In addition, if the GenCall score of a single SNP in an animal was less than 0.65, the SNP in this animal was treated as missing and the corresponding genotype coefficient was replaced by the expected genotype coefficient at this locus. After editing, there were 26,142 SNP markers available for 1,911 pigs .DG) from 30 kg to 100 kg. More details about the data can be found in the previous study allDATA) were used to calculate corrected phenotypic values of DG (see detail below). The corrected DG of the 1,911 genotyped pigs (genDATA) were used to estimate additive and non-additive genetic variances. The data were further divided into reference dataset  and test dataset  by a cut-off date June 1, 2008 (birth date). DATAref were used to predict breeding value of the genotyped pigs in DATAtest.The analyzed trait was average daily gain (cDG), instead of original observations, were used as response variables to estimate additive and non-additive genetic variances and to predict genetic effects using SNP markers. The reason for using DGc as response variables was to reduce noise by removing contemporary group effects which could be estimated much more accurately using a large dataset including all contemporaries and relatives, rather than using only genotyped animals. The contemporary group effects were estimated using a traditional pedigree-based linear model including sex, herd-week-section, pen, litter and additive genetic effects as well as random residuals. The DGc was defined as original observations of daily gain adjusted for all non-genetic effects except for litter effect because based on this model it included part of non-additive genetic effects that should not be corrected for.Corrected phenotypic values of daily gains .The analyses were carried out applying the average information restricted maximum likelihood algorithm gen. The superiority of an alternative model over model MA was tested using a likelihood ratio test. The predictive ability of the model (with respect to accuracy and unbiasedness) was evaluated by comparing predictions and DGc of animals in the test dataset (DATAtest). Prediction accuracy was measured as the correlation between predicted genetic values and DGc. Both predictions of additive genetic effect  and total genetic value (defined as the sum of the genetic effects in the model) were evaluated. Hotelling-Williams t-test c, divided by heritability of DGc, i.e., c on the genomic predictions. A necessary condition for unbiased predictions is that the regression coefficient does not deviate significantly from one.Goodness of fit for each model was evaluated by likelihood value based on the dataset DATAc (the corrected daily gain) were significantly different in the different datasets  and 0.985 (model MAED), the regression coefficients of DGc on predicted additive genetic values ranged from 0.927 (model MA) to 1.029 (model MAED). For all models, the regression coefficients were not significantly different from 1, indicating the predictions were not significantly biased. However, the regression coefficients for the predictions using the models that included the non-additive genetic effects were closer to 1, suggesting that these models slightly improved the unbiasedness of genomic predictions.As shown in This study described a novel approach to model additive, epistatic and dominance genetic effect. The approach, by constructing additive and non-additive genetic relationship matrices using information of genome-wide SNP markers, is generally applicable for different populations with or without pedigree information. The method was used to investigate the variance components of additive and non-additive genetic effects and the accuracy of genomic predictions for DG in Danish Duroc population. The estimates of dominance variance and epistatic variance indicated that non-additive genetic effects had a substantial contribution to total genetic variation of this trait. The models including non-additive genetic effects predicted breeding values more accurately and unbiasedly, compared with a model ignoring non-additive genetic effects.Genome-wide dense markers provide a new approach to detect non-additive genetic variation and predict genetic merit. Additive genomic relationship matrix G#G#D for additive by additive by dominance genomic relationship matrix, G#G#G for additive by additive by additive genomic relationship matrix. However, when higher order epistasis is involved in the analysis, more data information is required. A precise epistatic relationship matrix is difficult to be obtained when considering a large number of markers. Epistasis among a few specific markers can be modeled in a desired way The proposed method to construct epistatic genomic relationship matrix is an approximate approach and only second order epistasis (additive by additive interactions) is considered. Following previous report With additive and non-additive genetic relationship matrices, additive and non-additive genetic variances and genetic values can be easily estimated using a typical linear mixed model, such as a GBLUP model. A GBLUP model is equivalent to a linear random regression model assuming that effects of all SNP are normally distributed with equal variance. A GBLUP model may be not satisfactory in the situation that few markers have large effect whole the most markers have null effect or very small effect. However, experiences with real dairy cattle data and pig data indicate that a BLUP model performed well for most traits. Therefore, the approach proposed in this study could be not an optimal but a simple and feasible approach for estimating additive and non-additive variances and predicting genomic breeding values.Based on the present data, the estimated epistatic variance and dominance variance in proportion to additive genetic variance were about 26% and 15%, respectively. The estimates are in the range of those for complex traits reported in previous studies. Additive and non-additive variances are usually estimated using a model with a pedigree-based relationship matrix. In dairy cattle, it was reported that the ratios of dominance variance to additive genetic variance were 118% and 161%, and the ratios of epistatic variance to additive genetic variance were 59% and 2% for days open and service period, respectively, in US Holstein Several previous studies have reported non-additive genetic effects of detected QTL. In chicken, QTL analysis revealed that the non-additive genetic effect was more pronounced prior to 46 days of age, whereas additive genetic effect explained the major portion of the genetic variance later in life It has been observed that there is a large variation between the estimates of non-additive genetic variances in different studies, which may reflect the different features of various traits and populations. In addition, the large variation could be caused by large sampling error due to insufficient data information. As shown in this study, the standard errors of non-additive genetic variance were large. In fact, the estimated epistatic variance was not statistically significantly different from zero. The relative standard errors for the estimated dominance variance and epistatic variance (here defined as standard error/estimated variance) were 4.5 times and 6.5 times as large as that for additive genetic variance. Similarly, the relative standard error for dominance variance was 4.1 as large as that for additive genetic variance of stature in US Holstein The current and previous studies have shown that non-additive genetic variance is considerable in complex traits. Therefore, it is expected that a model including non-additive genetic effect would increase the prediction accuracy and improve the unbiasedness. In terms of practical animal breeding, focus is the prediction of breeding value  or the genetic improvement in next generation. In this study, reliabilities of genomic predicted breeding value using model MAE, MAD and MAED were 0.3%, 0.7% and 1.0% units higher than that using the additive genetic model. In addition the models including non-additive genetic effects slightly improved unbiasedness of genomic prediction.Compared to large non-additive genetic variances, the gain in reliability of genomic predictions by including non-additive genetic effects in the prediction model was relatively small. The small improvement indicates the difficulty to distinguish additive and non-additive genetic effects. Using traditional BLUP model with pedigree-based relationship matrix, a previous study on stature in US Holsteins In a genetic evaluation system, there are two limitations in practice for using a model with both additive and non-additive genetic effects for genomic prediction. The first limitation is that the computational demand for models with both additive and non-additive genetic effects are generally high, since both additive and non-additive genomic relationship matrices are dense. This requires more powerful computers and/or more efficient algorithms. Second, a reference population often consists of sires which have records of progeny performance, and popularly used response variables are conventional estimated breeding value (EBV), de-regressed EBV or mean of corrected progenies' performances which are more informative than individual observation. However, these pseudo observations are appropriate for an additive genetic model, but not appropriate for a model that includes non-additive genetics effects because the interaction effects observed in the offspring are not related to interactions of genes in the sire. A genomic prediction model including non-additive genetic effects requires that the response variable is individual record. An extension of single-step model The current study was based on data from purebred Duroc pigs. Non-additive genetic variation in a crossbred population is expected to be much larger than that in a purebred population. In farm animals such like poultry, pig, and beef cattle, crossbreds are usually the end product. Nonetheless, selection is carried out in the purebreds to improve performance of the crossbreds. The information from crossbred population will be useful for increasing accuracy of genetic evaluation of animals in the purebreds due to additional information of the relatives in crossbred. Moreover, the information of crossbred population allows selecting candidates in purebreds for the performance of their crossbred offspring. According to the results from the present study, when using data of crossbreds, a model including non-additive genetic effects, especially dominance effect, is expected to yield a substantial improvement of genetic evaluations.It can be concluded that the present method is a feasible approach to estimate additive and non-additive variances and predict genomic breeding values. Non-additive genetic effects are important sources of genetic variation for daily gain in pigs. Genomic prediction models including non-additive genetic effects can improve accuracy and unbiasedness of genomic predicted breeding value.


Multifactor Dimensionality Reduction (MDR) has been widely applied to detect gene-gene (GxG) interactions associated with complex diseases. Existing MDR methods summarize disease risk by a dichotomous predisposing model (high-risk/low-risk) from one optimal GxG interaction, which does not take the accumulated effects from multiple GxG interactions into account.dichotomous predisposing risk variable and provides higher resolution in the quantification of disease susceptibility. We evaluate this new A-MDR approach in a broad range of simulations. Also, we present the results of an application of the A-MDR method to a data set derived from Juvenile Idiopathic Arthritis patients treated with methotrexate (MTX) that revealed several GxG interactions in the folate pathway that were associated with treatment response. The epistasis enriched risk score that pooled information from 82 significant GxG interactions distinguished MTX responders from non-responders with 82% accuracy.We propose an Aggregated-Multifactor Dimensionality Reduction (A-MDR) method that exhaustively searches for and detects significant GxG interactions to generate an epistasis enriched gene network. An aggregated epistasis enriched risk score, which takes into account multiple GxG interactions simultaneously, replaces the The proposed A-MDR is innovative in the MDR framework to investigate aggregated effects among GxG interactions. New measures  are proposed to detect multiple GxG interactions. Human diseases usually have complex inheritance patterns, and single nucleotide polymorphisms (SNPs) has been utilized to explain the variation in susceptibility to many common complex diseases as well as the response to drug therapy. The advancement of genotyping technology has made genotypic data readily accessible to investigators at low cost. However, many challenges remain with regard to identifying genes that render people susceptible to non-Mendelian disorders and in understanding the associations and functional relationships among genes. More and more, researchers have been advocating for advanced statistical analysis to quantify complex and interactive biological and genetic relationships,2.Multifactor Dimensionality Reduction (MDR) is a statistical paradigm for characterizing and detecting nonlinear complex gene-to-gene interactions (epistasis) possibly associated with susceptibility to disease. When nuhttp://www.epistasis.org). In general, the MDR process can also be combined with a filter preprocess step by first applying global testing and filtration techniques to select the optimal number of SNPs for MDR analysis by searching for a subset of genes likely to interact with other genes using the ReliefF filtering process. Bec. Bec29].ATIC), which promotes the accumulation of AICAR ribotide, inhibiting adenosine deaminase and leading to a build up of adenosine, a potent anti-inflammatory agent[ATIC and MTHFD2 may indeed yield a more clinically apparent result by altering the anti-inflammatory effects of methotrexate. There is a potential to apply the proposed method to GWAS study by dissecting SNPs into pathways in order to detect GxG interactions in GWAS pathways. The major computational challenges from the proposed A-MDR and other approach in MDR framework are in the generation of p-values for MDR. MDR permutation computing time is largely dependent on the dimension of data sets. In other words, the computing time increases as the number of SNPs and/or the number of subjects increases. Several works have been devoted to improve the efficiency and shorten the computing time in MDR analysis in high-throughput data[We explore a radial network Figure to depicry agent. A disruput data,31,32. WIn summary, bioinformatics challenges remain in detecting and modeling epistasis in complex biological traits. We have developed a new A-MDR framework to interpret complex genetic variation and have proposed predicting an outcome using a continuous risk factor. Several other extensions and modifications of the original MDR have been proposed in the literature. Incorporation of valuable features from other MDR extension models into the A-MDR framework is worth further investigation. Prospective studies and validation in independent samples are needed to assess reliability of the A-MDR model’s predictive ability. Tools for statistical inference, including asymptotic distributions of the proposed test statistics, need to be developed to save computing time and improve reliability.Appendix I. Justification, 95% confidence intervals and permutation tests of pOR, pRR and pChi.0. To address this issue, we propose pOR, pRR and pChi by taking the null distribution of unadjusted statistics into account. Let x=pOR,pRR, or pChi, F(x) be the cumulative distribution function of the corresponding statistic under the alterative hypothesis (GxG interaction present) F0(x),be the cumulative distribution function of the corresponding statistic under the null hypothesis (GxG interaction absent) and F0− 1(x) be the inverse function of F0(x). The corrected pOR, pRR and pChi are then defined byH0, F(x)=F0(x), so pOR, pRR and pChi should equalH0. In this work, we evaluated pOR, pRR and pChi using a full data set while these methods can also be evaluated under the cross validation scheme typically used in MDR.Since the predisposing risk factor Table is condiF(x) and F0(x) can be estimated by the corresponding empirical distribution function. Permutation is applied to estimate F0(x) by reshuffling the relationship between SNPs and a phenotype, where SNPs for each individual in a system are maintained as a vector to preserve their correlation structure. For each permutation, we generated odds ratio (OR), relative risk (RR) and chi-square test statistic (Chi). Jackknife re-sampling was applied to estimate F(x) by generating random subsets of data, where 80% to 90% of subjects were randomly selected. SNPs and the phenotype from each subject are maintained as a vector to preserve the association between SNPs and the phenotype. Denote the OR, RR and Chi statistic from permutation or re-sampling as x1,x2,…xB where B is the number of permutation or resampling. The null distribution function F0(x) and F(x) can be estimated byz1,z2…,zB then the 95% confidence interval pOR, pRR and pChi is the interval from 2.5 to 97.5 percentile of z1,z2…,zB from resampling. The p-value for pOR, pRR and pChi for the ith GxG interaction, denoted by Pvaluei, will be calculated by the permutation testing, i.e.z1,z2…,zB are calculated from permutation samples and z is the pOR, pRR and pChi statistic calculated from the current data.The functions There are no competing interests to this work.HD conceived of the study. RC and AMR aided in study design and statistical method. HD performed the simulations and data analysis. MB and SL performed the clinical data collection, genotyping and interpretation of case study findings. All authors contributed to the manuscript writing. All authors read and approved the final manuscript.


Pathway analysis provides a powerful approach for identifying the joint effect of genes grouped into biologically-based pathways on disease. Pathway analysis is also an attractive approach for a secondary analysis of genome-wide association study (GWAS) data that may still yield new results from these valuable datasets. Most of the current pathway analysis methods focused on testing the cumulative main effects of genes in a pathway. However, for complex diseases, gene-gene interactions are expected to play a critical role in disease etiology. We extended a random forest-based method for pathway analysis by incorporating a two-stage design. We used simulations to verify that the proposed method has the correct type I error rates. We also used simulations to show that the method is more powerful than the original random forest-based pathway approach and the set-based test implemented in PLINK in the presence of gene-gene interactions. Finally, we applied the method to a breast cancer GWAS dataset and a lung cancer GWAS dataset and interesting pathways were identified that have implications for breast and lung cancers. Many genome-wide association studies (GWAS) have been conducted to identify markers associated with diseases over millions of SNPs. However, to survive the multiple testing correction over millions of tests, SNPs need to have strong marginal effects on the disease or a large sample size is required for SNPs with small effects. For a complex disease that is often caused by the joint effects of multiple genes with small marginal effects, considering the effects jointly will significantly increase the statistical power to identify these genes. Pathway analysis provides a powerful approach for identifying the joint effect of genes grouped into biologically-based pathways on disease. Promising pathway results have already been identified in GWAS datasets Recently, many statistical pathway analysis methods have been proposed. Most of them focused on testing the cumulative main effects of genes in a pathway Random Forest (RF) has been used as a tool for association tests RF has also been shown to be useful for pathway analysis due to its promising feature of considering both main effects and gene-gene interactions. Pang et al. identified candidate pathways by ranking the pathways using their prediction error rates calculated in RF for gene expression data Here we propose a powerful two-stage RF-based pathway test (TRF-pathway) based on SNP data, such as data from a GWAS. We used simulations to verify that the TRF-pathway has the correct type I error rates. We also compared the power of the TRF-pathway to the original RF-based pathway test used in Chang et al. Finally, we applied the TRF-pathway to a breast cancer GWAS dataset and a lung cancer GWAS dataset and the TRF-pathway identified candidate pathways that have implications for breast and lung cancer etiology.Z score.In the RF algorithm We incorporated the RF algorithm in the TRF-pathway test. SNPs in genes in a pathway are used as predictor variables to classify the case and control status in RF. For a large pathway, using SNPs in all the genes in the pathway may reduce the power for classification, because a majority of SNPs may not have effects on the disease. Therefore, we used a two-stage approach to eliminate the number of SNPs that may not have effects. The RF algorithm is performed on all SNPs in a pathway at the first stage. Then SNPs with variable importance scores greater than a user-specified threshold are selected at the second stage and the RF algorithm is performed only on the significant SNPs. The algorithm for the TRF-pathway is described as follows:For each pathway, we perform the following steps in the TRF-pathway algorithm:Select a set of SNPs within a user-specified distance to genes in a pathway.RF is performed based on the set of SNPs and the standardized variable importance score is calculated for each SNP.SNPs with importance scores greater than a user-specified threshold are selected as the important SNPs. RF is performed again based on the important SNPs.R for the pathway.The prediction error rate, which is the proportion of samples not correctly categorized, from the RF analysis in step 3 is used as a score K times. In each permutation i, the score iR is calculated.Permute the case-control affection status and repeat steps 2–4 I(S) = 1 when the statement S is true and I(S) = 0 when S is false. The null hypothesis for the TRF-pathway is that none of the SNPs in the pathway are associated with the disease.The p-value for the pathway is calculated as: Note that if the user does not specify a threshold for the variable importance scores, all the SNPs within genes in a pathway are used for RF analysis and the TRF-pathway algorithm is reduced to the RF-based pathway algorithm used in Chang et al. Unlike methods developed for GWAS data that compare test statistics of genes in a pathway with respect to statistics for background genes in the genome such as Wang's method 1X, 2X and 3X) with minor allele frequencies 0.25, 0.15 and 0.15, each in different genes, were used as disease loci. A penetrance function similar to the one used in X is a vector of 1X, 2X, and 3X, α is the parameter based on the disease prevalence, 1β, 2β, and 3β correspond to the conditional marginal effects for 1X, 2X, and 3X, 4β, 5β and 6β correspond to the conditional interaction effects for the second-order interactions, and 7β models the conditional interaction effects for the third-order interaction. iX is equal to 1 in the presence of at least one of the minor alleles at the locus i and equal to 0 if no minor allele is present. The disease prevalence was assumed to be 1%. We simulated 1,000 cases and 1,000 controls in each replicate of the simulations. We refer to the settings of these parameters  as Scenario 1.We conducted simulation studies to evaluate the type I error rates and power for the TRF-pathway. We used genomeSIMLA to simulate SNPs in genes in a pathway based on Affymetrix 550k chip Scenario 1, we also changed the parameters one at a time for a more comprehensive simulation study. For Scenario 2, we simulated 500 cases and 500 controls. For Scenario 3, we changed the disease prevalence to be 5%. For Scenario 4, we simulated a larger pathway with 100 genes. A total of 1,527 SNPs within 20 KB to the genes in the pathway were used. We simulated an additional disease locus with a minor allele frequency of 0.25 for the large pathway. The disease locus has only main effects on the disease and the model for the other three disease loci is the same as eq (1).In addition to K as 2000 in all of our simulations as well as in the real data analyses. The default bootstrap procedure in the RF algorithm was used to determine the relative proportions of the samples in the training and test sets. To evaluate the type I error rates for the TRF-pathway, the parameters (1–β7β) were all specified as 0. A total of 5,000 replicates of simulated datasets were used to calculate the type I error rates. For power simulations, we first simulated a model (Model 1) with main effects only. The parameters 1β, 2β, and 3β were specified as 0.92 and 4β–7β were specified as 0 in Model 1. Then we simulated a multiplicative model similar to the model used in Chatterjee et al. 1β, 2β, and 3β were specified as ϕ, 4β, 5β, and 6β were specified as 2ϕ, and 7β was specified as 3ϕ. Therefore, the joint effect of two or three markers was the product of the main effects of the individual markers. Model 2, Model 3, and Model 4 were simulated with ϕ equal to 0.18, 0.22, and 0.26, respectively. A total of 500 replicates of simulated datasets were used to calculate the power for each model.We downloaded the Random Jungle package 2r of 0.5 were tested for association using a standard chi-square test. The mean of the chi-square statistics for SNPs with p-values <0.05, which is the default setting in PLINK, was used as the statistic for the pathway in PLINK. A permutation procedure is used to create a null distribution for the statistic and estimate the p-value. Therefore, the set-based test in PLINK does not need background genes across the genome for the statistic. In the following text, we refer to PLINK as the set-based test implemented in PLINK. In all of the simulation models, we specified the threshold as 1.64 for the variable importance scores, which corresponds to p-value of 0.05 in a one-tailed Z-test, in step 3 in the TRF-pathway algorithm.We compared the power of the TRF-pathway with the RF-pathway and the set-based test in PLINK 1–β7β were all specified as 0. The power comparisons for the TRF-pathway with the RF-pathway and PLINK under the 4 scenarios were shown in Model 1 and Model 2. PLINK can also have more power than the RF-pathway in Models 1, 2, and 3 across the 4 scenarios. However, with the increased multiplicative effects of gene-gene interactions in Model 3 and Model 4, the TRF-pathway has significantly more power than PLINK. The results demonstrate that by using a two-stage approach, the TRF-pathway can improve power significantly when compared to the traditional RF-pathway approach. Moreover, the RF-based pathway methods can have more power than methods considering only main effects in the presence of strong gene-gene interaction effects.The type I error rates for the TRF pathway, RF-pathway and PLINK at the 0.05 and 0.01 levels under different scenarios are shown in We applied the TRF-pathway to a breast cancer GWAS dataset available at dbGaP Since Random Jungle assumes that there are no missing genotypes in the data, we imputed the missing genotypes in the sample using fastPHASE We also applied the TRF-pathway to a lung cancer GWAS dataset from the Cancer Prevention Study II Nutrition Cohort (CPS-II) We developed the TRF-pathway, which is a powerful two-stage RF-based pathway analysis method extended from the RF-pathway. Unlike many pathway analysis methods that consider only main effects of genes, the TRF-pathway considers both main effects of genes and gene-gene interactions. We used simulations to verify that both the TRF-pathway and the RF-pathway are valid tests for pathway association under the null hypothesis that none of the SNPs within a pathway are associated with the disease. We then used simulations to demonstrate that by employing a two-stage design, statistical power can be significantly increased in the TRF-pathway compared to the RF-pathway.Model 1 and Model 2), PLINK can have more power than the TRF-pathway and RF-pathway. This is not surprising as PLINK tests specifically for main effects. However, when the effects of gene-gene interaction are strong, the TRF-pathway has significantly more power than PLINK. Therefore, in practice, the TRF-pathway should be used as a tool that is complementary to the methods considering only main effects such as PLINK.Our power comparisons suggested that when there are only main effects or the effects of gene-gene interactions are not strong . Alternatively, an independent dataset can be used to calculate the unbiased prediction error rate for the significant SNPs in step 3 in the algorithm.The score The significant results shown in Unlike some pathway methods that calculate gene-specific scores for pathway statistics http://sourceforge.net/projects/trfpathway/.In summary, we used simulations and applications to breast cancer and lung cancer GWAS datasets to demonstrate that the TRF-pathway is a powerful pathway analysis tool. The TRF-pathway is implemented in a PERL script. The script uses the PLINK software to generate input files for Random Jungle (with –recodeA option) and uses Random Jungle for the RF calculations. A more efficient program of the TRF-pathway using C++ will be implemented so that the TRF-pathway can be applied to a large set of pathways. The script is freely available at


The analysis of complex diseases is an important problem in human genetics. Because multifactoriality is expected to play a pivotal role, many studies are currently focused on collecting information on the genetic and environmental factors that potentially influence these diseases. However, there is still a lack of efficient and thoroughly tested statistical models that can be used to identify implicated features and their interactions. Simulations using large biologically realistic data sets with known gene-gene and gene-environment interactions that influence the risk of a complex disease are a convenient and useful way to assess the performance of statistical methods.http://sourceforge.net/projects/gensim. The software is released under the GNU General Public License version 3.0.The Gene-Environment iNteraction Simulator 2 (GENS2) simulates interactions among two genetic and one environmental factor and also allows for epistatic interactions. GENS2 is based on data with realistic patterns of linkage disequilibrium, and imposes no limitations either on the number of individuals to be simulated or on number of non-predisposing genetic/environmental factors to be considered. The GENS2 tool is able to simulate gene-environment and gene-gene interactions. To make the Simulator more intuitive, the input parameters are expressed as standard epidemiological quantities. GENS2 is written in Python language and takes advantage of operators and modules provided by the simuPOP simulation environment. It can be used through a graphical or a command-line interface and is freely available from Data produced by GENS2 can be used as a benchmark for evaluating statistical tools designed for the identification of gene-gene and gene-environment interactions. Most of the common human diseases with high mortality rates  are classified as complex diseases ,2. By deGene-environment interactions (G×E) are expected to influence complex phenotypes, for example, disease risk. Hence individuals with predisposing genetics are more likely to develop a disease when exposed to a damaging environment than individuals, exposed to the same environment, without predisposing genetics ,6. The rComplex phenotypes are regulated by pathways and biochemical mechanisms that involve many genetic products. Hence, in addition to interactions among genes and environment, interactions among different genetic loci (G×G) can also influence disease risk. In particular, G×G are defined as epistatic when the allelic variations of one gene alters the effect of variations of another gene . Epistasr2 are generally inappropriate and tend to lead to an over/underestimation of disease risk , representing the dominance relation  of the risk related to one-unit increase in the exposure.KAPS2 also requires G×E and G×G models when two DPLs are provided. In particular: i) G×E models are chosen from a set of four predefined models, two models of interaction between DPLs and the environment, and two special models in which there is no gene-environment interaction but in which only one genetic or environmental factor contributes to the disease risk (see Table ii) G×G models (epistasis) are accepted in the form of percentage variations on the risk associated with a maximum number of three (out of the possible nine) combined genotypes.KAPS2 converts population features and G×E and G×G models into the corresponding parameters of the MLM in two steps. First, starting from the provided epidemiological parameters, KAPS2 calculates the penetrance of each combined genotype assuming no interaction between the genotypes of each locus. Epistasis (if defined) is then modeled through a deformation procedure, reflecting G×G variations, of the set of penetrance values that keeps it coherent with user defined epidemiological parameters. In this step, when there is more than one way to change the values of the set (i.e. less than three epistatic variations are provided), a mathematical optimization system is employed to find the deformation characterized by the smallest variation on the values not constrained by user defined epistatic variations. An example of the results of the epistasis application is presented in Figure Consequently, for each combined genotype, KAPS2 computes the coefficients of a penetrance function of the environmental exposure that is associated with the combined genotype in the MLM. In this step G×E are also modeled; the Additive model (ADD) assumes that combined genotypes with higher penetrance have a higher basal disease risk, while the risk associated with the environmental factor is just added. On the other hand, in the modulative model (GEM), combined genotypes with higher penetrance have the same basal risk although they are more ’sensitive’ to the effect of the environment (see the Methods section).In the final step the two branches of the procedure Figure  join. OnTo create simulated populations, we employed an existing tool, SimuPOP, and the implementation of the above described algorithm. Using SimuPOP it is possible to drive a forward-time simulation that results in a population composed by the desired number of individuals and having specified genotype frequencies for a set of markers. To be usable in GENS2, populations should be created in SimuPOP as described previously . This prids of the DPLs chosen from those present in the data set. For each DPL, the penetrance values are provided as relative risk (RR) and dominance model (see [GENS2 accepts as input a population produced by SimuPOP and the del (see ). RR is On the basis of the selected type of G×G and G×E, GENS2 calculates the coefficients of the MLM as described in the Method section.For each individual, GENS2 assigns the disease status (affected or unaffected) on the basis of its disease risk by applying the MLM and using a random process.The main output of the software can be either a single file or several files for a set of subpopulations of a given size produced by means of a subsampling procedure. Subsampling allows bootstrapping procedures to be executed on data sets produced with the same features. The output of GENS2 is in the form of a table in which each row represents an individual and the columns contain, from the left to the right, disease status, gender, environmental exposures and genotypes for each individual.Two possible formats for the genetics output are available: phased haplotypes or genotypes. In both output formats the initial columns are identical to those described above; however, they differ in the way the genetic information for each individual is represented. In the phased haplotype format, there are two columns for each SNP that report the allele status  on each chromosome. In the genotype format, each SNP is represented by one number , where 2 represents the heterozygote and 3 represents, for DLPs, the high risk homozygote or, for all the other SNPs, the lower frequency homozygote.ID and genomic position of the SNPs in the data set can be returned.In addition to the main output file, GENS2 outputs a log file that contains an extensive report of all the intermediate steps in the procedure and the values used to obtain the populations. Optionally, a file containing the GENS2 is designed to be used either from the command line as a Python script, or through a graphical user interface, similar to a wizard, that prompts the user in the specification of all required parameters [see Additional file Overall, the computational time complexity of the simulation procedure depends by both simuPOP and GENS2. For GENS2, the procedure is dominated by the assignment of the disease status to all individuals in the population. Indeed, after the KAPS2 module has performed the translation of user provided parameters into MLM parameters in bounded constant time, the time complexity becomes linear in the number of individuals and the number of represented variables  for each individual in the simulated population. On the other hand, the amount of time required to perform a simulation with simuPOP depends on the size of the simulation and scales roughly linearly with the number of markers and individuals that are used . GENS2 iHere we describe a method based on the MLM to simulate two genetic and one environmental factors interacting in the determination of a disease risk. The method is implemented in GENS2, a software that is freely available.glm function in R) with a different model for each test and considering the status as the dependent variable.To test populations produced by GENS2, we performed a set of analyses on some representative populations. The aim was to emulate a case in which GENS2 was used to assess the performances of a feature selection method. In particular, all the analyses were performed using a logistic regression (p<10−6). Furthermore, non-causative markers in LD with the two DPLs also showed a significant association that was roughly proportional to the value of r2 with the DPLs.The first test was a single-marker analysis on a population of 1,000 cases and 1,000 controls with two DPLs in two distinct genomic regions, with no epistasis and an additive G×E model. The association of each marker with the status was tested using logistic regression analyses with model: disease risk = genetic factor + environmental factor. As expected, the most significant associations were those of DPLs [see Additional file The second test was similar to the first, except that 10,000 cases and 10,000 controls and a modulative G×E model for the DPLs were used. For this test, the logistic regression was used by considering both an additive model  and a multiplicative model . None of the markers, when tested by additive model, reached a Bonferroni corrected significance level [see Additional file % penetrance for the  combined genotype) and an additive G×E model in a population of 5,000 cases and 5,000 controls. Because of the higher computational cost of this analysis, we performed the test on only a subset of about 1,200 markers surrounding the two DPLs. The results are displayed in Figure Finally, we tested an example of two DPLs with no marginal risk, an epistatic interaction  so that setting up the model does not become too complicated for the user. In this way, we reached a balance between the complexity of the represented phenomena and simplicity in the definition of the model. Moreover, the best strategy to identify even simple interactions as single G×G and G×E with binary environmental variables it is still debated . gs[gs the user can define G×G between two loci in two ways: one, by providing a penetrance matrix for combined genotypes or, two, by selecting a penetrance matrix from a predefined set of G×G models. However, both these approaches have some limitations. Although the first allows great control over the penetrance of each genotype, it easily leads to a loosening of control on marginal effects, making the replication of real populations difficult. The second approach, on the contrary, is too restrictive and does not allow any possible interaction to be simulated. For simulation of G×G, gs allows the user to input a list of rules regarding specific combinations of genotypes and levels of environmental values, and the corresponding risk levels. Again, this approach makes it very difficult to control the overall characteristics of populations in terms of marginal effects of genetic and environmental factors. The approaches described above can simulate complex interactions by loosening control on overall population characteristics or, alternatively, can keep the overall population characteristics under control by limiting the freedom to specify the interactions. Another strategy to simulate G×G and G×E is to manually write functions or sets of rules that associate each combination of genes and environmental factors to a risk value. Although this strategy provides more freedom, it is very difficult to set up when the control of marginal effects of single factors is desired. We believe that GENS2 provides a better balance between the freedom to define possible interactions among factors and the control of the overall population characteristics.Several methods simulating genetic data have been proposed, many of them also handle complex LD patterns and polygenic traits ,23. For The simulated populations produced with GENS2 can be thought of as a sampling of an ideal infinite population that has the characteristics specified by the user. From this point of view, it is easy to understand that fluctuations of observed values around the expected ones can occur. Among the elements that mostly affect these fluctuations, are sample size, allele frequencies, and penetrance values. In particular, small sample sizes increase the effect of sampling error and thus, as expected, these fluctuations tend to vanish as the sample size is increased [see Additional file Although the GENS2 part of the simulation process is reasonably fast, the procedure to simulate large populations using simuPOP takes time to complete. It would be difficult to simulate a large number of samples without a cluster system, unless multiple  samples are drawn from the same large population.GENS2 allows the simulation of gene-gene and gene-environment interactions among two genetic and one environmental factor in relation to the risk to develop a complex disease. It is based on data with a realistic pattern of LD and it has no limitations either on the number of individuals that can be simulated or on the number of genetic and environmental factors within a simulated data set. Furthermore, a large amount of effort has been channeled into allowing the input of parameters as standard epidemiological quantities so that the software is immediately usable by the biomedical community.GENS2 provides large biologically realistic data sets with known features that can be used to challenge, and eventually improve, the statistical tools that are designed to identify those interactions.Here we present the mathematical background underlying the extension of the earlier model  to the c1. the genetics can influence the disease risk either directly or by modifying the effect of the environment.2. the genetic loci can have independent effects (no epistasis) or can interact in an epistatic manner, and3. the DPLs are not in LD.gagb) (with ga∈GA={AAAaaa} and gb∈GB={BBBbbb}) who is exposed to the environmental level x, the disease risk is defined under the MLM by the conditional probability P(affected|gagbx), which is parameterized as: To model these situations we applied the MLM, here briefly summarized, that uses a different logistic function for each combination of the two genotypes . The depwhere To simulate a population, the coefficients m be the prevalence of the disease. Starting from these values and using the independence hypothesis for the variables A and B, the probability ga,gb) is the product ga,gb) the total risk for the disease insurgence ga,gb). The value of this parameter is obtained with the MLM as Let X is the domain of the environmental variable.where Because every logistic function in MLM is characterized by its own parameters, the 3×3 pairs of values  admits infinite solutions. However, the G×E model imposes some constraints on the coefficients. Thus, by fixing the value of one of the coefficients • Genetic effect (GEN): • Environmental effect (ENV): • Modulative effect (GEM): • Additive effect (ADD): βABhave been defined, a set of six transcendent equations can be written with the coefficients of the logistic functions (except βAB) as the unknown variables; these equations admit exactly one solution [When the interaction model, the matrix containing total risk values for each combination of genotypes, namely solution .m, namely To determine Moreover, the total risk values associated with the genotypes of a single locus are related to those of combined genotypes via marginalization: In general, once the marginals  Notice that the superscript “I” is a reminder that the independent polygenic model has been assumed.Using an independent polygenic model and a deformation procedure, epistatic interactions among DPLs can be modeled to obtain a matrix TRI, where each entry ga,gb). By definition Let Using the expressions in Eq.s (4) and ( 5) we get Because by construction, the matrix R3×3 in which the elements are assignments for the entries of matrix Once the quantities More precisely (in the two variables case), given the constraints of Eq.s (9) and (10) from one up to three entries for ga,gb) corresponding to non-user-assigned An objective function can be used to minimize the variance of the set of ratios Once the coefficients of the MLM are fixed, the disease risk for each individual in a population can be established by substituting the coefficients associated with the carried genotype into Eq. (1) and then by evaluating the resulting logistic function forthe exposure level of the environmental disease factor. Finally, to assign the disease status to each individual, the disease risk is compared with a random number drawn from a uniform distribution.Project name: Gene-Environment iNteraction Simulator 2Project home page:http://sourceforge.net/projects/gensim/Operating system(s): Platform independentProgramming language: PythonOther requirements: SimuPop, OpenOpt, wxPython License: GNU GPLv3The authors declare that they have no competing interests.MP conceived the model and the extensions, and drafted the manuscript; GS conceived and developed the extensions, implemented the software and drafted the manuscript; RA conceived the model and the extensions and drafted the manuscript; SC and GM conceived the study, and participated in its design and coordination and helped to draft the manuscript. All authors have read and approved the final manuscript.The GENS2 graphic user interface. Flowchart showing a typical way of using GENS2 through its graphical user interface. Portable Network Graphics (.png) image file.Click here for file10) of each marker when tested individually (each dot represents a different marker). The red dashed line represents the significance threshold (0.05 after Bonferroni correction) and the green dashed lines mark the position of the DPLs. In the bottom panel, the r2 for each marker with the DPL in the same region is shown. Portable Network Graphics (.png) image file.Association test in the case of additive G×E. The population comprised 1,000 cases and 1,000 controls. Two DPLs  in an additive G×E model (OR=1.2) with no epistatic interaction were present. The two DPLs are in two distinct genomic regions . In the upper panel, the Manhattan plot shows the significance of the association ) of each marker when tested individually (each dot represents a different marker), using a multiplicative and an additive model in the logistic regression. The red dashed line represents the significance threshold (0.05 after Bonferroni correction) and the green dashed lines mark the position of DPLs. In the bottom panel, the r2 of each marker with the DPL on the same region is shown. Portable Network Graphics (.png) image file.Association test in the case of modulative G×E. The population comprised 10,000 cases and 10,000 controls. Two DPLs  in a modulative G×E model (OR=1.2) with no epistatic interaction were present. The two DPLs are in two distinct genomic regions . In the upper panel, the two Manhattan plots show the significance of the association  are ordered according to their predicted affect on the overall disease risk. The x-axes show the sample size and the y-axes show the risk. The green lines represent the expected risk, the blue lines show the median observed risk, and the red dashed lines indicate the minimum and maximum observed disease risk in 100 replicates. Portable Network Graphics (.png) image file.Click here for file


Genomic imprinting refers to parent-of-origin dependent gene expression caused by differential DNA methylation of the paternally and maternally derived alleles. Imprinting is increasingly recognized as an important source of variation in complex traits, however, its role in explaining variation in muscle and physiological traits, especially those of commercial value, is largely unknown compared with genetic effects.We investigated both genetic and genomic imprinting effects on key muscle traits in mice from the Berlin Muscle Mouse population, a key model system to study muscle traits. Using a genome scan, we first identified loci with either imprinting or genetic effects on phenotypic variation. Next, we established the proportion of phenotypic variation explained by additive, dominance and imprinted QTL and characterized the patterns of effects. In total, we identified nine QTL, two of which show large imprinting effects on glycogen content and potential, and body weight. Surprisingly, all imprinting patterns were of the bipolar type, in which the two heterozygotes are different from each other but the homozygotes are not. Most QTL had pleiotropic effects and explained up to 40% of phenotypic variance, with individual imprinted loci accounting for 4-5% of variation alone.Surprisingly, variation in glycogen content and potential was only modulated by imprinting effects. Further, in contrast to general assumptions, our results show that genomic imprinting can impact physiological traits measured at adult stages and that the expression does not have to follow the patterns of paternal or maternal expression commonly ascribed to imprinting effects. Igf2) or vice versa . Differg. Igf2R;. At the g. Igf2R;.et al.)et al.)Mc12.1 and Mc19.1) are within regions identified by Gregg and colleagues in the hypothalamus and the medical prefrontal cortex . However, RNA-seqence analysis of brain tissues by Gregg and colleagues  revealedfat mass , and alss [et al.. The fins [et al., affectis [et al.,32 and ms [et al.. A mutats [et al..Results of our analysis of the proportion of phenotypic variance explained by additive, dominance and imprinting effects contrasts with results of weight traits in a different population generated from mouse lines divergent for body weight at day 60 . In thisOur results show that variation in key physiological traits such as glycogen content is modified by parent-of-origin-dependent gene expression or genomic imprinting but not by genetic variation. These results further demonstrate that imprinting effects are not limited to early developmental phenotypes nor that imprinting occurs predominantly as paternal or maternal expression.Cmptdl1Abc mutation. Two pairs of the Berlin Muscle Mouse inbred lines BMMI806 and BMMI816 were crossed reciprocally to generate the F1 generation [2 animals were produced that were then randomly mated [3 animals. For the analysis of imprinting effects, we used all 94 F2 animals and 331 F3 animals out of 345 for our QTL analysis because of genotyping errors in some individuals.For this study, we used mice of the Berlin Muscle Mouse (BMM) population, which has been selected for high body weight and muscle mass, primarily in order to investigate the selective mechanisms in livestock breeding . The selneration . 94 F2 aly mated  to produad libitum access to food and water. The animals were fed a standard diet  until they were 70 days old. This diet was composed of 27.0% crude protein, 5.0% crude fat, 4.5% crude fibre, 6.5% crude ash, 50.5% nitrogen free extract (starch and sugar), vitamins, trace elements and minerals .All experimental protocols were approved by the German Animal Welfare Authorities . The animals were maintained under standard conditions . Two to four animals of the same sex were put in cages with The mice, at 71 days of age, were anaesthetized by isoflurane and sacrificed after two hours of fasting. The Musculus longissimus (ML) and Musculus quadriceps (MQ) were dissected and weighed, and muscle mass (MM) was recorded as summed muscle weight of left and right M. longissimus and left and right M. quadriceps. The right muscles were promptly frozen in liquid nitrogen and then stored at −80°C. Carcasses were kept at 6 °C and pH values were measured within the M. biceps femoris at 1 and 24 hours post mortem . The glycolytic potential was measured from glycogen content and lactate content . We adjuThe two parental lines BMM806 and BMMI816 were generated from the same founder population that had been selected for high muscularity over many generations. Since we genotyped both lines with the Mouse Diversity SNP Array , compris2 and F3 generations were genotyped at KBiosciences . Genotypes were checked for errors and genetic distances with RQTL (http://cran.r-project.org/web/packages/RQTL/index.html). Genotype frequencies at every marker locus are given in Additional file All animals of the F2 generation and a set of ordered  haploytpes for the F3. We distinguish four ordered genotypes denoted AA, AB, BA, BB  with the A allele originating from BMMI806 and the B allele from BMMI816. In a first step we assigned the four ordered genotypes at the marker loci additive (a), dominance (d), and parent-of-origin (i) genotypic index scores following Wolf et al.[The data analysis for this paper was conducted using SAS 9.1  and SPSS  and PedPhase (version 2.0). We reconstructed haplotypes using Pedphase  to produThis can be solved to yield estimates of the genotypic valuesAA, AB, BA and BB are genotypic values and r is the reference point of the model, i.e. the midpoint between homozygotes, a gives the additive genotypic value, i.e. half of the difference between two homozygotes, d the dominance genotypic value, i.e. the difference between the heterozygote mean and the midpoint of the homozygotes and finally i denotes the imprinting value, i.e. half the difference between heterozygotes [ozygotes .a, locus d and locus i as fixed class variables [To detect QTL we performed a genome scan using a linear mixed model fitted by restricted maximum likelihood (REML) with locus ariables  and famia, d, i LOD scores exceeded the genome-wide threshold for a given trait.Significant genome-wide thresholds for each trait were calculated by 1000 permutations . In addiOur prior work demonstrated that parent-of-origin dependent effects on offspring phenotypes may be caused by either maternal genetic effects or genomic imprinting . DiffereR2, following Hager et al.[a2/Vp) × 100; for dominance effects this is given by (1/4 d2/Vp) × 100, and for imprinting effects by (1/2 i2/Vp) × 100.Finally, the proportion of phenotypic variance explained by a locus was calculated by dividing the genotypic variance (Vg) by the phenotypic variance (Vp) given that REML does not compute sums of squares and the corresponding er et al.. For addWe denote QTL as Mc, for mouse chromosome, followed by the number identifying the chromosome and the number of a particular locus on a chromosome.The authors declare that they have no competing interests.ARV, RH and SK conducted the analysis. SK collected pheno- and genotypes. RH, SK and GAB conceived the analysis. All authors contributed to writing the manuscript. All authors read and approved the final manuscript.Table S1. Number of individuals for each of the four genotypes at each of the markers. Frequencies of the four genotypes at each of the markers.Click here for fileTable S2. Comparison of iQTL with regions showing parent-origin specific gene expression in the hypothalamus (POA) and medical prefrontal cortex (mFPC), given by sex and expression pattern from [ern from .Click here for fileFigure S1. Map of reference single nucleotide polymorphisms used in this study.Positions are given in Mb. Bars indicate identified QTL with genome-wide significance.Click here for file