Mycobacterium avium subspecies paratuberculosis (MAP), causes economic losses in excess of $200 million annually to the US dairy industry. To identify genomic regions underlying susceptibility to MAP infection in Jersey cattle, a case-control genome-wide association study (GWAS) was performed. Blood and fecal samples were collected from ∼5,000 mature cows in 30 commercial Jersey herds from across the US. Discovery data consisted of 450 cases and 439 controls genotyped with the Illumina BovineSNP50 BeadChip. Cases were animals with positive ELISA and fecal culture (FC) results. Controls were animals negative to both ELISA and FC tests that matched cases on birth date and herd. Validation data consisted of 180 animals including 90 cases (positive to FC) and 90 controls (negative to ELISA and FC), selected from discovery herds and genotyped by Illumina BovineLD BeadChip (∼7K SNPs). Two analytical approaches were used: single-marker GWAS using the GRAMMAR-GC method and Bayesian variable selection (Bayes C) using GenSel software. GRAMMAR-GC identified one SNP on BTA7 at 68 megabases (Mb) surpassing a significance threshold of 5×10−5. ARS-BFGL-NGS-11887 on BTA23 (27.7 Mb) accounted for the highest percentage of genetic variance (3.3%) in the Bayes C analysis. SNPs identified in common by GRAMMAR-GC and Bayes C in both discovery and combined data were mapped to BTA23 , 3  and 17 (57 Mb). Correspondence between results of GRAMMAR-GC and Bayes C was high (70–80% of most significant SNPs in common). These SNPs could potentially be associated with causal variants underlying susceptibility to MAP infection in Jersey cattle. Predictive performance of the model developed by Bayes C for prediction of infection status of animals in validation set was low (55% probability of correct ranking of paired case and control samples).Paratuberculosis (Johne’s disease), an enteric disorder in ruminants caused by  Mycobacterium avium subspecies paratuberculosis (MAP). MAP is contagious; infected animals expose their cohorts to the pathogen by shedding bacterium into their colostrum, milk or feces MAP infection MAP transmission is fecal-oral MAP is a slow-growing intracellular bacterium; infected animals remain asymptomatic for 2 to 10 years before showing clinical signs of the infection. Clinical signs of JD in MAP-infected dairy cattle usually appear after 2nd or 3rd lactation and include poor nutrient uptake, severe diarrhea, progressive weight loss, low milk production and eventually death MAP-infected herds in the top 17 US dairy states to be at least 68% based on recovery of viable MAP in environmental fecal samples MAP infection in these herds was estimated to be 91.1% Paratuberculosis or Johne’s disease (JD) is a chronic bacterial infection of the gastrointestinal tract caused by JD, like most other complex diseases is multi-factorial i.e. under the influence of both genetic and environmental factors. Studies have shown that susceptibility to JD is heritable with the estimates ranging from 0.03 to 0.28 in cattle MAP has been found in some patients with CD Crohn’s disease (CD) is an inflammatory bowel disease (IBD) in humans with manifestations similar to those of JD in cattle. MAP in Holstein have been performed to date MAP infection in Holsteins. Jersey is the second most common dairy breed after Holstein in the United States. This is the first GWAS for susceptibility to paratuberculosis infection in the Jersey breed. Our objectives were to identify genomic regions that underlie susceptibility to infection with MAP as well as development of a multi-marker model to be used in genomic selection against susceptibility to MAP infection in Jersey cattle.In recent years, availability of the BovineSNP50 platform for genotyping ∼54,000 SNPs across the Bovine genome The University of Wisconsin-Madison College of Agricultural and Life Sciences Animal Care and Use Committee approved the procedures used with animals in this experiment.Blood and fecal samples were obtained from ∼5,000 mature cows (minimum age of 20 months) from 30 commercial Jersey herds throughout the US in a retrospective cross-sectional design. Nomination of herds for this study was based on the prevalence of JD evidenced by the herd’s owner or veterinarian. Sampling was performed selectively for three herds and completely (whole herd) for the remaining 27 herds. Samples were shipped in insulated containers with cold packs by overnight courier to the Johne’s Testing Center of the School of Veterinary Medicine at the University of Wisconsin-Madison and processed upon receipt . Serum was held at 4°C and fecal samples at –20°C until testing. All blood samples were tested within 7 days by a serum ELISA (JTC-ELISA) Sample – ODNegative control]/[ODPositive control – ODNegative control] MAP infection by fecal culture (FC) ELISA optical density (OD) values for each serum sample were converted to sample to positive ratios (S/P) using  were common between the 20 most significant SNPs in the two analyses  & 4. New and BTA1In a ten-fold cross-validation, the predictive abilities of models developed by training with 90% of combined data were evaluated in the remaining 10%. AUC ranged from 0.46 to 0.65 for the models  was in the range of pedigree-based heritability estimates . This suggests that markers captured most of the additive genetic variation through LD with QTL. However, the previous pedigree-based heritability estimates are from studies of Holstein cattle. No report is available for the heritability of susceptibility to paratuberculosis in Jersey cattle. Studies have shown that susceptibility to a priori be expected to have non-zero effect in 0.1% of MCMC samples. A model frequency of 0.23 indicates that particular SNP had a non-zero effect in 23% of MCMC samples. The relatively low significance of identified SNPs can be explained by the complex genetic architecture of susceptibility to paratuberculosis and that there are many genes with small effects influencing this disease. The power of GWAS to identify SNPs with small effect size is limited, however, this limitation may be overcome in studies of larger scale.The significance level in single-marker and Bayesian GWAS was generally low; no SNPs surpassed the threshold of strong association in separate or combined analyses. Likewise, the highest model frequency of SNPs in most significant windows for both discovery and combined analyses was 0.23. The low model frequencies resulted in higher PFP. For π = 0.999, a randomly chosen locus would The results of single-maker GWAS on discovery data showed evidence for association on BTA23 and BTA3 as multiple SNPs on these chromosomes in relatively close proximity were among the 20 most significant SNPs. There was a relatively high correspondence between the results of single-marker and Bayesian GWAS; 70% and 80% of the 20 most significant SNPs by GenABEL were also among 20 most significant SNPs by GenSel for discovery and combined analyses, respectively. The first three windows from Bayesian analysis of discovery data explained ∼10% of genetic variance while with combined data ∼7% of genetic variance was explained by the top three windows. From both GenABEL and GenSel analyses, combining data enhanced the significance of only a few SNPs.MAP infection across two breeds. Genes within 1 Mb of this location are four members of solute carrier family 17  located between 31.7– 31.8 Mb. The 1 Mb distance was chosen based on the extent of linkage disequilibrium in cattle. Kim and Kirkpatrick (2009) showed that for pairs of markers with relatively high LD (r2 = 0.4–0.6) the median physical distance was ∼1.1 Mb. However, for Jersey with smaller population size and higher inbreeding the extent of LD may be even higher. Association of SNPs in SLC11A1 (another member of SLC family) with MAP infection has been reported MAP infection in cattle.There is some correspondence between the results of this study and previous GWAS in Holsteins. The closest correspondence is for SNP ARS-BFGL-NGS-19381 . The nearest SNPs to this position that were identified in Holsteins were located at 32.1 and 32.2 Mb on BTA23 MAP shedding may be different from loci underlying humoral response to the pathogen. Another limitation is the small number of samples used in the validation data set.For validation of GWAS results the ideal situation is using samples from a population independent from the discovery population with the same phenotype that was used in the discovery stage. In this study, neither of these requirements was possible. Our validation data failed to replicate the results of discovery GWAS. One reason for this might be use of a different case definition; FC+ and ELISA+/FC+ may represent distinct phenotypes i.e. loci responsible for a cow’s ability for MAP infection in Holsteins was 0.73 by AUC in cross-validation analyses MAP infection.The predictive ability of the models developed by the Bayesian approach was low. Given that an AUC of 0.50 represents random guessing, an AUC of 0.55 or 0.56 is a weak classifier. The efficacy of the multi-marker model developed by Kirkpatrick et al. (2010) for prediction of susceptibility to Combining the results of GRAMMAR-GC and Bayes C for discovery and combined data, nine SNPs distributed on four chromosomes were commonly identified by all four analyses. These SNPs include ARS-BFGL-NGS-118877 (27.7 Mb), ARS-BFGL-BAC-35219 (29.3 Mb) and BTA-56690-no-rs (44.4 Mb) on BTA23; ARS-BFGL-NGS-109837 (100.9 Mb), Hapmap51790-BTA-103080 (101.1 Mb), BTB-00148619 (106.4 Mb) and Hapmap53765-ss46526662 (107 Mb) on BTA3; ARS-BFGL-NGS-100555 (57.1 Mb) on BTA17, and BTA-30686-no-rs (60.6 Mb) on BTA6. In most cases potentially relevant candidate genes are in near proximity (i.e. within 1 Mb). ARS-BFGL-NGS-11887 is located in the region of major histocompatibility complex (MHC) class I gene clusters (27.5–28.5 Mb) on BTA23. The role of multiple MHC genes  in the predisposition to CD in humans has been demonstrated or suggested Mycobacterium avium subsp. paratuberculosis in Jersey cattle. Two statistical approaches were used: single-marker regression (GRAMMAR-GC) and Bayesian methodology (Bayes C) for multi-marker regression. Nine SNPs representing four chromosomes  were identified by both GRAMMAR-GC and Bayes C analyses in discovery and combined  data. Multi-marker prediction models were developed and tested by both cross-validation and application to the validation data set; predictive ability of the models to correctly rank cases and controls was low (55-56%) based on the area under ROC curve. The application of these models to predict the phenotypic outcome of animals in regard to JD is limited, however, they can be used for prediction of genetic merit.We performed a case-control genome-wide association study for infection with Figure S1Multi-dimensional scaling plots. A) Discovery data (N = 889) B) Validation data (N = 180) and C) Combined data . Each animal is represented by one point. PC1 and PC2 are the first two principal components obtained from genomic kinship matrix. Distance between points represents the genetic distance between animals.(PDF)Click here for additional data file.Figure S2Quantile-quantile plots of P-values from genome-wide association analysis for susceptibility to MAP infection. A) Discovery data B) Validation data C) Combined data. Y-axis represents the observed P-values and X-axis the expected P-values under null hypothesis  of no association.(PDF)Click here for additional data file.


We generalize recent work on graphical models for linkage disequilibrium to estimate the conditional independence structure between all variables for individuals in the Genetic Analysis Workshop 17 unrelated individuals data set. Using a stepwise approach for computational efficiency and an extension of our previously described methods, we estimate a model that describes the relationships between the disease trait, all quantitative variables, all covariates, ethnic origin, and the loci most strongly associated with these variables. We performed our analysis for the first 50 replicate data sets. We found that our approach was able to describe the relationships between the outcomes and covariates and that it could correctly detect associations of disease with several loci and with a reasonable false-positive detection rate. As the search for genetic variation underlying complex diseases turns to rare variants of small effect ascertained from genome-wide or exome-wide sequence data, the need for methods of disease-gene association that account for the high degree of nonindependence between densely spaced genetic variants  becomes increasingly critical. Failure to consider LD may result in false-positive associations. We have recently shown  that graY is the disease phenotype; the n.A graphical model describes the joint distribution of a set of random variables as the product of conditional distributions on smaller subsets of the variables. This factorization has a corresponding Markov  graph, in which vertices represent variables and edges connect variables that appear together in the same conditional distribution  structure. We use a stepwise iterative approach to render the problem more computationally tractable.In the first step of our algorithm we estimate for all nonsynonymous variants on each chromosome the posterior most-probable population-dependent graphical model. That is, we empirically estimate a joint distribution on the ethnic origin variable and alleles at all loci with nonsynonymous variants . . The second and third steps of the algorithm are also carried out for each chromosome individually. In these steps, the haplotype phase is fixed according to the results of the first stage, and model fitting proceeds using Metropolis graph updates only. In the second step, a restricted graphical model is fitted to the full data set, including all loci, disease status, quantitative traits, and covariates. To reduce the model space and improve mixing, we enforce the restriction that edges can connect loci separated by at most 50 SNPs. We then prune the Markov graph to include only a subset of loci that are most strongly associated with any of the outcomes or covariates; that is, we restrict the subgraph to spanning all neighbors of all outcomes and covariates, with the exception of ethnic origin, up to third degree. At this stage we exclude the loci linked only to ethnic origin because they are unlikely to be relevant predictors of disease.The third step is similar to the second; at this stage, however, because of the decreased model space, we fit an unrestricted graphical model to the reduced data set. We then perform a second pruning step, this time restricting the new Markov graph to the subgraph of all first- and second-degree neighbors of all loci except ethnic origin.In the final stage of our algorithm, we combine the remaining loci from all chromosomes to determine the loci that are most strongly associated with any of the outcomes while controlling for ethic origin and all covariates. Again, we fit an unrestricted graphical model to the data. The nearest neighbors of the disease state or any of the quantitative traits are considered significantly associated (according to the information criterion given by Eq. (2)).We performed our analysis for the first 50 replicates of the GAW17 data set. In all cases, the estimated relationships between disease status, the quantitative traits, and the measured covariates closely approximate those specified in the model used to generate the data. To illustrate the method, we depict in Figures PIK3C2B, C6S2981 in VEGFA, and C10S3050 in SIRT1. All three of these variants are rare, with minor allele frequencies (MAF) of 0.0029, 0.0022, and 0.0022, respectively, and they were simulated to have additive mean effects β = 0.23, β = 1.2, and β = 0.97, respectively. We also detected two common variants, both in FLT1, associated with quantitative trait Q1: C13S522  and C13S523 . Our algorithm also returned 13 noncausal variants . We note that, as seen in Figure The final estimated graphical model for the first replicate is shown in Figure β < 0.5) or strong effect; the number of times (out of 50) that each variant was detected is indicated. Out of seven possible variants in the common and strong category, our algorithm detected three: C13S523 (FLT1) was seen in 25 replicates, C13S522 (FLT1) in 23 replicates, and C1S6533 (ARNT) in 2 replicates. Of the 57 variants in the strong and rare category, 6 were seen in at least one replicate. Of 17 common and weak variants, we observed only 1. Finally, out of 81 variants in the rare and weak category, only 1 was detected but in four distinct data replicates.Over the first 50 simulated data sets, our graphical modeling method identified, on average, 1.4 SNPs in causal genes per data set. Of these correctly identified SNPs, 90.2% were exactly the causal SNPs rather than SNPs located elsewhere in the causal gene. Our algorithm also identified an average of 15.8 noncausal variants per replicate. Table FLT1 gene in more than 70% of replicates. Furthermore, our approach shows some power to detect rare variants of large effect, detecting 6 of 57 such variants in at least one replicate. In its current implementation, our method has little power to detect variants with weak effect on phenotype; however, it did find the C10S3110 mutation in SIRT1  in four distinct replicates, probably because of LD between this locus and other rare mutations of stronger effect within the same gene. In general, however, our graphical modeling method is successful at controlling for LD, because it is able to pinpoint causal SNPs within causal genes more than 90% of the time.We have shown that a graphical modeling approach can detect loci associated with complex diseases. Our method also provides a good approximation of the true associations between outcomes and covariates specified by the model. Furthermore, our method has the ability to detect environmental effects on disease as well as gene-gene and gene-environment interactions. We have demonstrated that our method has reasonable power to detect common variants of strong effect, finding mutations in the Our graphical modeling approach, which controls for population structure, linkage disequilibrium, and relevant covariates, shows considerable promise for detection of both common and rare variants from exome sequence data. However, it does have some limitations that should be addressed in future work. First, our method is purely statistical and avoids biological assumptions (except in its exclusion of synonymous variants); hence it has only weak power to detect rare variants and also tends to detect associated but noncausal variants. Inclusion of, in particular, biological pathway information into the model would help to increase power and decrease detection of spurious associated SNPs. Second, our algorithm uses simulated annealing to find the single most likely graph to represent each data set; using the MCMC method to average over many models from the posterior distribution might help to decrease the number of false positives. By identifying only the nearest neighbors of the outcome variables in the most likely model, our algorithm sometimes excludes causal SNPs in favor of other tightly linked causal or noncausal SNPs. Model averaging and incorporation of, for instance, pathway information into the model would help to increase both sensitivity and specificity by appropriately constraining the model space and considering a larger number of possible solutions. Finally, our graphical models incorporate a single (given) ethnicity variable. In general, this could pose a problem because of admixture or possibly unreliable self-reported ethnicity. We are currently developing a method to extend our LD model  to allow the ethnicity variable to change between loci and small clusters of loci, thus allowing for admixed and unknown ethnic origins.Despite these limitations, the flexible method of graphical modeling provides a promising approach to the challenge of detecting the genetic variation that underlies complex disease. Given such a disease that involves environmental factors and many rare variants of small effect, inclusion of relevant biological information into the model would help to elucidate the true underlying factors. Graphical modeling might provide a framework for such a systems biological tack.The authors declare that there are no competing interests.HA performed the analysis and wrote the manuscript. Both authors created the software and read and approved the manuscript.


Gene-gene interaction in genetic association studies is computationally intensive when a large number of SNPs are involved. Most of the latest Central Processing Units (CPUs) have multiple cores, whereas Graphics Processing Units (GPUs) also have hundreds of cores and have been recently used to implement faster scientific software. However, currently there are no genetic analysis software packages that allow users to fully utilize the computing power of these multi-core devices for genetic interaction analysis for binary traits.Here we present a novel software package GENIE, which utilizes the power of multiple GPU or CPU processor cores to parallelize the interaction analysis. GENIE reads an entire genetic association study dataset into memory and partitions the dataset into fragments with non-overlapping sets of SNPs. For each fragment, GENIE analyzes: 1) the interaction of SNPs within it in parallel, and 2) the interaction between the SNPs of the current fragment and other fragments in parallel. We tested GENIE on a large-scale candidate gene study on high-density lipoprotein cholesterol. Using an NVIDIA Tesla C1060 graphics card, the GPU mode of GENIE achieves a speedup of 27 times over its single-core CPU mode run.http://www.cceb.upenn.edu/~mli/software/GENIE/.GENIE is open-source, economical, user-friendly, and scalable. Since the computing power and memory capacity of graphics cards are increasing rapidly while their cost is going down, we anticipate that GENIE will achieve greater speedups with faster GPU cards. Documentation, source code, and precompiled binaries can be downloaded from  The advent of high-throughput genotyping technologies has made it possible to study human genetic variation on a genome-wide scale. Recent years have seen an explosion of results generated from genome-wide association studies (GWAS). Most GWAS focus on single marker-based analysis in which each marker is analyzed individually, ignoring the dependence or interactions between markers. Although this approach has led to the discovery of disease susceptibility genes for many diseases, the identified markers often only explain a small fraction of the phenotypic variation, suggesting a large number of disease variants are yet to be discovered. It is becoming increasingly evident that gene-gene interactions play an important role in the etiology of complex diseases and traits -3, and lHowever, gene-gene interaction analysis is parallelizable in nature. Most of the current Central Processor Units (CPUs) have multiple cores. Parallel computing until recently meant using a computing cluster having multiple nodes with multi-core CPUs. The costs of building a computing cluster may run in hundreds of thousands of dollars, making it cost prohibitive. An emerging economic scientific computing paradigm is to use Graphics Processing Units (GPUs) that are present in graphic cards of most desktop computers or workstations for general purpose computing. A GPU is a processing unit that was traditionally used for accelerating graphical operations. The power of GPUs has been used to implement faster software solutions for biological problems -8. For eA typical graphics card has several processors as well as its own dedicated memory. We will be using the term "device memory" to refer to the built-in memory of the graphics card in the rest of this paper. Several vendors of graphics cards offer architectures and programming tools that enable GPU-based general purpose computing using high level programming language extensions. NVIDIA Common Unified Device Architecture (CUDA) is an example of a graphics card architecture for parallel general purpose computing. CUDA follows the Single Instruction Multiple Thread (SIMT) architecture that is similar to the Single Instruction Multiple Data (SIMD) architecture of parallel computing. In the case of CUDA, this means that multiple threads on the same instructions are executed simultaneously on different data. Programmers can exploit the CUDA architecture with relative ease to solve larger problems that can be decomposed into several sub-problems that can be solved in parallel. The computing power offered by the latest graphic cards is comparable to that of a computing cluster with hundreds of CPUs, but the GPU programming approach for parallel computing is much cheaper than using a traditional computing cluster.CUDA compatible graphics cards have several processors that are also known as multiprocessors (MP) that in turn have several stream/thread processors (SP) known as cores Figure . CUDA arCUDA provides a new scalable parallel programming model along with an instruction set. CUDA also provides several levels of abstraction in terms of threads and memory hierarchies. Such abstractions help CUDA provide data parallelism and thread parallelism. Programmers are exposed to these abstractions in the form of API. Figure Currently, there are no software packages that directly address the gene-gene interaction analysis problem for binary traits. Moreover, there are no gene-gene interaction analysis software packages that use multiple processor cores present in CPUs or GPU cards. Our software package, GENIE, utilizes multiple cores to parallelize gene-gene interaction analysis. As we show in the Results, GENIE achieves ~27 times speedup over a single-core CPU, and making large-scale gene-gene interaction analysis feasible even on a desktop computer.GENIE was implemented using C language and CUDA C language extension. It was developed on a Dell Precision T7500 Windows XP workstation that has 4 GB of main memory, a quad-core 2.13 GHz Intel Xeon processor, an NVIDIA Quadro FX1800 graphics card, and an NVIDIA Tesla C1060 graphics card with 4 GB of device memory. The NVIDIA C1060 card has 30 MPs and 240 cores. It has a CUDA compute capability of 1.3. We used CUDA toolkit version 3.1 and MS Visual C++ 2008 to compile the program under windows. To run GENIE, we used the NVIDIA Tesla C1060 card, whereas the Quadro FX1800 card was used only for display purposes.GENIE assumes the input genetic association study files contain case/control data with genotypes encoded under an additive model. Users can use the data converter tool that is a part of the GENIE software package to convert their data files to GENIE format. GENIE documentation includes detailed instructions and step-by-step examples that illustrate this process.The interaction between a pair of SNPs is tested using a logistic regression framework in which the SNP genotypes and their interaction terms are included as predictors. The model is expressed as the following:Y  is the disease status, and gj  is the genotype score that counts the number of minor alleles at SNP j. Interaction between the two SNPs is assessed by testing H0: β12 = 0 using a likelihood ratio test, which is asymptotically distributed as Chi-squared distribution with one degree of freedom. We note that this interaction test is implemented PLINK (http://pngu.mgh.harvard.edu/purcell/plink/) [where /plink/) , a populS SNPs each. This gives GENIE the ability to handle large datasets and maintain a small memory footprint in terms of output variables. We refer to S as the fragment size. As mentioned earlier, the interaction between a pair of SNPs is tested using a logistic regression framework. We briefly describe the algorithm below:We propose to parallelize the interaction analysis of different SNP pairs. Specifically, the GENIE algorithm partitions the input dataset into equally sized non-overlapping fragments with up to S be a non-zero integer.1) Read the genetic association study data file. Let fragment size S SNPs each. Let us assume that the fragments are numbered from 1 to F, where F is the total number of fragments. The partitioning of data is done in memory and no physical files need to be generated.2) Partition the data into non-overlapping fragments containing of i, where 1≤ i ≤ F:3) For each fragment S(S -1)/2 SNP pairs within fragment i in parallel.a. Test the interaction of j >i, test the interaction of S2 SNP pairs across fragment i and fragment j in parallel.b. For each of the other fragment P-value threshold.c. Output only those results that meet a prespecified S translates to the maximum x or y dimension of a thread block, which is 512 for cards with a compute capability of less than 2.0. As mentioned earlier, the size of a full warp is 32. We follow NVIDIA's performance recommendations [S must be a non-zero integer multiple of 32 that is less than or equal to 512.The fragment size ndations ,11 and rThe interaction analysis for each SNP pair is run on a separate GPU thread. The GPU kernel function of GENIE implements logistic regression. We modified PLINK's Newton's algorithm to fit the logistic regression model in (1). Specifically, we unrolled loops in the CUDA kernel. CUDA single precision floating point fast math functions were used in the CUDA kernel implementation. Due to the size of the input dataset, the current implementation of the GENIE algorithm makes use of local and global memory spaces. The input dataset as well as the intermediate output are stored in the global memory. We do not utilize any constant memory, shared memory, and texture memory spaces of the GPU device. We minimize the local memory utilization of the interaction analysis module/kernel to facilitate parallelization. Each GPU thread utilizes a maximum of 16 KB of the local memory. We use a single grid of two dimensional thread blocks. We enabled compiler code optimization. In order to reduce memory latency, we restrict the number of data transfers between main and device memories. Initially, GENIE reads the entire dataset into the main memory of the computer and then transfers it to the global memory space of the device memory. Intermediate results are periodically transferred from the global memory space of the device memory to the main memory of the workstation and then written to the output file. We also reduced the number of registers that are used by each kernel using the -maxrregcount compiler option in order to improve the CUDA utilization/occupancy.S, the last fragment has less than S total SNPs in it. In this case, we pad the rest of the fragment with missing values to maintain a fragment size of S.Due to CUDA local memory limitations of the NVIDIA Tesla C1060 card, we set the maximum number of individuals in a genetic association studies to 4,000, but this number can be increased for cards with larger local memory. When the total number of SNPs is not a multiple of The algorithm presented above can be easily extended to support multiple CPU cores. We implemented a simple serial single threaded version of the GENIE algorithm. We support multiple-cores using a script that first splits the input dataset into as many parts as there are cores. It then invokes a separate run of single threaded GENIE on each separate subset or part of the dataset.The current version of GENIE is a command line program and has a number of user input options. GENIE can be run in a GPU mode or in a CPU mode. The GENIE software package includes a GENIE executable, a data converter utility executable, and a script file. Multiple GPU cards, GPU clusters, multi-core CPUs, and CPU clusters are supported using the script file. This enables GENIE to handle datasets generated from larger genetic association studies in a scalable fashion. In the GPU mode, the interaction analysis is carried out using a CUDA compatible graphics card. If a machine has several GPU cards, GENIE allows users to choose a card that will be used to run the analysis. The script file can be used to distribute the analysis task across multiple GPU cards.In the CPU mode, the interaction analysis is carried out using a single processor core. If a machine has a multi-core CPU, the script file provided with GENIE package can be used to distribute the analysis task across multiple cores. Currently, the script distributes fragments among several separate parallel runs of a single threaded GENIE implementation for CPUs.th percentile for age and gender were considered as cases and subjects with HDL-C < 30th percentile for age and gender were considered as controls. 625 cases and 606 controls were genotyped using the IBC 50K SNP array [Data from the University of Pennsylvania High-Density Lipoprotein Cholesterol (HDL-C) Study was used to test GENIE. In this study, subjects of European ancestry with HDL-C > 90NP array . We extrWe tested GENIE on the same workstation that was used for its development. A fragment size of 256 was used during testing. Figure -9 involve rs1864163, a SNP in gene CETP that encodes for cholesteryl ester transfer protein. It is well known that CETP promotes the transfer of cholesteryl esters from HDL to low-density lipoprotein, and individuals that are genetically deficient for CETP often have extremely high HDL levels. Various genetics studies have shown that CETP SNPs are significantly associated with HDL-C level. In our analysis, we found that CETP may also interact with other genes in regulating the level of HDL-C.The GENIE output file contains 148,590 pairs that satisfy the MAF threshold of 0.15 and P-value threshold of 0.001. Table Recognizing the ongoing debate regarding using GPUs vs. CPUs for general purpose computing , we haveThe GENIE source code is optimized for the underlying GPU card and CPU. GENIE was implemented by following the NVIDIA CUDA C best programming practices . To makeS. Figure S can vary depending upon the graphics card that is used to run GENIE. The GPU computing field is growing rapidly. Faster and cheaper cards are being frequently released. NVIDIA recently released a faster CUDA based architecture called Fermi. The NVIDIA Tesla C2050 is an example of a Fermi based graphics card. A higher GPU mode speedup can be achieved by making use of the latest GPU computing capable graphics cards.The speedup achieved by the GPU mode of GENIE over its CPU implementation depends on the power of the graphics card and the fragment size The NVIDIA Tesla C1060 card that was provided by Dell with our workstation cost us $1,700, but it can be purchased online for a significantly cheaper price. The total cost of the workstation was $3,500. A CPU cluster with at least 27 processor cores is needed to match GPU-GENIE's performance. If each machine has a quad-core processor and costs $3,000, a CPU cluster based solution would cost at least $21,000. Thus, GPU computing offers an economical solution to computationally intensive problems.Future work will involve utilizing Message Passing Interface (MPI) to make GENIE's support of multiple graphics cards, GPU computing clusters, multi-core CPUs and CPU clusters more efficient. Since CUDA code can be easily ported to OpenCL using OpenCL translators such as Swan , we planGENIE is user friendly and is scalable and can handle large datasets as long as they fit the memory of the graphics card. As GPU cards with bigger memory capacities are entering the market at ever cheaper prices, it is now possible to analyze entire GWAS datasets economically with GENIE. An advantage of running GENIE in GPU mode is that a bulk of the computation is handled by the GPUs and the CPU is free to be utilized by other programs. Although GENIE was originally developed for GPU card, we have provided script that allows the users to run GENIE using CPUs. If the users do not have access to a GPU card, but have access to a CPU cluster, GENIE can still be easily used to perform interaction analysis.http://www.cceb.upenn.edu/~mli/software/GENIE/.GENIE is implemented using C and NVIDIA Common Unified Device Architecture (CUDA) C extension. The current version of GENIE is built for the windows operating system. GENIE needs to be recompiled to run on Unix and Linux. We recommend that users run GENIE on a CUDA compatible graphics card such as NVIDIA Tesla C1060 or better. NVIDIA recommends using a dedicated graphics card to perform GPU based computing, which means that all display related activities must be carried out using an additional graphics card. Documentation, source code, and precompiled binaries can be downloaded from The authors declare that they have no competing interests.SC, KW and ML designed the study. SC conducted the experiment and implemented the software package. SC and ML wrote the manuscript. All authors have read and approved the final manuscript.


The genetic parameters for growth, reproductive and maternal traits in a multibreed meat sheep population were estimated by applying the Average Information Restricted Maximum Likelihood method to an animal model. Data from a flock supported by the Programa de Melhoramento Genético de Caprinos e Ovinos de Corte (GENECOC) were used. The traits studied included birth weight (BW), weaning weight (WW), slaughter weight (SW), yearling weight (YW), weight gain from birth to weaning (GBW), weight gain from weaning to slaughter (GWS), weight gain from weaning to yearling (GWY), age at first lambing (AFL), lambing interval (LI), gestation length (GL), lambing date (LD - number of days between the start of breeding season and lambing), litter weight at birth (LWB) and litter weight at weaning (LWW). The direct heritabilities were 0.35, 0.81, 0.65, 0.49, 0.20, 0.15 and 0.39 for BW, WW, SW, YW, GBW, GWS and GWY, respectively, and 0.04, 0.06, 0.10, 0.05, 0.15 and 0.11 for AFL, LI, GL, LD, LWB and LWW, respectively. Positive genetic correlations were observed among body weights. In contrast, there was a negative genetic correlation between GBW and GWS (-0.49) and GBW and GWY (-0.56). Positive genetic correlations were observed between AFL and LI, LI and GL, and LWB and LWW. These results indicate a strong maternal influence in this herd and the presence of sufficient genetic variation to allow mass selection for growth traits. Additive effects were of little importance for reproductive traits, and other strategies are necessary to improve the performance of these animals. The Brazilian meat sheep industry has expanded in recent years mainly through an increase in the number of farmers. However, performance indexes are lower than those required to guarantee efficiency and competitiveness compared with other animal sectors. Correct selection is one of the most important strategies to maximize production in animal breeding. However, the lack of estimates for the genetic parameters necessary to predict genetic gains is commonly cited as an obstacle in the design and implementation of conservation-based selective breeding programs in the tropics. As a result, there are few reports on successful selective breeding programs in this region  and SarmAccording to The lack of systematic record keeping by Brazilian sheep breeders means that there have been few studies of the genetic parameters related to reproductive and maternal traits in the Brazilian flock. This lack of information greatly hinders adequate development of the meat sheep industry in which the reproductive and maternal efficiencies of ewes must be constantly evaluated in order to ensure profitable production. The aim of this study was therefore to estimate the additive genetic parameters for growth, reproductive and maternal traits in meat sheep of a multibreed population raised in Brazil.The data bank analyzed contained 12 years (1996-2007) of information from a flock maintained by Gaasa Agropecuária Ltda. and supported by the Programa de Melhoramento Genético de Caprinos e Ovinos de Corte (GENECOC) of Embrapa Caprinos e Ovinos. This herd is located at Inhumas in the state of Goiás  with a tropical semi-humid climate.®  at 50 and 80 days of age; all of the animals were vaccinated annually in March. Treatment for eimeriosis was done twice a year (March and October). Fecal egg counts and the Famacha method were used to control gastrointestinal nematodes. Footbaths  were used during the rainy season.The sheep underwent standard sanitary care and were vaccinated with Poli-StarThe breeding season was year round with ewes grouped by lots. The lambs were weaned at 60 days, confined, and fed with maize silage and corn and soy bran meal containing 21% crude protein (CP); they were slaughtered at 120-150 days. After weaning the lambs, the ewes underwent a 30-day breeding season in the presence of vasectomized rams. Ewes and rams were fed with Tifton 85 pasture, silage and meal containing 15% CP.To understand the flock formation, initially purebred Santa Inês, Poll Dorset, Hampshire Down, Suffolk, Île de France, Brazilian Somali and Texel rams were acquired with Santa Inês, Poll Dorset, Morada Nova, Brazilian Somali, Santa Inês x Morada Nova and Santa Inês x Brazilian Somali dams. Later, purebred Dorper, Primera and East Friesian rams were acquired. The matings were controlled but not designed. All rams of all breeds had the same opportunity to mate ewes of all genetic groups (purebred and crossbred). As a result, within a few years the flock was a mixture of crossbred dams with a varied contribution from the different breeds. Only the rams and some Santa Inês, Poll Dorset and Brazilian Somali dams were purebred.Data used in this study were obtained several years after formation of the flock, and already contained of the all breeds indicated above. However, many genetic groups were excluded from analysis because of insufficient information. The rams used included purebred Santa Inês (36), Poll Dorset (16), Hampshire Down (5), East Friesian (4), Dorper (4), Suffolk (3), Île de France (2), Brazilian Somali (2), Texel (1) and Primera (1). The dams used included purebred Santa Inês, Poll Dorset, Brazilian Somali and crossbreeds involving all of the breeds indicated above. The growth traits analyzed included birth weight (BW), weaning weight (WW), slaughter weight (SW), yearling weight (YW), weight gain from birth to weaning (GBW), weight gain from weaning to slaughter (GWS) and weight gain from weaning to yearling (GWY). The WW of lambs > 30 days old but weighing < 5 kg were excluded. Likewise, the SW of animals with a slaughter age > 365 days were also excluded. All weights from animals 330-395 days old were considered as YW. The YW of lambs > 20 kg were excluded.The reproductive and maternal traits analyzed included age at first lambing (AFL), lambing interval (LI), gestation length (GL), lambing date , litter weight at birth (LWB) and litter weight at weaning (LWW). Data for animals with an AFL > 800 days and an LI > 550 days were excluded. Lambing orders from 1 to 8 were analyzed.The MIXED procedure of SAS  was used to define fixed effects in the analysis. For each trait, many linear models were evaluated, with the use of contemporary groups or effects being analyzed individually. The logarithm of the Restricted Maximum Likelihood, Aikaikes's Information Criteria and Schwarz's Bayesian Information Criteria were the criteria for choice of the best fit.For growth traits, contemporary groups (CG) were formed by grouping animals born in the same year-season and with the same sex, genetic group and birth type . Only CGs with a minimum of five animals were considered. The birth seasons were: season 1 - January, February and March, season 2 - April, May and June, season 3 - July, August and September, and season 4 - October, November and December.For reproductive and maternal traits, two CGs were created: the first for AFL and the second for other traits. For AFL, the CG consisted of animals born in the same year-season, of the same genetic group, birth type  and interaction of birth type with lamb sex . For the other traits, the CGs consisted of animals of the same genetic group, with lambing in the same year-season, the same birth type  and the same lambing order. Only CGs with a minimum of five animals were considered.After preliminary analysis and based on criteria used to determine the best fit, the following fixed models were used, depending on the trait:BW - contemporary group and age class of the dam at lambing (1 to 8 years);WW and GBW - contemporary group, age class of the dam at lambing and lamb age (days) at weaning as linear and quadratic covariates;SW and GWS - contemporary group and animal age (days) at slaughter as the covariate (linear and quadratic);YW and GWY - contemporary group and animal age (days) on the date the animal was weighed as the covariate (linear and quadratic);AFL, LI, GL, LD, TBW and TWW - contemporary groups.-4, 2) a change in parameters of < 10-8 and 3) a gradient vector norm < 10-3.The (co)variances and genetic parameters were estimated by the Average Information Restricted Maximum Likelihood method (AI-REML) using the software WOMBAT , with siThe relationship matrix included 16,808 animals. Of these, 75 were rams and 4,272 were ewes, 13,124 animals had a complete pedigree (sire and dam known) and 1,251 were from the basal flock; 224 animals had an unknown sire. There was an average of 156 lambs per ram and 901 lambs per ram breed.The general model for growth traits was:Y is a (N x 1) vector of observations, β is the vector of fixed effects related to the incidence matrix X, a is the vector of direct genetic effects related to the incidence matrix Z1, m is the vector of maternal genetic effects related to the incidence matrix Z2, pe is the vector of permanent environmental maternal effects related to the incidence matrix Z3, and e is the vector of random residuals.where Z2m and Z3pe were not included in the analysis of SW, YW, GWS and GWY. BW, WW and SW were analyzed in a multiple trait model. YW was analyzed in a multiple trait model with BW and WW. GBW was analyzed with GWS in one analysis and with GWY in another analysis.The fixed effects have already been described previously. The general model presented above was used for BW, WW and GBW. The components Z2m component was not considered and the Z3pe component refers to permanent environmental effects on the animal and was included only for traits measured many times on the same animal . These two components were not included for AFL. AFL, LI and GL were analyzed in a multiple trait model, as were TBW and TWW. LD was analyzed in a single trait model.Using the same general model described previously for all reproductive and maternal traits, the i.e., there was no specific design in the breeding pattern used. As an initial simplification in the data analysis, the average effect (additive and non-additive) of the genetic groups was considered to be fixed.The crossbreeding in this population was completely random, The number of observations and the means for the traits analyzed are shown in The estimates of (co)variance, Log L value, heritabilities and genetic correlations for a multiple trait analysis for BW, WW and SW are summarized in vs. 0.35) but was lower for WW (0.60 vs. 0.81). The genetic correlations were high, with a maximum of 0.89 between BW and WW, as also observed in the analysis with BW, WW and SW. Comparison of Tables The (co)variances, Log L value, heritabilities and genetic correlations for multiple trait analysis with GBW and GWS are summarized in The (co)variances, Log L value, heritabilities and genetic correlation for the maternal traits LWB and LWW are shown in e.g., 0.12 and 0.21 for Poll Dorset sheep  of -0.47 and -0.24 between the direct and maternal effects for BW and WW, respectively, in purebred Santa Inês. These values were similar to those observed here, possibly because this breed provided the largest contribution to the flock in our study. am of -0.56 and -0.35 for BW in Hampshire Down and Poll Dorset sheep, respectively. am of 0.03 for Belgian Texel sheep and The (co)variance and correlation estimates between animal and maternal genetic effects were generally negative for all traits. There is a lack of information on the genetic parameters for YW in Brazilian flocks. The high correlations between BW and WW (0.83), BW and SW (0.70) and WW and SW (0.92) indicated the possibility of a correlated response with the selection for a given trait. The greater heritability of SW and the high correlation between WW and SW suggested that animals may be selected at an early age.Genetic correlations ranging from -0.33 to 0.81 have been reported between BW and WW in different breeds and at different ages  and maternal (0.06) heritabilities for GBW  were simvs. environmental interactions. In the pre-weaning period, the animals were maintained in confined conditions, whereas after weaning they had access to pasture and food supplementation.The negative correlations between GBW and GWS/GWY may reflect the limited number of observations for these parameters, partly because of the way the data were distributed. The animals in this flock were slaughtered between 120 and 150 days of age (128 on average), with only the males and females selected for reproduction reaching one year of age. Consequently, birth-to-weaning data were distributed into the weaning-to-slaughter and weaning-to-yearling groups. In addition, some lambs did not reach weaning. These negative correlations may also reflect differences in the animals responses to pre- and post-weaning environmental conditions, such as occurs in genotype AFL and LI were strongly influenced by environmental effects, whereas GL showed little inter-individual variation. Hence, the low heritabilities estimated for these traits were expected. A heritability of 0.27 for LI was estimated by According i.e., the ewes would be unable to express their full reproductive potential because the mating season was limited to a particular period. However, since in this flock the mating season was essentially continuous the year round, all of the ewes had an opportunity to mate at eight-month intervals.The similarity between the values for LD and LI suggests that these traits had essentially the same characteristics, possibly because of the reproductive management of the flock. The exposure of ewes to mating seasons meant that LI was probably not a good trait for selection because of possible bias. In this case, LD would probably be more efficient for selecting fertility because there would be no breeder-introduced bias, In conclusion, the results of this study show that there is strong maternal influence in this herd and the presence of sufficient genetic variation to allow mass selection for growth traits. Additive effects were of little importance for reproductive traits, and other strategies are necessary to improve the performance of these animals. However, the estimates presented here should be interpreted cautiously because of possible bias introduced by the absence of non-additive effects in the model used here. This model was chosen primarily because the lack of information on genetic parameters in Brazilian sheep precluded the choice of a more adequate one. Consequently, it is possible that the additive variance and heritability for some traits were overestimated. Nevertheless, the results described here provide a starting point for more detailed studies on the genetic parameters of multibreed sheep in Brazil.


MC1R locus, i.e. the extension locus, with alleles coding for black (ED), red (e), and wild-type (E+). In most mammals, these alleles are presumed to follow the dominance model of ED > E+ > e, although exceptions are found. In Bos indicus x Bos taurus F2 cattle, some EDE+ heterozygotes are discordant with the dominance series for MC1R and display various degrees of red pigmentation on an otherwise predicted black background. The objective of this study was to identify loci that modify black coat color in these individuals.In cattle, base color is assumed to depend on the enzymatic activity specified by the P < 0.05) and significant (P < 0.01) QTL on bovine chromosomes (BTA) 4 and 5, although these were not confirmed using single-marker association or Bayesian methods. Evidence of a major locus (F = 114.61) that affects reddening was detected between 60 and 73 Mb on BTA 6 (Btau4.0 build), and at 72 Mb by single-marker association and Bayesian methods. The posterior mean of the genetic variance for this region accounted for 43.75% of the genetic variation in reddening. This region coincided with a cluster of tyrosine kinase receptor genes . Fitting SNP haplotypes for a 1 Mb interval that contained all three genes and centered on KIT accounted for the majority of the variation attributed to this major locus, which suggests that one of these genes or associated regulatory elements, is responsible for the majority of variation in degree of reddening.Reddening was classified with a subjective scoring system. Interval analyses identified chromosome-wide suggestive (tyrosine kinase receptor genes implicated PDGFRA as the strongest positional candidate gene. A higher density marker panel and functional analyses will be required to validate the role of PDGFRA or other regulatory variants and their interaction with MC1R for the modification of black coat color in Bos indicus influenced cattle.Recombinants in a 5 Mb region surrounding the cluster of  Coat color phenotypes are used in breed recognition for cattle and other livestock species. Marketing strategies, such as the Certified Angus Beef program in the United States of America that only accepts cattle with at least 51% black coat color (and which exhibit Angus influence), have opened venues for price discrimination due to such qualifications. Availability of either premiums or discounts due to coat color phenotypes can have a direct impact on producers’ breeding strategies and an understanding of the genes and mechanisms involved in cattle pigmentation is desirable.MC1R gene, historically termed the extension locus, with alleles coding for black (ED), red (e), and wild-type (E+). These alleles were presumed to follow a dominance model, in which ED > E+ > e. The wild-type allele (E+) produces a functional receptor that responds to both the α-melanocyte-stimulating hormone (α-MSH) ligand and its antagonist, agouti-signaling protein (ASP). The ED allele, caused by a leucine to proline point mutation, creates a constitutively active receptor that results in eumelanin production most likely because of α-MSH ligand binding mimicry [e) leads to the production of only pheomelanin because of a frameshift mutation that creates a prematurely terminated, non-functional receptor [The melanocortin-1 receptor  is responsible for the primary switch of pheomelanin (red to yellow pigment) to eumelanin (black to brown pigment) in the melanogenesis pathway. In cattle, differences in base color are attributed to mutations in the receptor ,3.Bos indicus (Nellore) ½ Bos taurus (Angus) cattle with the EDE+ genotype at MC1R, which were expected to be black. This observation plus anecdotal evidence from other cattle crosses and mouse studies [2 Nellore-Angus cattle heterozygous (EDE+) for MC1R. We hypothesized that degree of reddening on a black background was a quantitative trait controlled by multiple genes of small effect.Various degrees of red pigmentation were observed in ½  studies  suggest 2 families produced by multiple ovulation and embryo transfer (ET) and founded by Nellore grandsires and Angus granddams. Blood samples were collected on all live-born animals in the population, and DNA was extracted from white blood cell pellets by proteinase K digestion. Parentage was assigned using a panel of 12 to 14 microsatellite markers. All procedures that involve animals were approved by the Texas A&M Institutional Care and Use Committee; AUP 2002–116, 2005–147 and 2008–234.Cattle used in this study were a subset of an experimental population, known as the Texas A&M McGregor Genomics Cycle 1 Population, located in central Texas  that consisted of 14 full-sibling FMC1R (RefSeq NM_174108.2) using the E3 and E4 primers designed by Klungland et al. [MC1R encompassing the SNP c.296C > T  and the deletion c.311delG .Calves were photographed at birth, steers were photographed in the feeding pens and females were photographed shortly after each calving. Because there was a wide spectrum of base colors observed in the population, all calves were genotyped for EDED homozygotes were black and E+E+ homozygotes ranged in color from yellow to black as expected. However, EDE+ heterozygotes often exhibited various degrees of reddening and were not black as predicted by the dominance series ED > E+ > e. Degree of black was classified from 1  to 9 (solid black) from photographs by three evaluators and a consensus score was determined .Comparison of resultant genotypes with photographs revealed that Bos indicus breeds and Bos indicus often darken during the fall and winter although this has not been documented experimentally. Some photographs were taken in the spring/summer whereas others were taken in the fall/winter, which may introduce bias because of lighting and seasonal effects. Because of this, dates of images, which were available for all photographs of the project, and season of photographs, were constructed as a two-level fixed effect to reduce bias. Photographs were classified as summer if taken from May to October and winter if taken from November to April due to the seasonal changes in Texas. To avoid further bias, each photograph was scored separately and independently. No comparisons were made between photos.Cattle influenced by MC1R of Bos indicus origin did not contribute to the reddening phenotype, the MC1-RF2 and MC1-RR1 primers from Graphodatskaya et al. [MC1R in a subset of the ET F2 and paternal half-siblings to the F2 animals (n = 31) produced by natural service that varied in their degree of black and their MC1R genotype. Straightbred Bos indicus and Bos taurus cattle and F1Bos indicus × Bos taurus cattle (n = 27) were also sequenced to further characterize the SNPs identified.To verify that a mutation in a et al.  were useEDE+ ET F2 cattle for this study.General linear model procedures of SAS®  were then used to evaluate fixed effects of sire, family-nested within sire, sex, year born or birth year-season combinations , and season photographed. The four bulls present in the dataset were excluded from subsequent analyses since they all ranged from 7 to 9 for degree of black and because color descriptions of bulls have shown that they can be darker than steers , which mP > 0.25), and were excluded from the final model. No interactions of sex, sire, and family-nested within sire were detected. Residuals from the final model, which included sire, family-nested within sire, sex and season photographed were used for interval and association analyses and full models were used to run a Bayesian association analysis. Genotypes for these analyses were produced using the BovSNP50v1 assay  and SNP coordinates from build Btau4.0 of the bovine genome sequence. Data were filtered to remove genotypes for SNPs with a minor allele frequency less than 0.05, SNPs with completion rates below 90%, and SNPs that deviated from Hardy-Weinberg equilibrium (P < 0.0001). Initially, a set of 7112 markers informative in every family was used for interval analyses by applying the full sibling additive and dominance models of GRIDQTL software [Neither birth year-season nor year of birth, when evaluated independently of each other, contributed to variation in degree of black (software  for 39 4w) for each window [th percentile  were said to be associated.Both interval and single-marker association analyses are known to not estimate the size of QTL effects precisely ,10, therg BayesC ,13 proceo BayesB  procedurh window ,16. Windhttp://www.ncbi.nlm.nih.gov). Genes within 5 Mb surrounding the QTL position were investigated based on available information in the NCBI database and reported results of other experiments. Functional homologs in mouse were also investigated.Coordinates of QTL were used to identify candidate genes concordant with the QTL region based on the Btau4.0 sequence assembly and genome maps at the National Center for Biotechnology Information  and c.663 T > C (ss784304485) of MC1R. These were the only additional coding variants that we identified in this intron-less gene and they were in complete phase with the Bos indicus E+ allele (c.296C > T). Detection of this single haplotype (TTC) in Bos indicus straightbred and crossbred cattle (n = 58) suggested that nucleotide variability in MC1R derived from Bos indicus did not contribute to variation in degree of black in EDE+ heterozygotes. Furthermore, we observed long-range haplotypes (due to linkage) that extended for more than 1 Mb either side of MC1R. All EDE+ heterozygotes were imputed from the haplotypes of their parents to be heterozygous for the CCT/TTC MC1R haplotypes. Within families, heterozygotes that shared the same pair of long-range haplotypes varied widely in their degree of black coat color; therefore, we think it is unlikely that a regulatory variant associated with MC1R could explain the observed reddening effect.Two coding SNPs of Bos taurus chromosome 6) at an average location of 71 Mb (95% CI from 60 to 73 Mb) by interval analysis (Table P < 0.05) and significant (P < 0.01) QTL were also identified on BTA4 and 5, respectively. When 10 of the 29 bovine chromosomes generated runtime errors due to long strings of heterozygous SNPs, single-marker association analysis was run and detected the major locus on BTA6 (Bonferroni corrected P = 3.88e-23), but did not confirm the QTL on BTA4 or 5. Bayesian association analysis also confirmed the major locus on BTA6 (but not those on BTA4 or 5) with PPAw = 1.00 , v-kit Hardy-Zuckerman 4 feline sarcoma viral oncogene homolog (KIT), and kinase insert domain receptor (KDR). Mutations in KIT cause pigmentation defects and patterns in several species  was fit as a fixed effect in the model. Although this effect was significant (P = 0.0025), the QTL for degree of black on BTA6 remained strongly significant. Second, the analysis was repeated using the subset of non-spotted animals (n = 167), and the QTL on BTA6 was detected as before. Both of these analyses confirm that degree of black and spotting are the result of separate loci and that these loci are probably tightly linked.The position of the major locus on BTA6 coincided with a cluster of tyrosine kinase receptor genes including i et al. ), and QTn cattle -20. The KIT and coat color, breed-of-origin of a 1 Mb region that spans these three genes and centered on KIT was assigned for 204 of the 208 F2EDE+ Nellore-Angus cattle. Breed-of-origin of this region and an interaction of breed-of-origin with sire was significant (P < 0.05). For all sires, NN homozygotes for the KIT region differed (P < 0.05) from AA homozygotes for degree of black, and red pigmentation appeared to be recessive and was associated with the Nellore allele. For three of the sires, the NN homozygotes had less black pigmentation as compared with any alternate breed-of-origin combination , whereas the remaining sire had NN homozygotes different from AA homozygotes, but not AN or NA genotypes. Fitting breed-of-origin of the KIT region into the original model accounted for the variation attributed to the major locus on BTA6  to be present in both corin serine peptidase (CORIN) is located 3.7 Mb upstream of PDGFRA and is associated with lighter coat colors in mice [KIT, KDR and CORIN as candidates. Although we cannot rule out possible effects of long-range regulatory elements associated with these genes, based on its location within the refined interval, PDGFRA appears to be the strongest positional candidate for the reddening phenotype.Because long-range haplotypes are recovered in this population due to linkage, we considered candidate genes up to 5 Mb from the leading SNP. The gene  in mice . HaplotyPDGFRA, which is a member of the platelet-derived growth factors (PDGF) family that consists of two receptors (α and β) and four ligands (A-D). Although we are not aware of any previously documented role for PDGFRA in cattle coat color, its function and cellular responses have been extensively studied in the developing mouse. To function as a tyrosine kinase receptor, the PDGF-receptor must first bind a ligand, then dimerize with another ligand-bound PDGF-receptor . SignalPDGFRA has been shown to have a role in mouse hair follicle development [PDGFRA and KIT is difficult. Based on its functions described in mouse  gene and influence Ca2+ levels, thereby directly impacting melanin synthesis.In relation to pigmentation, elopment  and posselopment  although Soriano ), PDGFRAogenesis . FurtherKIT drives melanocyte migration from the neural crest along the dorsolateral pathway. Mutations associated with KIT result in white coat color or white spotting in numerous species including cattle [KIT locus [During melanogenesis, g cattle . In 2012IT locus . AlthougPDGFRA is the gene CORIN that encodes a mosaic protein expressed primarily in the heart and necessary for pro-atrial natriuretic peptide (pro-ANP) activation. Enshell-Seijffers et al. [CORIN expression in the dermal papilla of the hair follicle and found that mice homozygous for the CORIN mutation (C–/C–) had lighter coat colors than mice with at least one functional CORIN. This observation depended on the genotype at the Agouti signaling protein (ASIP) gene, where at least one wild type or functional ASIP allele was necessary for homozygous CORIN mutants to express the altered pheomelanin production. The CORIN mutation was associated with an increase in the length of the basal band, which contains pheomelanin. Because of this, a larger proportion of the hair shaft contained pheomelanin and therefore produced a lighter coat color than wild types. A variant agouti protein was observed in only one family in our population (data not shown), which ruled out a similar interaction in cattle as the reason for the large effect observed.Located approximately 3.7 Mb upsteam of s et al.  charactesh in mice contains a genetic inversion that affects a regulatory element for KIT, downstream of PDGFRA in the 3’ breakpoint and disrupts CORIN expression due to the 5’ breakpoint that occurs between the fifth and sixth exons. Structural variation or other regulatory elements in the CORIN-PDGFRA interval cannot be ruled out based on these data and will be explored using higher density assays in the future.Nigrovic et al.  reportedtyrosine kinase receptor genes including PDGFRA, KIT, and KDR. Fitting breed-of-origin of the 1 Mb region that contains these three genes in the model accounted for the majority of variation attributed to the major locus. Therefore, one of these three coding genes, or an associated structural or regulatory variant, is probably responsible for the majority of the variation in the reddening phenotype. Recombinants within the F2EDE+ Nellore-Angus cattle identified PDGFRA as the strongest positional candidate gene for the reddening phenotype using the current marker panel, but additional positional candidates are present . Further research using higher density markers and functional analyses will be required to confirm the role of PDGFRA or other genes, and structural or regulatory variants, and their interaction with MC1R for the modification of black coat color in these cattle influenced by Bos indicus.Suggestive and significant QTL at the chromosome-wide level were identified on BTA4 and 5 using interval analysis for degree of black, but these QTL were not confirmed by single-marker association or Bayesian methods. The major locus at 72 Mb on BTA6 accounted for 43.75% of genetic variation using BayesB procedures with The authors declare that they have no competing interests.JOS, DGR, and CAG contributed to the conception and design of the study. JOS, DGR, CAG, and LLHH contributed to the statistical design and interpretation of the results. JOS and LLHH collected and formatted photographs for phenotype scoring, and LLHH, JOS, and DGR scored the phenotype. CAG contributed to data formatting and interpretation of results from QTL analyses and use of breed-of-origin data. CAA performed parentage testing, contributed to sequencing and interpretation of results. LLHH contributed to QTL analyses, breed-of-origin genotypes, and interpretation of results. All authors contributed to manuscript drafts and revision. All authors read and approved the final manuscript.


Two multivariate beta and multivariate beta-binomial processes are examined, and new distributions are introduced: the GMS-Sarmanov multivariate beta and its beta-binomial counterpart.Systems involving many variables are important in population and quantitative genetics, for example, in multi-trait prediction of breeding values and in exploration of multi-locus associations. We studied departures of the joint distribution of sets of genetic variables from independence. New measures of association based on notions of statistical distance between distributions are presented. These are more general than correlations, which are pairwise measures, and lack a clear interpretation beyond the bivariate normal distribution. Our measures are based on logarithmic (Kullback-Leibler) and on relative ‘distances’ between distributions. Indexes of association are developed and illustrated for quantitative genetics settings in which the joint distribution of the variables is either multivariate normal or multivariate- The analysis of systems involving several variables is of great importance in population and quantitative genetics, for example, in multi-trait prediction of breeding values or in the study of networks using multi-locus data. The problem addressed here is that of measuring departures of the joint distribution of sets of genetic variables from stochastic independence. The target variables could be, for instance, pleiotropic additive effects of many loci under the infinitesimal model of t-distribution with a diagonal covariance matrix in which variables are uncorrelated and, yet, stochastically dependent. This illustrates that lack of correlation does not provide definitive evidence of independence: in this example, variable X, say, informs about Y even though uncorrelated. A related question is the inability of correlation to measure association between, say, 10 loci or 20 pleiotropic effects. The measures of association developed in this study are based on departure between the distributions of the variables under independence and association assumptions. Indexes measuring association between random variables, irrespective of their number or of the form of the joint distribution, are suggested and illustrated.The objective of this work is to introduce new measures of association involving a system of variables based on notions of statistical distance between distributions. The resulting metrics are more general than correlations, which are pairwise measures only and lack a parametric interpretation beyond the bivariate normal distribution e.g., , especiat. The fourth section shows how these indexes can be used to study LD in a two-locus system with multiple alleles; here, a correlation would not have much meaning, if at all. The fifth section presents applications to systems of correlated beta distributions; all these are generalizations of the univariate beta distribution, which has been used extensively in population genetics to study evolution of gene frequencies. In particular, two multivariate beta and multivariate beta-binomial processes are discussed, and new distributions are introduced: the GMS-Sarmanov multivariate beta and its beta-binomial counterpart. The paper concludes with a discussion of the concepts and of other procedures that have been proposed for study of multi-locus LD. Mathematical details are relegated to an appendix.The article is organized as follows. Following this introduction, measures based on logarithmic and on relative ‘distances’ between distributions, as well as indexes of association, are presented. Subsequently, these indexes are developed and illustrated for quantitative genetics settings in which the joint distribution of the variables under consideration is either multivariate normal or multivariate-X, Y of random variables with realized values x, y; the ideas generalize readily to distributions of higher dimension simply by replacing scalars by vectors. Let p and p(x) × p(y) be the densities of the joint distributions under association and independence, respectively; p(x) and p(y) are the marginal densities of X and Y respectively. Association occurs whenever p(y|x) (or p(x|y)) differs from p(y) (or p(x)). Measures of dependence (association) can be derived from the density ratioTo motivate the approach used here, consider a pair αxy = 1 for all pairs of values, the ‘distance’ between distributions is 0 and independence holds. Values of αxy larger or smaller than 1 suggest statistical dependence. For example, αxy larger than 1 indicates ‘coupling’, i.e. the density of certain pairs of values is higher than expected under independence; values smaller than 1 would be indicative of ‘repulsion’. Likewise, strong departures of log αxy from 0 would be indicative of an association between x and y.If αxy and of αxy respectively. These measures are used subsequently to develop indexes of association, irrespective of the number of variables in a system.We discuss two measures of association based on either expected logarithmic distance between distributions or on expected relative distance, as conveyed by the average values of log X and Y are associated, andEp and Ep(x)p(y) denote expectations taken under the assumptions that  follow either a bivariate distribution (association) or that these variables are independently distributed.The Kullback-Leibler (KL) logarithmic distance  is definαxy, with the densities evaluated at the corresponding ML estimates would be the log of a ratio between maximized likelihoods. In DAI one would be taking expectations of the log-likelihood ratio over all possible realizations expected if the ‘full model’ (association) were true; on the other hand, in DIA the expectations of the reciprocal of the log-likelihood would be taken under the assumption of independence. In this context, DAI and DIA are averages of log-likelihood ratios, as opposed to an evaluation at a single realization (in ML estimation parameters vary whereas observations are taken as fixed). Note that DAI and DIA do not involve any requirement of ‘nesting’ of models, contrary to likelihood ratio tests under regularity conditions, but this is a different issue.A connection with the likelihood ratio statistic is fairly direct. Suppose that the parameters of the distributions under independence and association are estimated by maximum likelihood (ML), and that these estimates are regarded as if they were ‘true’ values. Then, log DAI ≠ DIA, and a unique, symmetric KL distance is defined (The discrepancies are such that  defined  as(2)DIA asDIA relative to DAI (or the opposite) is indicative of situations in which there is more association than what would be expected under independence so that θ or 1 − θ would be close to 1. Thus, values of θ or of 1 − θ away from x,y) pairs vs. associated ensembles.Then, an index of association  is obtained by normalizing DIA or DAI for many distributions, in which case these metrics must be calculated numerically or estimated using Monte Carlo methods, given some parameter values. If association is weak, the two logarithmic discrepancies and KL are nearly null, and this can lead to very unstable, even absurd, Monte Carlo estimates of θ. Also, the KL distance and the two discrepancies are logarithmic, and this perhaps provides a less intuitive notion of ‘distance’ than one based directly on αxy. An alternative ‘distance’ measure is presented immediately below.There is no closed form for Define the relative discrepanciesαxy under association and θ′ away from The metric In what follows, the proposed indexes are examined for some distributions that are used often in quantitative and population genetics research.K traits. Suppose that a matrix of estimates of additive genetic correlations is available (and taken as a ‘true’ matrix), and that we seek to arrive at a single measure of association among the K genetic values, as opposed to the K = 2, 3, 4 and of a bivariate t-distribution are considered to illustrate how θ behaves.In quantitative genetics, the analysis often focuses on a set of continuous correlated variables X1 and X2 be two standardized normally distributed genetic values with correlation ρ. For this distribution ∞. Thus, KL = ρ2(1 − ρ2)−1, which is 0 when the two random variables are independent and ∞ under a perfect correlation. Thenθ respectively.Let nd  e.g. . When thθ and of 1 − θ are plotted against ρ in θ) away from the independence models when this is true, and the dashed line (‘spills water’) depicts the trajectory of 1 − θ, i.e. the relative discrepancy under association; note that a given curve is a mirror image of the other. As the absolute value of the correlation increases θ → 1, as one would expect. Since an association index with values ranging between 0 and 1 may be easier to interpret, an alternative measure could beThe values of γ takes values in  provided that γ and ρ for this Gaussian model is also shown in θ than when assessed with the correlation ρ. For example, if θ is smaller than 0.15. In multivariate systems or in other distributions, θ can be much smaller than γ takes negative values, as illustrated below, so it turns out that θ often provides an easier to interpret measure of association than does γ.However, K-dimensional multivariate Gaussian distribution , where variates are in standard deviation units so that R is a correlation matrix; under independence . It can be shown tr(.) denotes the sum of the diagonal elements of a squared matrix such as R. Further,Where Examples of several multivariate normal distributions are given next.ρ, so thatLet three variables be equi-correlated with correlation θ < 1. For example, for ρ = 0.25, 0.50 and 0.75, the index of association θ takes values 0.49, 0.54 and 0.66 respectively. θ, 1 − θ and θ as a function of the coefficient of correlation. At ρ = 0, θ = 0.5, but then it decreases slightly as the correlation increases such that γ does not attain a positive value until γ reaches a value of about 0.3. Note that θ is not symmetric with respect to ρ (the same is true of |R|); for instance, when θ = 0.59 so that association would be viewed as stronger than when θ = 0.49 then. This is a consequence of the values that the discrepancies DIA and DAI take at varying values of ρ.With 0 < θ, derived from notions of statistical distance, does not have this limitation because KL discrepancies have a general meaning and can be calculated for any distribution, at least numerically, given values of the parameters. Naturally, any single measure of association in a multivariate distribution becomes less transparent as the system of variables increases in dimension.Alternative measures can be derived from eigenvalues of a correlation matrix; however, this is sensible only for distributions in which linear relationships between variables are meaningful. The index K = 4 and suppose the variables are equi-correlated with coefficient ρ. Then, Let θ and 1 − θ vary against ρ. At θ = 0 one has θ decreases somewhat subsequently and then increases, attaining 0.5 again at about ρ = 0.45. The values of the index of association suggest that the independence and association models are not ‘too distant’ unless the correlation is below −0.10, if negative, or above ρ = 0.50, if positive.K = 4 and take as correlation matrixτ ≤ 1, so that when t = 0 the four variables are independent. For t = 1, log |R| = −2.5459 and tr(R−1) = 24.8980, yielding θ = 0.8782. Likewise, for R| = −0.2960 and tr(R−1) = 4.6540, so that θ = 0.5473. This illustrates that the strength of the association is proportional to the strength of inter-correlation, and that θ provides a metric for measuring departure from independence.Let now θ say that is not informed by all possible pairwise correlations? Again, let K = 4 and take as correlation matricesWe finish this section by posing the following question: what does The salient point is that the correlation structure is such that while pairs of variables ,  and  are correlated,  are independent.θ1 = 0.54 and θ2 = 0.91 for the networks of variables characterized by θ gives an indication of the overall degree of association.What is the overall strength of the association in the system? Calculations yield X does not inform about Y. This highlights limitations of the correlation parameter as a metric for statistical dependence between sets of variables. One such case is the multivariate-t distribution, in which variables can be statistically dependent, yet uncorrelated. The univariate and multivariate t-distributions appear in robust linear regression models for quantitative genetic analysis based on pedigrees . As shown in D′AI, D′IA and θ′, the ‘distance’ D′AI from association to independence decreases as ν increases, approaching 1, whereas the distance from independence to association D′IA increases, approaching 1 as well. With ν = 100 000 the random variables are essentially independent since γ′ = 0, the setting ν = 1 produces the strongest possible association, with θ′ = 0 and both 1 − θ′ and θ′ equal to 1.The index of association θ′ in C and R alleles at locus A and locus B, respectively, such that data on gametic types can be arranged into an R×C contingency table. Let pij be the probability of observing a haplotype with configuration ‘ij’. If n gametes are screened and the observed number having such configuration is nij, the assumption of multinomial sampling leads toi’ and ‘j’at loci A and B respectively. Define parameters Dij = pij − pi.pj., measuring departure from independence . In a two-locus model a single D parameter is needed, and by using pij = D + pi.pj., then αxy would be the ratio between two likelihoods: the ‘unrestricted’ likelihood  and the ‘restricted’ likelihood . The asymptotic connection between the standard log-likelihood ratio statistic and the χ2 metrics used for multi-locus analysis of LD by Suppose there are Under this sampling scheme, the discrepancies becomeHenceθ is not defined when all Dij = 0 because then the two distributions would be identical and the KL distance is 0 in that case. Note that DAI is the expected value of the log-likelihood ratio, but under the assumption of association; on the other hand, when the sampling distribution of the log-likelihood statistic is considered, the reference distribution is the null model, i.e., linkage equilibrium. This illustrates an important conceptual difference, apart from the fact that DIA and DAI involve integration over the sampling space of allelic outcomes at the two loci in question.Naturally, HLA-A and HLA-B respectively; the data, after a modification explained subsequently, are shown in To illustrate, consider a hypothetical example, patterned after data in pij as true values, the discrepancy of the independence model away from association isTaking haplotype frequencies B locus to the 0.2928 in DAI above are (after rounding) 22.71%, 61.77%, 13.76% and 1.76% for B7, B8, B35 and B44 respectively, implying that associations with allele B8 are responsible for most of the departure of the independence model away from association, assuming this last one is ‘true’; note that the nij do not enter into the calculations, since the frequencies are viewed as true ones. For example, the relative contribution of B8 to DAI is arrived at as follows:B-locus alleles to DIA are 16.80%, 53.93%, 12.11% and 17.16% for B7, B8, B35 and B44 respectively, with B8 playing a major role again. Similar calculations can be done for the A locus. The KL distance is 31.6235 + 49.5987 = 81.2222, and the KL distance per haplotype scored (N = 108) is 81.2222/108 = 0.7521. The indexes of association are θ ≈ 0.39.The relative contributions of each of the alleles at the R×C table of alleles, a correlation has little meaning and is not invariant with respect to how alleles are scored  at each of the intervening loci. On the other hand, the chi-square statistics of θ is also score invariant, is independent with respect to the order of rows and columns, and applies to any distribution. For instance, if the simple multinomial sampling model is violated due to, e.g. over-dispersion, an appropriate θ could be developed under such model, whereas a ‘normalized’ χ2 would provide a crude, albeit probably useful, approximation.In an The ‘relative distance’ metric takes the formD′AI, D′IA and θ′ = D′IA/(D′IA + D′AI) using Monte Carlo sampling. In the calculation of D′AI one million random trials of size 108 each, with {nij} varying at random, were sampled from a multinomial distribution with probabilities pij; then, pi.pj. and the realizations of θ′ was estimated using the mean and median of the draws of θ′, yielding 0.9953 and 0.9998. The association (disequilibrium) between alleles at the A and B loci is patent, both when logarithmic (θ) and relative (θ′) distances are employed to measure it.We evaluated F-statistics  and Y: Beta respectively. Clearly, X and Y are positively correlated (through W), with a strength that depends on the values of a, b, c. X and Y have a bivariate beta distribution with density functionFirst, we review how a specific bivariate beta distribution arises. As in X < 1 and 0 < Y < 1. Moments E(XkYl) cannot be written in closed form but can be approximated numerically or using sampling methods. Large c and small a, b produce correlations close to 0, whereas large a, b or small c produce correlations close to 1  and p(y) are the beta densities of X and Y: Beta. ThenWhere D′AI, D′IA and KL′ can be calculated by taking expectations of density ratios as opposed to expectations of their logarithms.The corresponding D′AI, D′IA and KL′ were calculated for the following combinations of parameters, where ρ denotes the expected correlation: 1) a = 0.01, b = 0.01, c = 5 (ρ = 0.002); 2) a = 1, b = 0.5, c = 0.6 (ρ = 0.251), 3) a = 1, b = 1, c = 0.9 (ρ = 0.496) and 4) a = 2, b = 3, c = 0.4 (ρ = 0.750). Expectations under the independence model were approximated by drawing 1 million random numbers from independent X ∼ Beta and Y ∼ Beta distributions, and then averaging the evaluations of the appropriate expressions over the draws. In the association model, 1 million random triplets  were drawn from independent standard gamma variables with parameters a, b and c, to then form bivariate realizations as ρXY) are Monte Carlo estimates. The indexes of association θ′ and 1 − θ′ departed from 0.5 and approached 0 and 1, respectively, as the correlation in the bivariate beta distribution became stronger. Likewise, γ′ and γ′′ drifted away from 0, approaching |1|. In settings θ′ and 1 − θ′ suggest a stronger association than what would be indicated by ρ.Indexes of association based on l with allelic frequencies Pr(Al) = pl and Pr = 1 − pl; Al and al denote the two alleles at locus l. Suppose that a sample of N individuals is scored, and that the observed number of copies of Al and al are pl, the distribution of Binomial. Now, let the allelic frequency pl vary at random among clusters (e.g. sub-populations) according to a beta distribution with parameters . Then, the marginal distribution of the observed number of alleles is beta-binomial , where θ denotes its parameters, the density of the marginal distribution of the observed number of allelic counts under this association model would beSuppose now that this pair is a realization from a stochastic process where loci are sampled over a conceptual population of pairs, e.g. the two loci are drawn at random from a physical ‘bin’ of a certain length in kilobases. This process generates a covariance structure between allelic frequencies of pairs of loci within the ‘bin’. The alleles would be in LD, marginally, even though being in conditional  equilibrium. If the unobserved, ‘true’, allelic frequencies follow some bivariate distribution with density θ =  is assumed, the joint density of allelic frequencies p1 and p2 as in If a bivariate Olkin-Liu beta distribution with parameters This integral cannot be written in closed form, but it can be evaluated using Monte Carlo methods such as importance sampling e.g. . One posUnder this distributionpI(DATA) is the distribution under independence between observed allele numbers, and given by the product of two beta-binomial distributions, that isWhere N) as well, and the third one depends on observed allelic counts.ThenThe logarithmic and relative distances between distributions are calculated asθ and θ′ calculated as before. Calculations proceed on an entirely numerical basis.With K loci the joint density of the allelic frequencies takes the formA generalization of the beta process to a situation with more than two loci is in The marginal distribution of allelic counts is obtained by mixing a multinomial distribution over the multivariate beta density above, but results are not available in closed form, so a numerical solution is needed again.p1 and p2 has densityLee (1996) proposed a different bivariate beta distribution following B1(.) and B2(.) are beta densities as before and w is a parameter (w = 0 produces independence); θ* = . Above, p1 and p2 respectively. The marginal densities of p1 and p2 under this model are B1(.) and B2(.). Further,Where Eg denotes expectation under association andIn the Lee-Sarmanov bivariate beta distribution, a null correlation implies independence. Under this model the density ratio becomesθ′ was evaluated for three sets of values of the a,b parameters for each of the two intervening beta distributions while setting the correlation at D′AI in D′IA was estimated using 1 million random samples from each of the two beta distributions. For each set of a,b parameters w was calibrated viaThe index D′AI is solely a function of the correlation, so it was the same for the three pairs of beta distributions examined. Using D′IA was estimated by averaging θ′ and 1 − θ′ also drifted away monotonically from θ′ increased from 0.50 to 0.66 as the correlation increased from 0 to Beta distributions. This is shown in θ′. However, when the correlation increases by a factor of 3 from θ′ and 1 − θ′ move away from θ has an upper bound at 1, whereas KL takes any positive value in the real line. The bimodal shape of the density also suggests that more intensive sampling is needed to obtain reliable estimates of the indexes of association.Note that As shown in Using the fact that under independence the joint distribution is product beta-binomial as in A alleles at the two loci is  to name The problem addressed in this study was that of measuring departures of the joint distribution of genetic variables from independence. New measures of association based on notions of statistical distance were proposed and evaluated under several scenarios, spanning from multivariate Gaussian distributions modeling, say, pleiotropic effects, to systems of beta distributions describing association between allelic frequencies. Two hereto seemingly unreported probability distributions were derived, and termed GMS-Sarmanov multivariate beta and GMS-Sarmanov multivariate beta-binomial distributions. The standard LD problem was also dealt with to illustrate the generality of the approaches proposed.Y explained when X is observed) in a bivariate normal distribution, it is arguably less so when applied to discrete data, a problem that is well known in quantitative genetic analysis of discrete phenotypes  of the invariant KL distance, and express it relative to the maximum value it can take. Using their procedures with DIA would produce a different value of association, and maybe a different qualitative picture might emerge from analysis of genetic data. However, their ideas can be embedded in our approach, and expressing association relative to a maximum distance is a well taken point. It also turns out that these results can also be adapted to the continuous domain, with some care. To illustrate this, it suffices to consider two random vectors, x and y, as generalization to a higher-dimensional system is straightforward. Let H(x), H(y), and H be the entropies  of the marginal distributions of x, y, and x,y respectively, withCloser to the spirit of our procedures, The discrepancy away from the association model isNote thatH(y|x) is the entropy of the conditional distribution of y given x. This implies that H ≧ H(x), where x can also be any partition of x, e.g. its ith coordinate xi, say, and similarly H ≧ H(y). Thus, H ≧ maxiH(xi). Using this in Where z with n elements, the KL discrepancy away from the independence model isMore generally, for a vector DAI above, H(x) + H(y) ≧ maxi, so thatLikewisex: N1 and y: N2. One obtainsThis leads to additional indexes of association, such asρ, θ, θ** and q** produce a different value of the strength of association between random variables in a distribution. This suggests that probably there is no such thing as a universal measure, although it is clear that the coefficient of correlation lacks generality.The four indexes, In a nutshell, this paper presents measures of association for systems of genetic variables that go beyond standard two-dimensional statistics. The procedures apply to either continuous or discrete data, and typically require numerical implementation, because many of the expressions are not available in closed form, depending on the distribution assumed. Our procedures (like any other method) require knowledge of the parameters of the distributions under independence or association, and estimation of such parameters is not the objective of this paper. Today, computer-intensive approaches for parameter inference, such as Bayesian Markov chain Monte Carlo (Finally, since this volume is in honor of the contributions of Professor Moshe Soller, a relationship between this paper and his work should be established. As noted at the onset of this document, among many other accomplishments, he pioneered marker-assisted selection in animal and plants via exploitation of linkage and LD relationships between markers and unknown quantitative trait loci. Examples of his papers in this area include