Genome-wide Two-marker linkage disequilibrium mapping of quantitative trait loci
© Yang et al.; licensee BioMed Central Ltd. 2014
Received: 28 August 2013
Accepted: 31 January 2014
Published: 8 February 2014
In a natural population, the alleles of multiple tightly linked loci on the same chromosome co-segregate and are passed non-randomly from generation to generation. Capitalizing on this phenomenon, a group of mapping methods, commonly referred to as the linkage disequilibrium-based mapping (LD mapping), have been developed recently for detecting genetic associations. However, most current LD mapping methods mainly employed single-marker analysis, overlooking the rich information contained within adjacent linked loci.
We extend the single-marker LD mapping to include two linked loci and explicitly incorporate their LD information into genetic mapping models (tmLD). We establish the theoretical foundations for the tmLD mapping method and also provide a thorough examination of its statistical properties. Our simulation studies demonstrate that the tmLD mapping method significantly improves the detection power of association compared to the single-marker based and also haplotype based mapping methods. The practical usage and properties of the tmLD mapping method were further elucidated through the analysis of a large-scale dental caries GWAS data set. It shows that the tmLD mapping method can identify significant SNPs that are missed by the traditional single-marker association analysis and haplotype based mapping method. An R package for our proposed method has been developed and is freely available.
The proposed tmLD mapping method is more powerful than single marker mapping generally used in GWAS data analysis. We recommend the usage of this improved method over the traditional single marker association analysis.
KeywordsGenetic mapping Linkage disequilibrium mapping Linked loci Genome wide association study
Most economically, biologically and clinically important traits, such as those linked to poplar growth, cancer development and dental caries risk, are inherently complex in terms of their polygenic control and sensitivity to the environment . The number of genes involved in these traits is typically large, each exerting a small effect and acting singly or interactively with others in a complicated network. For this reason, the genetic analysis of complex traits has been very difficult. However, a profound understanding of the genetic control mechanisms of complex traits is crucial to economy and life. Therefore, the development of more powerful and complex genetic mapping methods has become increasingly urgent.
In recent years, with the advancement of new DNA-based biotechnologies, such as single-nucleotide polymorphism (SNP) arrays, genome-wide association studies (GWAS) have become feasible to dissect the phenotypic variation of a complex trait into individual genetic components. Particularly, SNP arrays have gained popularity due to their cost-effectiveness: in year 2011 alone, 1068 GWAS were performed, each with at least 100,000 SNPs genotyped (http://www.genome.gov/gwastudies). Based on the most recent summary data of dbSNP database (http://www.ncbi.nlm.nih.gov/projects/SNP), there are ~ $38 million (about 1 percent of the total genome) of validated SNPs in human genome. However, even the densest SNP array on the market can only accommodate ~1 million SNPs, and hence a great percentage of SNPs is not able to be sampled in a real genetic study. Fortunately, SNPs in the genome are not independent from each other, i.e. they are locally connected and form the so-called linkage disequilibrium (LD) blocks. Because of this unique correlation structure, the sampled genetic markers carry partial information about the unsampled SNPs and may be used for genomewide association analyses.
where D(t+1) is the LD value at generation t + 1 and r is the recombination rate between the two loci. Therefore, the LD value approaches to zero gradually at a geometric rate of 1-r. The larger the r, the faster the rate of convergence. According to Equation (), if a significant D(t+1) value can be detected in the current generation, it implies r must be very small, almost close to 0, under the assumption that the initial LD was generated long time ago (i.e. t is large). This assumption is plausible because it does take a long time for mutations/LD to be spread in a population. Therefore, the principle of linkage disequilibrium decaying with generation builds up an alternative mapping strategy [10, 11], which provides an important tool for the fine mapping of genes affecting a quantitative trait.
The LD mapping based on a single marker has been greatly studied [12–14]. However, little effort has been put on the LD mapping with multiple markers. Motivated by the seminal work of interval mapping proposed by Lander and Botstein in 1989 , in which genetic mapping was performed based on two neighboring genetic markers in controlled experiments, we propose to develop a new LD mapping framework that utilizes two SNP markers in a natural population. The new model explicitly incorporates the LD information between two markers into the mapping analysis, and thus we expect the analysis based on two markers is more powerful than that based on a single marker in a natural population just as Lander and Botstein have discovered in the controlled experiment. In the following sections, we first laid out the modeling framework for the two-marker LD mapping (tmLD), with details on parameter estimation and hypothesis testing. We then further elucidated our method through extensive simulation studies. Finally, we applied our method to a GWAS dental caries data set, followed by some discussions.
Two-marker LD (tmLD) mapping
and where i, j, k = 0, 1, D12, D23, D13 have exactly the same meaning as those in digenic disequilibria models for loci at positions 1/2, 2/3 and 1/3; and D123 is an additional trigenic disequilibria parameter for three loci together. Model (1) implies that D12, D23, D13 all geometrically decay with generations. It can be shown that with some reasonable assumptions, the D123 decreases with generations at a rate of (1-r13) and therefore also changes very slowly with time (Additional file 2). Hence, significant D12, D23, and D123 at current generation imply r12and r23 are very small, which form the basis for LD mapping using two genetic markers.
Joint zygote probabilities of the QTL genotypes at QTL Q and two-marker genotypes at markers M1 and M2, as expressed in terms of zygote configurations in a natural population
Joint marker-QTL genotype frequency
m 1 m 1 m 2 m 2
m 1 m 1 M 2 m 2
2p011p000 + 2p010p001
m 1 m 1 M 2 M 2
M 1 m 1 m 2 m 2
2p110p000 + 2p100p010
M 1 m 1 M 2 m 2
2p101p000 + 2p100p001
2p111p000 + 2p110p001
2p111p010 + 2p110p011
+ 2p101p010 + 2p100p011
M 1 m 1 M 2 M 2
2p111p001 + 2p101p011
M 1 M 1 m 2 m 2
M 1 M 1 M 2 m 2
2p111p100 + 2p110p101
M 1 M 1 M 2 M 2
Within the maximum likelihood estimation framework, an efficient EM algorithm can be implemented to obtain the MLEs of (Ω p , Ω q ), and is summarized into the following steps:
Step 1. Give initial values for the unknown parameters (Ω p , Ω q );
Step 2. E step – Calculate the posterior probabilities for each subject i to carry a particular QTL genotype j using the equation
Step 3. M step – Solve the log-likelihood equations for each parameter based on observed data and Πj|i to obtain its estimate. To estimate the quantitative genetic parameters (Ω q ), their expressions in closed forms can be derived based on the estimation equations. For the estimates of the population genetic parameters (Ω p ), another inner layer of EM algorithm can be employed.
Step 4. Repeat the E and M steps until the estimates converge to stable values. The estimates at convergence are the MLEs of parameters.
The detailed derivation for the EM algorithm is given in Additional file 3.
The estimates of the parameters under the null hypotheses can be obtained with the same EM algorithm derived for the alternative hypotheses, but with a constraint that all subjects have the same posterior probability. A likelihood ratio test (LRT) statistics can be constructed and computed to draw the inference about whether a QTL may be associated with given markers. Under the H0, the LRT statistics asymptotically follows a χ2-distribution with three degrees of freedom.
Let us randomly choose a sample of n subjects from a human population at Hardy-Weinberg equilibrium. In this population, one QTL is segregating and is inferred by a pair of markers. The allele frequencies of the markers (ℳ1 and ℳ2) and QTL () and their linkage disequilibria values are given as follows: p1 = 0.5 for allele M1 of ℳ1; p2 = 0.5 for allele Q of ; p3 = 0.5 for allele M2 of ℳ2. The LD parameters among the markers and QTL loci are given as: D12 = 0.05, D13 = 0.15, D23 = 0.05 and D123 = 0.04. For subjects who carry QTL genotype j, their phenotypic values were simulated based on Model (3), with μ2 = 10, μ1 = 5, μ0 = 0. The variances in phenotypic values were calculated based on different heritability values (H2). H2 quantifies the genetic contribution from the QTL to the overall trait and H2 = 0 implies that the means for three QTL genotype groups are the same, which are set to be 0. With the above given parameters and design, we simulated the phenotypic and marker information by assuming different sample sizes (N = 100, 250, 500, 1000, 1500, 2000, 2500, 3000), and different heritability values (H2 = 0, 0.05, 0.1, 0.2, 0.3, 0.4). Each simulation setting is carried out 1000 times for the evaluation of power and type I error.
Type I error evaluation and power comparison
Simulated data were used to compare our proposed tmLD method with single-marker based association analyses, including the single-marker LD mapping method (smLD) and single-marker based association test (smAT), and two-marker based haplotype analysis (haplo). The smLD was performed as described in Additional file 4. The smAT is a simple linear regression model with phenotypic trait as response variable and marker genotypes as categorical independent variable. The haplotype analysis was conducted as described in ; briefly, the haplotype that yields the best model fitting among those formed by two markers is used in comparison with tmLD.
Real data example
Dental caries or cavities, more commonly known as tooth decay, is one of the most common chronic disorders in humans, affecting approximately 40% children and adolescents and 90% adults in the US. The etiology and pathogenesis of dental caries have been determined to be multifactorial, such as environmental factors related to social behaviors . However, it is also apparent that some individuals are very susceptible to caries while some others are more resistant, almost irrelevant to the environmental risk factors they are exposed to, suggesting that genetic factors may play prominent roles in the caries development. Supported by evidence in both human and animal studies [18–21], the caries heritability has been estimated to be between 30-60%. The most compelling evidence come from the twin studies that the significant resemblance of dental caries lies within monozygotic but not dizygotic twin pairs [22, 23]. So it is without question that in addition to environmental factors, genetic components also profoundly influence the dental caries trait. To understand the genetic mechanisms of the dental caries, a GWAS study has been conducted and the dataset has been deposited in dbGaP (Study Accession: phs000095.v2.p1). Here we will apply our proposed model to analyze this caries GWAS dataset, in which 1843 adults were genotyped with a large panel of SNPs (610,000). We carried out the analysis using the caries outcomes that have been well defined in other GWAS studies, i.e. the D1MFT index which quantifies the total permanent tooth caries with white spots.
List of significant SNPs with p-value < 1.1e-7 in the Caries dataset
It is well recognized that naturally occurring variations in most complex disease traits have a genetic basis and consequently many GWAS studies have been conducted in the past few years. In analyzing these data, a phenomenon, called “missing heritability”, has been observed that the detected genetic variants can explain only a small portion of the heritability of phenotypic traits while a majority part remains mysterious . Part of the reason may be attributed to the lack of power in current methods. Thus, developing novel and powerful methods to better detect significant genes has been of great interest. Currently the routine GWAS analyses seek single-marker association between SNPs and phenotype, and when a significant association is detected, it implies that there might be some SNP(s) in linkage that are causal. Note that it cannot imply the test SNP itself is causal because there is no guarantee that the truly causal SNPs would have been genotyped. Since the interpretation of a significant association relies on the linkage concept, it is sensible to directly incorporate the LD information into association models. Additionally, due to the structure of LD blocks, a causal SNP is usually in linkage with multiple neighboring SNPs, all of which carry partial information about it. So in this sense, a new model that can incorporates more genetic information of linked SNPs should draw better inferences about the causal SNP.
In this article, we proposed a novel statistical method by considering two SNPs simultaneously. Our model is built upon the general LD mapping framework, and extends the previous methods based on single-marker LD. The simulation studies demonstrated that our new methods dramatically improved the detection power of the underlying QTLs. This is intuitively reasonable since our model can capture the linkage information between SNP markers, and hence has more power to detect the particular QTL that are in LD with both markers. Furthermore, the simulation studies indicated that even when the underlying QTL is indeed genotyped and is one of the markers, the performance of the tmLD analysis is nearly identical to that of the optimal test resulting from the causal SNP, suggesting the robustness of our model.
We applied our model to a GWAS date set that aimed to understand the genetic mechanisms of the dental caries. The data set contains a large cohort of 1,843 subjects as well as a very large number of SNPs (443,175). This shows that both our proposed method and the corresponding software package in R can be well applied to a typical GWAS data set. In addition, we also observed that the association analyses based on the single-marker and the two-marker models yielded different profiles of significant SNPs. This is somewhat expected since their assumptions are different. For the tmLD method, we assume that both markers must obey HWE and have to be in LD with the casual SNP. It might be possible that some SNPs would violate these assumptions and become unsuitable to the tmLD. In this sense, the single and two-marker analyses may be complementary to each other, and therefore it might be beneficial to use both methods in analyzing a real data set.
Sometimes population structure may be a concern in a GWAS analysis if subpopulations indeed exist in the sample, as it may lead to spurious associations. Several well-known methods developed to account for population structure  can be incorporated into our LD mapping framework to address this issue. For instance, the principal component analysis (PCA) can be applied to correct for stratifications . That is, we may first apply PCA on the genotype data and then choose the first few large principal components to be included in the Model (3) as additional covariates. With slight modifications, the computation algorithms and hypothesis testing described in the Method section can be readily applied.
In this work, we generalized the single marker LD analysis to a more general LD mapping framework using two adjacent markers. There are several ongoing works worthy of further investigation. First, the model can be easily extended to other types of phenotypic data, such as case–control binary and count data. Second, currently the two adjacent markers were used for the analysis; however, it is possible that another two markers in the same LD block might have better power, so it would be very interesting to determine how to choose the best SNP pair. Third, typically, one LD block may contain several SNPs, and if there exists one causal SNP within the LD block, it would be very interesting to see if we can summarize all SNPs in one LD block to make even better inference about the unobserved QTL.
The proposed tmLD model is a novel mapping method that can simultaneously consider two linked SNPs in a natural population. Through the extensive simulation studies, the tmLD method demonstrates better power than single-marker mapping strategies traditionally used in GWAS association analysis. The practical usage of the tmLD method was also shown in the analysis of a large-scale dental GWAS dataset. Hence, we recommend the usage of this improved method over the traditional single-marker association analysis.
Quantitative trait loci
Genome-wise association study
Single-marker association test
Single-marker linkage disequilibrium method
Two-marker linkage disequilibrium method
Two-marker based haplotype analysis
Minor allele frequency
The authors would like to thank the anonymous reviewers for their valuable comments and suggestions that have helped improve the quality of the paper significantly. This work is partly supported by the FUSION award from the Stony Brook University to SW.
The dataset used in the real data example was obtained from dbGaP through dbGaP accession number [phs000095]. Funding support for collecting this dataset was provided by the National Institute of Dental and Craniofacial Research (NIDCR, grant number U01-DE018903). Data and samples were provided by: (1) the Center for Oral Health Research in Appalachia (NIDCR R01-DE 014899); (2) the University of Pittsburgh School of Dental Medicine (SDM) DNA Bank and Research Registry (NIH/NCRR/CTSA Grant UL1-RR024153); (3) the Iowa Fluoride Study and the Iowa Bone Development Study (NIDCR R01-DE09551and R01-DE12101); and (4) the Iowa Comprehensive Program to Investigate Craniofacial and Dental Anomalies (NIDCR, P60-DE-013076).
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