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Sequencing Complex Diseases With HapMap

Tian Liu^*, Julie A. Johnson, George Casella^* and Rongling Wu^*,1

^* Department of Statistics, University of Florida, Gainesville, Florida 32611
Department of Pharmacy Practice, University of Florida, Gainesville, Florida 32611

¹ Corresponding author: Department of Statistics, 533 McCarty Hall C, University of Florida, Gainesville, FL 32611.
E-mail: rwu{at}stat.ufl.edu

	ABSTRACT

TOP ABSTRACT THEORY RESULTS DISCUSSION ACKNOWLEDGEMENTS LITERATURE CITED

 
Determining the patterns of DNA sequence variation in the human genome is a useful first step toward identifying the genetic basis of a common disease. A haplotype map (HapMap), aimed at describing these variation patterns across the entire genome, has been recently developed by the International HapMap Consortium. In this article, we present a novel statistical model for directly characterizing specific sequence variants that are responsible for disease risk based on the haplotype structure provided by HapMap. Our model is developed in the maximum-likelihood context, implemented with the EM algorithm. We perform simulation studies to investigate the statistical properties of this disease-sequencing model. A worked example from a human obesity study with 155 patients was used to validate this model. In this example, we found that patients carrying a haplotype constituted by allele Gly16 at codon 16 and allele Gln27 at codon 27 genotyped within the ß2AR candidate gene display significantly lower body mass index than patients carrying the other haplotypes. The implications and extensions of our model are discussed.

The basic principle for QTL mapping is the cosegregation of the alleles at a QTL with those at one or a set of known polymorphic markers genotyped on a genome. If a QTL is cosegregating with molecular markers, the genetic effects of QTL and their genomic positions can be estimated from the markers. This approach is robust and powerful for the detection of major QTL and presents the most efficient way to utilize marker information when marker maps are sparse. However, this approach is limited in two aspects. First, because the markers and QTL bracketed by them are located at different genomic positions, the significant linkage of a QTL detected with given markers cannot give any information about the sequence structure and organization of QTL. Second, the inference of the QTL positions using the nearby markers is sensitive to marker informativeness, marker density, and mapping population type. As a result of these, only a few QTL detected from markers have been successfully cloned (FRARY et al. 2000), despite a considerable number of QTL reported in the literature.

A more accurate and useful approach for the characterization of genetic variants contributing to quantitative variation is to directly analyze DNA sequences associated with a particular disease. If a string of DNA sequence is known to increase disease risk, this risk can be reduced by the alteration of this string of DNA sequence using a specialized drug. The control of this disease can be made more efficient if all possible DNA sequences determining its variation are identified in the entire genome.

With the recent development of the human genome project, massive amounts of DNA sequence data have been available across the human genome (INTERNATIONAL HAPMAP CONSORTIUM 2003). In particular, single-nucleotide polymorphisms (SNPs), being the most common type of variant in the DNA sequence, provide a powerful means for genotyping the whole genome. This facilitates the complete identification of specific sequence variants responsible for complex diseases. A linear arrangement of alleles (i.e., nucleotides) at different SNPs on a single chromosome, or part of a chromosome, is called a haplotype. The cosegregation of SNP alleles on haplotypes leads to nonrandom association, i.e., linkage disequilibrium (LD), between these alleles in the population. Empirical analyses of LD for SNPs have shown that nearby SNPs in the human genome tend to display highly significant levels of LD and are often distributed in block-like patterns, rather than displaying random- or even-spaced distribution as originally predicted (PATIL et al. 2001; DAWSON et al. 2002; GABRIEL et al. 2002). SNPs within haplotype blocks are much more strongly associated with each other than are those between different blocks. Haplotype diversity within each block can be well explained by only a finite number of SNPs, called tag SNPs or representative SNPs. The existence of these tag SNPs means that it is not necessary to associate a disease with all SNPs in the DNA sequence to understand the complete genetic control of the disease.

In this article, we present a novel statistical model for determining specific DNA sequences that are associated with the phenotypic variation of disease risk. This model is derived on the basis of multilocus haplotype analysis using a finite number of tag SNPs. We derived a closed-form solution for estimating the effects of haplotypes, haplotype frequencies, allele frequencies, and the degrees of LD of various orders among tag SNPs underlying the disease. We performed simulation studies to test the statistical behavior of this haplotype-based sequence-mapping model. A worked example is used to validate our model, in which a DNA sequence variant is detected to significantly reduce human obesity.

	THEORY

TOP ABSTRACT THEORY RESULTS DISCUSSION ACKNOWLEDGEMENTS LITERATURE CITED

 
Suppose there is a random sample of size n drawn from a natural human population at Hardy-Weinberg equilibrium. In this sample, a number of SNPs are genotyped genome-wide, aimed at the identification of DNA sequences responsible for a complex disease. Consider R (R > 1) tag SNPs for a haplotype block. Each of these R SNPs has two alleles denoted by A^r_kr (k_r = 1, 2; r = 1, ... R), with allele frequencies denoted by p_kr for the rth SNP. We use superscripts and subscripts to distinguish between different SNPs and different alleles within SNPs, respectively. These SNPs form 2^R possible haplotypes expressed as A¹_k1A²_k2 ... A^R_kR, whose frequencies are denoted by p_k1k2...kR. The haplotype frequencies are composed of allele frequencies at each SNP and linkage disequilibria of different orders among SNPs (LOU et al. 2003). The random combination of maternal and paternal haplotypes generates 2^R–1(2^R + 1) diplotypes expressed as (1 k₁ l₁ 2, ... , 1 k_R l_R 2). These R SNPs form 3^R observable multilocus zygotic genotypes, generally expressed as A¹_k1A¹_l1/A²_k2A²_l2/.../A^R_kRA^R_lR. When at most one SNP is heterozygous, the diplotype is consistent to its zygotic genotype. However, when two or more SNPs are heterozygous, the genotype will have different diplotypes and, therefore, the number of multilocus genotypes will be less than the number of diplotypes. Let P and P_{k1l1/k2l2/ .../kRlR} denote the diplotype and genotype frequencies, respectively, and n_{k1l1/k2l2/ .../kRlR} denote the observations of various genotypes.

Our interest is to search for the haplotype diversity that can explain phenotypic variation in a complex disease. The association between haplotype diversity and phenotypic variation has been detected in several studies of drug responses (JUDSON et al. 2000; BADER 2001). This allows us to assume that a particular haplotype is different from other haplotypes for a given disease. Because haplotypes (comprising diplotypes) cannot be directly observed, the effects of different haplotypes on the phenotype need be postulated from observed zygotic genotypes. The inference of diplotypes for a particular genotype is statistically a missing data problem that can be formulated by a finite mixture model (WU and CASELLA 2005).

Likelihood function:
In this study, the complete data are diplotype configurations at a given set of SNPs for each genotype and for disease outcomes of subjects, whereas the observed data are the genotypes of these SNPs and the disease outcomes. The missing data are at the connection from the genotypes to diplotypes. For any given genotype, all possible diplotypes can be written out. For example, genotype A¹₁A¹₁/A²₁A²₁ has one diplotype , as does genotype A¹₁A¹₁/A²₁A²₂, the diplotype . This is because in these situations where at most one SNP is heterozygous, the genotype and diplotype are identical. However, for a double-heterozygous genotype A¹₁A¹₂/A²₁A²₂, we have two diplotypes, and .

Table 1 lists all possible genotypes and diplotypes at two SNPs genotyped from a sample of size n. Each genotype (and therefore each diplotype) is composed of two haplotypes, one from the mother and the other from the father. Two haplotypes composing a diplotype come from four possible haplotypes, A¹₁A²₁, A¹₁A²₂, A¹₂A²₁, and A¹₂A²₂, whose frequencies are expressed as p₁₁, p₁₂, p₂₁, and p₂₂, respectively. The diplotype frequencies can be expressed in terms of the haplotype frequencies (Table 1). Two diplotypes and of a double heterozygote A¹₁A¹₂/A²₁A²₂ have frequencies p₁₁p₂₂ and p₁₂p₁₂, respectively. Thus, the relative frequencies of these two diplotypes for this double heterozygote are a function of haplotype frequencies, which can be expressed as p₁₁p₂₂/(p₁₁p₂₂ + p₁₂p₂₁) and p₁₂p₂₁/(p₁₁p₂₂ + p₁₂p₂₁), respectively.

The haplotype frequencies, arrayed by _p = (p₁₁, p₁₂, p₂₁, p₂₂), that belong to population genetic parameters can be estimated using the nine observed genotypes (G) for two SNPs (Table 1). The log-likelihood function of unknown haplotype frequencies given observed genotypes can be written as a multinomial form, i.e.,

(1)

Assuming that diplotypes are associated with phenotypic variation in a disease, we formulate a likelihood for unknown population (_p) and quantitative genetic parameters (_q) given observed phenotypes (y) and SNP genotypes (G). Generally speaking, a given 2-SNP genotype, A¹_k1A¹_l1/A²_k2A²_l2, can be partitioned into two possible diplotypes, and . Thus, such a log-likelihood function can be formulated on the basis of a two-component mixture model, i.e.,

(2)

where the mixture proportion,

represents the relative frequency of subject i whose diplotype is , and

are the probability density functions for subject i who has two possible diplotypes, respectively, with the genotypic means of µ for diplotype and µ for diplotype and the common residual variance of ². Note that subscript i is used to describe the genotypic means in the two density functions above because these means are diplotype- (and therefore subject-) dependent although there are only a total of 10 genotypic means for two SNPs. These means and variance are contained in vector .

Suppose there is a particular haplotype, A¹_k1A²_k2, which is different from the other three haplotypes, A¹_k1A²₂, A¹₁A²_k2, and A¹₁A²₂ (where a bar means the alternative of that allele), in its effect on a complex disease. The phenotypic means of the three genotypes that contain these two distinct groups of haplotypes are denoted as µ₁ for A¹_k1A¹_k1/A²_k2A²_k2, µ₂ for A¹_k1A¹₁/A²_k2A²₂, and µ₃ for A¹₁A¹₁/A²₂A²₂. On the basis of quantitative genetic theory, these three µ's can be partitioned into the overall mean (µ), the additive effect (a) due to the substitution of different haplotypes, and the dominant effect (d) due to the interaction between different haplotypes, i.e., µ₁ = µ + a, µ₂ = µ + d, and µ₃ = µ – a. In Table 1, the genotypic means are also given for different genotypes by assuming that haplotype A¹₁A²₁ is different from the rest of the haplotypes.

The log-likelihood function described by Equation 2 can be expanded to include all possible SNP genotypes, which is now expressed as

(3)

given that these SNP genotypes are independent. It can be seen from the above likelihood function that, although most zygote genotypes contain a single component (diplotype), the double heterozygote is the mixture of two possible diplotypes weighted by and 1 – . Thus, an advanced statistical method should be implemented to obtain the MLEs of the underlying population and quantitative genetic parameters.

An integrative EM algorithm:
We derived a closed-form solution for estimating the unknown parameters with the EM algorithm (DEMPSTER et al. 1977). Haplotype frequencies can be expressed as a function of allelic frequencies and LD. For a two-SNP haplotype, we have

(4)

where D is the linkage disequilibrium between the two SNPs. Thus, once haplotype frequencies are estimated, we can estimate allelic frequencies and LD by solving Equation 3. The estimates of haplotype frequencies are based on the log-likelihood function of Equation 1, whereas the estimates of diplotype genotypic means (and therefore the overall mean, haplotype additive, and dominant effects) and residual variance are based on the log-likelihood function of Equation 2. These two different types of parameters can be estimated using an integrative EM algorithm.

In the E step, the expected number (_i) and probabilities (_i) of subject i of a double-heterozygous genotype who carries diplotype are calculated by

(5)

(6)

Note that for all the other genotypes, such probabilities do not exist.

In the M step, the probabilities calculated in the previous iteration are used to estimate the haplotype frequencies using

(7)

(8)

(9)

(10)

Assuming that haplotype A¹₁A¹₁ has an effect different from the other three haplotypes, these probabilities are used to estimate the haplotype additive and dominant effects and the residual variance by

(11)

(12)

(13)

(14)

where = n_11/12 + n_12/11 and = n_11/22 + n_12/22 + n_21/21 + n_21/22 + n_22/22. Iterations including the E and M steps are repeated among Equations 5–14 until the estimates of the parameters converge to stable values. The sampling errors of these parameters can be estimated by calculating LOUIS's (1982) observed information matrix.

Hypothesis tests:
We can test two major hypotheses in the following sequence: (1) the association between two SNPs by testing their LD and (2) the difference of a given haplotype from the rest of the haplotypes in its effect on the disease outcome by testing the significance of haplotype additive and dominant effects. The LD between two given SNPs can be tested using two alternative hypotheses:

(15)

The log-likelihood-ratio test statistic (LR) for the significance of LD is calculated by comparing the likelihood values under H₁ (full model) and H₀ (reduced model) using

(16)

where the tilde and hat denote the maximum-likelihood estimates (MLEs) of unknown parameters under H₀ and H₁, respectively. The LR₁ is considered to asymptotically follow a ² distribution with 1 d.f. The MLEs of allelic frequencies under H₀ can be estimated using the EM algorithm described above, but with the constraint p₁₁p₂₂ = p₁₂p₂₁.

Diplotype or haplotype effects on a complex trait can be tested using two alternative hypotheses expressed as

(17)

The log-likelihood-ratio test statistic (LR₂) under these two hypotheses can be similarly calculated. The LR₂ may asymptotically follow a ² distribution with 2 d.f. However, the approximation of a ² distribution may be inappropriate when some regularity conditions, such as normal and uncorrelated residuals, are violated. The permutation test approach proposed by CHURCHILL and DOERGE (1994), which does not rely upon the distribution of the LR₂, may be used to determine the critical threshold for determining the existence of a QTL.

R-SNP sequence model:
The idea for sequencing a complex trait is described for a two-SNP model. It is possible that the two-SNP model is too simple to characterize genetic variants for quantitative variation. With the analytical line for the two-SNP sequencing model, we can readily extend our model to include an arbitrary number of SNPs whose sequences are associated with the phenotypic variation.

Consider R SNPs that form 3^R observable multilocus zygotic genotytpes, generally expressed as A¹_k1A¹_l1/A²_k2A²_l2/... /A^R_kRA^R_lR. These genotypes are collapsed from a total of 2^R–1(2^R + 1) diplotypes expressed as (1 k₁ l₁ 2, ... , 1 k_R l_R 1). A key issue for the multi-SNP sequencing model is how to distinguish among 2^r–1 different diplotypes for the same genotype heterozygous at r loci. The relative frequencies of these diplotypes can be expressed in terms of haplotype frequencies. The integrative EM algorithm can be employed to estimate the MLEs of haplotype frequencies. LOU et al. (2003) provided a general formula for expressing haplotype frequencies in terms of allele frequencies and linkage disequilibria of different orders. The MLEs of the latter can be obtained by solving a system of equations.

In the multi-SNP sequencing model, we face many haplotypes and haplotype pairs. An Akaike information criterion- or Bayes information criterion-based model selection strategy (BURNHAM and ANDERSSON 1998) has been framed to determine the haplotype that is most distinct from the rest of the haplotypes in explaining quantitative variation. However, in practice, a simultaneous analysis of too many SNPs will encounter considerable computational load and, also, may not be necessary for the explanation of disease variation. These two factors should be considered in a further study of the determination of a maximal number of SNPs for sequencing mapping.

	RESULTS

TOP ABSTRACT THEORY RESULTS DISCUSSION ACKNOWLEDGEMENTS LITERATURE CITED

 
Simulation:
We performed Monte Carlo simulation experiments to examine the statistical properties of the model proposed for association studies between variation in DNA sequence and a complex trait. Our simulation studies were designed to consider the effects of different heritability levels (H² = 0.1 vs. 0.4), different sample sizes (n = 100 vs. 400), and different gene action modes (additive vs. dominant vs. overdominant) on the estimation precision of parameters from our model. The ratio of dominant (d) over additive effect (a) is used to determine the degree of dominant. A haplotype is regarded as additive, dominant, or overdominant if this ratio is zero, one, and greater than one, respectively (LYNCH and WALSH 1998).

Suppose two SNPs are segregating in a natural population at Hardy-Weinberg equilibrium. The allele frequencies and linkage disequilibrium between these two SNPs are given in Table 2. We assume that one of the four possible haplotypes for the two SNPs is different from the rest of the haplotypes in their effects on the phenotype of a complex trait, which leads to three distinct groups of diplotypes. By assigning the genetic values for these two contrasting haplotypes under different gene action modes (Table 2), we calculate the genotypic values for all possible diplotypes and further simulate their phenotypic values on the basis of normal distribution. The residual variance is determined according to different heritability levels (0.1 vs. 0.4) expressed as the relative proportion of genetic variance to total observed variance. The genetic variance is determined on the basis of the genotypic values of all diplotypes and their frequencies.

The estimation precision of quantitative parameters is dependent on heritability level, sample size, and gene action mode. In almost all situations, the additive effects can be estimated better than the dominant effect. The estimation of additive and dominant effects responds differently to genetic action mode, depending on the degree of dominance. For example, given the same heritability H² = 0.1 and same sample size n = 100, the sampling error of the additive effect estimation is increased by 121% from additive to dominant modes, but is stable from dominant to overdominant modes. These two percentage values are 1 and 42% for the sampling errors of the dominant effect estimation. It appears that the additive effect estimation is more sensitive than the dominant effect estimation to the changes of heritability and sample size.

We analyze the power to detect a significant diplotype difference from our model under different combinations of H² value, sample size, and gene action mode (Table 2). To do so, we first obtain empirical critical threshold values for declaring the significance of diplotype difference through simulations. The thresholds at the significance level = 0.05 or 0.01 are estimated as the 95th and 99th percentile for the simulated test statistics calculated from simulated data involving no diplotype difference over 1000 LR₂ values. It is clear that increased H² levels and increased sample sizes can increase the power to detect diplotype differences. When H² = 0.4 or n = 400, such power is 100% (Table 2).

An additional simulation study was performed to investigate the statistical behavior of the multi-SNP sequence model. The results from the 3-SNP sequence model are summarized in Table 3. First, all genetic parameters can be reasonably estimated from the 3-SNP sequence model, although the estimation precision of the dominance effect is reduced compared to that from the 2-SNP model. Second, the power to detect significant sequence variants and the estimation precision of different genetic parameters from the 3-SNP model can be improved with increased heritability levels, increased sample sizes, and decreased gene interactions.

To determine whether sequence variants at these two polymorphisms of BAR-2 are associated with obesity phenotypes, we investigated a group of 155 women of ages 32 to 86 years old with a large variation in body fat mass. Each of these patients was determined for her genotypes at codon 16 with two alleles, Arg16 (A) and Gly16 (G), and at codon 27 with two alleles, Gln27 (C) and Glu27 (G), within the BAR-2 gene and measured for body mass index (BMI). These two SNPs form four haplotypes designated as AC, AG, GC, and GG, which lead to nine genotypes, AA/CC, AA/CG, AA/GG, AG/CC, AG/CG, AG/GG, GG/CC, GG/CG, and GG/GG, and the 10 corresponding diplotypes, [AC][AC], [AC][AG], [AG][AG], [AC][GC], and [AC][GG] or [AG][GC], [AG][GG], [GC][GC], [GC][GG] and [GG][GG]. Our model is used to associate diplotype differences with variation in BMI. The MLEs of the haplotype frequencies, allele frequencies, and linkage disequilibrium between the two SNPs were obtained (Table 4). These two SNPs are highly associated with each other, whose LD is estimated as 0.1261. The allele frequencies are 0.61 for allele G at codon 16 and 0.38 for allele G at codon 27, respectively, suggesting that both of them have fairly high heterozygosity.

	DISCUSSION

TOP ABSTRACT THEORY RESULTS DISCUSSION ACKNOWLEDGEMENTS LITERATURE CITED

 
The elucidation of the entire human genome using SNPs will make it possible to develop a haplotype map of the human genome. Although such a HapMap has been recently available due to international collaborations (INTERNATIONAL HAPMAP CONSORTIUM 2003), there is a serious lack of powerful statistical tools for specific utility of this expensive HapMap to detect genes and genetic variations that affect health and disease. Such a statistical model has been developed and presented in this article. The idea behind our statistical modeling is that differences between SNP-constructed haplotypes may be associated with differing susceptibility to disease (INTERNATIONAL HAPMAP CONSORTIUM 2003).

SNPs reside at sites in the DNA sequence where individuals differ at a single DNA base. Sets of nearby SNPs on the same chromosome are inherited in blocks. Blocks may contain a large number of SNPs, but a few SNPs are enough to uniquely identify the haplotypes in a block (GABRIEL et al. 2002). The HapMap is a map of these haplotype blocks constructed by tag SNPs, those that explain most of haplotype diversity. The HapMap should be valuable by reducing the number of SNPs required to examine the entire genome for association with a phenotype from the 10 million SNPs that exist to 500,000 tag SNPs. This will make genome scan approaches to finding regions with genes that affect diseases much more efficient and comprehensive, since effort will not be wasted typing more SNPs than necessary and all regions of the genome can be included.

Our model is founded on the discovery of tag SNPs in the genome, thus allowing for a fast scan for the association between variation in DNA sequence and traits. This model has three advantages. First, it can materialize the genetic basis for quantitative variation by directly characterizing specific DNA sequences predisposing to a certain disease. The traditional statistical models for genetic mapping attempt to postulate the position of hypothesized QTL that are linked with known markers genotyped from the genome (LANDER and BOTSTEIN 1989; LOU et al. 2003). The QTL detected from these models are regarded as "hypothesized" because it is not possible to know their DNA sequences and, therefore, physiological function. As opposed to the traditional "indirect" approach, our model presents a "direct" approach. At present, the utility of the direct approach is limited to sequencing the functional parts of candidate genes with known biochemical or physiological function. With the release of HapMap, our model will make the direct approach more useful and more efficient in searching for causal variants throughout the whole genome. Our model is also different from traditional association studies (e.g., ZAYKIN et al. 2002) whose aims are the significance test for the association between haplotypes and phenotypes (mostly discrete traits) rather than the precise estimation of haplotype effects. Second, our model is statistically simple and computationally fast. The most difficult part for the estimation from our model is to construct diplotype configurations for heterozygous genotypes at two or more SNPs. The estimation of population genetic parameters is based on a multinomial-likelihood function of the observed genotype data, whereas the estimation of quantitative genetic parameters is based on a mixture-based likelihood function including different diplotypes. These two likelihood functions can be easily integrated to a unified estimation framework implemented with the EM algorithm.

Finally, our model is flexible to different genetic and experimental settings. The results from a simulation study indicate that the association between DNA sequence and phenotype can be well detected when the trait has a modest heritability level (0.1) or a modest sample size (100) is used. Our model can also obtain fairly precise estimation of parameters when diplotypes display overdominance in the situation with modest heritability and sample size. The specific utility of our model to a real example from an obesity study leads to the successful detection of a DNA sequence (haplotype) at codons 16 and 27 genotyped within the ß2AR candidate gene (CHAGNON et al. 2003) for its significant impact on human obesity. This haplotype, composed of the Gly16 form of codon 16 and the Gln27 form of codon 27, tends to reduce BMI when it is combined with itself or any other haplotypes and accounts for 6% of the total observed variation in BMI for 155 patient samples.

Although our simulation and example were based on 2- or 3-SNP analyses, our sequencing model has been developed to allow for the detection of sequence variants involving any number of SNPs within a haplotype block. In addition to its use in studying genetic associations with disease, our sequencing model can be extended to study the genetic factors contributing to variation in response to environmental factors, in susceptibility to infection, and in the effectiveness of and adverse responses to drugs and vaccines (EVANS and MCLEOD 2003; WEINSHILBOUM 2003). It can also be modified to estimate the effects of sequence-sequence interaction on a complex trait. It is possible that a haplotype within a candidate gene interacts with haplotypes from other candidate genes. The characterization of specific DNA sequence variants for diseases should allow the development of tests to predict which drugs or vaccines would be most effective in individuals with particular genotypes for genes affecting drug metabolism.

	ACKNOWLEDGEMENTS

TOP ABSTRACT THEORY RESULTS DISCUSSION ACKNOWLEDGEMENTS LITERATURE CITED

 
We thank two anonymous referees for their constructive comments on this manuscript. This work is supported by an Outstanding Young Investigator Award of the National Natural Science Foundation of China (30128017), a University of Florida Research Opportunity Fund (02050259), and a University of South Florida biodefense grant (7222061-12) to R.W. The publication of this manuscript was approved as journal series no. R-10063 by the Florida Agricultural Experiment Station.

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TOP ABSTRACT THEORY RESULTS DISCUSSION ACKNOWLEDGEMENTS LITERATURE CITED

 

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