Modeling RV association

We define a locus as a genomic region of fixed size (nucleotides). Let denote the set of RVs in the locus. We abuse notation slightly by using to also denote the locus itself. A case-control study at includes a set of individual genotypes. For genotype , and RV , let denote the number of minor alleles that genotype carries for variant . Extending the notation to subsets, of RVs, define . For a subset , denote a union-variant as follows: individual has the allele if and only if . Otherwise, . The union-variant is a virtual construct that helps combine the effect of multiple RVs. Let (respectively, ) represent the case (respectively, control) status of an individual.

For an individual chosen at random, and , let Xcorr denote an association test statistic between the union-variant and the disease status . Here, we will use Pearson's as the test-statistic, but the method remains unchanged for other measures. Using this notation, the collapsing strategy described by Li and Leal [15] uses the test-statistic Xcorr to associate a locus with the disease. Instead, we define the association statistic for locus by(1)

The RareCover method

Our method, RareCover, accepts a locus containing a set of RVs in a window of fixed size (nucleotides). It returns the test-statistic, Xcorr , a -value on the statistic, and the subset of RVs that contribute to the union variant. The window size is a parameter. When the input locus is larger than the window size, RareCover looks at overlapping windows of size , where each window is shifted one RV away from the previous window. For each window, the Xcorr statistic is output, along with a non-adjusted -value.

The computation for the Xcorr statistic on a single window is described in the Algorithm below. Given a set of RVs over individuals, the naive computation for computing Xcorr needs computations. A reduction from the MAX-COVER problem can be used to show that the problem is NP-hard, indicating that no provably efficient algorithm is likely [28]. Similar reductions can also be used to show the hardness result for a variety of other proposed association statistics. Therefore, we employ a greedy heuristic that is fast ( computations), and does well in practice. In each step, (see Algorithm), we select the variant that adds the most to the statistic, until no further improvement is possible. On a standard Linux workstation, the computation is fast, about windows per second.

Computing significance.

While the test-statistic is a test on the union-variant , significance cannot be computed directly, as is optimized over many possibilities. The multiple-tests will increase the statistic for non-associated loci as well. We compute significance by applying RareCover to permutations of the case-control genotypes.

The number of permuted trials required to achieve genome-wide significance can be large. We make the computation tractable using two ideas: first, empirical tests show that the statistic on correlates tightly with the permutation -value (Figure 1). Note that the saturation at the end is due to limited trials. Let denote the value of Xcorr for a window, . When is less than a pre-determined threshold (), no permutation test is done, as the window is unlikely to be significant. When , the statistic is recomputed after permuting case and control labels (default permutations), and a -value is computed as the fraction of the permuted samples whose Xcorr value matches or exceeds . To save time on this computation, we run permutations in a data-driven fashion. We run a maximum of permutations, but stop as soon as we obtain samples for which the RareCover statistic exceeds .(2)Both, the parameter , and the maximum number of iterations can be adjusted, based on desired level of significance, and the size of the genomic region. Here, we set parameter , which corresponds to a permutation adjusted -value of in Figure 1. This ensures fast computation with no loss of power (See Results).

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Figure 1. Permutation -values versus the statistic value on the union-variant .

The mean of the empirical -values (obtained by permuting cases and controls) were plotted against each value of the statistic obtained over many tests over the entire range of simulation parameters, by varying sample size , locus PAR, and penetrance. As is the most significant subset among many possible subsets, the theoretical -value suggested by the distribution cannot be used directly. However, the plot shows that the locus value correlates tightly with the -value, implying that the union statistic can be used to filter the significant windows with no loss of power. The saturation at the ends is due to the number of trial being limited to .

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Parameters for RV simulation

Consider a locus with a set of rare-variants. Let a subset of RVs be causal, in the sense that a mutation at any increases the likelihood of disease. For an individual, , we use and as short-forms of the events , and , respectively. Similarly, events reflect case-control status for the individual. We work with the following parameters for power calculations:

1. Disease prevalence in the population, denoted by .
2. Penetrance of the locus, denoted by
3. Locus-PAR, denoted by

Note that the PAR for a variant is often described by the following (Ex:Bodmer and Bonilla, 1999 [14])(3)where is the number of individuals with the phenotype, and is the number of individuals that show the phenotype, but do not have the variant allele. In our terminology

The choice of these parameters is intuitive as we expect an RV to have moderate penetrance, but very low PAR . However, the multiple RVs in have roughly additive effect, leading to moderate locus-PARs. These parameters are tightly related to other, more common measures of locus association, such as the Odds-Ratio (OR), as shown below:To compute , we start with a Bayesian relation for computing the likelihood of a genotype containing a causal RV as(4)Then,(5)and,(6)

Simulating constant sized populations (CP)

We simulate multiple case-control studies over a range . A simulation of individuals begins with the division of the individuals into cases and controls. Once this is done two additional steps take place.

1. Generate a set of RVs for the simulated locus containing causal, and neutral RVs.
2. Simulate the genotypes for each individual.

We start by generating causal RVs. As RVs do not show high LD, we can model the population by generating each RV independently. We adapt Pritchard's argument that the frequency distribution of rare, deleterious, RVs must follow Wright's model under purifying selection [29]. Therefore, the allele frequencies are sampled according to:(7)where,

• , allelic frequency
• , selection coefficient
• , rate of mutation from normal allele to causal
• , rate of repair from causal allele to normal

We choose [29]. Note that we do not control the number of causal RVs, , directly, in our simulation. Recall thatFurther,Therefore, setting a value for R limits the size of the causal RV.(8)Further, the sampling procedure occasionally generates SNPs with a high individual PAR. These variants would show up as being significant even with a single marker analysis. Therefore, these are discarded. The procedure SimulateRV describes the method to generate causal RVs. To generate neutral RVs, we use Fu's model of allele distributions [30] on a coalescent, which suggests that the number of mutations that affect individuals in a population with mutation rate is given by . For the purposes of our simulation we use .

Simulating genotypes

For both cases and controls, each RV is sampled independently. For non-causal variants , the probability of picking a minor allele is , for both case and control individuals. To sample alleles from causal SNPs, recall that under the union model, for all . Therefore, the minor allele frequencies are given byWe assume HW equilibrium to sample genotypes for case and control individuals.

Simulating populations with bottleneck and recent expansion (BRE)

Recently, Kryukov and colleagues [31] described a demographic model that explicitly models European ancestry. The population is assumed to be relatively stable for a long period, but is followed by a bottleneck, and rapid expansion after the bottleneck (about 7500–9000 years ago, with 20–25 years per generation). They validate their model by comparing observed versus predicted allelic frequencies. To this model, they add ‘causal’ (mostly deleterious) mutations using a distribution of selection coefficients from a gamma distribution. The causal alleles are associated with a change in a quantitative trait (QT). The QT values are normally distributed. Individuals carrying any causal mutation have QT values drawn from a Normal distribution with a shifted mean. For Rare variant analysis, individuals are chosen from the lower (Control) and upper (Case) tails of the QT distribution.

For our study, the authors provided us with individual genotypes simulated according to their demographic model, with causal mutations contributing to the following shifts: (Low), (Medium), (High). The highest and lowest , and % of the QT distributions were used for the Case and Control populations. For the % population (500 controls, 500 cases), the locus PAR varied as –. For the % populations, the number of individuals is larger (1000 controls, 1000 cases), but the PAR values decrease to (Low), (Medium), and (High).

Reimplementing alternative strategies

For the purposes of comparison, we reimplemented the collapsing statistic proposed by Li and Leal [15] as well as the weighted-sum statistic used by Madsen and Browning [16]. Both publications discuss the separation of variants into groups based upon function (i.e. non-synonymous coding SNPs) or other property. However, because we are performing our studies on model free, unannotated data, we do not perform any such grouping.

As a result, the CMC approach proposed by Li and Leal [15] is equivalent to collapsing all variants in the locus and calculating the association. Li and Leal show that the assignment of variants to functional groups, separately collapsing these groups, and finally performing a multivariate analysis improves power to detect causal loci. However, separation of variants into groups is inexact and the authors show that errors in group assignment can confound tests for significance. Additionally, performing this separation on a genome wide scale may be intractable.

The weighted-sum statistic proposed by Madsen and Browning [16] is used to detect association between a pre-defined group and a disease state. To compare fairly we defined the group of mutations as all mutations at a locus. We reimplemented the weighting approach based upon allele frequency as well as the sum and ranking approach to determine a score. Finally, we implemented a single-marker test as a bi-allelic -statistic with 1df. The tests were used to score windows over a wide range of simulation parameters to better understand how RareCover performed in comparison to the collapsing, and weighted strategies. For each strategy, a -value of significance was established by doing randomized trials using permuted case and control data. All three methods were run on the same sets of permuted data, and the -values were used to compare. Code for all methods is available upon request from the authors.

MSMB gene resequencing

Recently, Yeager and colleagues [32] resequenced a Kbp region including the micro-seminoprotein- (MSMB) gene (chr10:–) for prostate cancer cases, controls, plus another CEPH individuals. While the number of individuals is too small to derive rare variants, we used the prediced genotypes supplied by the authors for RV analysis. For this analysis, we used individuals together as controls, and all 284 variants with MAF % were used as input to RareCover.

CRESCENDO data

In a recently submitted study LR-PCR amplicons (Harismendy et al., unpublished) were used to re-sequence Kbp from the FAAH locus (NCBI36 chr1:46621328–46653043) and Kbp from the MGLL locus (NCBI36 chr3:128880456–129037011). A total of individuals were selected for sequencing from two tails of the BMI distribution of the CRESCENDO cohort (http://clinicaltrials.gov/ct/show/NCT00263042). individuals had BMI lower than kg/ and individuals a BMI greater than kg/. DNA sequencing libraries were prepared, and sequenced as previously described in Harismendy, 2009 [33] with the following modifications: sequencing libraries were indexed by nt barcode located downstream of the adapter [34] and between one and six libraries were loaded per lane of the Illumina GAII instrument. The reads obtained from several lanes were merged, aligned and the variant called using MAQ mapmerge, map and cnsview+SNPfilter options respectively [35]. All samples had an average coverage greater than . Allowing for a minimum coverage of reads and a minimum base quality (Phred ), a raw set of single nucleotides variants (SNVs) were identified in the population. The SNVs were filtered for Hardy Weinberg Equilibrium in the controls () and genotyping rate % of the samples to obtain a final set of SNVs (220 FAAH, 1173 MGLL). Of these, SNVs had MAF , and were selected for RV analysis. The list and location of the RVs identified by RareCover as supporting the association is available in Supplemental Table S1.

Simulations (CP)

We simulated cases and controls for a collection of sample sizes, ranging from to over individuals with equal numbers of cases and controls. The MAF for rare variants ranged from to . Throughout, we assume the disease prevalence in the population to be . The PAR for the locus was set to . The penetrance, was varied in the interval , corresponding to OR values of –. The dependence on parameters is somewhat non-trivial. To see this, note that is a lower bound on relative risk. Reducing would increase the relative risk, only making association easier. In other words if the disease incidence is low, and a causal variant is low frequency, then the presence of the causal variant is a strong indicator of disease status.

The results of the simulation are shown in Figure 2. For each choice of parameters (, and ), case-control studies were sampled as described in Methods. The data-set was tested using RareCover, collapsing, weighted-sum heuristics, as well as single-marker tests.

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Figure 2. Power of RV analyses, tested over different values of penetrance , PAR , and individuals (cases+controls).

For each choice of parameters, test cases were simulated. Each test-case was analyzed using methods, and the -value computed using permutations of cases and controls. The score is considered significant only if it is higher than all permuted values. The power of the test is the fraction of test-cases that had a significant score. RareCover dominates the other methods implying greater power over all choice of parameters. For all methods, power increases with an increase in , or sample size.

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To enable a fair comparison, the methods were applied to randomizations of the same data-set, obtained by permuting cases and controls. The -value is similar to a False Discovery Rate calculation. The span of a typical human gene is about Kbp, and will contain about rare-variants, implying fewer than distinct windows per gene. If we assume candidate genes for a phenotype, we would have candidate windows. A -value, or FDR of could therefore be considered significant at the genome-wide level. A test score was considered significant if it was higher than each of the permutations, giving the 95% confidence interval of the p-value as . The power of a test for a specific choice of parameters is the fraction of () tests that had a significant score. Consider the sample point in Figure 2, with , and a sample of individuals. The power of RareCover is over , which can be contrasted with the low power of the weighted-sum [16], and collapsing heuristics. For any choice of parameters, RareCover shows better performance than the other methods.

Our phenotypic model differs somewhat from the one proposed by Madsen and Browning. In their model, the PAR for each causal variant is assumed to be equal, and is equal to the groupwise PAR divided by the number of causal variants. We also applied the tests to this model, using cases, and controls, and groupwise PAR values at . The power of the MB test at these values was computed to be respectively, while the power of RareCover on the data sets is (Supplemental Figure S1).

An advantage of the RareCover approach is that it does not depend upon MAF, or the density of RVs in a region. This is partly because it combines the effects of multiple associating RVs that associate, and discards the RVs that do not associate. By contrast, other methods combine all RVs, albeit with different weights. While RareCover does not recover all of the causal RVs, it always recovered a significant subset of the causal RVs in our simulations. See Figure 3, which summarizes the results for . Let correspond to the simulated, causal RVs, while corresponds to the set returned by RareCover. Thus, corresponds to the fraction of causal RVs recovered. With modest sample size, more than of the RVs are recovered, and help provide a direct interpretation of the genetic association. A somewhat unexpected aspect is that the number of causal RVs, , (and also, ) increases with an increasing sample size. For larger samples, we can recover a larger number of the low frequency variants, and the causal set has a larger mix of low frequency RVs. As we only consider RVs with MAF , the number saturates by K individuals.

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Figure 3. Comparisons between causal RVs, and RVs recovered by RareCover.

The -axis describes the raw number of causal RVs (), RVs recovered (), their intersection, and the fraction recovered (, scaled for exposition). Close to of the causal RVs are recovered over a wide range of sample populations.

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Simulated BRE populations

The methods were also applied to the data sets provided by Kryukov et al., as explained in Methods. The cases and controls are chosen from the extremes of a population of phenotyped individuals to reflect current population cohorts. As the locus PARs are very small (–), we work with a nominal -value cut-off of . As before, power is defined as the fraction of 1000 simulations on which the test met the -value cut-off. In addition to the methods, we also plot the power of the true causal mutations to illustrate their small effect.

Figure 4 shows the results upon choosing the %, and % extremes for different levels of phenotypic association. RareCover outperforms other methods over the different tests, and is comparable to the results of selecting the true causal mutations. As suggested previously, increasing PAR values, and population sizes, increase the power of RareCover, as with all methods. However, the power of RareCover does not appear to be affected by the specifics of the demographic simulation.

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Figure 4. Power calculations on populations with bottleneck, and recent expansion.

Simulated population data with quantitative trait (QT) values was provided by Kryukov et al. The QT values are normally distributed. Individuals carrying any causal mutation have QT values drawn from a Normal distribution with a shifted mean. The shift is characterized as Low (), Medium (), and High (). As the locus PAR values are low, power is computed as the fraction of simulations that showed significance at -value . Individuals were chosen from the lower (Control) and upper (Case) tails of the QT distribution. The power of all methods is compared using the % extremes ( cases, controls), and the % ( cases, and controls). RareCover is shown to have the highest power, comparable to the power of the causal mutations.

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The allele frequency spectrum of the CP and BRE models is shown in Figure 5. There are significant differences in allele frequency spectra in the two cases. In the CP case, there is a bias in the cases among alleles with lower frequency. High frequency causal variants represent an easier case, as they can be detected by single marker analysis. To eliminate these cases, we discarded high frequency causal variants from the simulations, which partly explains the bias in CP, relative to BRE. In the CP (respectively, BRE) models, the average number of variants per 5Kbp window was (respectively, ), with (respectively, ) causal variants. The performance of RareCover is robust against different demographic models, and depends mainly upon locus PAR, and sample size.

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Figure 5. Allele frequency spectra in various demographic models.

BRE refers to the simulation of population under bottleneck followed by recent expansion from Kryukov et al.; CP refers to the simulation under a constant population size. The allele frequencies in CP are biased toward rare variants in cases, while there is little bias in BRE. The performance of RareCover is robust to data sets with different allele frequency spectra.

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Running time

The running time of RareCover increases linearly with the number of SNPs, and the number of individuals, as shown in Figure 6. For a population of individuals, the running time time goes from 80ms to 311ms on a standard Linux desktop, as the number of SNPs in the window increases from to . The times shown here do not include the cost of reading and writing the data, which incurs a fixed additional cost (about ms. See Supplemental Figure S2). The total running time is at most that of a single marker test.

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Figure 6. Running time of RareCover as a function of sample size, and number of SNPs.

As RareCover is a greedy approach, the running time increases linearly with an increase in number of SNPs, and individuals. The running time shown here does not include the time for disk input and output of the data, which incurs a fixed additional cost of ms to each run. The total running time is about twice that of single marker tests.

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On the FAAH data ( individuals), the running time for a window of Kbp was computed to be seconds. Consider a genome-wide scan with windows. To achieve genome-wide significance, we would need randomizations for each window, which could be computationally intensive.

However, we run RareCover in two passes, using the Xcorr statistic on the union-variant as a filter (Figure 1). The permutation test is only applied to the fraction of the windows for which the Xcorr statistic exceeds a threshold ().

Therefore the RareCover computation is executed on windows. As discussed in the methods, . For (corresponding to in Figure 1), the total time isIf the number of candidate windows is larger (), and a conservative filter is chosen (corresponding to ), the running time increases to hrs., easily accomplished on a small cluster.

Results on resequencing data

CRESCENDO cohort data.

As described earlier, the CRESCENDO cohort subjects selected for sequencing were individuals at the extremes of BMI distribution. We applied RareCover to overlapping windows of length Kbp over the region (as described in Methods), to analyze the impact of RVs (For the purposes of comparison the performance of all methods can be seen in Supplemental Figure S4). The permutation based -values for two genes are plotted in Figure 7. For both loci, we found a single region of approximately bp, that was enriched in strongly associating variants.

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Figure 7. FAAH locus association.

RareCover was used to analyze overlapping windows of Kbp in the re-sequenced region around FAAH. A -value was computed for each window using permutations of cases and controls. Each point corresponds to the -value of a single window starting at that location. The most significant window (described by the box) is Kbp upstream of the FAAH transcription start site. The region is part of an LTR element, which are known to carry regulatory signals, and is enriched in transcription factor binding sites, suggesting a regulatory role for the rare variants.

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The FAAH enzyme (1p33) is known to hydrolyze anandamide (AEA), and other fatty acid amides. The region with the most significant association, located upstream of the FAAH transcription start site (TSS), contains RVs. RareCover selected a subset of RVs, with a union-variant that appears in cases, and controls (nominal permutation -value ). The locus specific -value for the window is . Analyzing the locus for functional significance, the locus falls within a retroviral Long-Terminal-Repeat (LTR). Insertion of retroviral elements, followed by adaptation of the viral regulatory elements is a well known mechanism for gene regulation. A recent analysis of the FAAH core promoter (bp upstream) in human T-cells identified a C/EBP site which (through a STAT3 tethering) mediated the leptin regulation of FAAH expression [36]. Surprisingly, the leptin mediated regulation of FAAH was observed in immune cells, but not a model of neuronal cells [37]. Our results suggest that an alternative regulatory region Kbp upstream of the TSS is disrupted by RVs in obese individuals. A scan for transcription factor binding sites reveals many relevant transcription factor binding sites, including one for C/EBP (data not shown).

The enzyme monoacylglycerol lipase (MGLL), encoded by the MGLL gene located on chromosome 3q21.3, is a presynaptic enzyme that hydrolyzes 2-arachidonoylglycerol (2-AG), the most abundant endocannabinoid found in the brain. The RareCover scan on RVs identified a single window enriched with associated RVs. The window lies immediately upstream of the gene, suggesting that the causal RVs have a regulatory function. At the most significant locus (chr3:–, upstream of MGLL TSS), of RVs were selected, with the union RV present in cases, and controls. While the nominal -value is , the locus adjusted -value is at the margin of significance, at . The locus contains a known LINE element and a promoter for RNA polymerase II. Mutations in this promoter could easily interfere with binding affinity for RNA Polymerase II and affect subsequent transcription/translation.

In our analysis, we only considered RVs (MAF ). Harismendy et al. have reported on the connection between common variant, and RV associations in a recently submitted study. Their results suggest that common variants do not identify significant associations in FAAH, but identify regions with significantly associated SNPs at the MGLL locus. LD between the significant common variants and the MGLL union-variant is low, with the highest value of . This suggests that the rare and common variations might have independent, additive effects on the phenotype.