Recent analyses have suggested a strong heritable component to circulating fatty acid (FA) levels; however, only a limited number of genes have been identified which associate with FA levels. In order to expand upon a previous genome wide association study done on participants in the Framingham Heart Study Offspring Cohort and FA levels, we used data from 2,400 of these individuals for whom red blood cell FA profiles, dietary information and genotypes are available, and then conducted a genome-wide evaluation of potential genetic variants associated with 22 FAs and 15 FA ratios, after adjusting for relevant dietary covariates. Our analysis found nine previously identified loci associated with FA levels (FADS, ELOVL2, PCOLCE2, LPCAT3, AGPAT4, NTAN1/PDXDC1, PKD2L1, HBS1L/MYB and RAB3GAP1/MCM6), while identifying four novel loci. The latter include an association between variants in CALN1 (Chromosome 7) and eicosapentaenoic acid (EPA), DHRS4L2 (Chromosome 14) and a FA ratio measuring delta-9-desaturase activity, as well as two loci associated with less well understood proteins. Thus, the inclusion of dietary covariates had a modest impact, helping to uncover four additional loci. While genome-wide association studies continue to uncover additional genes associated with circulating FA levels, much of the heritable risk is yet to be explained, suggesting the potential role of rare genetic variation, epistasis and gene-environment interactions on FA levels as well. Further studies are needed to continue to understand the complex genetic picture of FA metabolism and synthesis.
Given a nontrivial homogeneous ideal I ⊆ k[x 1 , x 2 , . . . , x d ], a problem of great recent interest has been the comparison of the rth ordinary power of I and the mth symbolic power I (m) . This comparison has been undertaken directly via an exploration of which exponents m and r guarantee the subset containment I (m) ⊆ I r and asymptotically via a computation of the resurgence ρ(I), a number for which any m/r > ρ(I) guarantees I (m) ⊆ I r . Recently, a third quantity, the symbolic defect, was introduced; as I t ⊆ I (t) , the symbolic defect is the minimal number of generators required to add to I t in order to get I (t) .We consider these various means of comparison when I is the edge ideal of certain graphs by describing an ideal J for which I (t) = I t + J. When I is the edge ideal of an odd cycle, our description of the structure of I (t) yields solutions to both the direct and asymptotic containment questions, as well as a partial computation of the sequence of symbolic defects.
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In the search for an understanding of how genetic variation contributes to the heritability of common human disease, the potential role of epigenetic factors, such as methylation, is being explored with increasing frequency. Although standard analyses test for associations between methylation levels at individual cytosine-phosphate-guanine (CpG) sites and phenotypes of interest, some investigators have begun testing for methylation and how methylation may modulate the effects of genetic polymorphisms on phenotypes. In our analysis, we used both a genome-wide and candidate gene approach to investigate potential single-nucleotide polymorphism (SNP)–CpG interactions on changes in triglyceride levels. Although we were able to identify numerous loci of interest when using an exploratory significance threshold, we did not identify any significant interactions using a strict genome-wide significance threshold. We were also able to identify numerous loci using the candidate gene approach, in which we focused on 18 genes with prior evidence of association of triglyceride levels. In particular, we identified GALNT2 loci as containing potential CpG sites that moderate the impact of genetic polymorphisms on triglyceride levels. Further work is needed to provide clear guidance on analytic strategies for testing SNP–CpG interactions, although leveraging prior biological understanding may be needed to improve statistical power in data sets with smaller sample sizes.
Although methylation data continues to rise in popularity, much is still unknown about how to best analyze methylation data in genome-wide analysis contexts. Given continuing interest in gene-based tests for next-generation sequencing data, we evaluated the performance of novel gene-based test statistics on simulated data from GAW20. Our analysis suggests that most of the gene-based tests are detecting real signals and maintaining the Type I error rate. The minimum p value and threshold-based tests performed well compared to single-marker tests in many cases, especially when the number of variants was relatively large with few true causal variants in the set.
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Rounding has proven to be a fundamental tool in theoretical computer science. By observing that rounding and partitioning of R d are equivalent, we introduce the following natural partition problem which we call the secluded hypercube partition problem: Given k ∈ N (ideally small) and > 0 (ideally large), is there a partition of R d with unit hypercubes such that for every point p ∈ R d , its closed -neighborhood (in the ∞ norm) intersects at most k hypercubes?We undertake a comprehensive study of this partition problem. We prove that for every d ∈ N, there is an explicit (and efficiently computable) hypercube partition of R d with k = d + 1 and = 1 2d . We complement this construction by proving that the value of k = d + 1 is the best possible (for any ) for a broad class of "reasonable" partitions including hypercube partitions. We also investigate the optimality of the parameter and prove that any partition in this broad class that has k = d + 1, must have ≤ 1 2 √ d . These bounds imply limitations of certain deterministic rounding schemes existing in the literature. Furthermore, this general bound is based on the currently known lower bounds for the dissection number of the cube, and improvements to this bound will yield improvements to our bounds.While our work is motivated by the desire to understand rounding algorithms, one of our main conceptual contributions is the introduction of the secluded hypercube partition problem, which fits well with a long history of investigations by mathematicians on various hypercube partitions/tilings of Euclidean space.
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