Eigenvalues of Dirichlet Laplacian within the class of open sets with constant diameter

This paper is about a shape optimization problem related to the Dirichlet Laplacian eingevalues in the Euclidean plane. More precisely we study the shape of the minimizer in the class of open sets of constant width. We prove that the disk is not a local minimizer except for a limited number of eigenvalues

Comparison of results from stochastic GUESSFM and stepwise searches.

<p><i>p</i> values are shown for the stepwise search and compare the listed model to the model above (or the null model, for single SNPs). Bayesian Information Criterion (BIC) is shown for stepwise models and index models found through GUESSFM search.</p

Haplotype analysis of MS association using the index SNPs for the one and two SNP models selected.

<p>Minor alleles are indicated by bold font. Fq = haplotype frequency.</p

The proportion of naive CD4⁺ T cells that express CD25 (log scale) increases with age.

<p>The MS protective allele for the M2 SNP rs41295055:C > T associates with fewer CD4⁺ T cells expressing CD25 across all ages (<i>p</i> = 3.45 × 10⁻⁸), and is statistically preferred to the previously reported M1 SNP, rs2104286:T > C (<i>p</i> = 2.56 × 10⁻⁶; Δ BIC = 8.43). S and P are used to represent the (common) MS-susceptible and (rare) MS-protective alleles respectively at each SNP. These SNPs are in limited LD (<i>r</i>² = 0.3).</p

Six sets of SNPs can best explain the association of T1D and MS in the chromosome 10p15 region.

<p><b>LD</b>: a heatmap indicating the <i>r</i>² between SNPs. <b>Assoc</b>: MPPI for MS and T1D the SNPs in a group, with total MPPI across a SNP group, gMPPI, indicated by the height of the shaded rectangle (see <a href="http://www.plosgenetics.org/article/info:doi/10.1371/journal.pgen.1005272#pgen.1005272.t005" target="_blank">Table 5</a> for numerical details). SNP groups are labelled by the letters A-F for reference. SNPs in this track are ordered by SNP group for ease of visualisation. <b>Genes</b>: SNPs are mapped back to physical position and shown in relation to genes in the region. <b>RNAseq</b>: read counts in two pooled replicates of resting (“rest1” and “rest2”) and anti-CD3/CD28 stimulated (“stim1” and “stim2”) CD4⁺ T cells; y axes were truncated to allow visualization of intronic read counts. Note the different limits for resting and stimulated cells, which show greater transcription of all protein coding genes in the region. <b>DNase</b>: DNase hypersensitivity measured in CD4 cells by the Roadmap consortium. Replicate 1 (“rest1”) is RO_01689; replicate 2 (“rest2”) is RO_01736; y axes were truncated again to improve visualization.</p

Comparison of of several multivariate methods for fine mapping using simulated data.

<p>We simulated quantitative phenotype data with between two and five causal variants using genotype data from the T1D dataset for the <i>IL2RA</i> region. The simulated data sets were analysed using forward stepwise regression, GUESSFM, the lasso, the group lasso and the elastic net. GUESSFM produces credible sets for each variant chosen using the snp.picker algorithm described in Materials and Methods. We defined pseudo “credible sets” for the other approaches as the set of SNPs with <i>r</i>² > 0.8 with a selected SNP. We calculated the discovery rate (the proportion of causal variants within at least one credible set, y axis) and false discovery rate (proportion of detected variants whose credible sets did not contain any causal variant, x axis) at different thresholds for the stepwise <i>p</i> value, the group marginal posterior probability of inclusion (gMPPI) for GUESSFM and the regularization parameter(s) across simulated datasets (see <a href="http://www.plosgenetics.org/article/info:doi/10.1371/journal.pgen.1005272#sec004" target="_blank">Methods</a> for details). GUESSFM-3 and GUESSFM-5 refer to GUESSFM run with a prior expectation of three or five causal variants per region, respectively. Results are averaged over 1000 replicates.</p

Overview of the fine mapping tailored stochastic search strategy in GUESSFM.

<p>1. SNPs are clustered based on genotype data. Tagging is used to remove cases of extreme LD (<i>r</i>² > 0.99) by selecting one SNP from each cluster (“tag set”), that which is in highest average <i>r</i>² with all other SNPs. 2. All possible models that can be formed from the tag SNPs may be considered by GUESS. Here, all seven possible models are considered but, in practice, with larger numbers of tags than shown here, GUESS employs a stochastic search strategy to consider only a subset of models, prioritising those with greatest statistical support. 3. GUESS selects the most likely models amongst those it has visited. Here, it selects two of the seven, but in larger data sets we retain the 30,000 most likely. 4. Each of these selected models is expanded by considering all possible substitutions of tags by other members of their tag set. Each expanded model is then assessed again individually, using an approximate Bayes factor [<a href="http://www.plosgenetics.org/article/info:doi/10.1371/journal.pgen.1005272#pgen.1005272.ref014" target="_blank">14</a>].</p

LD (<i>r</i>²) between SNPs in selected models for T1D using stochastic GUESSFM and stepwise searches.

<p>Each GUESSFM search signal has a correspondng SNP found by conditional stepwise regression in moderate to strong LD.</p