Projections of determinantal point processes

Let $\mathbf x=\{x^{(1)},\dots,x^{(n)}\}$ be a space filling-design of $n$ points defined in $[0{,}1]^d$. In computer experiments, an important property seeked for $\mathbf x$ is a nice coverage of $[0{,}1]^d$. This property could be desirable as well as for any projection of $\mathbf x$ onto $[0{,}1]^\iota$ for $\iota<d$ . Thus we expect that $\mathbf x_I=\{x_I^{(1)},\dots,x_I^{(n)}\}$, which represents the design $\mathbf x$ with coordinates associated to any index set $I\subseteq\{1,\dots,d\}$, remains regular in $[0{,}1]^\iota$ where $\iota$ is the cardinality of $I$. This paper examines the conservation of nice coverage by projection using spatial point processes, and more specifically using the class of determinantal point processes. We provide necessary conditions on the kernel defining these processes, ensuring that the projected point process $\mathbf{X}_I$ is repulsive, in the sense that its pair correlation function is uniformly bounded by 1, for all $I\subseteq\{1,\dots,d\}$. We present a few examples, compare them using a new normalized version of Ripley's function. Finally, we illustrate the interest of this research for Monte-Carlo integration

Projections of determinantal point processes

In computer experiments setting, space-filling designs are used to produce inputs, viewed as point patterns. A first important property of the design is that the point pattern covers regularly the input space. A second property is the conservation of this regular covering if the point pattern is projected onto a lower dimensional space. According to the first requirement, it seems then natural to consider classes of spatial point process which generate repulsive patterns. The class of determinantal point processes (DPPs) is considered in this paper. In particular, we address the question: Can we construct a DPP such that any projection on a lower-dimensional space remains a DPP, or at least remains repulsive? By assuming a particular form for the kernel defining the DPP, we prove rigorously that the answer is positive. We propose several examples of models, and in particular stationary models, achieving this property. These models defined on a compact set of $\mathbb{R}^d$ are shown to be efficient for Monte-Carlo integration problems; we show that the same initial spatial design, defined in $\mathbb{R}^d$, can be used to efficiently estimate integrals of $\mathbb{R}^\omega$-valued for any $\omega=1,\dots,d$

Genetic Variants For Head Size Share Genes and Pathways With Cancer

The size of the human head is highly heritable, but genetic drivers of its variation within the general population remain unmapped. We perform a genome-wide association study on head size (N = 80,890) and identify 67 genetic loci, of which 50 are novel. Neuroimaging studies show that 17 variants affect specific brain areas, but most have widespread effects. Gene set enrichment is observed for various cancers and the p53, Wnt, and ErbB signaling pathways. Genes harboring lead variants are enriched for macrocephaly syndrome genes (37-fold) and high-fidelity cancer genes (9-fold), which is not seen for human height variants. Head size variants are also near genes preferentially expressed in intermediate progenitor cells, neural cells linked to evolutionary brain expansion. Our results indicate that genes regulating early brain and cranial growth incline to neoplasia later in life, irrespective of height. This warrants investigation of clinical implications of the link between head size and cancer

Identification of a BRCA2-Specific modifier locus at 6p24 related to breast cancer risk

Common genetic variants contribute to the observed variation in breast cancer risk for BRCA2 mutation carriers; those known to date have all been found through population-based genome-wide association studies (GWAS). To comprehensively identify breast cancer risk modifying loci for BRCA2 mutation carriers, we conducted a deep replication of an ongoing GWAS discovery study. Using the ranked P-values of the breast cancer associations with the imputed genotype of 1.4 M SNPs, 19,029 SNPs were selected and designed for inclusion on a custom Illumina array that included a total of 211,155 SNPs as part of a multi-consortial project. DNA samples from 3,881 breast cancer affected and 4,330 unaffected BRCA2 mutation carriers from 47 studies belonging to the Consortium of Investigators of Modifiers of BRCA1/2 were genotyped and available for analysis. We replicated previously reported breast cancer susceptibility alleles in these BRCA2 mutation carriers and for several regions (including FGFR2, MAP3K1, CDKN2A/B, and PTHLH) identified SNPs that have stronger evidence of association than those previously published. We also identified a novel susceptibility allele at 6p24 that was inversely associated with risk in BRCA2 mutation carriers (rs9348512; per allele HR = 0.85, 95% CI 0.80-0.90, P = 3.9×10−8). This SNP was not associated with breast cancer risk either in the general population or in BRCA1 mutation carriers. The locus lies within a region containing TFAP2A, which encodes a transcriptional activation protein that interacts with several tumor suppressor genes. This report identifies the first breast cancer risk locus specific to a BRCA2 mutation background. This comprehensive update of novel and previously reported breast cancer susceptibility loci contributes to the establishment of a panel of SNPs that modify breast cancer risk in BRCA2 mutation carriers. This panel may have clinical utility for women with BRCA2 mutations weighing options for medical prevention of breast cancer

Genome-Wide Association Study in BRCA1 Mutation Carriers Identifies Novel Loci Associated with Breast and Ovarian Cancer Risk

BRCA1-associated breast and ovarian cancer risks can be modified by common genetic variants. To identify further cancer risk-modifying loci, we performed a multi-stage GWAS of 11,705 BRCA1 carriers (of whom 5,920 were diagnosed with breast and 1,839 were diagnosed with ovarian cancer), with a further replication in an additional sample of 2,646 BRCA1 carriers. We identified a novel breast cancer risk modifier locus at 1q32 for BRCA1 carriers (rs2290854, P = 2.7×10-8, HR = 1.14, 95% CI: 1.09-1.20). In addition, we identified two novel ovarian cancer risk modifier loci: 17q21.31 (rs17631303, P = 1.4×10-8, HR = 1.27, 95% CI: 1.17-1.38) and 4q32.3 (rs4691139, P = 3.4×10-8, HR = 1.20, 95% CI: 1.17-1.38). The 4q32.3 locus was not associated with ovarian cancer risk in the general population or BRCA2 carriers, suggesting a BRCA1-specific associat

An original phylogenetic approach identified mitochondrial haplogroup T1a1 as inversely associated with breast cancer risk in BRCA2 mutation carriers

Introduction: Individuals carrying pathogenic mutations in the BRCA1 and BRCA2 genes have a high lifetime risk of breast cancer. BRCA1 and BRCA2 are involved in DNA double-strand break repair, DNA alterations that can be caused by exposure to reactive oxygen species, a main source of which are mitochondria. Mitochondrial genome variations affect electron transport chain efficiency and reactive oxygen species production. Individuals with different mitochondrial haplogroups differ in their metabolism and sensitivity to oxidative stress. Variability in mitochondrial genetic background can alter reactive oxygen species production, leading to cancer risk. In the present study, we tested the hypothesis that mitochondrial haplogroups modify breast cancer risk in BRCA1/2 mutation carriers. Methods: We genotyped 22,214 (11,421 affected, 10,793 unaffected) mutation carriers belonging to the Consortium of Investigators of Modifiers of BRCA1/2 for 129 mitochondrial polymorphisms using the iCOGS array. Haplogroup inference and association detection were performed using a phylogenetic approach. ALTree was applied to explore the reference mitochondrial evolutionary tree and detect subclades enriched in affected or unaffected individuals. Results: We discovered that subclade T1a1 was depleted in affected BRCA2 mutation carriers compared with the rest of clade T (hazard ratio (HR) = 0.55; 95% confidence interval (CI), 0.34 to 0.88; P = 0.01). Compared with the most frequent haplogroup in the general population (that is, H and T clades), the T1a1 haplogroup has a HR of 0.62 (95% CI, 0.40 to 0.95; P = 0.03). We also identified three potential susceptibility loci, including G13708A/rs28359178, which has demonstrated an inverse association with familial breast cancer risk. Conclusions: This study illustrates how original approaches such as the phylogeny-based method we used can empower classical molecular epidemiological studies aimed at identifying association or risk modification effects.Peer reviewe

Common breast cancer susceptibility alleles are associated with tumor subtypes in BRCA1 and BRCA2 mutation carriers : results from the Consortium of Investigators of Modifiers of BRCA1/2.

Abstract Introduction Previous studies have demonstrated that common breast cancer susceptibility alleles are differentially associated with breast cancer risk for BRCA1 and/or BRCA2 mutation carriers. It is currently unknown how these alleles are associated with different breast cancer subtypes in BRCA1 and BRCA2 mutation carriers defined by estrogen (ER) or progesterone receptor (PR) status of the tumour. Methods We used genotype data on up to 11,421 BRCA1 and 7,080 BRCA2 carriers, of whom 4,310 had been affected with breast cancer and had information on either ER or PR status of the tumour, to assess the associations of 12 loci with breast cancer tumour characteristics. Associations were evaluated using a retrospective cohort approach. Results The results suggested stronger associations with ER-positive breast cancer than ER-negative for 11 loci in both BRCA1 and BRCA2 carriers. Among BRCA1 carriers, single nucleotide polymorphism (SNP) rs2981582 (FGFR2) exhibited the biggest difference based on ER status (per-allele hazard ratio (HR) for ER-positive = 1.35, 95% CI: 1.17 to 1.56 vs HR = 0.91, 95% CI: 0.85 to 0.98 for ER-negative, P-heterogeneity = 6.5 &#215; 10-6). In contrast, SNP rs2046210 at 6q25.1 near ESR1 was primarily associated with ER-negative breast cancer risk for both BRCA1 and BRCA2 carriers. In BRCA2 carriers, SNPs in FGFR2, TOX3, LSP1, SLC4A7/NEK10, 5p12, 2q35, and 1p11.2 were significantly associated with ER-positive but not ER-negative disease. Similar results were observed when differentiating breast cancer cases by PR status. Conclusions The associations of the 12 SNPs with risk for BRCA1 and BRCA2 carriers differ by ER-positive or ER-negative breast cancer status. The apparent differences in SNP associations between BRCA1 and BRCA2 carriers, and non-carriers, may be explicable by differences in the prevalence of tumour subtypes. As more risk modifying variants are identified, incorporating these associations into breast cancer subtype-specific risk models may improve clinical management for mutation carriers