A Survey on the Krein-von Neumann Extension, the corresponding Abstract Buckling Problem, and Weyl-Type Spectral Asymptotics for Perturbed Krein Laplacians in Nonsmooth Domains

In the first (and abstract) part of this survey we prove the unitary equivalence of the inverse of the Krein--von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, $S\geq \varepsilon I_{\mathcal{H}}$ for some $\varepsilon >0$ in a Hilbert space $\mathcal{H}$ to an abstract buckling problem operator. This establishes the Krein extension as a natural object in elasticity theory (in analogy to the Friedrichs extension, which found natural applications in quantum mechanics, elasticity, etc.). In the second, and principal part of this survey, we study spectral properties for $H_{K,\Omega}$, the Krein--von Neumann extension of the perturbed Laplacian $-\Delta+V$ (in short, the perturbed Krein Laplacian) defined on $C^\infty_0(\Omega)$, where $V$ is measurable, bounded and nonnegative, in a bounded open set $\Omega\subset\mathbb{R}^n$ belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class $C^{1,r}$, $r>1/2$.Comment: 68 pages. arXiv admin note: extreme text overlap with arXiv:0907.144

Investigating the Causal Relationship of C-Reactive Protein with 32 Complex Somatic and Psychiatric Outcomes: A Large-Scale Cross-Consortium Mendelian Randomization Study

BACKGROUND: C-reactive protein (CRP) is associated with immune, cardiometabolic, and psychiatric traits and diseases. Yet it is inconclusive whether these associations are causal. METHODS AND FINDINGS: We performed Mendelian randomization (MR) analyses using two genetic risk scores (GRSs) as instrumental variables (IVs). The first GRS consisted of four single nucleotide polymorphisms (SNPs) in the CRP gene (GRSCRP), and the second consisted of 18 SNPs that were significantly associated with CRP levels in the largest genome-wide association study (GWAS) to date (GRSGWAS). To optimize power, we used summary statistics from GWAS consortia and tested the association of these two GRSs with 32 complex somatic and psychiatric outcomes, with up to 123,865 participants per outcome from populations of European ancestry. We performed heterogeneity tests to disentangle the pleiotropic effect of IVs. A Bonferroni-corrected significance level of less than 0.0016 was considered statistically significant. An observed p-value equal to or less than 0.05 was considered nominally significant evidence for a potential causal association, yet to be confirmed. The strengths (F-statistics) of the IVs were 31.92-3,761.29 and 82.32-9,403.21 for GRSCRP and GRSGWAS, respectively. CRP GRSGWAS showed a statistically significant protective relationship of a 10% genetically elevated CRP level with the risk of schizophrenia (odds ratio [OR] 0.86 [95% CI 0.79-0.94]; p < 0.001). We validated this finding with individual-level genotype data from the schizophrenia GWAS (OR 0.96 [95% CI 0.94-0.98]; p < 1.72 × 10-6). Further, we found that a standardized CRP polygenic risk score (CRPPRS) at p-value thresholds of 1 × 10-4, 0.001, 0.01, 0.05, and 0.1 using individual-level data also showed a protective effect (OR < 1.00) against schizophrenia; the first CRPPRS (built of SNPs with p < 1 × 10-4) showed a statistically significant (p < 2.45 × 10-4) protective effect with an OR of 0.97 (95% CI 0.95-0.99). The CRP GRSGWAS showed that a 10% increase in genetically determined CRP level was significantly associated with coronary artery disease (OR 0.88 [95% CI 0.84-0.94]; p < 2.4 × 10-5) and was nominally associated with the risk of inflammatory bowel disease (OR 0.85 [95% CI 0.74-0.98]; p < 0.03), Crohn disease (OR 0.81 [95% CI 0.70-0.94]; p < 0.005), psoriatic arthritis (OR 1.36 [95% CI 1.00-1.84]; p < 0.049), knee osteoarthritis (OR 1.17 [95% CI 1.01-1.36]; p < 0.04), and bipolar disorder (OR 1.21 [95% CI 1.05-1.40]; p < 0.007) and with an increase of 0.72 (95% CI 0.11-1.34; p < 0.02) mm Hg in systolic blood pressure, 0.45 (95% CI 0.06-0.84; p < 0.02) mm Hg in diastolic blood pressure, 0.01 ml/min/1.73 m2 (95% CI 0.003-0.02; p < 0.005) in estimated glomerular filtration rate from serum creatinine, 0.01 g/dl (95% CI 0.0004-0.02; p < 0.04) in serum albumin level, and 0.03 g/dl (95% CI 0.008-0.05; p < 0.009) in serum protein level. However, after adjustment for heterogeneity, neither GRS showed a significant effect of CRP level (at p < 0.0016) on any of these outcomes, including coronary artery disease, nor on the other 20 complex outcomes studied. Our study has two potential limitations: the limited variance explained by our genetic instruments modeling CRP levels in blood and the unobserved bias introduced by the use of summary statistics in our MR analyses. CONCLUSIONS: Genetically elevated CRP levels showed a significant potentially protective causal relationship with risk of schizophrenia. We observed nominal evidence at an observed p < 0.05 using either GRSCRP or GRSGWAS-with persistence after correction for heterogeneity-for a causal relationship of elevated CRP levels with psoriatic osteoarthritis, rheumatoid arthritis, knee osteoarthritis, systolic blood pressure, diastolic blood pressure, serum albumin, and bipolar disorder. These associations remain yet to be confirmed. We cannot verify any causal effect of CRP level on any of the other common somatic and neuropsychiatric outcomes investigated in the present study. This implies that interventions that lower CRP level are unlikely to result in decreased risk for the majority of common complex outcomes

Meta-analysis of GWAS of over 16,000 individuals with autism spectrum disorder highlights a novel locus at 10q24.32 and a significant overlap with schizophrenia.

To access publisher's full text version of this article, please click on the hyperlink in Additional Links field or click on the hyperlink at the top of the page marked FilesOver the past decade genome-wide association studies (GWAS) have been applied to aid in the understanding of the biology of traits. The success of this approach is governed by the underlying effect sizes carried by the true risk variants and the corresponding statistical power to observe such effects given the study design and sample size under investigation. Previous ASD GWAS have identified genome-wide significant (GWS) risk loci; however, these studies were of only of low statistical power to identify GWS loci at the lower effect sizes (odds ratio (OR) <1.15).We conducted a large-scale coordinated international collaboration to combine independent genotyping data to improve the statistical power and aid in robust discovery of GWS loci. This study uses genome-wide genotyping data from a discovery sample (7387 ASD cases and 8567 controls) followed by meta-analysis of summary statistics from two replication sets (7783 ASD cases and 11359 controls; and 1369 ASD cases and 137308 controls).We observe a GWS locus at 10q24.32 that overlaps several genes including PITX3, which encodes a transcription factor identified as playing a role in neuronal differentiation and CUEDC2 previously reported to be associated with social skills in an independent population cohort. We also observe overlap with regions previously implicated in schizophrenia which was further supported by a strong genetic correlation between these disorders (Rg = 0.23; P = 9 × 10(-6)). We further combined these Psychiatric Genomics Consortium (PGC) ASD GWAS data with the recent PGC schizophrenia GWAS to identify additional regions which may be important in a common neurodevelopmental phenotype and identified 12 novel GWS loci. These include loci previously implicated in ASD such as FOXP1 at 3p13, ATP2B2 at 3p25.3, and a 'neurodevelopmental hub' on chromosome 8p11.23.This study is an important step in the ongoing endeavour to identify the loci which underpin the common variant signal in ASD. In addition to novel GWS loci, we have identified a significant genetic correlation with schizophrenia and association of ASD with several neurodevelopmental-related genes such as EXT1, ASTN2, MACROD2, and HDAC4.National Institutes of Mental Health (NIMH, USA) ACE Network Autism Genetic Resource Exchange (AGRE) is a program of Autism Speaks (USA) The Autism Genome Project (AGP) from Autism Speaks (USA) Canadian Institutes of Health Research (CIHR), Genome Canada Health Research Board (Ireland) Hilibrand Foundation (USA) Medical Research Council (UK) National Institutes of Health (USA) Ontario Genomics Institute University of Toronto McLaughlin Centre Simons Foundation Johns Hopkins Autism Consortium of Boston NLM Family foundation National Institute of Health grants National Health Medical Research Council Scottish Rite Spunk Fund, Inc. Rebecca and Solomon Baker Fund APEX Foundation National Alliance for Research in Schizophrenia and Affective Disorders (NARSAD) endowment fund of the Nancy Pritzker Laboratory (Stanford) Autism Society of America Janet M. Grace Pervasive Developmental Disorders Fund The Lundbeck Foundation universities and university hospitals of Aarhus and Copenhagen Stanley Foundation Centers for Disease Control and Prevention (CDC) Netherlands Scientific Organization Dutch Brain Foundation VU University Amsterdam Trinity Centre for High Performance Computing through Science Foundation Ireland Autism Genome Project (AGP) from Autism Speak

Analysis Of Shared Heritability In Common Disorders Of The Brain

Disorders of the brain exhibit considerable epidemiological comorbidity and frequently share symptoms, provoking debate about the extent of their etiologic overlap. Here we apply linkage disequilibrium score regression (LDSC) to quantify the extent of shared genetic contributions across 23 brain disorders (n=842,820), 11 quantitative and four dichotomous traits of interest (n=722,125) based on genome-wide association meta-analyses. Psychiatric disorders show substantial sharing of common variant risk, while many neurological disorders appear more distinct from one another, suggesting substantive differences in the specificity of the genetic etiology of these disorders. Further, we observe little evidence of widespread sharing of the common genetic risk between neurological and psychiatric disorders studied. In addition, we identify significant sharing of genetic influences between the certain quantitative measures and brain disorders, including major depressive disorder and neuroticism personality score. These results highlight the importance of common genetic variation as a source of risk for brain disorders and the potential of using heritability methods to obtain a more comprehensive view of the genetic architecture of brain phenotypes

Rediscovering the value of families for psychiatric genetics research

As it is likely that both common and rare genetic variation are important for complex disease risk, studies that examine the full range of the allelic frequency distribution should be utilized to dissect the genetic influences on mental illness. The rate limiting factor for inferring an association between a variant and a phenotype is inevitably the total number of copies of the minor allele captured in the studied sample. For rare variation, with minor allele frequencies of 0.5% or less, very large samples of unrelated individuals are necessary to unambiguously associate a locus with an illness. Unfortunately, such large samples are often cost prohibitive. However, by using alternative analytic strategies and studying related individuals, particularly those from large multiplex families, it is possible to reduce the required sample size while maintaining statistical power. We contend that using whole genome sequence (WGS) in extended pedigrees provides a cost-effective strategy for psychiatric gene mapping that complements common variant approaches and WGS in unrelated individuals. This was our impetus for forming the “Pedigree-Based Whole Genome Sequencing of Affective and Psychotic Disorders” consortium. In this review, we provide a rationale for the use of WGS with pedigrees in modern psychiatric genetics research. We begin with a focused review of the current literature, followed by a short history of family-based research in psychiatry. Next, we describe several advantages of pedigrees for WGS research, including power estimates, methods for studying the environment, and endophenotypes. We conclude with a brief description of our consortium and its goals.This research was supported by National Institute of Mental Health grants U01 MH105630 (DCG), U01 MH105634 (REG), U01 MH105632 (JB), R01 MH078143 (DCG), R01 MH083824 (DCG & JB), R01 MH078111 (JB), R01 MH061622 (LA), R01 MH042191 (REG), and R01 MH063480 (VLN).UCR::Vicerrectoría de Investigación::Unidades de Investigación::Ciencias Básicas::Centro de Investigación en Biología Celular y Molecular (CIBCM)UCR::Vicerrectoría de Docencia::Ciencias Básicas::Facultad de Ciencias::Escuela de Biologí