Visualizing Spacetime Curvature via Frame-Drag Vortexes and Tidal Tendexes II. Stationary Black Holes

When one splits spacetime into space plus time, the Weyl curvature tensor (which equals the Riemann tensor in vacuum) splits into two spatial, symmetric, traceless tensors: the tidal field $E$, which produces tidal forces, and the frame-drag field $B$, which produces differential frame dragging. In recent papers, we and colleagues have introduced ways to visualize these two fields: tidal tendex lines (integral curves of the three eigenvector fields of $E$) and their tendicities (eigenvalues of these eigenvector fields); and the corresponding entities for the frame-drag field: frame-drag vortex lines and their vorticities. These entities fully characterize the vacuum Riemann tensor. In this paper, we compute and depict the tendex and vortex lines, and their tendicities and vorticities, outside the horizons of stationary (Schwarzschild and Kerr) black holes; and we introduce and depict the black holes' horizon tendicity and vorticity (the normal-normal components of $E$ and $B$ on the horizon). For Schwarzschild and Kerr black holes, the horizon tendicity is proportional to the horizon's intrinsic scalar curvature, and the horizon vorticity is proportional to an extrinsic scalar curvature. We show that, for horizon-penetrating time slices, all these entities ($E$, $B$, the tendex lines and vortex lines, the lines' tendicities and vorticities, and the horizon tendicities and vorticities) are affected only weakly by changes of slicing and changes of spatial coordinates, within those slicing and coordinate choices that are commonly used for black holes. [Abstract is abbreviated.]Comment: 19 pages, 7 figures, v2: Changed to reflect published version (changes made to color scales in Figs 5, 6, and 7 for consistent conventions). v3: Fixed Ref

Frame-Dragging Vortexes and Tidal Tendexes Attached to Colliding Black Holes: Visualizing the Curvature of Spacetime

When one splits spacetime into space plus time, the spacetime curvature (Weyl tensor) gets split into an "electric" part E_{jk} that describes tidal gravity and a "magnetic" part B_{jk} that describes differential dragging of inertial frames. We introduce tools for visualizing B_{jk} (frame-drag vortex lines, their vorticity, and vortexes) and E_{jk} (tidal tendex lines, their tendicity, and tendexes), and also visualizations of a black-hole horizon's (scalar) vorticity and tendicity. We use these tools to elucidate the nonlinear dynamics of curved spacetime in merging black-hole binaries.Comment: 4 pages, 5 figure

Visualizing Spacetime Curvature via Frame-Drag Vortexes and Tidal Tendexes III. Quasinormal Pulsations of Schwarzschild and Kerr Black Holes

In recent papers, we and colleagues have introduced a way to visualize the full vacuum Riemann curvature tensor using frame-drag vortex lines and their vorticities, and tidal tendex lines and their tendicities. We have also introduced the concepts of horizon vortexes and tendexes and 3-D vortexes and tendexes (regions where vorticities or tendicities are large). Using these concepts, we discover a number of previously unknown features of quasinormal modes of Schwarzschild and Kerr black holes. These modes can be classified by mode indexes (n,l,m), and parity, which can be electric [(-1)^l] or magnetic [(-1)^(l+1)]. Among our discoveries are these: (i) There is a near duality between modes of the same (n,l,m): a duality in which the tendex and vortex structures of electric-parity modes are interchanged with the vortex and tendex structures (respectively) of magnetic-parity modes. (ii) This near duality is perfect for the modes' complex eigenfrequencies (which are well known to be identical) and perfect on the horizon; it is slightly broken in the equatorial plane of a non-spinning hole, and the breaking becomes greater out of the equatorial plane, and greater as the hole is spun up; but even out of the plane for fast-spinning holes, the duality is surprisingly good. (iii) Electric-parity modes can be regarded as generated by 3-D tendexes that stick radially out of the horizon. As these "longitudinal," near-zone tendexes rotate or oscillate, they generate longitudinal-transverse near-zone vortexes and tendexes, and outgoing and ingoing gravitational waves. The ingoing waves act back on the longitudinal tendexes, driving them to slide off the horizon, which results in decay of the mode's strength. (iv) By duality, magnetic-parity modes are driven in this same manner by longitudinal, near-zone vortexes that stick out of the horizon. [Abstract abridged.]Comment: 53 pages with an overview of major results in the first 11 pages, 26 figures. v2: Very minor changes to reflect published version. v3: Fixed Ref

Frame-Dragging Vortexes and Tidal Tendexes Attached to Colliding Black Holes: Visualizing the Curvature of Spacetime

When one splits spacetime into space plus time, the spacetime curvature (Weyl tensor) gets split into an "electric" part E_{jk} that describes tidal gravity and a "magnetic" part B_{jk} that describes differential dragging of inertial frames. We introduce tools for visualizing B_{jk} (frame-drag vortex lines, their vorticity, and vortexes) and E_{jk} (tidal tendex lines, their tendicity, and tendexes), and also visualizations of a black-hole horizon's (scalar) vorticity and tendicity. We use these tools to elucidate the nonlinear dynamics of curved spacetime in merging black-hole binaries.Comment: 4 pages, 5 figure

Visualizing Spacetime Curvature via Frame-Drag Vortexes and Tidal Tendexes I. General Theory and Weak-Gravity Applications

When one splits spacetime into space plus time, the Weyl curvature tensor (vacuum Riemann tensor) gets split into two spatial, symmetric, and trace-free (STF) tensors: (i) the Weyl tensor's so-called "electric" part or tidal field, and (ii) the Weyl tensor's so-called "magnetic" part or frame-drag field. Being STF, the tidal field and frame-drag field each have three orthogonal eigenvector fields which can be depicted by their integral curves. We call the integral curves of the tidal field's eigenvectors tendex lines, we call each tendex line's eigenvalue its tendicity, and we give the name tendex to a collection of tendex lines with large tendicity. The analogous quantities for the frame-drag field are vortex lines, their vorticities, and vortexes. We build up physical intuition into these concepts by applying them to a variety of weak-gravity phenomena: a spinning, gravitating point particle, two such particles side by side, a plane gravitational wave, a point particle with a dynamical current-quadrupole moment or dynamical mass-quadrupole moment, and a slow-motion binary system made of nonspinning point particles. [Abstract is abbreviated; full abstract also mentions additional results.]Comment: 25 pages, 20 figures, matches the published versio

Nucleic Acid-Sensing and Interferon-Inducible Pathways Show Differential Methylation in MZ Twins Discordant for Lupus and Overexpression in Independent Lupus Samples: Implications for Pathogenic Mechanism and Drug Targeting.

Systemic lupus erythematosus (SLE) is a chronic, multisystem, autoimmune inflammatory disease with genomic and non-genomic contributions to risk. We hypothesize that epigenetic factors are a significant contributor to SLE risk and may be informative for identifying pathogenic mechanisms and therapeutic targets. To test this hypothesis while controlling for genetic background, we performed an epigenome-wide analysis of DNA methylation in genomic DNA from whole blood in three pairs of female monozygotic (MZ) twins of European ancestry, discordant for SLE. Results were replicated on the same array in four cell types from a set of four Danish female MZ twin pairs discordant for SLE. Genes implicated by the epigenetic analyses were then evaluated in 10 independent SLE gene expression datasets from the Gene Expression Omnibus (GEO). There were 59 differentially methylated loci between unaffected and affected MZ twins in whole blood, including 11 novel loci. All but two of these loci were hypomethylated in the SLE twins relative to the unaffected twins. The genes harboring these hypomethylated loci exhibited increased expression in multiple independent datasets of SLE patients. This pattern was largely consistent regardless of disease activity, cell type, or renal tissue type. The genes proximal to CpGs exhibiting differential methylation (DM) in the SLE-discordant MZ twins and exhibiting differential expression (DE) in independent SLE GEO cohorts (DM-DE genes) clustered into two pathways: the nucleic acid-sensing pathway and the type I interferon pathway. The DM-DE genes were also informatically queried for potential gene-drug interactions, yielding a list of 41 drugs including a known SLE therapy. The DM-DE genes delineate two important biologic pathways that are not only reflective of the heterogeneity of SLE but may also correlate with distinct IFN responses that depend on the source, type, and location of nucleic acid molecules and the activated receptors in individual patients. Cell- and tissue-specific analyses will be critical to the understanding of genetic factors dysregulating the nucleic acid-sensing and IFN pathways and whether these factors could be appropriate targets for therapeutic intervention

A molecular biomarker for prediction of clinical outcome in children with ASD, constipation, and intestinal inflammation

Abstract In children with autism spectrum disorder (ASD) who present to the gastroenterologist with chronic constipation on a background of colonic inflammation, we have identified two distinct clinical subtypes: (1) patients who experience a sustained state of GI symptomatic remission while on maintenance anti-inflammatory therapy (fast responders) and, (2) those with recurrent right-sided fecal loading requiring regular colon cleanouts during treatment for enterocolitis (slow responders). We hypothesized that a detailed molecular analysis of tissue from the affected region of the colon would provide mechanistic insights regarding the fast versus slow response to anti-inflammatory therapy. To test this, ascending colon biopsy tissues from 35 children with ASD (20 slow responders and 15 fast responders) were analyzed by RNAseq. Hierarchical cluster analysis was performed to assign samples to clusters and gene expression analysis was performed to identify differentially expressed transcripts (DETs) between samples within the clusters. Significant differences were found between the two clusters with fast responder-predominant cluster showing an upregulation of transcripts involved in the activation of immune and inflammatory response and the slow responder-predominant cluster showing significant over-representation of pathways impacting colonic motility (e.g. genes involved in tryptophan and serotonin degradation and mitochondrial dysfunction). Regression analysis identified a single long non-coding RNA that could predict cluster assignment with a high specificity (0.88), sensitivity (0.89) and accuracy (0.89). Comparison of gene expression profiles in the ascending colon from a subset of patients with ASD, chronic right-sided fecal loading constipation and a slow versus fast response to therapy has identified molecular mechanisms that likely contribute to this differential response following the primary therapeutic intervention (i.e. treatment for colonic inflammation with brief induction immunosuppression followed by maintenance non-steroidal anti-inflammatory therapy). Importantly, we have identified a transcript that, if validated, may provide a biomarker that can predict from the outset which patients will be slow responders who would benefit from an alternate therapeutic strategy in treating their constipation

An Isobaric Labeling Approach to Enhance Detection and Quantification of Tissue-Derived Plasma Proteins as Potential Early Disease Biomarkers

The proteomic analysis of plasma holds great promise to advance precision medicine and identify biomarkers of disease. However, it is likely that many potential biomarkers circulating in plasma originate from other tissues and are only present in low abundances in the plasma. Accurate detection and quantification of low abundance proteins by standard mass spectrometry approaches remain challenging. In addition, it is difficult to link low abundance plasma proteins back to their specific tissues or organs of origin with confidence. To address these challenges, we developed a mass spectrometry approach based on the use of tandem mass tags (TMT) and a tissue reference sample. By applying this approach to nonhuman primate plasma samples, we were able to identify and quantify 820 proteins by using a kidney tissue homogenate as reference. On average, 643 ± 16 proteins were identified per plasma sample. About 58% of proteins identified in replicate experiments were identified both times. A ratio of 50 μg kidney protein to 10 μg plasma protein, and the use of the TMT label with the highest molecular weight (131) for the kidney reference yielded the largest number of proteins in the analysis, and identified low abundance proteins in plasma that are prominently found in the kidney. Overall, this methodology promises efficient quantification of plasma proteins potentially released from specific tissues, thereby increasing the number of putative disease biomarkers for future study

Optimization of Imputation Strategies for High-Resolution Gas Chromatography–Mass Spectrometry (HR GC–MS) Metabolomics Data

Gas chromatography–coupled mass spectrometry (GC–MS) has been used in biomedical research to analyze volatile, non-polar, and polar metabolites in a wide array of sample types. Despite advances in technology, missing values are still common in metabolomics datasets and must be properly handled. We evaluated the performance of ten commonly used missing value imputation methods with metabolites analyzed on an HR GC–MS instrument. By introducing missing values into the complete (i.e., data without any missing values) National Institute of Standards and Technology (NIST) plasma dataset, we demonstrate that random forest (RF), glmnet ridge regression (GRR), and Bayesian principal component analysis (BPCA) shared the lowest root mean squared error (RMSE) in technical replicate data. Further examination of these three methods in data from baboon plasma and liver samples demonstrated they all maintained high accuracy. Overall, our analysis suggests that any of the three imputation methods can be applied effectively to untargeted metabolomics datasets with high accuracy. However, it is important to note that imputation will alter the correlation structure of the dataset and bias downstream regression coefficients and p-values