Grassmannians,Calibrations and Five-Brane Intersections

We present a geometric construction of a new class of hyper-Kahler manifolds with torsion. This involves the superposition of the four-dimensional hyper-Kahler geometry with torsion associated with the NS-5-brane along quaternionic planes in quaternionic k-space, \bH^k. We find the moduli space of these geometries and show that it can be constructed using the bundle space of the canonical quaternionic line bundle over a quaternionic projective space. We also investigate several special cases which are associated with certain classes of quaternionic planes in \bH^k. We then show that the eight-dimensional geometries we have found can be constructed using quaternionic calibrations. We generalize our construction to superpose the same four-dimensional hyper-Kahler geometry with torsion along complex planes in \bC^{2k}. We find that the resulting geometry is Kahler with torsion. The moduli space of these geometries is also investigated. In addition, the applications of these new geometries to M-theory and sigma models are presented. In particular, we find new solutions of IIA supergravity with the interpretation of intersecting NS-5-branes at Sp(2)-angles on a string and show that they preserve 3/32, 1/8, 5/32 and 3/16 of supersymmetry. We also show that two-dimensional sigma models with target spaces the above manifolds have (p,q) extended supersymmetry.Comment: 39 pages, phyzzx; a previously undetermined fraction of supersymmetry has now been fixed; a table has been replaced; version submitted for publication in CM

Instantons at Angles

We interpret a class of 4k-dimensional instanton solutions found by Ward, Corrigan, Goddard and Kent as four-dimensional instantons at angles. The superposition of each pair of four-dimensional instantons is associated with four angles which depend on some of the ADHM parameters. All these solutions are associated with the group $Sp(k)$ and are examples of Hermitian-Einstein connections on \bE^{4k}. We show that the eight-dimensional solutions preserve 3/16 of the ten-dimensional N=1 supersymmetry. We argue that under the correspondence between the BPS states of Yang-Mills theory and those of M-theory that arises in the context of Matrix models, the instantons at angles configuration corresponds to the longitudinal intersecting 5-branes on a string at angles configuration of M-theory.Comment: 17 pages, phyzzx, many changes and a new section was adde

Intra-tumour signalling entropy determines clinical outcome in breast and lung cancer.

The cancer stem cell hypothesis, that a small population of tumour cells are responsible for tumorigenesis and cancer progression, is becoming widely accepted and recent evidence has suggested a prognostic and predictive role for such cells. Intra-tumour heterogeneity, the diversity of the cancer cell population within the tumour of an individual patient, is related to cancer stem cells and is also considered a potential prognostic indicator in oncology. The measurement of cancer stem cell abundance and intra-tumour heterogeneity in a clinically relevant manner however, currently presents a challenge. Here we propose signalling entropy, a measure of signalling pathway promiscuity derived from a sample's genome-wide gene expression profile, as an estimate of the stemness of a tumour sample. By considering over 500 mixtures of diverse cellular expression profiles, we reveal that signalling entropy also associates with intra-tumour heterogeneity. By analysing 3668 breast cancer and 1692 lung adenocarcinoma samples, we further demonstrate that signalling entropy correlates negatively with survival, outperforming leading clinical gene expression based prognostic tools. Signalling entropy is found to be a general prognostic measure, valid in different breast cancer clinical subgroups, as well as within stage I lung adenocarcinoma. We find that its prognostic power is driven by genes involved in cancer stem cells and treatment resistance. In summary, by approximating both stemness and intra-tumour heterogeneity, signalling entropy provides a powerful prognostic measure across different epithelial cancers

Approximate entropy of network parameters

We study the notion of approximate entropy within the framework of network theory. Approximate entropy is an uncertainty measure originally proposed in the context of dynamical systems and time series. We firstly define a purely structural entropy obtained by computing the approximate entropy of the so called slide sequence. This is a surrogate of the degree sequence and it is suggested by the frequency partition of a graph. We examine this quantity for standard scale-free and Erd\H{o}s-R\'enyi networks. By using classical results of Pincus, we show that our entropy measure converges with network size to a certain binary Shannon entropy. On a second step, with specific attention to networks generated by dynamical processes, we investigate approximate entropy of horizontal visibility graphs. Visibility graphs permit to naturally associate to a network the notion of temporal correlations, therefore providing the measure a dynamical garment. We show that approximate entropy distinguishes visibility graphs generated by processes with different complexity. The result probes to a greater extent these networks for the study of dynamical systems. Applications to certain biological data arising in cancer genomics are finally considered in the light of both approaches.Comment: 11 pages, 5 EPS figure

Cell-type deconvolution in epigenome-wide association studies: a review and recommendations

A major challenge faced by epigenome-wide association studies (EWAS) is cell-type heterogeneity. As many EWAS have already demonstrated, adjusting for changes in cell-type composition can be critical when analyzing and interpreting findings from such studies. Because of their importance, a great number of different statistical algorithms, which adjust for cell-type composition, have been proposed. Some of the methods are ‘reference based’ in that they require a priori defined reference DNA methylation profiles of cell types that are present in the tissue of interest, while other algorithms are ‘reference free.’ At present, however, it is unclear how best to adjust for cell-type heterogeneity, as this may also largely depend on the type of tissue and phenotype being considered. Here, we provide a critical review of the major existing algorithms for correcting cell-type composition in the context of Illumina Infinium Methylation Beadarrays, with the aim of providing useful recommendations to the EWAS community

Single-cell entropy for accurate estimation of differentiation potency from a cell's transcriptome

The ability to quantify differentiation potential of single cells is a task of critical importance. Here we demonstrate, using over 7,000 single-cell RNA-Seq profiles, that differentiation potency of a single cell can be approximated by computing the signalling promiscuity, or entropy, of a cell's transcriptome in the context of an interaction network, without the need for feature selection. We show that signalling entropy provides a more accurate and robust potency estimate than other entropy-based measures, driven in part by a subtle positive correlation between the transcriptome and connectome. Signalling entropy identifies known cell subpopulations of varying potency and drug resistant cancer stem-cell phenotypes, including those derived from circulating tumour cells. It further reveals that expression heterogeneity within single-cell populations is regulated. In summary, signalling entropy allows in silico estimation of the differentiation potency and plasticity of single cells and bulk samples, providing a means to identify normal and cancer stem-cell phenotypes

Epigenetic and genetic deregulation in cancer target distinct signaling pathway domains

Cancer is characterized by both genetic and epigenetic alterations. While cancer driver mutations and copy-number alterations have been studied at a systems-level, relatively little is known about the systems-level patterns exhibited by their epigenetic counterparts. Here we perform a pan-cancer wide systems-level analysis, mapping candidate cancer-driver DNA methylation (DNAm) alterations onto a human interactome. We demonstrate that functional DNAm alterations in cancer tend to map to nodes of lower connectivity and inter-connectivity, compared to the corresponding alterations at the genomic level. We find that epigenetic alterations are relatively over-represented in extracellular and transmembrane signaling domains, whereas cancer genes undergoing amplification or deletion tend to be enriched within the intracellular domain. A pan-cancer wide meta-analysis identifies WNT and chemokine signaling, as two key pathways where epigenetic deregulation preferentially targets extracellular components. We further pinpoint specific chemokine ligands/receptors whose epigenetic deregulation associates with key epigenetic enzymes, representing potential targets for epigenetic therapy. Our results suggest that epigenetic deregulation in cancer not only targets tissue-specific transcription factors, but also modulates signaling within the extra-cellular domain, providing novel system-level insight into the potential distinctive role of genetic and epigenetic alterations in cancer

Increased signaling entropy in cancer requires the scale-free property of protein interaction networks

One of the key characteristics of cancer cells is an increased phenotypic plasticity, driven by underlying genetic and epigenetic perturbations. However, at a systems-level it is unclear how these perturbations give rise to the observed increased plasticity. Elucidating such systems-level principles is key for an improved understanding of cancer. Recently, it has been shown that signaling entropy, an overall measure of signaling pathway promiscuity, and computable from integrating a sample's gene expression profile with a protein interaction network, correlates with phenotypic plasticity and is increased in cancer compared to normal tissue. Here we develop a computational framework for studying the effects of network perturbations on signaling entropy. We demonstrate that the increased signaling entropy of cancer is driven by two factors: (i) the scale-free (or near scale-free) topology of the interaction network, and (ii) a subtle positive correlation between differential gene expression and node connectivity. Indeed, we show that if protein interaction networks were random graphs, described by Poisson degree distributions, that cancer would generally not exhibit an increased signaling entropy. In summary, this work exposes a deep connection between cancer, signaling entropy and interaction network topology.Comment: 20 pages, 5 figures. In Press in Sci Rep 201

Prognostic gene network modules in breast cancer hold promise

A substantial proportion of lymph node-negative patients who receive adjuvant chemotherapy do not derive any benefit from this aggressive and potentially toxic treatment. However, standard histopathological indices cannot reliably detect patients at low risk of relapse or distant metastasis. In the past few years several prognostic gene expression signatures have been developed and shown to potentially outperform histopathological factors in identifying low-risk patients in specific breast cancer subgroups with predictive values of around 90%, and therefore hold promise for clinical application. We envisage that further improvements and insights may come from integrative expression pathway analyses that dissect prognostic signatures into modules related to cancer hallmarks