Compact smallest eigenvalue expressions in Wishart-Laguerre ensembles with or without fixed-trace

The degree of entanglement of random pure states in bipartite quantum systems can be estimated from the distribution of the extreme Schmidt eigenvalues. For a bipartition of size M\geq N, these are distributed according to a Wishart-Laguerre ensemble (WL) of random matrices of size N x M, with a fixed-trace constraint. We first compute the distribution and moments of the smallest eigenvalue in the fixed trace orthogonal WL ensemble for arbitrary M\geq N. Our method is based on a Laplace inversion of the recursive results for the corresponding orthogonal WL ensemble by Edelman. Explicit examples are given for fixed N and M, generalizing and simplifying earlier results. In the microscopic large-N limit with M-N fixed, the orthogonal and unitary WL distributions exhibit universality after a suitable rescaling and are therefore independent of the constraint. We prove that very recent results given in terms of hypergeometric functions of matrix argument are equivalent to more explicit expressions in terms of a Pfaffian or determinant of Bessel functions. While the latter were mostly known from the random matrix literature on the QCD Dirac operator spectrum, we also derive some new results in the orthogonal symmetry class.Comment: 25 pag., 4 fig - minor changes, typos fixed. To appear in JSTA

Entangled random pure states with orthogonal symmetry: exact results

We compute analytically the density $\varrho_{N,M}(\lambda)$ of Schmidt eigenvalues, distributed according to a fixed-trace Wishart-Laguerre measure, and the average R\'enyi entropy $\langle\mathcal{S}_q\rangle$ for reduced density matrices of entangled random pure states with orthogonal symmetry $(\beta=1)$. The results are valid for arbitrary dimensions $N=2k,M$ of the corresponding Hilbert space partitions, and are in excellent agreement with numerical simulations.Comment: 15 pages, 5 figure

Design of a novel quantitative PCR (QPCR)-based protocol for genotyping mice carrying the neuroprotective Wallerian degeneration slow (Wlds) gene

<p>Abstract</p> <p>Background</p> <p>Mice carrying the spontaneous genetic mutation known as Wallerian degeneration slow (<it>Wld</it>^<it>s</it>) have a unique neuroprotective phenotype, where axonal and synaptic compartments of neurons are protected from degeneration following a wide variety of physical, toxic and inherited disease-inducing stimuli. This remarkable phenotype has been shown to delay onset and progression in several mouse models of neurodegenerative disease, suggesting that <it>Wld</it>^<it>s</it>-mediated neuroprotection may assist in the identification of novel therapeutic targets. As a result, cross-breeding of <it>Wld</it>^{<it>s </it>}mice with mouse models of neurodegenerative diseases is used increasingly to understand the roles of axon and synapse degeneration in disease. However, the phenotype shows strong gene-dose dependence so it is important to distinguish offspring that are homozygous or heterozygous for the mutation. Since the <it>Wld</it>^{<it>s </it>}mutation comprises a triplication of a region already present in the mouse genome, the most stringent way to quantify the number of mutant <it>Wld</it>^{<it>s </it>}alleles is using copy number. Current approaches to genotype <it>Wld</it>^{<it>s </it>}mice are based on either Southern blots or pulsed field gel electrophoresis, neither of which are as rapid or efficient as quantitative PCR (QPCR).</p> <p>Results</p> <p>We have developed a rapid, robust and efficient genotyping method for <it>Wld</it>^{<it>s </it>}using QPCR. This approach differentiates, based on copy number, homozygous and heterozygous <it>Wld</it>^{<it>s </it>}mice from wild-type mice and each other. We show that this approach can be used to genotype mice carrying the spontaneous <it>Wld</it>^{<it>s </it>}mutation as well as animals expressing the <it>Wld</it>^{<it>s </it>}transgene.</p> <p>Conclusion</p> <p>We have developed a QPCR genotyping method that permits rapid and effective genotyping of <it>Wld</it>^{<it>s </it>}copy number. This technique will be of particular benefit in studies where <it>Wld</it>^{<it>s </it>}mice are cross-bred with other mouse models of neurodegenerative disease in order to understand the neuroprotective processes conferred by the <it>Wld</it>^{<it>s </it>}mutation.</p

An updated PREDICT breast cancer prognostication and treatment benefit prediction model with independent validation

BACKGROUND PREDICT is a breast cancer prognostic and treatment benefit model implemented online. The overall fit of the model has been good in multiple independent case series, but PREDICT has been shown to underestimate breast cancer specific mortality in women diagnosed under the age of 40. Another limitation is the use of discrete categories for tumour size and node status resulting in 'step' changes in risk estimates on moving between categories. We have refitted the PREDICT prognostic model using the original cohort of cases from East Anglia with updated survival time in order to take into account age at diagnosis and to smooth out the survival function for tumour size and node status. METHODS Multivariable Cox regression models were used to fit separate models for ER negative and ER positive disease. Continuous variables were fitted using fractional polynomials and a smoothed baseline hazard was obtained by regressing the baseline cumulative hazard for each patients against time using fractional polynomials. The fit of the prognostic models were then tested in three independent data sets that had also been used to validate the original version of PREDICT. RESULTS In the model fitting data, after adjusting for other prognostic variables, there is an increase in risk of breast cancer specific mortality in younger and older patients with ER positive disease, with a substantial increase in risk for women diagnosed before the age of 35. In ER negative disease the risk increases slightly with age. The association between breast cancer specific mortality and both tumour size and number of positive nodes was non-linear with a more marked increase in risk with increasing size and increasing number of nodes in ER positive disease. The overall calibration and discrimination of the new version of PREDICT (v2) was good and comparable to that of the previous version in both model development and validation data sets. However, the calibration of v2 improved over v1 in patients diagnosed under the age of 40. CONCLUSIONS The PREDICT v2 is an improved prognostication and treatment benefit model compared with v1. The online version should continue to aid clinical decision making in women with early breast cancer

Large Deviations of the Maximum Eigenvalue in Wishart Random Matrices

We compute analytically the probability of large fluctuations to the left of the mean of the largest eigenvalue in the Wishart (Laguerre) ensemble of positive definite random matrices. We show that the probability that all the eigenvalues of a (N x N) Wishart matrix W=X^T X (where X is a rectangular M x N matrix with independent Gaussian entries) are smaller than the mean value =N/c decreases for large N as $\sim \exp[-\frac{\beta}{2}N^2 \Phi_{-}(\frac{2}{\sqrt{c}}+1;c)]$, where \beta=1,2 correspond respectively to real and complex Wishart matrices, c=N/M < 1 and \Phi_{-}(x;c) is a large deviation function that we compute explicitly. The result for the Anti-Wishart case (M < N) simply follows by exchanging M and N. We also analytically determine the average spectral density of an ensemble of constrained Wishart matrices whose eigenvalues are forced to be smaller than a fixed barrier. The numerical simulations are in excellent agreement with the analytical predictions.Comment: Published version. References and appendix adde

Phosphoinositide-dependent kinase 1 controls migration and malignant transformation but not cell growth and proliferation in PTEN-null lymphocytes

In normal T cell progenitors, phosphoinositide-dependent kinase l (PDK1)–mediated phosphorylation and activation of protein kinase B (PKB) is essential for the phosphorylation and inactivation of Foxo family transcription factors, and also controls T cell growth and proliferation. The current study has characterized the role of PDK1 in the pathology caused by deletion of the tumor suppressor phosphatase and tensin homologue deleted on chromosome 10 (PTEN). PDK1 is shown to be essential for lymphomagenesis caused by deletion of PTEN in T cell progenitors. However, PTEN deletion bypasses the normal PDK1-controlled signaling pathways that determine thymocyte growth and proliferation. PDK1 does have important functions in PTEN-null thymocytes, notably to control the PKB–Foxo signaling axis and to direct the repertoire of adhesion and chemokine receptors expressed by PTEN-null T cells. The results thus provide two novel insights concerning pathological signaling caused by PTEN loss in lymphocytes. First, PTEN deletion bypasses the normal PDK1-controlled metabolic checkpoints that determine cell growth and proliferation. Second, PDK1 determines the cohort of chemokine and adhesion receptors expressed by PTEN-null cells, thereby controlling their migratory capacity

In silico assessment of potential druggable pockets on the surface of α1-Antitrypsin conformers

The search for druggable pockets on the surface of a protein is often performed on a single conformer, treated as a rigid body. Transient druggable pockets may be missed in this approach. Here, we describe a methodology for systematic in silico analysis of surface clefts across multiple conformers of the metastable protein α1-antitrypsin (A1AT). Pathological mutations disturb the conformational landscape of A1AT, triggering polymerisation that leads to emphysema and hepatic cirrhosis. Computational screens for small molecule inhibitors of polymerisation have generally focused on one major druggable site visible in all crystal structures of native A1AT. In an alternative approach, we scan all surface clefts observed in crystal structures of A1AT and in 100 computationally produced conformers, mimicking the native solution ensemble. We assess the persistence, variability and druggability of these pockets. Finally, we employ molecular docking using publicly available libraries of small molecules to explore scaffold preferences for each site. Our approach identifies a number of novel target sites for drug design. In particular one transient site shows favourable characteristics for druggability due to high enclosure and hydrophobicity. Hits against this and other druggable sites achieve docking scores corresponding to a Kd in the µM–nM range, comparing favourably with a recently identified promising lead. Preliminary ThermoFluor studies support the docking predictions. In conclusion, our strategy shows considerable promise compared with the conventional single pocket/single conformer approach to in silico screening. Our best-scoring ligands warrant further experimental investigation

ProteoLens: a visual analytic tool for multi-scale database-driven biological network data mining

Background New systems biology studies require researchers to understand how interplay among myriads of biomolecular entities is orchestrated in order to achieve high-level cellular and physiological functions. Many software tools have been developed in the past decade to help researchers visually navigate large networks of biomolecular interactions with built-in template-based query capabilities. To further advance researchers' ability to interrogate global physiological states of cells through multi-scale visual network explorations, new visualization software tools still need to be developed to empower the analysis. A robust visual data analysis platform driven by database management systems to perform bi-directional data processing-to-visualizations with declarative querying capabilities is needed. Results We developed ProteoLens as a JAVA-based visual analytic software tool for creating, annotating and exploring multi-scale biological networks. It supports direct database connectivity to either Oracle or PostgreSQL database tables/views, on which SQL statements using both Data Definition Languages (DDL) and Data Manipulation languages (DML) may be specified. The robust query languages embedded directly within the visualization software help users to bring their network data into a visualization context for annotation and exploration. ProteoLens supports graph/network represented data in standard Graph Modeling Language (GML) formats, and this enables interoperation with a wide range of other visual layout tools. The architectural design of ProteoLens enables the de-coupling of complex network data visualization tasks into two distinct phases: 1) creating network data association rules, which are mapping rules between network node IDs or edge IDs and data attributes such as functional annotations, expression levels, scores, synonyms, descriptions etc; 2) applying network data association rules to build the network and perform the visual annotation of graph nodes and edges according to associated data values. We demonstrated the advantages of these new capabilities through three biological network visualization case studies: human disease association network, drug-target interaction network and protein-peptide mapping network. Conclusion The architectural design of ProteoLens makes it suitable for bioinformatics expert data analysts who are experienced with relational database management to perform large-scale integrated network visual explorations. ProteoLens is a promising visual analytic platform that will facilitate knowledge discoveries in future network and systems biology studies