Invariant integration on orthosymplectic and unitary supergroups

The orthosymplectic supergroup OSp(m|2n) and unitary supergroup U(p|q) are studied following a new approach that starts from Harish-Chandra pairs and links the sheaf-theoretical supermanifold approach of Berezin and others with the differential geometry approach of Rogers and others. The matrix elements of the fundamental representation of the Lie supergroup G are expressed in terms of functions on the product supermanifold G_0 x R^{0|N}, with G_0 the underlying Lie group and N the odd dimension of G. This product supermanifold is isomorphic to the supermanifold of G. This leads to a new expression for the standard generators of the corresponding Lie superalgebra g as invariant derivations on G. Using these results a new and transparent formula for the invariant integrals on OSp(m|2n) and U(p|q) is obtained

Protein Co-Expression Analysis as a Strategy to Complement a Standard Quantitative Proteomics Approach:Case of a Glioblastoma Multiforme Study

Although correlation network studies from co-expression analysis are increasingly popular, they are rarely applied to proteomics datasets. Protein co-expression analysis provides a complementary view of underlying trends, which can be overlooked by conventional data analysis. The core of the present study is based on Weighted Gene Co-expression Network Analysis applied to a glioblastoma multiforme proteomic dataset. Using this method, we have identified three main modules which are associated with three different membrane associated groups; mitochondrial, endoplasmic reticulum, and a vesicle fraction. The three networks based on protein co-expression were assessed against a publicly available database (STRING) and show a statistically significant overlap. Each of the three main modules were de-clustered into smaller networks using different strategies based on the identification of highly connected networks, hierarchical clustering and enrichment of Gene Ontology functional terms. Most of the highly connected proteins found in the endoplasmic reticulum module were associated with redox activity while a core of the unfolded protein response was identified in addition to proteins involved in oxidative stress pathways. The proteins composing the electron transfer chain were found differently affected with proteins from mitochondrial Complex I being more down-regulated than proteins from Complex III. Finally, the two pyruvate kinases isoforms show major differences in their co-expressed protein networks suggesting roles in different cellular locations