Polyhedra in loop quantum gravity

Interwiners are the building blocks of spin-network states. The space of intertwiners is the quantization of a classical symplectic manifold introduced by Kapovich and Millson. Here we show that a theorem by Minkowski allows us to interpret generic configurations in this space as bounded convex polyhedra in Euclidean space: a polyhedron is uniquely described by the areas and normals to its faces. We provide a reconstruction of the geometry of the polyhedron: we give formulas for the edge lengths, the volume and the adjacency of its faces. At the quantum level, this correspondence allows us to identify an intertwiner with the state of a quantum polyhedron, thus generalizing the notion of quantum tetrahedron familiar in the loop quantum gravity literature. Moreover, coherent intertwiners result to be peaked on the classical geometry of polyhedra. We discuss the relevance of this result for loop quantum gravity. In particular, coherent spin-network states with nodes of arbitrary valence represent a collection of semiclassical polyhedra. Furthermore, we introduce an operator that measures the volume of a quantum polyhedron and examine its relation with the standard volume operator of loop quantum gravity. We also comment on the semiclassical limit of spinfoams with non-simplicial graphs.Comment: 32 pages, many figures. v2 minor correction

The POM Monoclonals: A Comprehensive Set of Antibodies to Non-Overlapping Prion Protein Epitopes

PrPSc, a misfolded and aggregated form of the cellular prion protein PrPC, is the only defined constituent of the transmissible agent causing prion diseases. Expression of PrPC in the host organism is necessary for prion replication and for prion neurotoxicity. Understanding prion diseases necessitates detailed structural insights into PrPC and PrPSc. Towards this goal, we have developed a comprehensive collection of monoclonal antibodies denoted POM1 to POM19 and directed against many different epitopes of mouse PrPC. Three epitopes are located within the N-terminal octarepeat region, one is situated within the central unstructured region, and four epitopes are discontinuous within the globular C-proximal domain of PrPC. Some of these antibodies recognize epitopes that are resilient to protease digestion in PrPSc. Other antibodies immunoprecipitate PrPC, but not PrPSc. A third group was found to immunoprecipitate both PrP isoforms. Some of the latter antibodies could be blocked with epitope-mimicking peptides, and incubation with an excess of these peptides allowed for immunochromatography of PrPC and PrPSc. Amino-proximal antibodies were found to react with repetitive PrPC epitopes, thereby vastly increasing their avidity. We have also created functional single-chain miniantibodies from selected POMs, which retained the binding characteristics despite their low molecular mass. The POM collection, thus, represents a unique set of reagents allowing for studies with a variety of techniques, including western blotting, ELISA, immunoprecipitation, conformation-dependent immunoassays, and plasmon surface plasmon resonance-based assays

MCL1 is Required for Maintenance of Intestinal Homeostasis and Prevention of Carcinogenesis in Mice

BACKGROUND & AIMS Intestinal epithelial homeostasis depends on a tightly regulated balance between intestinal epithelial cell (IEC) death and proliferation. Disruption of factors that promote IEC death result in intestinal inflammation, whereas loss of anti-apoptotic proteins, such as BCL2 or its family member BCL2L1, has no effect on intestinal homeostasis in mice. We investigated the functions of the anti-apoptotic protein MCL1, another member of the BCL2 family, in intestinal homeostasis in mice. METHODS We generated mice with IEC-specific disruption of Mcl1 (Mcl1$^{ΔIEC}$ mice) or tamoxifen-inducible IEC-specific disruption of Mcl1 (i-Mcl1$^{ΔIEC}$ mice); these mice and mice with full-length Mcl1 (controls) were raised under normal or germ-free conditions. Some mice were given antibiotics in their drinking water or the PORCUPINE WNT inhibitor WNT974. Mice were analyzed by endoscopy and for intestinal epithelial barrier permeability. Intestinal tissues were analyzed by histology, in situ hybridization, proliferation assays, and immunoblots. Levels of calprotectin, a marker of intestinal inflammation, were measured in intestinal tissues and feces. RESULTS Mcl1$^{ΔIEC}$ mice spontaneously developed apoptotic enterocolopathy, characterized by increased IEC apoptosis, hyperproliferative crypts, epithelial barrier dysfunction, and chronic inflammation. Loss of MCL1 retained intestinal crypts in a hyperproliferated state and prevented the differentiation of intestinal stem cells. Proliferation of intestinal stem cells in MCL1-deficient mice required WNT signaling and was associated with DNA damage accumulation. By 1 year of age, Mcl1$^{ΔIEC}$ mice developed intestinal tumors with morphologic and genetic features of human adenomas and carcinomas. Germ-free housing of Mcl1$^{ΔIEC}$ mice reduced markers of microbiota-induced intestinal inflammation but not tumor development. CONCLUSION The anti-apoptotic protein MCL1, a member of the BCL2 family, is required for maintenance of intestinal homeostasis and prevention of carcinogenesis in mice. Loss of MCL1 results in development of intestinal carcinomas, even under germ-free conditions, and therefore does not involve microbe-induced chronic inflammation. Mcl1$^{ΔIEC}$ mice might be used to study apoptotic enterocolopathy and inflammatory bowel diseases

The prion protein is an agonistic ligand of the G protein-coupled receptor Adgrg6

Ablation of the cellular prion protein PrPC leads to a chronic demyelinating polyneuropathy affecting Schwann cells. Neuron-restricted expression of PrPC prevents the disease1, suggesting that PrPC acts in trans through an unidentified Schwann cell receptor. Here we show that the cAMP concentration in sciatic nerves from PrPC-deficient mice is reduced, suggesting that PrPC acts via a G protein-coupled receptor (GPCR). The amino-terminal flexible tail (residues 23–120) of PrPC triggered a concentration-dependent increase in cAMP in primary Schwann cells, in the Schwann cell line SW10, and in HEK293T cells overexpressing the GPCR Adgrg6 (also known as Gpr126). By contrast, naive HEK293T cells and HEK293T cells expressing several other GPCRs did not react to the flexible tail, and ablation of Gpr126 from SW10 cells abolished the flexible tail-induced cAMP response. The flexible tail contains a polycationic cluster (KKRPKPG) similar to the GPRGKPG motif of the Gpr126 agonist type-IV collagen2. A KKRPKPG-containing PrPC-derived peptide (FT23–50) sufficed to induce a Gpr126-dependent cAMP response in cells and mice, and improved myelination in hypomorphic gpr126 mutant zebrafish (Danio rerio). Substitution of the cationic residues with alanines abolished the biological activity of both FT23–50 and the equivalent type-IV collagen peptide. We conclude that PrPC promotes myelin homeostasis through flexible tail-mediated Gpr126 agonism. As well as clarifying the physiological role of PrPC, these observations are relevant to the pathogenesis of demyelinating polyneuropathies—common debilitating diseases for which there are limited therapeutic options

The toxicity of antiprion antibodies is mediated by the flexible tail of the prion protein

Prion infections cause lethal neurodegeneration. This process requires the cellular prion protein (PrP(C); ref. 1), which contains a globular domain hinged to a long amino-proximal flexible tail. Here we describe rapid neurotoxicity in mice and cerebellar organotypic cultured slices exposed to ligands targeting the α1 and α3 helices of the PrP(C) globular domain. Ligands included seven distinct monoclonal antibodies, monovalent Fab1 fragments and recombinant single-chain variable fragment miniantibodies. Similar to prion infections, the toxicity of globular domain ligands required neuronal PrP(C), was exacerbated by PrP(C) overexpression, was associated with calpain activation and was antagonized by calpain inhibitors. Neurodegeneration was accompanied by a burst of reactive oxygen species, and was suppressed by antioxidants. Furthermore, genetic ablation of the superoxide-producing enzyme NOX2 (also known as CYBB) protected mice from globular domain ligand toxicity. We also found that neurotoxicity was prevented by deletions of the octapeptide repeats within the flexible tail. These deletions did not appreciably compromise globular domain antibody binding, suggesting that the flexible tail is required to transmit toxic signals that originate from the globular domain and trigger oxidative stress and calpain activation. Supporting this view, various octapeptide ligands were not only innocuous to both cerebellar organotypic cultured slices and mice, but also prevented the toxicity of globular domain ligands while not interfering with their binding. We conclude that PrP(C) consists of two functionally distinct modules, with the globular domain and the flexible tail exerting regulatory and executive functions, respectively. Octapeptide ligands also prolonged the life of mice expressing the toxic PrP(C) mutant, PrP(Δ94-134), indicating that the flexible tail mediates toxicity in two distinct PrP(C)-related conditions. Flexible tail-mediated toxicity may conceivably play a role in further prion pathologies, such as familial Creutzfeldt-Jakob disease in humans bearing supernumerary octapeptides