Pruning Algorithms for Pretropisms of Newton Polytopes

Pretropisms are candidates for the leading exponents of Puiseux series that represent solutions of polynomial systems. To find pretropisms, we propose an exact gift wrapping algorithm to prune the tree of edges of a tuple of Newton polytopes. We prefer exact arithmetic not only because of the exact input and the degrees of the output, but because of the often unpredictable growth of the coordinates in the face normals, even for polytopes in generic position. We provide experimental results with our preliminary implementation in Sage that compare favorably with the pruning method that relies only on cone intersections.Comment: exact, gift wrapping, Newton polytope, pretropism, tree pruning, accepted for presentation at Computer Algebra in Scientific Computing, CASC 201

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

Shadoo (Sprn) and prion disease incubation time in mice

Prion diseases are transmissible neurodegenerative disorders of mammalian species and include scrapie, bovine spongiform encephalopathy (BSE), and variant Creutzfeldt-Jakob disease (vCJD). The prion protein (PrP) plays a key role in the disease, with coding polymorphism in both human and mouse influencing disease susceptibility and incubation time, respectively. Other genes are also thought to be important and a plausible candidate is Sprn, which encodes the PrP-like protein Shadoo (Sho). Sho is expressed in the adult central nervous system and exhibits neuroprotective activity reminiscent of PrP in an in vitro assay. To investigate the role of Sprn in prion disease incubation time we sequenced the open reading frame (ORF) in a diverse panel of mice and saw little variation except in strains derived from wild-trapped mice. Sequencing the untranslated regions revealed polymorphisms that allowed us to carry out an association study of incubation period in the Northport heterogeneous stock of mice inoculated with Chandler/RML prions. We also examined the expression level of Sprn mRNA in the brains of normal and prion-infected mice and saw no correlation with either genotype or incubation time. We therefore conclude that Sprn does not play a major role in prion disease incubation time in these strains of mice

Functionally Relevant Domains of the Prion Protein Identified In Vivo

The prion consists essentially of PrPSc, a misfolded and aggregated conformer of the cellular protein PrPC. Whereas PrPC deficient mice are clinically healthy, expression of PrPC variants lacking its central domain (PrPΔCD), or of the PrP-related protein Dpl, induces lethal neurodegenerative syndromes which are repressed by full-length PrP. Here we tested the structural basis of these syndromes by grafting the amino terminus of PrPC (residues 1–134), or its central domain (residues 90–134), onto Dpl. Further, we constructed a soluble variant of the neurotoxic PrPΔCD mutant that lacks its glycosyl phosphatidyl inositol (GPI) membrane anchor. Each of these modifications abrogated the pathogenicity of Dpl and PrPΔCD in transgenic mice. The PrP-Dpl chimeric molecules, but not anchorless PrPΔCD, ameliorated the disease of mice expressing truncated PrP variants. We conclude that the amino proximal domain of PrP exerts a neurotrophic effect even when grafted onto a distantly related protein, and that GPI-linked membrane anchoring is necessary for both beneficial and deleterious effects of PrP and its variants

The Comprehensive Native Interactome of a Fully Functional Tagged Prion Protein

The enumeration of the interaction partners of the cellular prion protein, PrPC, may help clarifying its elusive molecular function. Here we added a carboxy proximal myc epitope tag to PrPC. When expressed in transgenic mice, PrPmyc carried a GPI anchor, was targeted to lipid rafts, and was glycosylated similarly to PrPC. PrPmyc antagonized the toxicity of truncated PrP, restored prion infectibility of PrPC-deficient mice, and was physically incorporated into PrPSc aggregates, indicating that it possessed all functional characteristics of genuine PrPC. We then immunopurified myc epitope-containing protein complexes from PrPmyc transgenic mouse brains. Gentle differential elution with epitope-mimetic decapeptides, or a scrambled version thereof, yielded 96 specifically released proteins. Quantitative mass spectrometry with isotope-coded tags identified seven proteins which co-eluted equimolarly with PrPC and may represent component of a multiprotein complex. Selected PrPC interactors were validated using independent methods. Several of these proteins appear to exert functions in axomyelinic maintenance

N-Glycans and Glycosylphosphatidylinositol-Anchor Act on Polarized Sorting of Mouse PrPC in Madin-Darby Canine Kidney Cells

The cellular prion protein (PrPC) plays a fundamental role in prion disease. PrPC is a glycosylphosphatidylinositol (GPI)-anchored protein with two variably occupied N-glycosylation sites. In general, GPI-anchor and N-glycosylation direct proteins to apical membranes in polarized cells whereas the majority of mouse PrPC is found in basolateral membranes in polarized Madin-Darby canine kidney (MDCK) cells. In this study we have mutated the first, the second, and both N-glycosylation sites of PrPC and also replaced the GPI-anchor of PrPC by the Thy-1 GPI-anchor in order to investigate the role of these signals in sorting of PrPC in MDCK cells. Cell surface biotinylation experiments and confocal microscopy showed that lack of one N-linked oligosaccharide leads to loss of polarized sorting of PrPC. Exchange of the PrPC GPI-anchor for the one of Thy-1 redirects PrPC to the apical membrane. In conclusion, both N-glycosylation and GPI-anchor act on polarized sorting of PrPC, with the GPI-anchor being dominant over N-glycans

Protease-Resistant Prions Selectively Decrease Shadoo Protein

The central event in prion diseases is the conformational conversion of the cellular prion protein (PrPC) into PrPSc, a partially protease-resistant and infectious conformer. However, the mechanism by which PrPSc causes neuronal dysfunction remains poorly understood. Levels of Shadoo (Sho), a protein that resembles the flexibly disordered N-terminal domain of PrPC, were found to be reduced in the brains of mice infected with the RML strain of prions [1], implying that Sho levels may reflect the presence of PrPSc in the brain. To test this hypothesis, we examined levels of Sho during prion infection using a variety of experimental systems. Sho protein levels were decreased in the brains of mice, hamsters, voles, and sheep infected with different natural and experimental prion strains. Furthermore, Sho levels were decreased in the brains of prion-infected, transgenic mice overexpressing Sho and in infected neuroblastoma cells. Time-course experiments revealed that Sho levels were inversely proportional to levels of protease-resistant PrPSc. Membrane anchoring and the N-terminal domain of PrP both influenced the inverse relationship between Sho and PrPSc. Although increased Sho levels had no discernible effect on prion replication in mice, we conclude that Sho is the first non-PrP marker specific for prion disease. Additional studies using this paradigm may provide insight into the cellular pathways and systems subverted by PrPSc during prion disease

The N-Terminal, Polybasic Region Is Critical for Prion Protein Neuroprotective Activity

Several lines of evidence suggest that the normal form of the prion protein, PrPC, exerts a neuroprotective activity against cellular stress or toxicity. One of the clearest examples of such activity is the ability of wild-type PrPC to suppress the spontaneous neurodegenerative phenotype of transgenic mice expressing a deleted form of PrP (Δ32–134, called F35). To define domains of PrP involved in its neuroprotective activity, we have analyzed the ability of several deletion mutants of PrP (Δ23–31, Δ23–111, and Δ23–134) to rescue the phenotype of Tg(F35) mice. Surprisingly, all of these mutants displayed greatly diminished rescue activity, although Δ23–31 PrP partially suppressed neuronal loss when expressed at very high levels. Our results pinpoint the N-terminal, polybasic domain as a critical determinant of PrPC neuroprotective activity, and suggest that identification of molecules interacting with this region will provide important clues regarding the normal function of the protein. Small molecule ligands targeting this region may also represent useful therapeutic agents for treatment of prion diseases

The Mitochondrial Chaperone Protein TRAP1 Mitigates α-Synuclein Toxicity

Overexpression or mutation of α-Synuclein is associated with protein aggregation and interferes with a number of cellular processes, including mitochondrial integrity and function. We used a whole-genome screen in the fruit fly Drosophila melanogaster to search for novel genetic modifiers of human [A53T]α-Synuclein–induced neurotoxicity. Decreased expression of the mitochondrial chaperone protein tumor necrosis factor receptor associated protein-1 (TRAP1) was found to enhance age-dependent loss of fly head dopamine (DA) and DA neuron number resulting from [A53T]α-Synuclein expression. In addition, decreased TRAP1 expression in [A53T]α-Synuclein–expressing flies resulted in enhanced loss of climbing ability and sensitivity to oxidative stress. Overexpression of human TRAP1 was able to rescue these phenotypes. Similarly, human TRAP1 overexpression in rat primary cortical neurons rescued [A53T]α-Synuclein–induced sensitivity to rotenone treatment. In human (non)neuronal cell lines, small interfering RNA directed against TRAP1 enhanced [A53T]α-Synuclein–induced sensitivity to oxidative stress treatment. [A53T]α-Synuclein directly interfered with mitochondrial function, as its expression reduced Complex I activity in HEK293 cells. These effects were blocked by TRAP1 overexpression. Moreover, TRAP1 was able to prevent alteration in mitochondrial morphology caused by [A53T]α-Synuclein overexpression in human SH-SY5Y cells. These results indicate that [A53T]α-Synuclein toxicity is intimately connected to mitochondrial dysfunction and that toxicity reduction in fly and rat primary neurons and human cell lines can be achieved using overexpression of the mitochondrial chaperone TRAP1. Interestingly, TRAP1 has previously been shown to be phosphorylated by the serine/threonine kinase PINK1, thus providing a potential link of PINK1 via TRAP1 to α-Synuclein