Polyhedral products, flag complexes and monodromy representations

This article presents a machinery based on polyhedral products that produces faithful representations of graph products of finite groups and direct products of finite groups into automorphisms of free groups $\rm Aut(F_n)$ and outer automorphisms of free groups $\rm Out(F_n)$, respectively, as well as faithful representations of products of finite groups into the linear groups $\rm SL(n,\mathbb Z)$ and $\rm GL(n,\mathbb Z)$. These faithful representations are realized as monodromy representations.Comment: 20 page

Homological stability for spaces of commuting elements in Lie groups

In this paper we study homological stability for spaces ${\rm Hom}(\mathbb{Z}^n,G)$ of pairwise commuting $n$-tuples in a Lie group $G$. We prove that for each $n\geqslant 1$, these spaces satisfy rational homological stability as $G$ ranges through any of the classical sequences of compact, connected Lie groups, or their complexifications. We prove similar results for rational equivariant homology, for character varieties, and for the infinite-dimensional analogues of these spaces, ${\rm Comm}(G)$ and ${\rm B_{com}} G$, introduced by Cohen-Stafa and Adem-Cohen-Torres-Giese respectively. In addition, we show that the rational homology of the space of unordered commuting $n$-tuples in a fixed group $G$ stabilizes as $n$ increases. Our proofs use the theory of representation stability - in particular, the theory of ${\rm FI}_W$-modules developed by Church-Ellenberg-Farb and Wilson. In all of the these results, we obtain specific bounds on the stable range, and we show that the homology isomorphisms are induced by maps of spaces.Comment: 56 pages, accepted versio

On spaces of commuting elements in Lie groups

The main purpose of this paper is to introduce a method to stabilize certain spaces of homomorphisms from finitely generated free abelian groups to a Lie group $G$, namely $Hom(\mathbb Z^n,G)$. We show that this stabilized space of homomorphisms decomposes after suspending once with summands which can be reassembled, in a sense to be made precise below, into the individual spaces $Hom(\mathbb Z^n,G)$ after suspending once. To prove this decomposition, a stable decomposition of an equivariant function space is also developed. One main result is that the topological space of all commuting elements in a compact Lie group is homotopy equivalent to an equivariant function space after inverting the order of the Weyl group. In addition, the homology of the stabilized space admits a very simple description in terms of the tensor algebra generated by the reduced homology of a maximal torus in favorable cases. The stabilized space also allows the description of the additive reduced homology of the individual spaces $Hom(\mathbb Z^n,G)$, with the order of the Weyl group inverted.Comment: 27 pages, with an appendix by Vic Reine

Hilbert-Poincare series for spaces of commuting elements in Lie groups

In this article we study the homology of spaces ${\rm Hom}(\mathbb{Z}^n,G)$ of ordered pairwise commuting $n$-tuples in a Lie group $G$. We give an explicit formula for the Poincare series of these spaces in terms of invariants of the Weyl group of $G$. By work of Bergeron and Silberman, our results also apply to ${\rm Hom}(F_n/\Gamma_n^m,G)$, where the subgroups $\Gamma_n^m$ are the terms in the descending central series of the free group $F_n$. Finally, we show that there is a stable equivalence between the space ${\rm Comm}(G)$ studied by Cohen-Stafa and its nilpotent analogues.Comment: 20 pages, journal versio

Functional interaction of Parkinson's disease-associated LRRK2 with members of the dynamin GTPase superfamily

Mutations in LRRK2 cause autosomal dominant Parkinson's disease (PD). LRRK2 encodes a multi-domain protein containing GTPase and kinase domains, and putative protein-protein interaction domains. Familial PD mutations alter the GTPase and kinase activity of LRRK2 in vitro. LRRK2 is suggested to regulate a number of cellular pathways although the underlying mechanisms are poorly understood. To explore such mechanisms, it has proved informative to identify LRRK2-interacting proteins, some of which serve as LRRK2 kinase substrates. Here, we identify common interactions of LRRK2 with members of the dynamin GTPase superfamily. LRRK2 interacts with dynamin 1-3 that mediate membrane scission in clathrin-mediated endocytosis and with dynamin-related proteins that mediate mitochondrial fission (Drp1) and fusion (mitofusins and OPA1). LRRK2 partially co-localizes with endosomal dynamin-1 or with mitofusins and OPA1 at mitochondrial membranes. The subcellular distribution and oligomeric complexes of dynamin GTPases are not altered by modulating LRRK2 in mouse brain, whereas mature OPA1 levels are reduced in G2019S PD brains. LRRK2 enhances mitofusin-1 GTP binding, whereas dynamin-1 and OPA1 serve as modest substrates of LRRK2-mediated phosphorylation in vitro. While dynamin GTPase orthologs are not required for LRRK2-induced toxicity in yeast, LRRK2 functionally interacts with dynamin-1 and mitofusin-1 in cultured neurons. LRRK2 attenuates neurite shortening induced by dynamin-1 by reducing its levels, whereas LRRK2 rescues impaired neurite outgrowth induced by mitofusin-1 potentially by reversing excessive mitochondrial fusion. Our study elucidates novel functional interactions of LRRK2 with dynamin-superfamily GTPases that implicate LRRK2 in the regulation of membrane dynamics important for endocytosis and mitochondrial morpholog

PARK9-associated ATP13A2 localizes to intracellular acidic vesicles and regulates cation homeostasis and neuronal integrity

Mutations in the ATP13A2 gene (PARK9, OMIM 610513) cause autosomal recessive, juvenile-onset Kufor-Rakeb syndrome and early-onset parkinsonism. ATP13A2 is an uncharacterized protein belonging to the P5-type ATPase subfamily that is predicted to regulate the membrane transport of cations. The physiological function of ATP13A2 in the mammalian brain is poorly understood. Here, we demonstrate that ATP13A2 is localized to intracellular acidic vesicular compartments in cultured neurons. In the human brain, ATP13A2 is localized to pyramidal neurons within the cerebral cortex and dopaminergic neurons of the substantia nigra. ATP13A2 protein levels are increased in nigral dopaminergic and cortical pyramidal neurons of Parkinson's disease brains compared with normal control brains. ATP13A2 levels are increased in cortical neurons bearing Lewy bodies (LBs) compared with neurons without LBs. Using short hairpin RNA-mediated silencing or overexpression to explore the function of ATP13A2, we find that modulating the expression of ATP13A2 reduces the neurite outgrowth of cultured midbrain dopaminergic neurons. We also find that silencing of ATP13A2 expression in cortical neurons alters the kinetics of intracellular pH in response to cadmium exposure. Furthermore, modulation of ATP13A2 expression leads to reduced intracellular calcium levels in cortical neurons. Finally, we demonstrate that silencing of ATP13A2 expression induces mitochondrial fragmentation in neurons. Oppositely, overexpression of ATP13A2 delays cadmium-induced mitochondrial fragmentation in neurons consistent with a neuroprotective effect. Collectively, this study reveals a number of intriguing neuronal phenotypes due to the loss- or gain-of-function of ATP13A2 that support a role for this protein in regulating intracellular cation homeostasis and neuronal integrit

Production of an Oily Adjuvant Vaccine against the Swine Erysipelothrix

Infection with Erysipelothrix rhusiopathiae in swine has a significant economic impact on pig production systems worldwide. Since a character zoonotic disease, it poses a problem for public health and veterinary services. the clinical and pathological features of the disease have been well-described. Inactivated and attenuated vaccines are available to prevent development of clinical signs of swine erysipelas. In Albania, repeatedly has had outbreaks in swine erysipelas. For too long in our country has been produced a liquid alive attenuated vaccine with strains VR2. This vaccine has a short validity and cannot be used in any epizootic situation. For these reasons we proposed to produce a oily inactivated vaccine, against erysipelas in pigs with a long validity and can be used in any situation epizootic. This constitutes also main purpose of this paper.Keywords: swine, erysipelas, oily vaccine