Minimal Delaunay Triangulations of Hyperbolic Surfaces

Motivated by recent work on Delaunay triangulations of hyperbolic surfaces, we consider the minimal number of vertices of such triangulations. First, we show that every hyperbolic surface of genus g has a simplicial Delaunay triangulation with O(g) vertices, where edges are given by distance paths. Then, we construct a class of hyperbolic surfaces for which the order of this bound is optimal. Finally, to give a general lower bound, we show that the ?(?g) lower bound for the number of vertices of a simplicial triangulation of a topological surface of genus g is tight for hyperbolic surfaces as well

Delaunay triangulations of generalized Bolza surfaces

The Bolza surface can be seen as the quotient of the hyperbolic plane, represented by the Poincar\'e disk model, under the action of the group generated by the hyperbolic isometries identifying opposite sides of a regular octagon centered at the origin. We consider generalized Bolza surfaces $\mathbb{M}_g$, where the octagon is replaced by a regular $4g$-gon, leading to a genus $g$ surface. We propose an extension of Bowyer's algorithm to these surfaces. In particular, we compute the value of the systole of $\mathbb{M}_g$. We also propose algorithms computing small sets of points on $\mathbb{M}_g$ that are used to initialize Bowyer's algorithm.Comment: 50 pages, 28 figure

A novel mutation and first report of dilated cardiomyopathy in ALG6-CDG (CDG-Ic): a case report

Congenital disorders of glycosylation (CDG) are an expanding group of inherited metabolic diseases with multisystem involvement. ALG6-CDG (CDGIc) is an endoplasmatic reticulum defect in N-glycan assembly. It is usually milder than PMM2-CDG (CDG-Ia) and so is its natural course. It is characterized by psychomotor retardation, seizures, ataxia, and hypotonia. In contrast to PMM2-CDG (CDGIa), there is no cerebellar hypoplasia. Cardiomyopathy has been reported in a few CDG types and in a number of patients with unexplained CDG. We report an 11 year old Saudi boy with severe psychomotor retardation, seizures, strabismus, inverted nipples, dilated cardiomyopathy, and a type 1 pattern of serum transferrin isoelectrofocusing. Phosphomannomutase and phosphomannose isomerase activities were normal in fibroblasts. Full gene sequencing of the ALG6 gene revealed a novel mutation namely c.482A>G (p.Y161C) and heterozygosity in the parents. This report highlights the importance to consider CDG in the differential diagnosis of unexplained cardiomyopathy

Differential effects of lobe A and lobe B of the conserved oligomeric golgi complex on the stability of β1,4-galactosyltransferase 1 and α2,6-sialyltransferase 1

Erworben im Rahmen der Schweizer Nationallizenzen (http://www.nationallizenzen.ch)Initially described by Jaeken et al. in 1980, congenital disorders of glycosylation (CDG) is a rapidly expanding group of human multisystemic disorders. To date, many CDG patients have been identified with deficiencies in the conserved oligomeric Golgi (COG) complex which is a complex involved in the vesicular intra-Golgi retrograde trafficking. Composed of eight subunits that are organized in two lobes, COG subunit deficiencies have been associated with Golgi glycosylation abnormalities. Analysis of the total serum N-glycans of COG-deficient CDG patients demonstrated an overall decrease in terminal sialylation and galactosylation. According to the mutated COG subunits, differences in late Golgi glycosylation were observed and led us to address the question of an independent role and requirement for each of the two lobes of the COG complex in the stability and localization of late terminal Golgi glycosylation enzymes. For this, we used a small-interfering RNAs strategy in HeLa cells stably expressing green fluorescent protein (GFP)-tagged β1,4-galactosyltransferase 1 (B4GALT1) and α2,6-sialyltransferase 1 (ST6GAL1), two major Golgi glycosyltransferases involved in late Golgi N-glycosylation. Using fluorescent lectins and flow cytometry analysis, we clearly demonstrated that depletion of both lobes was associated with deficiencies in terminal Golgi N-glycosylation. Lobe A depletion resulted in dramatic changes in the Golgi structure, whereas lobe B depletion severely altered the stability of B4GALT1 and ST6GAL1. Only MG132 was able to rescue their steady-state levels, suggesting that B4GALT1- and ST6GAL1-induced degradation are likely the consequence of an accumulation in the endoplasmic reticulum (ER), followed by a retrotranslocation into the cytosol and proteasomal degradation. All together, our results suggest differential effects of lobe A and lobe B for the localization/stability of B4GALT1 and ST6GAL1. Lobe B would be crucial in preventing these two Golgi glycosyltransferases from inappropriate retrograde trafficking to the ER, whereas lobe A appears to be essential for maintaining the overall Golgi structure

Differential effects of lobe A and lobe B of the Conserved Oligomeric Golgi complex on the stability of β1,4-galactosyltransferase 1 and α2,6-sialyltransferase 1

Initially described by Jaeken et al. in 1980, congenital disorders of glycosylation (CDG) is a rapidly expanding group of human multisystemic disorders. To date, many CDG patients have been identified with deficiencies in the conserved oligomeric Golgi (COG) complex which is a complex involved in the vesicular intra-Golgi retrograde trafficking. Composed of eight subunits that are organized in two lobes, COG subunit deficiencies have been associated with Golgi glycosylation abnormalities. Analysis of the total serum N-glycans of COG-deficient CDG patients demonstrated an overall decrease in terminal sialylation and galactosylation. According to the mutated COG subunits, differences in late Golgi glycosylation were observed and led us to address the question of an independent role and requirement for each of the two lobes of the COG complex in the stability and localization of late terminal Golgi glycosylation enzymes. For this, we used a small-interfering RNAs strategy in HeLa cells stably expressing green fluorescent protein (GFP)-tagged β1,4-galactosyltransferase 1 (B4GALT1) and α2,6-sialyltransferase 1 (ST6GAL1), two major Golgi glycosyltransferases involved in late Golgi N-glycosylation. Using fluorescent lectins and flow cytometry analysis, we clearly demonstrated that depletion of both lobes was associated with deficiencies in terminal Golgi N-glycosylation. Lobe A depletion resulted in dramatic changes in the Golgi structure, whereas lobe B depletion severely altered the stability of B4GALT1 and ST6GAL1. Only MG132 was able to rescue their steady-state levels, suggesting that B4GALT1- and ST6GAL1-induced degradation are likely the consequence of an accumulation in the endoplasmic reticulum (ER), followed by a retrotranslocation into the cytosol and proteasomal degradation. All together, our results suggest differential effects of lobe A and lobe B for the localization/stability of B4GALT1 and ST6GAL1. Lobe B would be crucial in preventing these two Golgi glycosyltransferases from inappropriate retrograde trafficking to the ER, whereas lobe A appears to be essential for maintaining the overall Golgi structur

Delaunay triangulations of symmetric hyperbolic surfaces

International audienceOf the several existing algorithms for computing Delaunay triangulations of point sets in Euclidean space, the incremental algorithm has recently been extended to the Bolza surface, a hyperbolic surface of genus 2. We will generalize this algorithm to so called symmetric hyperbolic surfaces of arbitrary genus. Delaunay triangulations of point sets on hyperbolic surfaces can be constructed by using the fact that such point sets can be regarded as periodic point sets in the hyperbolic plane. However, one of the main issues is then that the result might contain 1-or 2-cycles, which means that the triangulation is not simplicial. As the incremental algorithm that we use can only work with simplicial complexes, this situation must be avoided. In this work, we will first compute the systole of the symmetric hyperbolic surfaces, i.e., the length of the shortest non-contractible loop. The value of the systole is used in a condition to ensure that the triangulations will be simplicial. Secondly, we will show that it is sufficient to consider only a finite subset of the infinite periodic point set in the hyperbolic plane. Finally, we will algorithmically construct a point set with which we can initialize the algorithm

Delaunay triangulations of regular hyperbolic surfaces

International audienceThe talk presents work on computing Delaunay triangulations of some symmetric hyperbolic surfaces of genus at least 2

Delaunay triangulations of generalized Bolza surfaces

The Bolza surface can be seen as the quotient of the hyperbolic plane, represented by the Poincaré disk model, under the action of the group generated by the hyperbolic isometries identifying opposite sides of a regular octagon centered at the origin. We consider _generalized_ Bolza surfaces Mg, where the octagon is replaced by a regular 4g-gon, leading to a genus g surface.We propose an extension of Bowyer's algorithm to these surfaces. In particular, we compute the value of the systole of Mg. We also propose algorithms computing small sets of points on Mg that are used to initialize Bowyer's algorithm