Topological Signals of Singularities in Ricci Flow

We implement methods from computational homology to obtain a topological signal of singularity formation in a selection of geometries evolved numerically by Ricci flow. Our approach, based on persistent homology, produces precise, quantitative measures describing the behavior of an entire collection of data across a discrete sample of times. We analyze the topological signals of geometric criticality obtained numerically from the application of persistent homology to models manifesting singularities under Ricci flow. The results we obtain for these numerical models suggest that the topological signals distinguish global singularity formation (collapse to a round point) from local singularity formation (neckpinch). Finally, we discuss the interpretation and implication of these results and future applications.Comment: 24 pages, 14 figure

Metagenomic Analysis of Respiratory Tract DNA Viral Communities in Cystic Fibrosis and Non-Cystic Fibrosis Individuals

The human respiratory tract is constantly exposed to a wide variety of viruses, microbes and inorganic particulates from environmental air, water and food. Physical characteristics of inhaled particles and airway mucosal immunity determine which viruses and microbes will persist in the airways. Here we present the first metagenomic study of DNA viral communities in the airways of diseased and non-diseased individuals. We obtained sequences from sputum DNA viral communities in 5 individuals with cystic fibrosis (CF) and 5 individuals without the disease. Overall, diversity of viruses in the airways was low, with an average richness of 175 distinct viral genotypes. The majority of viral diversity was uncharacterized. CF phage communities were highly similar to each other, whereas Non-CF individuals had more distinct phage communities, which may reflect organisms in inhaled air. CF eukaryotic viral communities were dominated by a few viruses, including human herpesviruses and retroviruses. Functional metagenomics showed that all Non-CF viromes were similar, and that CF viromes were enriched in aromatic amino acid metabolism. The CF metagenomes occupied two different metabolic states, probably reflecting different disease states. There was one outlying CF virome which was characterized by an over-representation of Guanosine-5′-triphosphate,3′-diphosphate pyrophosphatase, an enzyme involved in the bacterial stringent response. Unique environments like the CF airway can drive functional adaptations, leading to shifts in metabolic profiles. These results have important clinical implications for CF, indicating that therapeutic measures may be more effective if used to change the respiratory environment, as opposed to shifting the taxonomic composition of resident microbiota

Simplicial Ricci flow: an example of a neck pinch singularity in 3D

We examine a Type-1 neck pinch singularity in simplicial Ricci flow (SRF) for an axisymmetric piecewise flat 3-dimensional geometry with topology S³. SRF was recently introduced as an unstructured mesh formulation of Hamilton’s Ricci flow (RF). It describes the RF of a piecewise-flat simplicial geometry. In this paper, we apply the SRF equations to a representative double-lobed axisymmetric piecewise flat geometry with mirror symmetry at the neck similar to the geometry studied by Angenent and Knopf (A-K). We choose a specific radial profile and compare the SRF equations with the corresponding finite-difference solution of the continuum A-K RF equations. The piecewise-flat 3-geometries considered here are built of isosceles-triangle-based frustum blocks. The axial symmetry of this model allows us to use frustum blocks instead of tetrahedra. The S² cross-sectional geometries in our model are regular icosahedra. We demonstrate that, under a suitably-pinched initial geometry, the SRF equations for this relatively low-resolution discrete geometry yield the canonical Type-1 neck pinch singularity found in the corresponding continuum solution. We adaptively remesh during the evolution to keep the circumcentric dual lattice wellcentered. Without such remeshing, we cannot evolve the discrete geometry to neck pinch. We conclude with a discussion of future generalizations and tests of this SRF model

Functions of Type II Pneumocyte-Derived Vascular Endothelial Growth Factor in Alveolar Structure, Acute Inflammation, and Vascular Permeability

Vascular endothelial growth factor-A (VEGF) is a potent regulator of vascular permeability, inflammatory response, and cell survival in the lung. To explore the functions of VEGF produced locally in type II pneumocytes, we generated mice with a conditional deletion of VEGF-A using Cre recombinase driven by the human surfactant protein C (SPC) promoter. In 7- to 10-week-old VEGF-knockout (SPC-VEGF-KO) mice, lung histology and physiology were essentially normal, except for higher dynamic lung compliance and lower pulmonary vascular permeability. Emphysema was seen in 28- to 32-week-old animals. To investigate the role of type II pneumocyte-derived VEGF in acute lung injury, we challenged 7- to 10-week-old SPC-VEGF-KO mice and their wild-type littermates with intestinal ischemia-reperfusion. Bronchoalveolar lavage fluid total cell count, pulmonary permeability, and lung injury score were significantly attenuated, and total lung VEGF levels were significantly lower in SPC-VEGF-KO mice compared with wild-type controls. In SPC-VEGF-KO mice, activated caspase 3-positive type II epithelial cells were increased after intestinal ischemia-reperfusion, even though there was no significant difference in the total number of cells positive for terminal deoxynucleotidyl transferase dUTP nick-end labeling. We conclude that VEGF in type II cells helps protect alveolar epithelial cells from caspase-dependent apoptosis. However, VEGF produced from type II cells may contribute to increased vascular permeability during acute lung injury