Lie and Noether symmetries of geodesic equations and collineations

The Lie symmetries of the geodesic equations in a Riemannian space are computed in terms of the special projective group and its degenerates (affine vectors, homothetic vector and Killing vectors) of the metric. The Noether symmetries of the same equations are given in terms of the homothetic and the Killing vectors of the metric. It is shown that the geodesic equations in a Riemannian space admit three linear first integrals and two quadratic first integrals. We apply the results in the case of Einstein spaces, the Schwarzschild spacetime and the Friedman Robertson Walker spacetime. In each case the Lie and the Noether symmetries are computed explicitly together with the corresponding linear and quadratic first integrals.Comment: 19 page

Architecture of the NADPH oxidase family of enzymes

The NADPH Oxidases (NOX) catalyze the deliberate production of reactive oxygen species (ROS) and are established regulators of redox-dependent processes across diverse biological settings. Proper management of their activity is controlled through a conserved electron transfer (ET) cascade from cytosolic NADPH substrate through the plasma membrane to extracellular O-2. After decades-long investigations of their biological functions, including potential as drug targets, only very recently has atomic-resolution information of NOX enzymes been made available. In this graphical review, we summarize the present structural biology understanding of the NOX enzymes afforded by X-ray crystallography and cryo-electron microscopy. Combined molecular-level insights predominantly informed by DUOX1 full-length Cryo-EM structures suggest a general structural basis for the control of their catalytic activity by intracellular domain-domain stabilization