Betanin and betanidin are compounds with extensive interest; they are effectively free radical scavengers. The present work aims to elucidate the differences between the mechanism of the antioxidant activity of betanin, betanidin, and their respective C15-epimers. Shape Theory establishes comparisons between the molecules’ geometries and determines parallelisms with the descriptors BDE, PA, ETE IP, PDE, and infrared spectra (IR) obtained from the molecule simulations. Furthermore, the molecules were optimized using the B3LYP/6-31+G(d,p) protocol. Finally, the molecular docking technique analyzes the antioxidant activity of the compounds in the complex with the therapeutic target xanthine oxidase (XO), based on a new proposal for the geometrical arrangement of the ligand atoms in the framework of Shape Theory. The results obtained indicate that the SPLET mechanism is the most favorable in all the molecules studied and that the first group that loses the hydrogen atom in the four molecules is the C17COOH, presenting less PA the isobetanidin. Furthermore, regarding the molecular docking, the interactions of these compounds with the target were favorable, standing out to a greater extent the interactions of isobetanidin with XO, which were analyzed after applying molecular dynamics.

The following explicit model, valid for high aperture refraction with homogenous and isotropic materials, encompasses all explicit solutions of the first-order nonlinear differential equation representing the perfect image-forming process of any axial object point into its axial image point. Solutions include well-known cases, such as flats, spheres, prolate ellipsoids, prolate hyperboloids, and other sections of nondegenerate Cartesian ovals of revolution, now classified according to the recurrent explicit solution introduced herein. We also present some series expansions, given in cylindrical coordinates $z(r)$, for more efficient computation. Explicit solutions allow accurate and expedite thickness calculation as compared to the regular series, parametric, or implicit solutions commonly used. The results of this study are useful in the design of centered optical systems that are perfectly aligned.

Automatic search of cavities and binding mode analysis between a ligand and a 3D protein receptor are challenging problems in drug design or repositioning. We propose a solution based on a shape theory theorem for an invariant coupled system of ligand-protein. The theorem provides a matrix representation with the exact formulas to be implemented in an algorithm. The method involves the following results: (1) exact formulae for the shape coordinates of a located-rotated invariant coupled system; (2) a parameterized search based on a suitable domain of van der Waals radii; (3) a scoring function for the discrimination of sites by measuring the distance between two invariant coupled systems including the atomic mass; (4) a matrix representation of the Lennard-Jones potential type 6-12 and 6-10 as the punctuation function of the algorithm for a molecular docking; and (5) the optimal molecular docking as a solution of an optimization problem based on the exploration of an exhaustive set of rotations. We apply the method in the xanthine oxidase protein with the following ligands: hypoxanthine, febuxostat, and chlorogenic acid. The results show automatic cavity detection and molecular docking not assisted by experts with meaningful amino acid interactions. The method finds better affinities than the expert software for known published cavities.

We report tunable RF-sputtered AZO nanofilms through of deposition conditions. Perfect adsorption of light was simulated in ENZ mode. The coupling condition of thickness and wavelength in perfect absorption was obtained by a Drude model. The fitted parameters range as follows: charge density, from 1.6 × 10²⁰ to 4.4 × 10²⁰ cm⁻³; mobilities, from 8.23 to 12.76 cm² V⁻¹ s⁻¹; and resistivities, from 14.20 to 34.98 Ω cm. The so-called spectroscopy shape analysis is introduced for automatic detection of elusive XPS peaks and sample surface-etching classification.