The β-shape and β-complex for analysis of molecular structures

Abstract

The topology among particles frequently plays a core role in many applications. One of the emerging application areas of particle systems is the analysis of molecular structures since the morphology of a molecule has been recognized as one of the most important factors which determines the functions of the molecule. To understand the morphology of molecules, various computational methodologies have been extensively investigated like the Voronoi diagram of the centers of atoms in the molecule, the power diagram for the weighted points where the weights are related to the radii of the atoms, etc. For a more improved efficiency, constructs like an β-shape or a weighted β-shape have been developed and used frequently in a systematic analysis of the morphology of molecules. However, it has been recently shown that β-shapes and weighted β-shapes lack the fidelity to Euclidean distance for molecules with polysized spherical atoms. We will present the theory, as well as the corresponding algorithms, of β-shape and β-complex in R3 which reflects the size difference among atoms in their full Euclidean metric. We show that these new concepts are more natural for most applications and therefore will have a significant impact on applications based on particles, in particular in molecular biology. The theory will be equivalently useful for other application areas such as computer graphics, geometric modeling, chemistry, physics, material science, etc.