MGOS: A library for molecular geometry and its operating system

The geometry of atomic arrangement underpins the structural understanding of molecules in many fields. However, no general framework of mathematical/computational theory for the geometry of atomic arrangement exists. Here we present "Molecular Geometry (MG)'' as a theoretical framework accompanied by "MG Operating System (MGOS)'' which consists of callable functions implementing the MG theory. MG allows researchers to model complicated molecular structure problems in terms of elementary yet standard notions of volume, area, etc. and MGOS frees them from the hard and tedious task of developing/implementing geometric algorithms so that they can focus more on their primary research issues. MG facilitates simpler modeling of molecular structure problems; MGOS functions can be conveniently embedded in application programs for the efficient and accurate solution of geometric queries involving atomic arrangements. The use of MGOS in problems involving spherical entities is akin to the use of math libraries in general purpose programming languages in science and engineering. (C) 2019 The Author(s). Published by Elsevier B.V

Voronoi diagrams, quasi-triangulations, and beta-complexes for disks in R2: the theory and implementation in BetaConcept

Voronoi diagrams are powerful for solving spatial problems among particles and have been used in many disciplines of science and engineering. In particular, the Voronoi diagram of three-dimensional spheres, also called the additively-weighted Voronoi diagram, has proven its powerful capabilities for solving the spatial reasoning problems for the arrangement of atoms in both molecular biology and material sciences. In order to solve application problems, the dual structure, called the quasi-triangulation, and its derivative structure, called the beta-complex, are frequently used with the Voronoi diagram itself. However, the Voronoi diagram, the quasi-triangulation, and the beta-complexes are sometimes regarded as somewhat difficult for ordinary users to understand. This paper presents the twodimensional counterparts of their definitions and introduce the BetaConcept program which implements the theory so that users can easily learn the powerful concept and capabilities of these constructs in a plane. The BetaConcept program was implemented in the standard C++ language with MFC and OpenGL and freely available at Voronoi Diagram Research Center (http://voronoi.hanyang.ac.kr)

Abstract Euclidean Voronoi diagram of 3D balls and its computation via tracing edges

known as an additively weighted Voronoi diagram, in 3D space has not been studied as much as it deserves. In this paper, we present an algorithm to compute the Euclidean Voronoi diagram for 3D spheres with different radii. The presented algorithm follows Voronoi edges one by one until the construction is completed in O(mn) time in the worst-case, where m is the number of edges in the Voronoi diagram and n is the number of spherical balls. As building blocks, we show that Voronoi edges are conics that can be precisely represented as rational quadratic Bézier curves. We also discuss how to conveniently represent and process Voronoi faces which are hyperboloids of two sheets

Pocket extraction on proteins via the Voronoi diagram of spheres.

Proteins consist of atoms. Given a protein, the automatic recognition of depressed regions, called pockets, on the surface of proteins is important for protein-ligand docking and facilitates fast development of new drugs. Recently, computational approaches have emerged for recognizing pockets from the geometrical point of view. Presented in this paper is a geometric method for the pocket recognition which is based on the Voronoi diagram for atoms. Given a Voronoi diagram, the proposed algorithm transforms the atomic structure to meshes which contain the information of the proximity among atoms, and then recognizes depressions on the surface of a protein using the meshes. (c) 2007 Elsevier Inc. All rights reservedclose192

Euclidean Voronoi diagrams of 3D sphereds and applications to protein structure analysis.

Despite its many important applications in various disciplines in sciences and engineering, the Euclidean Voronoi diagram for spheres in 3D space has not been studied as much as it deserves. In this paper, we present an algorithm to compute a Euclidean Voronoi diagram for 3D spheres and show how the diagram can be used in the analysis of protein structures. Given an initial Voronoi vertex, the presented edge-tracing algorithm follows Voronoi edges until the construction is completed in O(mn) time in the worst-case, where m and n are the numbers of edges and spheres, respectively. Once a Voronoi diagram for 3D atoms of a protein is computed, it is shown that the diagram can be used to efficiently and precisely analyze the spatial structure of the protein. It turns out that this capability of a Voronoi diagram can be crucial to solving several important problems remaining to be solved in structural biologyclose172

BetaDock: Shape-Priority Docking Method Based on Beta-Complex

This paper presents an approach and a software, BetaDock, to the docking problem by putting the priority on shape complementarity between a receptor and a ligand. The approach is based on the theory of the ??-complex. Given the Voronoi diagram of the receptor whose topology is stored in the quasi-triangulation, the ??-complex corresponding to water molecule is computed. Then, the boundary of the ??-complex defines the ??-shape which has the complete proximity information among all atoms on the receptor boundary. From the ??-shape, we first compute pockets where the ligand may bind. Then, we quickly place the ligand within each pocket by solving the singular value decomposition problem and the assignment problem. Using the conformations of the ligands within the pockets as the initial solutions, we run the genetic algorithm to find the optimal solution for the docking problem. The performance of the proposed algorithm was verified through a benchmark test and showed that BetaDock is superior to a popular docking software AutoDock 4.close131

Multi-resolution protein model.

The area of molecular biology opens new applications for the communities of computer graphics, geometric modeling and computational geometry. It has been a usual understanding that the structure of a molecule is one of the major factors determining the functions of the molecule and therefore the efforts to better understand the molecular structure have been made. It turns out that the analysis and the prediction of the spatial structure of a molecule usually takes a significant amount of computation even though the number of atoms involved in the molecule is relatively small. Examples are the protein-ligand docking, protein folding, etc. In many molecules, however, the number of atoms is quite large. The number of atoms in the system varies from hundreds to thousands of thousand. The problem size gets even larger by both incorporating more details of a model and expanding the scope of the model from a single protein to a whole cell. This trend will continue as the computational resource gets more powerful and therefore the computational requirement will always remain critical. In this paper, we propose a multi-resolution model for a protein (MRPM) to find a seemingly optimal trade-off between the computational requirement and the solution quality. There are two aspects of the proposal: The avoidance of computation and the delay of computation until it is really necessary