Youngsong Cho

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

Despite its important applications in various disciplines in science and engineering, the Euclidean Voronoi diagram for spheres, also 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 Bezier curves. We also discuss how to conveniently represent and process Voronoi faces which are hyperboloids of two sheets.

Three-dimensional beta-shapes and beta-complexes via quasi-triangulation

The proximity and topology among particles are often the most important factor for understanding the spatial structure of particles. Reasoning the morphological structure of molecules and reconstructing a surface from a point set are examples where proximity among particles is important. Traditionally, the Voronoi diagram of points, the power diagram, the Delaunay triangulation, and the regular triangulation, etc. have been used for understanding proximity among particles. In this paper, we present the theory of the @b-shape and the @b-complex and the corresponding algorithms for reasoning proximity among a set of spherical particles, both using the quasi-triangulation which is the dual of the Voronoi diagram of spheres. Given the Voronoi diagram of spheres, we first transform the Voronoi diagram to the quasi-triangulation. Then, we compute some intervals called @b-intervals for the singular, regular, and interior states of each simplex in the quasi-triangulation. From the sorted set of simplexes, the @b-shape and the @b-complex corresponding to a particular value of @b can be found efficiently. Given the Voronoi diagram of spheres, the quasi-triangulation can be obtained in O(m) time in the worst case, where m represents the number of simplexes in the quasi-triangulation. Then, the @b-intervals for all simplexes in the quasi-triangulation can also be computed in O(m) time in the worst case. After sorting the simplexes using the low bound values of the @b-intervals of each simplex in O(mlogm) time, the @b-shape and the @b-complex can be computed in O(logm+k) time in the worst case by a binary search followed by a sequential search in the neighborhood, where k represents the number of simplexes in the @b-shape or the @b-complex. The presented theory of the @b-shape and the @b-complex will be equally useful for diverse areas such as structural biology, computer graphics, geometric modelling, computational geometry, CAD, physics, and chemistry, where the core hurdle lies in determining the proximity among spherical particles.

Quasi-worlds and quasi-operators on quasi-triangulations

Quasi-triangulation is the dual structure of the Voronoi diagram of spheres, and it has been used as a convenient and powerful geometric construct for representing the proximity among spherical particles with different radii. In this paper, we present the formalism of the quasi-triangulation based on a quasi-world model and define primitive query operators called quasi-operators for correct and efficient topology traversal on the quasi-triangulation. Algorithms for the quasi-operators are also presented based on the extended inter-world data structure. The proposed quasi-operators have the potential to be a fundamental platform on which efficient algorithms for application problems on quasi-triangulation can be correctly and easily developed. The recently announced powerful constructs of the @b-complex and the @b-shape are such examples.

Recognition of docking sites on a protein using β-shape based on Voronoi diagram of atoms

A protein consists of atoms. Given a protein, the automatic recognition of depressed regions on the surface of the protein, often called docking sites or pockets, is important for the analysis of interaction between a protein and a ligand and facilitates fast development of new drugs. Presented in this paper is a geometric approach for the detection of docking sites using @b-shape which is based on the Voronoi diagram for atoms in Euclidean distance metric. We first propose a geometric construct called a @b-shape which represents the proximity among atoms on the surface of a protein. Then, using the @b-shape, which takes the size differences among different atoms into account, we present an algorithm to extract the pockets for the possible docking site on the surface of a protein.

Euclidean Voronoi diagrams of 3D spheres 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 andn 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 biology.

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

Abstract 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 β-complex. Given the Voronoi diagram of the receptor whose topology is stored in the quasi-triangulation, the β-complex corresponding to water molecule is computed. Then, the boundary of the β-complex defines the β-shape which has the complete proximity information among all atoms on the receptor boundary. From the β-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.

Interaction interfaces in proteins via the Voronoi diagram of atoms

Abstract A protein consists of one or more chains where each chain is a linear sequence of amino acids bonded by peptide bonds. Chains in a protein interact with each other and the interaction is known to be one of the most fundamental factors which determine important physiological phenomena in the body. Hence, biologists have been investigating protein interactions since the early days of life science. While the studies on the interactions by biologists have emphasized the biological, chemical and/or physical aspects of the interactions, we approach the interactions from a geometric point of view. In this paper, we define an interaction interface using the Voronoi diagram of atoms in proteins. Based on a mathematical definition of the interaction interface, we provide a set of measures to characterize the inter- and intra-protein interactions. Given a Voronoi diagram of atoms in a protein consisting of a number of chains, we compute a geometric mid-surface, called an interaction interface, between each pair of chains in the protein. The interface consists of a set of faces where each face in the interface is a Voronoi face defined by two atoms belonging to different chains. Hence, the interface can be found in O ( m ) time in the worst case for m Voronoi faces in the Voronoi diagram. Then, a number of geometric and topological measures are derived from the interface to characterize the interaction.

An efficient algorithm for three-dimensional β-complex and β-shape via a quasi-triangulation

The concept of a β-shape has been recently proposed by extending the concept of the well-known α-shape. Since the β-shape takes full consideration of the Euclidean geometry of spherical particles, it is better suited than the (weighted) α-shape for applications using spatial queries on the system of variable sized spheres based on the Euclidean distance metric. In this paper, we present an efficient and elegant algorithm which computers a β-shape from a quasi-triangulation in O(log m + k) time in the worst case, where the quasi-triangulation has m simplicies and the boundary of β-shape consists of k simplicies. We believe that the β-shape and β-complex for a set of variable sized spheres (such as the atoms in a protein) will be very useful in the near future since the precise and efficient analysis of molecular structure can be conveniently facilitated by using these structures.

Innovative platform for transmission localized surface plasmon transducers and its application in detecting heavy metal Pd(II).

Transmission localized surface plasmon resonance (T-LSPR) transducers based on the characteristic surface plasmon absorption band of Au island films have become increasingly attractive. The first and main bottleneck hampering the development of T-LSPR sensors is instability, manifested as change in the surface plasmon absorbance band following immersion in organic solvents and aqueous solutions. In this paper, we innovate the platform for T-LSPR transducer by using remarkably stable and highly adhesive Au/Al(2)O(3) nanocomposite film. Isolated Au nanoparticles embedded in dielectric matrix Al(2)O(3) were prepared by a simple one-step radio frequency magnetron cosputtering technique. The obtained nanocomposite film is exceedingly stable during immersion in solvents, drying, and binding of different molecules; it successfully passes the adhesive tape test and sonication treatment. The superior stability and adhesion, obtained without the use of any intermediate adhesion layer or protective overlayer, is attributed to (1) the Au nanoparticles embedment and Al(2)O(3) rim formation during the sputtering process and (2) the resistance of element Al in matrix to the nucleophilic attack by the solvent molecules. Given this success, we believe that the Au/Al(2)O(3) nanocomposite film holds promise as an innovative sensing platform in T-LSPR detection technology, as demonstrated here for the Pd(II) sensing process with excellent sensitivity and low detection limit.

Querying simplexes in quasi-triangulation

Given a quasi-triangulation, the dual structure of the Voronoi diagram of a molecule, querying its simplexes of a particular bounding state for the spherical probe of a given radius occurs frequently. The @b-complex and the @b-shape are such examples with various applications for reasoning the spatial structure of molecules. While such simplexes can be found by linearly scanning all simplexes in the quasi-triangulation, it is desirable to do the query faster. This paper introduces the Q2P-transformation that transforms the simplex query problem in the three-dimensional space to the orthogonal range search problem of points in another three-dimensional space. Based on this observation, we show that the kd-tree data structure suits very well for developing efficient query algorithms for the quasi-triangulation of molecules using the set of sixteen primitive and the set of ten high-level query operators. In particular, the proposed method shows an extremely efficient performance for incremental queries. A subset of the presented observations facilitates efficient simplex queries for any simplicial complexes including the Delaunay and regular triangulations.