Joonghyun Ryu

Molecular surfaces on proteins via beta shapes

A protein consists of linearly combined amino acids via peptide bonds, and an amino acid consists of atoms. It is known that the geometric structure of a protein is the primary factor which determines the functions of the protein. Given the atomic complex of a protein, one of the most important geometric structures of a protein is its molecular surface since this distinguishes between the interior and exterior of the protein and plays an important role in protein folding, docking, interactions between proteins, and other functions. This paper presents an algorithm for the precise and efficient computation of the molecular surface of a protein, using a recently proposed geometric construct called the @b-shape based on the Voronoi diagram of atoms in a protein. Given a Voronoi diagram of atoms, based on the Euclidean distance from the atom surfaces, the proposed algorithm first computes the @b-shape with an appropriate sized probe. Then, the molecular surface is computed by employing a blending operation on the atomic complex of the protein. In this paper, it is also shown that for a given Voronoi diagram of atoms, the multiple molecular surfaces can be computed by using various sized probes.

Three-dimensional beta shapes

The Voronoi diagram of a point set has been extensively used in various disciplines ever since it was first proposed. Its application realms have been even further extended to estimate the shape of point clouds when Edelsbrunner and Mucke introduced the concept of @a-shape based on the Delaunay triangulation of a point set. In this paper, we present the theory of @b-shape for a set of three-dimensional spheres as the generalization of the well-known @a-shape for a set of points. The proposed @b-shape fully accounts for the size differences among spheres and therefore it is more appropriate for the efficient and correct solution for applications in biological systems such as proteins. Once the Voronoi diagram of spheres is given, the corresponding @b-shape can be efficiently constructed and various geometric computations on the sphere complex can be efficiently and correctly performed. It turns out that many important problems in biological systems such as proteins can be easily solved via the Voronoi diagram of atoms in proteins and @b-shapes transformed from the Voronoi diagram.

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.

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.

Beta-decomposition for the volume and area of the union of three-dimensional balls and their offsets†

Given a set of spherical balls, called atoms, in three‐dimensional space, its mass properties such as the volume and the boundary area of the union of the atoms are important for many disciplines, particularly for computational chemistry/biology and structural molecular biology. Despite many previous studies, this seemingly easy problem of computing mass properties has not been well‐solved. If the mass properties of the union of the offset of the atoms are to be computed as well, the problem gets even harder. In this article, we propose algorithms that compute the mass properties of both the union of atoms and their offsets both correctly and efficiently. The proposed algorithms employ an approach, called the Beta‐decomposition, based on the recent theory of the beta‐complex. Given the beta‐complex of an atom set, these algorithms decompose the target mass property into a set of primitives using the simplexes of the beta‐complex. Then, the molecular mass property is computed by appropriately summing up the mass property corresponding to each simplex. The time complexity of the proposed algorithm is O(m) in the worst case where m is the number of simplexes in the beta‐complex that can be efficiently computed from the Voronoi diagram of the atoms. It is known in ℝ3 that m = O(n) on average for biomolecules and m = O(n2) in the worst case for general spheres where n is the number of atoms. The theory is first introduced in ℝ2 and extended to ℝ3. The proposed algorithms were implemented into the software BetaMass and thoroughly tested using molecular structures available in the Protein Data Bank. BetaMass is freely available at the Voronoi Diagram Research Center web site.

Anomalies in quasi-triangulations and beta-complexes of spherical atoms in molecules

The beta-complex is the most compact and efficient representation of molecular structure as it stores the precise proximity among spherical atoms in molecules. Thus, the beta-complex is a powerful tool for solving otherwise difficult shape-related problems in molecular biology. However, to use the beta-complex properly, it is necessary to correctly understand the anomalies of both the quasi-triangulation and the beta-complex. In this paper, we present the details of the anomaly of the beta-complex in relation to the quasi-triangulation. With a proper understanding of anomaly theory, seemingly complicated application problems related to the geometry and topology among spherical balls can be correctly and efficiently solved in rather straightforward computational procedures. We present the theory with examples in both R^2 and R^3.

BetaSCPWeb: side-chain prediction for protein structures using Voronoi diagrams and geometry prioritization

Many applications, such as protein design, homology modeling, flexible docking, etc. require the prediction of a proteins optimal side-chain conformations from just its amino acid sequence and backbone structure. Side-chain prediction (SCP) is an NP-hard energy minimization problem. Here, we present BetaSCPWeb which efficiently computes a conformation close to optimal using a geometry-prioritization method based on the Voronoi diagram of spherical atoms. Its outputs are visual, textual and PDB file format. The web server is free and open to all users at http://voronoi.hanyang.ac.kr/betascpweb with no login requirement.

Protein structure optimization by side-chain positioning via beta-complex

A molecular structure determines a molecular function(s) and a correct understanding of molecular structure is important for biotechnology. The computational prediction of molecular structure is a frequent requirement for important biomolecular applications such as a homology modeling, a docking simulation, a protein design, etc. where the optimization of molecular structure is fundamental. One of the core problems in the optimization of protein structure is the optimization of side-chains called the side-chain positioning problem. The side-chain positioning problem, assuming the rigidity of backbone and a rotamer library, attempts to optimally assign a rotamer to each residue so that the potential energy of protein is minimized in its entirety. The optimal solution approach using (mixed) integer linear programming, with the dead-end elimination technique, suffers even for moderate-sized proteins because the side-chain positioning problem is NP-hard. On the other hand, popular heuristic approaches focusing on speed produce solutions of low quality. This paper presents an efficient algorithm, called the BetaSCP, for the side-chain positioning problem based on the beta-complex which is a derivative geometric construct of the Voronoi diagram. Placing a higher priority on solution quality, the BetaSCP algorithm produces a solution very close to the optima within a reasonable computation time. The effectiveness and efficiency of the BetaSCP are experimentally shown via a benchmark test against well-known algorithms using twenty test models selected from Protein Data Bank.

QTF: Quasi-triangulation file format

Abstract A quasi-triangulation is the dual structure of the Voronoi diagram of spherical balls and its properties and algorithms are well-studied in three-dimensional space. Quasi-triangulation has been used for efficiently solving various structure/shape related problems for biomolecules. The computation of the quasi-triangulation directly from an input file can take a significant amount of time. If the quasi-triangulation is computed a priori and stored in a file, an application software can directly load the file for solving application problems. In this paper, we propose a neutral file format, called the quasi-triangulation file format QTF, so that users can use the quasi-triangulation more effectively and efficiently by focusing more on his or her own application problems than the Voronoi diagram or the quasi-triangulation itself. The proposed QTF file format was thoroughly validated through an extensive experiment by computing the molecular volumes of one hundred molecular models in the Protein Data Bank. This approach has an important consequence: The QTF file format separates the computation of the Voronoi diagram from its applications.

\beta-shape Based Computation of Blending Surfaces on a Molecule

It has been generally accepted that the structure of molecule is one of the most important factors which determine the functions of a molecule. Hence, studies have been conducted to analyze the structure of a molecule. Molecular surface is an important example of molecular structure. Given a molecular surface, the area and volume of the molecule can be computed to facilitate problems such as protein docking and folding. Therefore, it is important to compute a molecular surface precisely and efficiently. This paper presents an algorithm for correctly and efficiently computing the blending surfaces of a protein which is an important part of the molecular surface. Assuming that the Voronoi diagram of atoms of a protein is given, we first compute the beta-shape of the protein corresponding to a solvent probe. Then, we use a search space reduction technique for the intersection tests while the link blending surface is computed. Once a beta-shape is obtained, the blending surfaces corresponding to a given solvent probe can be computed in O(n) in the worst case, where n is the number of atoms. The correctness and efficiency of the algorithm stem from the powerful properties of beta-shape, quasi-triangulation, and the interworld data structure.