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

Kim, Deok-Soo; Cho, Youngsong; Sugihara, Kōkichi; Ryu, Joonghyun; Kim, Donguk

doi:10.1016/j.cad.2010.06.004

Cited by 50 publications

(54 citation statements)

References 21 publications

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“…The recently developed theory of the beta-complex overcomes this drawback of the weighted alpha-complex so that the Voronoi diagram is computed only once for the entire lifetime of a given atom set and stored in the quasi-triangulation format for any type of application problems for biomolecules. [26,27] The beta-complex not only efficiently solves geometry/topology problems in computational chemistry/biology but also brings otherwise computationally infeasible problems into a feasible space.…”

Section: Combinatorial Structure Of the Moleculementioning

confidence: 99%

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

et al. 2012

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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 ℝ(3) that m = O(n) on average for biomolecules and m = O(n(2)) in the worst case for general spheres where n is the number of atoms. The theory is first introduced in ℝ(2) and extended to ℝ(3). 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.

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Section: Combinatorial Structure Of the Moleculementioning

confidence: 99%

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

et al. 2012

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“…Authors' group has proposed the geometric constructs called the beta-complex and betashape along with their computational algorithms (23) . Consider a three-dimensional space filled with a soft matter with some spherical atoms of varying radii scattered within the matter.…”

Section: Beta-complex and Beta-shapementioning

confidence: 99%

“…Given the quasi-triangulation, the betacomplex and the beta-shape can be computed very efficiently via a pre-processing of the analysis of the quasi-triangulation simplexes with respect to the size of atoms and the probe radius. The algorithm for searching simplexes of the beta-complex from the quasi-triangulation taking O(log n + k) time in the worst case, where there are k simplexes in the beta-complex, were reported in (23). Recently, a more generalized query optimization has been reported (25) .…”

Section: Beta-complex and Beta-shapementioning

confidence: 99%

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BetaMol: A Molecular Modeling, Analysis and Visualization Software Based on the Beta-Complex and the Quasi-Triangulation

Cho

Kim

Ryu

et al. 2012

JAMDSM

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Molecular shape is one of the most critical factors that determines molecular function. Therefore, it is frequently desirable to understand geometric characteristics of a molecule more precisely and efficiently. In this paper, we introduce the BetaMol, a molecular modeling, analysis, and visualization software based on the recent theory of the beta-complex and the quasi-triangulation that are derived from the Voronoi diagram of three-dimensional spherical atoms. The powerful features of the BetaMol are solely based on a unified, single framework of the mathematically rigorous and computationally efficient beta-complex theory. The BetaMol is implemented in the standard C++ language with OpenGL graphics library and freely available at Voronoi Diagram Research Center web site (http://voronoi.hanyang.ac.kr).

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“…ernet.in on the theory of alpha shapes and discrete flows in Delaunay triangulations. Kim et al [11,13] propose a definition of cavities based on an alternate representation of a set of atoms called beta shapes that faithfully captures proximity. Tools based on the above approach are available and widely used [2,14,12].…”

Section: Introductionmentioning

confidence: 99%

Extraction of Robust Voids and Pockets in Proteins

Sridharamurthy

Masood

Doraiswamy³

et al. 2016

Visualization in Medicine and Life Sciences III

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Voids and pockets in a protein refer to empty spaces that are enclosed by the protein molecule. Existing methods to compute, measure, and visualize the voids and pockets in a protein molecule are sensitive to inaccuracies in the empirically determined atomic radii. This paper presents a topological framework that enables robust computation and visualization of these structures. Given a fixed set of atoms, voids and pockets are represented as subsets of the weighted Delaunay triangulation of atom centers. A novel notion of (ε, π)-stable voids helps identify voids that are stable even after perturbing the atom radii by a small value. An efficient method is described to compute these stable voids for a given input pair of values (ε, π).

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

Cited by 50 publications

References 21 publications

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

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

BetaMol: A Molecular Modeling, Analysis and Visualization Software Based on the Beta-Complex and the Quasi-Triangulation

Extraction of Robust Voids and Pockets in Proteins

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