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.

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Molecular shape is essential in understanding molecular function, and understanding molecular shape requires definition of molecular boundaries. In this paper, we review the conceptual evolution of three molecular boundary types: the van der Waals surface, the Connolly surface, and the Lee-Richards (accessible) surface. Then, we point out the confusion among the names of these surfaces existing in the literature. Since it is desirable to have a well-defined terminology in a discipline, we propose the standard names of the three molecular boundary types and their corresponding volumes in order to maximize consistency among researchers, respect the first individual who defined or computed a surface type, and promote collaboration between biologists and geometers.

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).