Computer-Aided Design 39 (2007) 1042–1057 www.elsevier.com/locate/cad Molecular surfaces on proteins via beta shapes Joonghyun Ryu a , Rhohun Park a , Deok-Soo Kim b,∗ a Voronoi Diagram Research Center, Hanyang University, 17 Haengdang-dong, Seongdong-gu, Seoul 133-791, South Korea b Department of Industrial Engineering, Hanyang University, 17 Haengdang-dong, Seongdong-gu, Seoul 133-791, South Korea Received 7 September 2005; accepted 31 October 2006 Abstract 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 β -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 β -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. c  2006 Elsevier Ltd. All rights reserved. Keywords: Protein; Molecular surface; β-shape; Voronoi diagram of atoms; Blending surface 1. Introduction A protein consists of linearly combined amino acids via peptide bonds, and an amino acid consists of atoms. Hence, a protein usually consists of hundreds of thousands of atoms. When a peptide bond is created between two consecutive amino acids, a water molecule H 2 O is always removed. The atomic structure of a protein is usually determined using either X-ray crystallography or nuclear magnetic resonance (NMR) spectroscopy. Once the atomic structure is determined, the new structure is uploaded to the Protein Data Bank (PDB) [4,34] with a unique identification. One of the main contents of protein data in PDB is the coordinates of atoms constituting the protein. Fig. 1(a) shows a transcription regulation complex, encoded as 1bh8 in PDB, consisting of 1074 atoms (680 C’s, 181 N’s, 203 O’s, and 10 S’s). This protein is an example of a dimer, which consists of two disconnected chains of amino acids. In this figure, each atom ∗ Corresponding author. Tel.: +82 2 2220 0472; fax: +82 2 2292 0472. E-mail addresses: jhryu@voronoi.hanyang.ac.kr (J. Ryu), rhpark@voronoi.hanyang.ac.kr (R. Park), dskim@hanyang.ac.kr (D.-S. Kim). is shown as a hard sphere with a corresponding van der Waals radius. It is well known that the geometric structure of a protein is one of the most important factors which determine the functions of the protein. In particular, the atoms located at the boundary of a protein mainly determine the functions of a protein [9,8]. Hence, correctly locating the atoms constituting the protein boundary efficiently becomes very important. One of the approaches for doing this is to compute a compact smooth surface on the boundary atoms of a protein, and this surface is called a molecular surface or Connolly surface after the researcher who first analytically defined the surface [8]. Shown in Fig. 1(b) is such a molecular surface covering the boundary atoms of protein in Fig. 1(a). There are many important uses of the molecular surface of a protein [35,9,8,39,10,31]. An important example is the computation of the volume and density of a protein. Unless a molecular surface is clearly defined, the computations of the surface area and the volume of a protein are not possible. The molecular surface is also very useful in the study of protein docking, folding, and inter- and intra-protein interactions since it defines the boundary of free space where other atoms or 0010-4485/$ - see front matter c  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.cad.2006.10.008