Interaction interfaces in proteins via the Voronoi ... - Semantic Scholar

Computer-Aided Design 38 (2006) 1192–1204 www.elsevier.com/locate/cad

Interaction interfaces in proteins via the Voronoi diagram of atoms Chong-Min Kim a , Chung-In Won a , Youngsong Cho b , Donguk Kim b , Sunghoon Lee c , Jonghwa Bhak c , Deok-Soo Kim a,b,∗ a Department of Industrial Engineering, Hanyang University, 17 Haengdang-dong, Seongdong-gu, Seoul 133-791, South Korea b Voronoi Diagram Research Center, Hanyang University, 17 Haengdang-dong, Seongdong-gu, Seoul 133-791, South Korea c Korean Bioinformation Center, Korea Research Institute of Bioscience and Biotechnology, 52 Yuseong-gu, Eoeun-dong, Daejeon 305-333, South Korea

Received 25 February 2006; accepted 26 July 2006

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. c 2006 Elsevier Ltd. All rights reserved.

Keywords: Interaction interface; Protein interaction; Voronoi diagram

1. Introduction In this era of biotechnology, a significant amount of research is being devoted to the study of proteins since they are related to the critical functions of the body. Hence, there have been many studies on proteins and the amount of related data is enormous and available in various public repositories including the Protein Data Bank (PDB) [6,13,21,51,60,62]. It is known that the interactions among proteins affect various functions in a living cell. Therefore, biologists have been giving a lot of attention to the analysis of the interactions between proteins [31,45,53–56,63,65,68]. ∗ Corresponding author at: Department of Industrial Engineering, Hanyang University, 17 Haengdang-dong, Seongdong-gu, Seoul 133-791, South Korea. Tel.: +82 2 2220 0472; fax: +82 2 2292 0472. E-mail addresses: [email protected] (C.-M. Kim), [email protected] (C.-I. Won), [email protected] (Y. Cho), [email protected] (D. Kim), [email protected] (S. Lee), [email protected] (J. Bhak), [email protected] (D.-S. Kim).

c 2006 Elsevier Ltd. All rights reserved. 0010-4485/$ - see front matter doi:10.1016/j.cad.2006.07.007

A large portion of these studies has focused on the biological, chemical and/or physical aspects of these interactions. However, many recent studies are examining the interactions from a geometric perspective [1,4,22,30,46,50,57,64,66]. Since the role of geometry in biological systems is becoming more important, studies on the geometry in these systems will provide new challenges as well as opportunities for the CAD and CAGD communities. One of the most fundamental geometric issues in biology is the interaction interface among proteins. In 1995, Varshney et al. defined an interface between proteins from a geometric perspective. They used the power diagram of the atoms in proteins and called the interface a molecular interface surface [66]. Each face in the interface is defined by a bisector face between two nearby atoms of different proteins. They also presented an algorithm to extract the interface from the power diagram and visualize the interface. In 2004, Ban et al. also used the power diagram of atoms in proteins and provided a set of measures to characterize

C.-M. Kim et al. / Computer-Aided Design 38 (2006) 1192–1204

1193

Fig. 1. Examples of protein complexes: (a) a dimer (1A0M), (b) a trimer (1CE0), and (c) a tetramer (1CSK). PDB codes of the proteins are in the parentheses.

the interaction interface between proteins [4]. These measures include the genus and the number of holes which are used for determining the global measure, and the distances between atoms in different proteins which is the local measure. Later, Cazals and Proust worked again on the definition and computation of the interaction interface based on the power diagram [14]. In 2004, Kim proposed another approach using the Voronoi diagram of atoms, instead of a power diagram, to define and analyze the interaction interface in proteins [35,39]. In this study, the distance metric is the minimum Euclidean distance from the boundary of an atom. Kim also showed that the Voronoi diagram of atoms is a better fit for analyzing the structure of proteins for other geometric problems as well [40–42]. Recently, a thesis fully dedicated to the definition and the analysis of interaction interfaces using a Voronoi diagram of atoms has been published [33]. In addition, a database of the computed interaction interfaces has been also proposed in this thesis. In this paper, we define an interaction interface in protein complexes and present an algorithm to extract the interaction interface from the Voronoi diagram of atoms. We also provide detailed measures for the analysis of extracted interfaces so that the interfaces can be classified. The remainder of the paper is organized as follows: Section 2 explains the basics of protein structure and the interactions between proteins. Section 3 briefly discusses the Voronoi diagram of atoms since it is the most fundamental computational tool used in this paper. Section 4 provides the definition of the interaction interface in protein complexes using the Voronoi diagram. Section 5 provides the definition of a base surface which is the minimum bending energy surface transformation of an interaction interface. Section 6 presents various measures for analyzing the geometric characteristics of interaction interfaces. Section 7 presents the experimental results for the analysis of the computed interaction interfaces for proteins from PDB. The conclusion is provided in Section 8. 2. Interactions in proteins One of the most fundamental goals of protein analysis is to understand how proteins carry out various essential processes for life. One of the main efforts of the analysis is to identify the protein structure of proteins and the interactions among proteins [10].

Currently, the three-dimensional coordinates of atoms for more than 30,000 proteins are known and available through PDB [60]. However, the properties and functions of many proteins have yet to be studied/determined. One way of determining a protein’s function is by investigating its interactions with other proteins. This approach is based on the premise that the function of a protein can be discovered in part via its interactions with other proteins with known functions. Hence, the study of interactions can be vital to the understanding of proteins [31]. A protein is a macromolecule consisting of a linear sequence of up to 20 different amino acids, and the distinct sequence of the amino acids determines the unique three-dimensional structure of a protein. It is believed that the function of a protein is mostly determined by its unique three-dimensional structure. An amino acid consists of dozens of atoms. For example, glycine, the smallest among the twenty amino acids, consists of 10 atoms: 2 C’s, 5 H’s, 2 O’s and a single N. On the other hand, tryptophan, the largest amino acid, consists of 27 atoms: 11 C’s, 12 H’s, 2 O’s and 2 N’s. An amino acid has a carbon called an α-carbon Cα at its topological center. Around the Cα , a hydrogen (H), an amino-group (NH2 ), a carboxyl-group (COOH) and a side-chain (R) are attached. Note that a sidechain is also called an R-group. Most of the 20 amino acids have an identical global topology structure except for the sidechain. Therefore, the side-chain determines the unique threedimensional structure and function of a protein. Consecutive amino acids are connected to each other via a peptide bond between their respective carboxyl- and amino-groups [5]. The structure of a protein is usually viewed from four different hierarchical levels. The linear sequence of amino acids is called the primary structure of a protein. Due to the interactions between the atoms in the primary structure, some amino acids form α-helices or β-sheets in three-dimensional space and these descriptors are denoted as the secondary structure of a protein. The fold of secondary structures in the three dimensional space is called a tertiary structure of the protein. The linear sequence of amino acids connected via peptide bonds forms a chain. Many proteins exist in nature as a set of more than one chain called the quaternary structure of a protein [12]. A protein consisting of a single chain is called a monomer. When a protein consists of two, three, or four chains, it is called a dimer, a trimer, or a tetramer, respectively. Fig. 1 shows examples of a dimer, a trimer, and a tetramer.

1194

C.-M. Kim et al. / Computer-Aided Design 38 (2006) 1192–1204

An interaction defined between two (or more) proteins is called an inter-protein interaction, and an interaction between chains in a protein is called an intra-protein interaction [9,32]. From a geometric point of view, the interaction interfaces of both types of interactions are identical. Even though we will focus the discussion on the intra-protein interaction interface, the concept and techniques can be directly applied to the interprotein interaction interface as well. 3. Voronoi diagram of atoms in protein Suppose that a protein A = {a1 , . . . , an } is a set of van der Waals atoms ai = (ci , ri ) with a center ci and van der Waals radius ri . Let K = {c1 , . . . , cn } be the centers of atoms in A. Then, it is important to have a compact and efficient representation of the topology among atoms in the protein to process various queries about the spatial structure of the protein. For this purpose, the ordinary point set Voronoi diagram VD(K ) for atom centers K was first used by Richards in 1974 to study protein structures [61]. However, Richards immediately noticed that the ordinary Voronoi diagram of points does not account for the size variation among atoms in a system [61]. Richards, therefore, proposed to translate the bisector between two atoms according to the size difference between the two atoms. However, the translations of bisectors caused so-called vertex error since this transformation does not guarantee a correct tessellation of space. In 1982, Gellatly and Finney proposed a method using radical planes to make sure that no vertex error occurs while the size variation of atoms is somewhat reflected [7]. This radical plane approach is in fact equivalent to the approach using the power diagram PD(A) of A, as named by Aurenhammer in 1987 [2]. Since then, the power diagram has been frequently used for biological problems since it reflects the size differences among atoms at a certain level [3,4,24,26,66]. Note that the theory of the power diagram is also well-established and efficient and robust codes are available [15]. However, PD(A) does not fully reflect the size differences among the atoms in the problems requiring Euclidean distance metric and therefore the need for the Voronoi diagram of atoms arises [27,58]. Recently, Kim et al. proposed to use the Voronoi diagram VD(A) of atoms A for the structural analysis of proteins and showed its effectiveness and efficiency for structural biology problems [33,35,37,38,40,42]. The Voronoi diagram VD(A) for an atomic complex A is defined as follows. Given a set of atoms A, we call a region V Ri the Voronoi region of an atom ai where V Ri = { p ∈ R 3 | dist( p, ci ) − ri ≤ dist( p, c j ) − r j , i 6= j}. Note that dist( p, q) denotes an ordinary Euclidean distance between two points p and q. Then, the set VD(A) = {V R1 , V R2 , . . . , V Rn } is the Voronoi diagram for the atom set A. A Voronoi vertex v is the center of an empty sphere tangent to four nearby atoms, and a Voronoi edge e is defined as a locus of points equi-distant from the surfaces of three surrounding atoms. In addition, a Voronoi face f is a hyperbolic surface equi-distant from the boundaries of two nearby atoms. In this research, VD(A) is represented as G = (V V , E V , F V ) where V V =

{v1 , v2 , . . .}, E V = {e1 , e2 , . . .} and F V = { f 1 , f 2 , . . .} are sets of Voronoi vertices, edges and faces, respectively. For more details of algorithms used in this research as well as other related algorithms, readers are referred to [34,37,38]. Once the Voronoi diagram is constructed, then it is stored in an appropriate topology data structure such as a radial data structure for the efficient processing of various queries [16–18, 47,49]. We assume that the degree of a Voronoi vertex is always four for all vertices in the Voronoi diagram. The treatment of degeneracy is not within the scope of this paper and will be discussed separately. 4. Interaction interface via a Voronoi diagram of atoms In this section, we provide the formal definition of an interaction interface from a geometric point of view, and present an algorithm to compute the interaction interface from the Voronoi diagram VD(A) of an atom set A. 4.1. Untrimmed interaction interface: IIF ∞ Suppose that we are given a proper tessellation of space where an atom set A is defined. Examples of such tessellation may be a Voronoi diagram of atom centers, a power diagram of the atoms, or a Voronoi diagram of the atoms. In these tessellations, each face in the tessellation is defined by two nearby atoms in the set using a distance definition approximation for the tessellation. Suppose that set A consists of two or more groups where the unit of the group may be either a chain or a protein. Then, we collect a set F of faces where each face f ∈ F is defined by two atoms from different groups in the set. Then, set F is called an interaction interface among the groups in set A. Note that this definition of the interaction interface applies to any of the previously described schemes of the space tessellation of set A. For example, the subset of Voronoi faces of the ordinary Voronoi diagram of atom centers may be used as an interaction interface. In some of the previous studies of the interaction interface, the faces of the power diagram were used to compute the interaction interface [4,28,66]. In this paper, we instead define the interaction interface using the Voronoi diagram of atoms to make it more precise by fully reflecting the size differences among atoms in the set. Let A = {a1 , a2 , . . . , am }, and B = {b1 , b2 , . . . , bn } be two chains in a protein, where ai and b j are atoms with appropriate centers and radii. The interaction interface IIF ∞ (A, B) between chains A and B is defined as IIF ∞ (A, B) = { p | dist( p, A) = dist( p, B)}

(1)

where dist( p, A) denotes the minimum Euclidean distance from p to the surfaces of all van der Waals atoms in set A. Then, it can be easily shown that IIF ∞ (A, B) is the subset of Voronoi faces in the Voronoi diagram VD(A ∪ B) of all atoms of A ∪ B, where each face f ∈ IIF ∞ (A, B) is defined by two atoms from two different chains. Hence, the following definition of IIF ∞ (A, B) is equivalent to the definition in Eq. (1) IIF ∞ (A, B) = { f ∈ F V | dist( p, A) = dist( p, B), ∀ p ∈ f } (2)

C.-M. Kim et al. / Computer-Aided Design 38 (2006) 1192–1204

1195

three types of trimming primitives in the computation of IIF: circumscribing, voiding, and depilating.

Fig. 2. A dimer(1R95) and the corresponding IIF ∞ : (a) the red spheres represent chain A and the blue spheres represent chain B, and (b) the pink surface represents IIF ∞ . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

where F V is the set of Voronoi faces in VD(A ∪ B). Therefore, IIF ∞ (A, B) can be computed by collecting the appropriate faces from all the Voronoi faces in the Voronoi diagram VD(A∪ B) by simply checking where each Voronoi face has two defining atoms from different groups. Hence, it takes O(m) time in the worst case, where m is the number of Voronoi faces in the Voronoi diagram. Note that IIF ∞ (A, B) expands to infinity. Hereafter, we will omit (A, B) whenever possible for notational convenience. Fig. 2 shows dimer 1R95 and the corresponding interaction interface IIF ∞ . The interaction interface IIF ∞ computed from the Voronoi diagram of atoms extends to infinity as shown in Fig. 2(b) and therefore is called an untrimmed interaction interface. Protein 1R95 is found in an important bacterium called “Escherichia coli”, and is a dimer consisting of 2 groups, 194 amino acids, and 1468 atoms (except hydrogens) [8]. 4.2. Trimmed interaction interface: IIF While an interaction interface is important for the understanding of a protein’s function, the points on IIF ∞ which are far away from the protein, are biologically less significant. This is because proteins usually exist in a solvent which is mostly water [20]. Hence, we define a biologically more meaningful interaction interface which is relatively near to the atoms of both chains. A solvent molecule interacts with a protein in a rather complicated fashion if we consider its detailed threedimensional structure. To simplify the model and the calculation, the usual practice is to approximate the solvent molecule with a small sphere, called a probe, enclosing the molecule. Note that the water molecule, H2 O, has an internal angle of 104.5◦ at 6 HOH and the common probe for the water ˚ [5]. molecule is a sphere with a radius 1.4 A Using a probe, we trim off the meaningless parts of IIF ∞ . Suppose that a probe has a radius ρ. Then, the trimmed interaction interface IIF(A, B) is defined as follows IIF(A, B) = { p | dist( p, A) = dist( p, B) ≤ ρ}.

(3)

Hence, IIF can be computed by trimming off the less meaningful parts of the surface from IIF ∞ . This trimming is done using a spherical probe as a virtual trimmer, and the edges created by the trimming are called trimming edges. There are

4.2.1. Circumscribing A circumscribing is a trimming operation primitive on the Voronoi faces of IIF ∞ where at least one of the trimming edges lies on a Voronoi face emanating to infinity. Fig. 3(a) is a schematic diagram of a circumscribing operation for a dimer in the plane. The green solid line shows the resulting interaction interface IIF after a circumscribing operation is applied and the green dotted lines show the trimmed-off part of IIF ∞ . The first step for a circumscribing operation is to choose an infinite Voronoi edge on IIF ∞ . Once an unbounded and infinite edge is located, we place the probe P at infinity and move it towards the protein until P touches three atoms simultaneously and computes the center of P at this instance. Then, we move P around the protein with its center on IIF ∞ while keeping tangential contact with some of the atoms in the protein. Then, the trajectory of the center of P defines an edge loop called a trimming edge loop and the part of IIF ∞ exterior to the trimming edge loop is trimmed off. Hence, this edge loop is called an exterior edge loop. Note that a circumscribing process may transform a single IIF ∞ to a number of components in IIF. It should be noted that the number of trimming edge loops due to circumscribing operations is identical to the number of components in IIF. 4.2.2. Voiding A voiding refers to a trimming operation primitive on the Voronoi faces of IIF ∞ which are buried in a protein with a sufficient free space for a trimming probe P to move around. The green dotted poly-line in the middle of Fig. 3(b) is an example of a voiding operation. The first step for the voiding operation is to choose a Voronoi vertex on IIF ∞ with a sufficient distance from the atoms defining the vertex and then place P at the vertex. Then, we move P around with its center on IIF ∞ while keeping tangential contact with atoms. The resulting trajectory of the center of P is then another trimming edge loop called an interior edge loop which defines a topological hole inside the interaction interface IIF. The number of edge loops due to voiding operations is identical to the number of holes in the IIF. 4.2.3. Depilating A depilating is a trimming operation primitive on the Voronoi faces of IIF ∞ which are bounded but not buried inside the protein. Illustrated in Fig. 3(c) is an example for a depilating operation where the dotted green poly-line represents the trimmed-off part of IIF ∞ . A depilating operation is similar to a voiding in that the depilating starts by locating a Voronoi vertex which is sufficiently far away from nearby defining atoms and trims off the extraneous portion of the IIF ∞ . This operation also leaves a topological hole on the IIF and therefore the corresponding edge loop is also called an interior edge loop. Suppose that the rightmost blue atom in Fig. 3(c) is moved towards the right a little so that the corresponding Voronoi

1196

C.-M. Kim et al. / Computer-Aided Design 38 (2006) 1192–1204

Fig. 3. The trimming process of IIF ∞ . The sphere P denotes a probe for a solvent molecule. (a) Circumscribing, (b) voiding, and (c) depilating. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

region becomes unbounded. Then, the current depilating operation becomes a circumscribing operation. The trimming of IIF ∞ consists of the subset of the three trimming operation primitives depending on the configurations of the atoms in the molecule. For example, consider a dimer where chain A is completely buried in chain B. Then, IIF ∞ itself does not extend to infinity because it is already bounded. Hence, a circumscribing is not necessary at all. Depending on the relative locations of atoms in both chains, a depilating may not be necessary as well. In addition, a voiding may not be necessary if there is no free space in the molecule for the probe to fit while keeping its center on IIF ∞ . Note that IIF ∞ ≡ IIF in such a case. 4.3. Interaction interface for three or more chains The previous discussion on the IIF can be extended to trimers, tetramers, pentamers, and so on. Suppose that a given protein is an m-mer which has m chains. Then, a Voronoi face in the Voronoi diagram of the whole protein can be defined by two atoms from either the same chain or two distinct chains. Suppose that we assign an attribute to a Voronoi face. A distinct attribute corresponds to a distinct combination of chains where the two atoms define the Voronoi face. Therefore,
a Voronoi face is assigned with a unique attribute out of m2 different attributes. Let F0 be the set of Voronoi faces where f ∈ F0 is defined by two atoms from the same chain. In addition, let Fi be
the set of Voronoi faces with i-th attribute where 1 ≤ i ≤ m2 . Then, the IIF of a m-mer can be now defined as follows n  m o IIF(m-mer) = Fi | 1 ≤ i ≤ . (4) 2 Each Fi in Eq. (4) is called a pairwise-chain interface and may consist of multiple components. Therefore, there is only one pairwise-chain interface for a dimer and itself becomes the whole IIF. A trimer may contain up to three pairwise-chain interfaces and a tetramer has the possibility of six at most. Note that three pairwise-chain interfaces may share one or more edges in common and such an edge is called a tri-edge.

Fig. 4. The magnification of IIF for a protein 1R95.

4.4. Topology data structure for interaction interfaces The design of an efficient data structure for appropriately storing the topology of the IIF as well as the B S is another important issue to be discussed. As was discussed previously, a pairwise-chain interface may consist of multiple components and each component is a 2-manifold surface possibly with one or more holes. Each face in the pairwise-chain interface is a polygon on a hyperboloid surface with an arbitrary number of boundary edges. Let F be a pairwise-chain interface. It is obvious that the graph defined by the vertices and edge of F is planar and the e of F can be defined. Then, the topology plane-embedding F e of F can be stored in the winged-edge data structure or its e is three variation [47,52,59]. If the degree of the vertices of F e or less, the dual structure D(F) maps to a set of triangles in the e can be compactly represented by an array. plane. Hence, D(F) At first, intuition tells us that the degree of the Voronoi vertices on IIF should be always three unless it is on the boundary of F. According to our experiment, however, it turns out that approximately 30% of whole vertices on the IIF have a degree four. Fig. 4 shows a subset of IIF of a protein 1R95. Vertices marked by triangles and rectangles are the ones with degree three and four, respectively. Fig. 5 illustrates the situations when the degrees of three and four occur on IIF. The four circles on the vertices of the

C.-M. Kim et al. / Computer-Aided Design 38 (2006) 1192–1204

1197

Fig. 5. The cases for the degree three and four of vertices on IIF: (a) the case of degree three and (b) the case of degree four.

tetrahedra are atom centers that define the Voronoi vertex v on IIF. The white circle corresponds to an atom in chain A and the black circle corresponds to an atom in chain B. The edges e1 , e2 , e3 and e4 are the Voronoi edges defined by each triplet of appropriate atoms in the tetrahedron. The IIF in Fig. 5(a) is defined as a surface between chains A and B. In this figure, the three Voronoi faces sharing edges e1 e2 , e2 e3 and e3 e1 constitute the IIF. Note that edge e4 does not contribute to the interaction interface. Hence, vertex v has a degree three on IIF in this case. On the other hand, Fig. 5(b) shows a case that the degree of v is four on IIF. In this figure, two atoms from the A chain and two atoms from chain B constitute a tetrahedron, and the IIF consists of four Voronoi faces sharing the Voronoi edges e1 e2 , e2 e3 , e3 e4 , and e4 e1 . Hence, all four Voronoi edges incident to the vertex v contribute to the faces in IIF, and therefore the degree of v is four. e may consist of Due to this anomaly, the direct dual D(F) triangles as well as rectangles and therefore the dual cannot be easily stored in a simple array. To obtain topological homogeneity, however, the rectangle can be split into two triangles by arbitrarily defining a diagonal in the rectangle so that the dual of the IIF can be compactly stored in an array. We want to note here that a vertex on the boundary of IIF may have the degree two or three depending on the condition. The topology structure of the inter-pairwise interface is another issue. Note that a triplet of pairwise-chain interfaces may define tri-edges, and these tri-edges define an edge-graph. Therefore, the topology of the whole IIF can be stored by having three pointers to the incident pairwise-chain interfaces from each tri-edge. Note the whole IIF for an m-mer, where m > 2, is in general a non-manifold model and the details of the data structure for such models are well-described in [47–49]. A more compact representation of a whole IIF for an m-mer is an issue for further study.

Fig. 6. IIF for a dimer (1R95) and the corresponding B S: (a) the IIF, and (b) the B S.

meaningful information on the interaction. Fig. 6(a) shows the IIF for a protein 1R95 and Fig. 6(b) shows the corresponding B S. B S can be computed from IIF via Laplacian smoothing [23, 44] applied to the vertices of the IIF. As a preprocessing step, we first triangulate each Voronoi face in F to obtain a triangulated face set F 4 . Since F 4 is 2-manifold and consists of only triangles, it can be efficiently stored in a compact data structure using a simple array. Laplacian smoothing is then applied to a triangular mesh F 4 as follows. Let 41 , 42 , . . . , 4m be triangles which share a vertex v ∗ , and let v1 , v2 , . . . , vm be the remaining vertices of 41 , 42 , . . . , 4m other than v ∗ . Then, Laplacian smoothing defines a new coordinate for v ∗ according to the following equation v ∗ = (v1 + · · · + vm )/m.

(5)

The above equation is applied to all vertices in F repeatedly, except those on the edge loops, until the geometry of a new surface formed by new vertices sufficiently converges. Then, this surface is considered to be a base surface B S. The proper resolution for the triangulation in order to obtain an appropriate F 4 is another topic for further study. 6. Analysis of interaction interface

5. Base surface: B S Suppose that a pair-wise chain interface F has a single component though it may consist of multiple components. Hence, F has an exterior edge loop and possibly one or more interior edge loops. We define a base surface B S as the surface interpolating all edge loops on the F with the minimum bending energy. B S can then provide more

Once an IIF is constructed, various structural characteristics of the interaction among chains in the protein can be analyzed. The analyses can be divided into two groups depending on the nature of the analysis: geometrical measures and topological measures. We want to note here that the measures provided in the following are just a few examples that can be defined by our definition of an IIF. There can be more measures of higher

1198

C.-M. Kim et al. / Computer-Aided Design 38 (2006) 1192–1204

significance that can be derived from the Voronoi diagram of atoms. In addition, we want to refer readers to [4] for the initial definitions of some of these measures. 6.1. Geometrical measures The geometrical measures of IIF as follows are defined to characterize the structure and shape of the IIF. 6.1.1. Areas of interaction interface and base surface Since a B S can be regarded as the minimum energy surface for a given IIF, comparison of various geometric properties for both B S and IIF can provide a meaningful insight into how to characterize the interaction between two chains. One such measure is the area of both surfaces. If the interface IIF is computed from the point set Voronoi diagram of atom centers, the precise value of the area of IIF can be easily computed since IIF consists of planar faces. In the case of Voronoi diagram of atoms, we approximate both the IIF and B S with triangular faces for computational convenience in our current implementation. Let Area(IIF) and Area(B S) be the areas of IIF and B S, respectively. Then, Area(B S) ≤ Area(IIF) since B S is the minimum energy surface computed from IIF. 6.1.2. Ratio of areas The ratio between Area(IIF) and Area(B S) can give additional meaningful information about the geometric complexity of the interaction between chains in a protein. Let Rarea = Area(B S)/Area(IIF)

(6)

be the ratio between two areas. When Rarea gets close to 1, the shape of IIF becomes closer to the shape of B S. On the other hand, when the value Rarea gets close to 0, the shape of IIF will be quite different from the shape of B S. Hence, the prediction of smoothness of IIF against B S is possible with the measure of Rarea . 6.2. Topological measures Topological measures can also be defined for better analysis of the interaction among chains through an IIF. Provided in this section are the familiar concepts of component, hole, genus of IIF, and the adjacency among chains. However, we can easily define other useful notions for the topological measures once the Voronoi diagram of atoms is available. 6.2.1. Components An IIF may consist of several components. There are two causes for this multiple component property. The first case is due to the circumscribing operation which may change an IIF ∞ to an IIF with multiple components even in a dimer as shown in Fig. 7. The other case is for an m-mer, m > 2 where each pair of chains defines a separate pair-wise chain interface but no triedge exists. The number of components in an IIF is identical to the number of exterior boundary loops. Provided that an IIF is stored in an appropriate data structure, locating the loops is not a difficult task.

Fig. 7. IIF of dimers with multiple components: (a) two components (PDB ID: 1RGC), and (b) four components (PDB ID: 1F1C).

Fig. 8. An IIF with topological holes: (a) the IIF of 1A8V with a hole, and (b) the IIF of 1A78 with two holes.

6.2.2. Holes A topological hole can be created on an IIF by either a depilating or a voiding operation of the trimming. Proteins usually exist in some solvent, usually water [20]. A depilating trimming operation, may trim off some inner edges of IIF ∞ and creates an interior loop on an IIF defining a topological hole on the IIF. If a protein folds into a structure containing a void in the interior of the protein, applying a voiding operation may create one or more topological holes as shown in Fig. 8. The number of holes is determined by counting the number of inner loops in each component of an IIF. 6.2.3. Genus Fig. 9(a)–(c) show a dimer and the corresponding IIF from two different views. Note that IIF a surface of genus one. Depending on the interaction between two chains, an IIF may be a surface with the genus g ≥ 1 depending on the nature of the interaction. The strength of the interaction and bonding between two chains may be influenced by the value of genus g; the larger g is, the greater the strength. Once an IIF is appropriately computed, g can be counted by a simple calculation using the Euler characteristic of IIF [52]. Suppose that |V |, |E|, |F| and |B| are the number of vertices, edges, faces and the exterior boundary loops on IIF, respectively. Then, g in IIF can be easily calculated by the following equation χ = 2(s − g)

(7)

where s denotes the number of components and χ denotes the Euler characteristic of IIF. Note that χ = |V | − |E| + |F| if IIF

C.-M. Kim et al. / Computer-Aided Design 38 (2006) 1192–1204

1199

Fig. 9. A dimer (1AOJ) and the corresponding IIF with a genus of one: (a) two chains, (b) the IIF from one view point, and (c) the IIF from another view point.

Fig. 10. A trimer(2RDV) and the corresponding IIF. (a) Three chains with the embedded IIF. The red, green, and blue spheres denote the chains A, B, and C, respectively. (b) Corresponding IIF without chains. (c) Corresponding chain adjacency matrix. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

is a closed surface without a boundary, and χ = |V | − |E| + |F| + |B| if IIF has a boundary. The model shown in Fig. 9 is for an IIF with g = 1. 6.2.4. Chain adjacency Suppose that two chains define an IIF. Then, we say that the chains are adjacent to each other. In a dimer consisting of two chains A and B with an IIF, the two chains are always adjacent to each other. However, if there are three or more chains in a protein, the adjacency relationship among the chains can be more complicated. Hence, we present a concept of chain adjacency to denote this property among chains in a protein. Once an IIF is computed, it is easy to determine adjacent chains. Note that each face in IIF is defined by atoms in different chains and this information can be used to locate the chain adjacency. For the representation of chain adjacency, we use a square matrix where the number of dimension is identical to the number of chain in the protein. In other words, the chain adjacency for a protein consisting of chains A, B, C can be denoted as a matrix M = {ai j }, where i, j ∈ {A, B, C}. ai j = 1

if the chains i and j are adjacent to each other, and ai j = 0 if they are not. For example, a matrix M = {(0, 0, 1), (0, 0, 1), (1, 1, 0)}, where a pair of parentheses denotes a row in the matrix, represents the chain adjacency of the trimer 2RDV shown in Fig. 10. The red, green, and blue colors in Fig. 10(a) denote the chains A, B, and C, respectively. Note that A and B are adjacent to each other in Fig. 10(b). Similarly, B and C are also adjacent to each other. However, A and C are not adjacent to each other. The matrix in Fig. 10(c) stores this adjacency relations in a compact representation. 6.2.5. The number of atoms in an interface Let A = {a1 , . . . , am } and B = {b1 , . . . , bn } be the e = {a e , . . . , a e } be the interacting chains in a dimer. Let A A1 Ai atoms in A which contribute to the faces in IIF. Then, the ratio e R AA defined as R AA = e

e | A| |A|

(8)

1200

C.-M. Kim et al. / Computer-Aided Design 38 (2006) 1192–1204

can also be another meaningful measure for inferencing the structural characteristics of IIF with respect to chain A. e e ⊆ A. If R Ae ≈ 1 and IIF is Note that R AA ≤ 1 since A A bounded, then IIF looks like a thin plate with a possible twist e If R Ae ≈ 0 while R Ae ≥ 0 and IIF is bounded, a since A ≈ A. A A relatively small number of the atoms in A contribute to IIF. On e the other hand, if IIF is bounded while R AA ≈ 0, A is buried in B and therefore the overall shape of A is rather globular. Hence, e R AA is a meaningful descriptor for the interaction interface. 7. Experiments and discussions In this paper, we implemented the construction of the Voronoi diagram of atoms, the extraction of IIF ∞ , the trimming of IIF ∞ to get IIF, and the various measures for both geometry and topology. Then, we tested all proteins in the Protein Data Bank (PDB) [60]. The software was developed using Microsoft Visual C++ and tested on the platform of Intel Pentium IV 3.0 GHz processor with 1 GB. For each protein selected from PDB, we computed IIFs using both the Voronoi diagram VD(A) of atoms and the Voronoi diagram VD(K ) of the centers K of atoms in the protein. The interaction interface IIF A computed from VD(A) was trimmed by the standard probe for the water molecule. Note that the van der Waals radius of a probe for a water ˚ [25]. The interaction interface IIF K molecule used is 1.41 A is computed from VD(K ) by trimming using a probe with a ˚ to account for the size of atoms in the protein radius of 3.04 A considering the frequencies of atom types in the protein [5] and their standard van der Waals radii [11]. In our experiment, VD(A) is computed using the in-house software that was developed by the Voronoi Diagram Research Center at Hanyang University [67] and the ordinary point set Voronoi diagram of atom centers was computed using CGAL [15]. The experimental results are shown in Table 1 for a sample set of nine proteins: three dimers, trimers, and tetramers. The first column, PDB ID, shows the identification code in PDB and denotes whether the protein is a dimer, a trimer or a tetramer. The second column indicates the number of residues and atoms (except hydrogen) in the protein. For example, 1AOJ consists of 120 residues and 984 atoms. Starting from the third column, there are two rows: one for the Voronoi diagram VD(K ) of atom centers and another for the Voronoi diagram VD(A) of atoms. Hence, the columns Area (IIF), Area (B S), and Rarea denote the areas of IIF, B S, and their ratios for each protein based on both VD(K ) and VD(A). The columns for Component, Hole and Genus denote the numbers of those topological characteristics in IIF based on A both VD(K ) and VD(A). Ratom denotes the ratio defined in Eq. (8) based on VD(A). For example, refer to the first entry 1AOJ whose threedimensional structure is shown in Fig. 9(a). Area (IIF) and Area (B S) computed using the Voronoi diagram VD(K ) of the atom ˚ 2 and 1405 A ˚ 2 , respectively. centers are approximately 1761 A The same areas computed using the Voronoi diagram VD(A) ˚ 2 and 1423 A ˚ 2 . The of the atoms are approximately 1774 A

ratio Rarea for both VD(K ) and VD(A) are both 0.80 and it can be concluded that the interaction interface is relatively smooth without significant bumps on the surface. Hence, this protein is not too sensitive to the difference between the two types of Voronoi diagrams. This protein consists of 2 components with 10 holes inside. Note that one of the components has a genus meaning that the chains of this protein are intertwined with each other and are fused together very strongly. While Rarea of 1AOJ is identical for both VD(A) and VD(K ), some proteins may have significantly different values of Rarea . For example, refer to dimer 1GDQ, the third entry in Table 1, whose three-dimensional structure is shown in Fig. 11(a). Fig. 11(b) shows the interaction interface computed using the Voronoi diagram VD(A) of atoms and Fig. 11(c) shows the corresponding base surface. Similarly, Fig. 11(d) and Fig. 11(e) illustrate the interaction interface and the base surface computed using the Voronoi diagram VD(K ) of atom centers. Note the big difference between two base surfaces in Fig. 11(c) and (e), while the two interaction interfaces shown in Fig. 11(b) and (d) look quite similar. This big difference in the base surfaces is due to the topological holes which exist at the upper corners on the interface of VD(A) which do not exist on the interface of VD(K ). This characteristic is reflected in the definition of Rarea as the big difference between the Rarea values corresponding to VD(A) and VD(K ): 0.11 and 0.36, respectively. e e The ratio R AA and R BB of 1GDQ are 0.03 and 1, respectively. e The fact R BB = 1 indicates that all atoms in the chain B e contribute to the interaction while R AA = 0.03 says that a few number of atoms in the chain A contribute to the interaction. From these two numbers, we can conclude that the chain B is either completely buried in chain A or the complete chain B is attached to one side of chain A. As shown in Fig. 11(a), the current example is the former case. Consider the Chain adjacency column in Table 1. A trimer 1SWI, the first entry in the trimer, is shown in Fig. 12(a) with three chains A, B and C. The main component of IIF of this protein, shown in Fig. 12(b), is defined by all three chains simultaneously. Therefore, the corresponding adjacency matrix has all 1’s except for the diagonal elements. Note that the corresponding base surface is illustrated in Fig. 12(c). The other two trimers (2RDV and 1GG3) in Table 1 show different adjacency relationships among the chains. The above analysis was done for tetramers as well. Fig. 13(a) shows a tetramer 6RLX, the first entry in the tetramer section in Table 1, and its interaction interface is shown in Fig. 13(b). Fig. 13(c), the base surface, clearly shows that all four chains simultaneously share the main component of the interface. Hence, its corresponding adjacency matrix also has all 1’s except for the diagonal elements. The other two tetramers, 1UMR and 1IEB, also show different types of adjacency among chains. Lastly, we want to briefly discuss the computation and memory requirements for the proposed approach to compute the interaction interface. Given the Voronoi diagram of atoms for a protein, the only computation necessary to decide if a

Tetramer

Trimer

Dimer

PDB ID

1760.90 1773.81 1428.69 1459.05 214.38 224.79 1315.06 1373.21 643.19 622.72 1177.04 1168.55 1843.77 1858.58

3844.77 3835.80

6984.85 6956.32

VD(K ) VD(A)

VD(K ) VD(A)

VD(K ) VD(A)

VD(K ) VD(A)

VD(K ) VD(A)

VD(K ) VD(A)

VD(K ) VD(A)

VD(K ) VD(A)

VD(K ) VD(A)

134 1074

215 3180

91 754

156 1173

837 6837

100 788

520 4318

792 6428

1BH8

1GDQ

1SWI

2RDV

1GG3

6RLX

1UMR

1IEB

5323.42 5320.94

2730.72 2702.55

1515.17 1535.67

962.90 974.38

510.96 499.92

1105.04 1123.37

22.93 80.21

1119.55 1103.20

1404.68 1423.08

0.76 0.76

0.71 0.70

0.82 0.83

0.82 0.83

0.79 0.80

0.84 0.82

0.11 0.36

0.78 0.76

0.80 0.80

12 11

4 4

1 1

14 13

3 2

2 3

1 1

2 2

2 2

Component

Rarea

Topology

˚2 Area (IIF) A

˚2 Area (BS) A

Geometry

120 984

VD

1AOJ

# of residue # of atom

Attributes

Table 1 Analysis of interaction interfaces for various proteins in PDB

31 30

15 13

8 7

8 6

2 3

13 11

3 2

16 8

6 10

Hole

4 3

0 0

0 0

0 0

0 0

0 0

0 0

0 0

1 1

Genus

0 1 0 1

0 1 1 1

0 1 1 1

0 1 1

0 0 1

0 1 1

0 1

0 1

0 1

1 0 1 1

1 0 0 1

1 0 1 1

1 0 0

0 0 1

1 0 1

1 0

1 0

1 0

0 1 0 1

1 0 0 1

1 1 0 1

1 0 0

1 1 0

1 1 0

1 1 1 0

1 1 1 0

1 1 1 0

Chain adjacency

0.58

0.24

0.28

0.13

0.39

0.07

0.03

0.50

0.48

e

R AA

0.48

0.22

0.30

0.11

0.40

0.03

1.00

0.26

0.49

e

R BB

0.57

0.25

0.28

0.25

0.43

0.04

–

–

–

e

C RC

0.58

0.28

0.29

–

–

–

–

–

–

e

D RD

C.-M. Kim et al. / Computer-Aided Design 38 (2006) 1192–1204 1201

1202

C.-M. Kim et al. / Computer-Aided Design 38 (2006) 1192–1204

Fig. 11. A dimer (1GDQ): (a) the configuration of two chains, (b) IIF A , (c) B S A , (d) IIF K , and (e) B S K . Note that two B S’s in (c) and (e) are quite different.

Fig. 12. A trimer (1SWI): (a) three chains with the embedded IIF A , (b) the chains removed and IIF A remains, and (c) the corresponding B S A .

Fig. 13. A tetramer (6RLX): (a) four chains with the embedded IIF A , (b) the chains removed and IIF A remains, and (c) the corresponding B S A .

Voronoi face constitutes an interaction interface is to check if the two nearby atoms are from different atom sets. This computation is repeated for all faces in the Voronoi diagram. Hence, the time complexity is linear to the number of Voronoi

faces where the constant factor is tiny. In addition, it is known that the number of Voronoi faces of a protein is also linear to the number of atoms [29]. Given a Voronoi diagram, we determined the computation times to obtain the interaction interface for

1203

C.-M. Kim et al. / Computer-Aided Design 38 (2006) 1192–1204 Table 2 The computation time of the interaction interfaces for 1R95, 1SW1 and 6RLX PDB ID

# of atoms

# of faces in VD

# of faces on IIF ∞

# of faces on IIF

Computation time (s)

1R95 (dimer) 1SWI (trimer) 6RLX (tetramer)

1468 754 788

11 163 5 750 5 972

449 851 1163

159 503 840

0.203 0.437 0.625

the proteins shown in Fig. 2 (a dimer), Fig. 12 (a trimer), and Fig. 13 (a tetramer). These results are summarized in Table 2. The first column shows the number of atoms in the proteins, and the second, the third and the fourth columns show the number of faces in the Voronoi diagram, and the number of faces in IIF ∞ and IIF. The last column of the table shows the computation time in seconds taken to compute the interaction interface for a given Voronoi diagram. The storage requirement is also linear since the number of faces in the interaction interface is linear to the number of atoms and the interaction interface can be stored in the winged-edge data structure which requires linear storage in terms of the number of edges and faces [47]. 8. Conclusions A Voronoi diagram is an important mathematical and computational tool used in many disciplines. While the Voronoi diagram has been generalized to various directions, the computation of the Voronoi diagram for spheres has not been quite satisfactory in many aspects. Recently, the algorithms and their implementations have been known for the successful computation of the Voronoi diagram of spheres [35,37–40]. Since then, the Voronoi diagram of spheres has been applied to solve various problems in structural molecular biology [33, 35,36,39–41,43]. Since the investigation of molecular structure is interdisciplinary in its nature, collaborations between geometers and biologists are essential [19,40,41]. In this paper, we have shown that the Voronoi diagram of spheres can be used for the analysis of the interactions between chains in a protein, the interactions between proteins, and the interactions between any sets of atomic structures. Compared to previous studies which used the power diagram of atoms [4], this paper presents an approach to study the interactions based on the Voronoi diagram of atoms where the distance is defined as the minimum Euclidean distance from the boundary of an atom. The analysis is precise, efficient and robust once the Voronoi diagram is available. The results of analysis are currently being stored in a database and will be available from the Voronoi Diagram Research Center at Hanyang University [67]. We have also presented some geometrical as well as topological measures for the proper analysis of interactions. The formal study of the interaction interface from a geometric point of view for both inter-protein and intra-proteins is in its infancy. Like other applications in structural biology, this problem domain opens a new, interesting and important areas of application for geometers in the CAD and CAGD communities. We expect more significant studies by the members of the CAD and CAGD communities in this area to follow and hope that the

Voronoi diagram of spheres will be one of the most fundamental computational tools used in this research. Acknowledgements The authors thank the anonymous reviewers for their constructive comments which have improved the quality of this paper. This research was supported by a Creative Research Initiative from the Ministry of Science and Technology, Korea. Y. Cho was supported by BK21 of Korea, JB is supported by the Biogreen21 and MOST funds and SL is supported by the KRIBB Research Initiative Program. References [1] Agarwal PK, Edelsbrunner H, Harer J, Wang Y. Extreme elevation on a 2-manifold. In: Proceedings of the 20th annual ACM symposium on computational geometry. 2004. p. 357–65. [2] Aurenhammer F. Power diagrams: Properties, algorithms and applications. SIAM Journal on Computing 1987;16:78–96. [3] Bajaj CL, Pascucci V, Shamir A, Holt RJ, Netravali AN. Dynamic maintenance and visualization of molecular surfaces. Discrete Applied Mathematics 2003;127(1):23–51. [4] Ban Y-EA, Edelsbrunner H, Rudolph J. Interface surfaces for proteinprotein complexes. In: Proceedings of the 8th annual international conference on research in computational molecular biology. 2004. p. 205–12. [5] Berg JM, Tymoczko JL, Stryer L. Biochemistry. 5th ed. W.H. Freeman; 2002. [6] Berman H, Westbrook J, Feng Z, Gilliland G, Bhat T, Weissig H, et al. The protein data bank. Nucleic Acids Research 2000;28:235–42. [7] Bernal JD, Finney JL. Random close-packed hard-sphere model ii. Geometry of random packing of hard spheres. Discussions of the Faraday Society 1967;43:62–9. [8] Bilder PW, Ding H, Newcomer ME. Crystal structure of the ancient, FeS scaffold IscA reveals a novel protein fold. Biochemistry 2004;43(1): 133–9. [9] Bolser D, Dafas P, Harrington R, Park J, Schroeder M. Visualisation and graph-theoretic analysis of a large-scale protein structural interactome. BMC Bioinformatics 2003;4(45):10. [10] Bolser D, Park J. Biological network evolution hypothesis applied to protein structural interactome. Genomics & Informatics 2003;1(1):7–19. [11] Bondi A. van der Waals volumes and radii. Journal of Physical Chemistry 1964;68:441–51. [12] Bourne PE, Weissig H. Structural bioinformatics. Wiley-Liss; 2003. [13] CATH Database. 2005. http://www.biochem.ucl.ac.uk/bsm/cath/. [14] Cazals F, Proust F. Revisiting the description of protein-protein interfaces. Part I: Algorithms. 2004. http://www-sop.inria.fr/geometrica/ team/Frederic.Cazals/intervor/. [15] CGAL Library Homepage. http://www.cgal.org/. [16] Cho Y, Kim D, Kim D-S. Topology representation for Euclidean Voronoi diagram of spheres in 3D. In: Proceedings of the digital engineering workshop and 5th Japan–Korea CAD/CAM Workshop. Tokyo (Japan): University of Tokyo; 2005. p. 121–6.

1204

C.-M. Kim et al. / Computer-Aided Design 38 (2006) 1192–1204

[17] Cho Y, Kim D, Kim D-S. Topology representation for the Voronoi diagram of 3D spheres. International Journal of CAD/CAM 2005;5(1): 59–68. [18] Choi Y. Vertex-based boundary representation of non-manifold geometric models. Ph.D. thesis. USA: Carnegie-Mellon University; 1989. [19] Choset H. Sensor based motion planning: The hierarchical generalized Voronoi graph. Ph.D. thesis. Pasadena (CA, USA): California Institute of Technology; 1996. [20] Creighton TE. Protein folding. W.H. Freeman; 1999. [21] DALI Domain Dictionary. 2005. http://www.ebi.ac.uk/dali/domain/. [22] Edelsbrunner H, Facello M, Liang J. On the definition and the construction of pockets in macromolecules. Discrete Applied Mathematics 1998;88: 83–102. [23] Field DA. Laplacian smoothing and Delaunay triangulations. Communications in Applied Numerical Methods 1988;4:709–12. [24] Fischer W, Koch E. Geometrical packing analysis of molecular compounds. Zeitschrift f¨ur Kristallographie 1979;150:245–60. [25] Franks F. Water: A matrix of life. 2nd ed. Royal Society of Chemistry; 2000. [26] Gellatly BJ, Finney JL. Calculation of protein volumes: An alternative to the Voronoi procedure. Journal of Molecular Biology 1982;161(2): 305–22. [27] Goede A, Preissner R, Fr¨ommel C. Voronoi cell: New method for allocation of space among atoms: Elimination of avoidable errors in calculation of atomic volume and density. Journal of Computational Chemistry 1997;18(9):1113–23. [28] Gong S, Park C, Choi H, Ko J, Jang I, Lee J, et al. A protein domain interaction interface database: Interpare. BMC Bioinformatics 2005; 6(207):8. [29] Halperin D, Overmars MH. Spheres, molecules, and hidden surface removal. In: Proceedings of the 10th ACM symposium on computational geometry. 1994. p. 113–22. [30] Heifets A, Eisenstein M. Effect of local shape modifications of molecular surfaces on rigid-body protein–protein docking. Protein Engineering 2003;16(3):179–85. [31] Jones S, Thornton JM. Principles of protein–protein interactions. Proceedings of the National Academy of Sciences of the United States of America 1996;93:13–20. [32] Ju B-H, Park B, Park JH, Han K. Visualization and analysis of protein interactions. Bioinformatics 2003;19(2):317–8. [33] Kim C-M. Interaction interfaces in proteins using Voronoi diagram of atoms and their database. Master’s thesis. Korea: Hanyang University; 2005. [34] Kim D, Kim D-S. Region-expansion for the Voronoi diagram of 3D spheres. Computer-Aided Design 2006;38(5):417–30. [35] Kim D-S. Euclidean Voronoi diagram of atoms and protein. In: The 1st biogeometry meeting — in conjunction with the ACM symposium on computational geometry. New York (USA): The Technical University of New York; 2004. http://biogeometry.cs.duke.edu/meetings/ITR/04jun12/. [36] Kim D-S, Cho C-H, Kim D, Cho Y. Recognition of docking sites on a protein using β-shape based on Voronoi diagram of atoms. ComputerAided Design 2006;38(5):431–43. [37] Kim D-S, Cho Y, Kim D. Edge-tracing algorithm for Euclidean Voronoi diagram of 3D spheres. In: Proceedings of the 16th Canadian conference on computational geometry. 2004. p. 176–9. [38] Kim D-S, Cho Y, Kim D. Euclidean Voronoi diagram of 3D balls and its computation via tracing edges. Computer-Aided Design 2005;37(13): 1412–24. [39] Kim D-S, Cho Y, Kim D, Cho C-H. Protein sructure analysis using Euclidean Voronoi diagram of atoms. In: Proceedings of the international workshop on biometric technologies. 2004. p. 125–9. [40] Kim D-S, Cho Y, Kim D, Kim S, Bhak J, Lee S-H. Euclidean Voronoi diagrams of 3D spheres and applications to protein structure analysis. In: Proceedings of the 1st international symposium on Voronoi diagram in science and engineering. 2004. p. 137–44. [41] Kim D-S, Cho Y, Kim D, Kim S, Bhak J, Lee S-H. Euclidean Voronoi diagrams of 3D spheres and applications to protein structure analysis. Japan Journal of Industrial and Applied Mathematics 2005;22(2):251–65.

[42] Kim D-S, Kim D, Cho Y. Euclidean Voronoi diagrams of 3d spheres: Their construction and related problems from biochemistry. Lecture notes in computer science, vol. 3604. Springer-Verlag; 2005. p. 255–71. [43] Kim D-S, Seo J, Kim D, Ryu J, Cho C-H. Three dimensional beta shapes. Computer-Aided Design 2006. doi:10.1016/j.cad.2006.07.002. [44] Kobbelt L, Campagna S, Vorsatz J, Seidel H-P. Interactive multiresolution modeling on arbitrary meshes. In: Proceedings of the 25th annual conference on computer graphics and interactive techniques. New York (USA): ACM Press; 1998. p. 105–14. [45] Lappe M, Park J, Niggemann O, Holm L. Generating protein interaction maps from incomplete data: Application to fold assignment. Bioinformatics 2001;1(1):1–9. [46] Lee B, Richard FM. The interpretation of protein structures: Estimation of static accessibility. Journal of Molecular Biology 1971;55:379–400. [47] Lee K. Principles of CAD/CAM/CAE systems. Addison Wesley; 1999. [48] Lee SH, Lee K. Partial entity structure: A compact boundary representation for non-manifold geometric modeling. ASME Journal of Computing & Information Science in Engineering 2001;1(4):356–65. [49] Lee SH, Lee K. Partial entity structure: A compact non-manifold boundary representation based on partial topological entities. In: Proceedings of the 6th ACM symposium on solid modeling and applications. 2001. p. 159–70. [50] Liang J, Edelsbrunner H, Woodward C. Anatomy of protein pockets and cavities: Measurement of binding site geometry and implications for ligand design. Protein Science 1998;7:1884–97. [51] Macromolecular Structure Database. 2005. http://www.ebi.ac.uk/msd/. [52] M¨antyl¨a M. An introduction to solid modeling. Computer Science Press; 1988. [53] Nooren IMA, Thornton JM. Diversity of protein–protein interactions. The EMBO Journal 2003;22(14):3486–92. [54] Ofran Y, Rost B. Analysing six types of protein–protein interfaces. Journal of Molecular Biology 2003;325:377–87. [55] Park J, Bolser D. The conservation of protein interaction network in evolution. Genome Informatics 2001;12:135–40. [56] Park J, Lappe M, Teichmann SA. Mapping protein family interactions: Intramolecular and intermolecular protein family interaction repertoires in the pdb and yeast. Journal of Molecular Biology 2001;30(307):929–38. [3]. [57] Peters KP, Fauck J, Fr¨ommel C. The automatic search for ligand binding sites in protein of known three dimensional structure using only geometric criteria. Journal of Molecular Biology 1996;256:201–13. [58] Poupon A. Voronoi and Voronoi-related tessellations in studies of protein structure and interaction. Current Opinion in Structural Biology 2004;14: 233–41. [59] Preparata FP, Shamos MI. Computational geometry: An introduction. Springer-Verlag; 1985. [60] RCSB Protein Data Bank Homepage. http://www.rcsb.org/pdb/. [61] Richards FM. The interpretation of protein structures: Total volume, group volume distributions and packing density. Journal of Molecular Biology 1974;82:1–14. [62] SCOP Database. 2005. http://scop.berkeley.edu/. [63] Sheinerman FB, Honig B. On the role of electrostatic interactions in the design of protein–protein interfaces. Journal of Molecular Biology 2002; 318:161–77. [64] Shoichet B, Kunts I. Protein docking and complementarity. Journal of Molecular Biology 1991;221:327–46. [65] Tsai C-J, Lin SL, Wolfon HJ, Nussinov R. Studies of protein–protein interfaces: A statistical analysis of the hydrophobic effect. Protein Science 1997;6:53–4. [66] Varshney A, Brooks Jr FP, Richardson DC, Wright WV, Manocha D. Defining, computing, and visualizing molecular interfaces. In: Proceedings of the IEEE visualization ’95. 1995. p. 36–43. [67] Voronoi Diagram Research Center. http://voronoi.hanyang.ac.kr/. [68] Xu D, Tsai C-J, Nussinov R. Hydrogen bonds and salt bridges across protein–protein interfaces. Protein Engineering 1997;10(9):999–1012.