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Triangulation of Analytic Molecular Surface Masaryk University Faculty of Informatics Triangulation of analytic molecular surface Bachelor’s Thesis Radoslav Mráz Brno, Spring 2018 Masaryk University Faculty of Informatics Triangulation of analytic molecular surface Bachelor’s Thesis Radoslav Mráz Brno, Spring 2018 This is where a copy of the official signed thesis assignment and a copy of the State- ment of an Author is located in the printed version of the document. Declaration Hereby I declare that this paper is my original authorial work, which I have worked out on my own. All sources, references, and literature used or ex- cerpted during elaboration of this work are properly cited and listed in com- plete reference to the due source. Radoslav Mráz Advisor: RNDr. Adam Jurčík Ph.D. i Acknowledgements I would like to thank my advisor RNDr. Adam Jurčík Ph.D. for his consulta- tions and insights throughout writing of this thesis. Moreover I would like to thank my family for supporting me in my studies. iii Abstract The goal of this thesis is to implement an algorithm for triangulating ana- lytically represented solvent-excluded molecular surface. The first part of this thesis describes related molecular surfaces, the solvent-excluded sur- face itself and significant works in this field. The second part presents an algorithm for generating a triangular mesh of the solvent-excluded surface. The last part contains results from testing of the presented algorithm. iv Keywords molecular surface, SES, solvent-excluded surface, triangulation v Contents Introduction 1 1 Molecular surfaces 3 1.1 Van der Waals surface .......................3 1.2 Solvent-accessible surface ......................3 1.3 Solvent-excluded surface ......................4 2 Analytic representation and methods of computation of SES 7 2.1 Connoly’s definition of the SES ...................7 2.2 Reduced surface ........................... 10 2.3 Contour-buildup method ...................... 10 3 Algorithm overview 13 4 Boundaries of spherical patches 15 4.1 Construction of a boundary ..................... 15 4.1.1 Convex spherical patches . 15 4.1.2 Concave spherical patches . 16 4.1.3 Nested boundaries . 17 4.2 Intersections of concave patches ................... 18 4.2.1 Finding points of intersection . 19 4.2.2 Trimming a boundary . 21 5 Triangulation of SES 23 5.1 Meshing toroidal patches ...................... 23 5.1.1 Toroidal patches . 23 5.1.2 Meshing algorithm . 23 5.1.3 Resulting mesh . 24 5.1.4 Free toroidal patches . 26 5.1.5 Self-intersecting toroidal patches . 26 5.2 Meshing spherical patches ..................... 28 5.2.1 Advancing front approach . 28 5.2.2 Initialization of the front . 29 5.2.3 Advancing of the front . 29 6 Results and testing 35 6.1 Testing ................................ 35 6.2 Results ............................... 36 vii 6.3 Limitations ............................. 37 Conclusion 41 Bibliography 43 Appendices 45 A Triangulation statistics 47 viii Introduction Visualization of molecules such as proteins is an important aspect of many scientific and industry fields. By visualizing molecules we can either study their structure, bonds between individual atoms or analyze interactions with other molecules. Scientists can use such information in order to better under- stand the function of a molecule or to design new ones. There are two types of models of molecular representations. The first type, atomistic model, represents molecules by directly depicting atoms of a molecule [1]. The second type, abstract model, shows information that might not be obvious in atomistic models. One example of an abstract model is the cartoon representation proposed by Richardson in [2]. The cartoon represen- tation shows the secondary structure of a protein. The atomistic models are further divided into two branches: bond-centric and molecular surface models [1]. Bond-centric models, such as licorice or ball-and-stick models [3], depict chemical bonds between atoms. The other branch, molecular surfaces, represent molecules by depicting the atoms that constitute a molecule. This kind of representation allows scientists to see the overall shape of the molecule, compute the surface area and volume of the molecule, but also to predict secondary structure of proteins [4]. There are two approaches of visualizing molecular surfaces. The first ap- proach uses a triangular mesh to approximate the surface of a molecule. Quality of the triangular mesh can be controlled which in turn has a di- rect impact on the computation time and memory demands. Such triangular mesh, besides viewing, can be exported into other 3D software or 3D printed. The other approach, instead of a triangular mesh, directly renders the sur- face of a molecule, e.g by ray-casting of analytically represented patches. The aim of this thesis is to implement an algorithm that generates a tri- angular mesh in order to approximate analytically represented molecular surface. Along with generating the mesh, the algorithm should also be able to handle singularities that often occur in molecular surface. The resulting triangular mesh then can be viewed directly or exported in Wavefront OBJ and STL format. This thesis consists of the following chapters. Chapter 1 describes three types of molecular surfaces—van der Waals, solvent accessible and solvent ex- cluded surface, the latter being the subject of this thesis. Chapter 2 describes the analytical representation of solvent excluded surface (SES) showing var- ious types of surface patches and how they are defined. At the end of the Chapter 2, methods of computing the SES are briefly described. Chapter 3 1 offers a short overview of the algorithm implemented in later chapters and discusses a possibility of parallelization of the algorithm. In Chapter 4 I de- scribe boundaries, elements that are used to delimit patches of the SES, and discuss their special cases. Chapter 5 consists of two sections that describe al- gorithms used to generate triangular mesh of all types of patches of the SES. Finally, the results of the implemented algorithm are presented in Chapter 6. 2 1 Molecular surfaces Molecular surfaces represent molecules either by showing the space that they occupy or the space that is accessible or inaccessible to a solvent in- teracting with them. There are various types of molecular surface represen- tations, but probably the most used ones are van der Waals(vdW), solvent- accessible(SAS) and solvent-excluded surface(SES). 1.1 Van der Waals surface The first surface I describe is the van der Waals surface (Figure 1.1a). Itisa type of space-filling model. In this model, atoms are represented as spheres with radii proportional to their van der Waals radii. The surface can then be defined as a topological boundary of the union of these spheres [5]. Thevan der Waals surface is suitable for visualizing the overall shape of the molecule and its volume. 1.2 Solvent-accessible surface Solvent-accessible surface or SAS was first defined by Lee and Richards in 1971 [6]. It is very similar to van der Waals surface with a little difference. Rather than showing how much space the molecule occupies, the purpose is to show how accessible the molecule is to other molecules. This is done by approximating the solvent that interacts with the molecule by a probe— a sphere with a certain radius (Figure 1.1a). The probe then rolls over the atoms, always maintaining contact with at least one atom, and its center traces the accessible surface from the solvent’s point of view. In other words, the purpose of SAS is to show all regions around the molecule that can be accessed by the solvent molecule [1]. Solvent-accessible surface in general is computed in the same way as computing van der Waals surface but each atom is assigned radius equal to the sum of its radius and the radius of the probe [6]. Therefore, van der Waals surface and solvent-accessible surface are in a sense identical and both can be visualized using the same algorithm. 3 1. Molecular surfaces (a) Van der Waals surface in red color (b) Solvent-accessible surface in blue and solvent-accessible surface in blue and solvent-excluded surface in green color. color. Figure 1.1: Three types of molecular surfaces. 1.3 Solvent-excluded surface Solvent-excluded surface, or the molecular surface was first introduced by Richards in 1977 [7]. The term solvent-excluded surface was coined by Greer and Bush in [8]. As with SAS, the purpose of SES is to show the surface of the molecule in relation to the solvent molecule that interacts with it. In [7], Richards pro- posed two surface types that SES consists of. The first is contact surface and it occurs by rolling the probe over the sur- face of atom. That means the probe has contact with only one atom at a moment. Contact surface is, in fact, subset of the van der Waals surface. The second type of surface is called reentrant surface. This type of sur- face does not occur in the vdW surface or SAS. It occurs when the probe is tangent to more than one atom, and it represents surface beyond which the probe cannot penetrate more "into" the molecule (see Figure 1.1b). There are two types of patches contributing to reentrant surface: concave spherical triangles—occur when the probe has contact with 3 atoms at a mo- ment; and toroidal patches—occur when the probe has contact with two atoms at a moment(see Figure 1.2). Computation and subsequent visualization of SES is substantially more complex because of the presence of more than one type of surface patches and also because special cases often occur in this surface and handling them requires additional effort. More specifically, the special cases, also called sin- gularities, are intersections of adjacent concave triangles and self-intersecting 4 1. Molecular surfaces Figure 1.2: Solvent-excluded surface of crambin molecule(PDB ID: 1CRN).
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