Computer Graphics Applications on Molecular Biology and Drug Design Katerina PERDIKURI 1, Athanasios TSAKALIDIS 1 1 Department of Computer Engineering and Informatics University of Patras,26500 Patras, Greece Abstract. Molecular structure-based drug design is an art and a science. As graphics hardware has matured and software continues to mature, the applications of computer algorithms on molecular biology provides a rigorous basis for many drug design efforts. In this paper we try to present the basic principles in molecular graphics as long as it concerns molecular representations and modelling, and outline some important challenges and open problems in the field of computer aided drug design. 1. Introduction During the last 20 years, the process of drug discovery and drug design has been enormously affected from the increased available information, concerning 3D structures of biomolecules, determined by X-ray crystallography and NMR structure determination techniques. Exploiting this structural information, scientists can design novel pharmaceutical molecules (often referred to as ligands), which bind tightly and selectively to a target macromolecule, such as a protein (often referred to as receptor). The binding of a ligand to a specified receptor is based on the lock-and-key principle, which was firstly recognized by Emil Fischer more than 100 years ago. Along with this principle, the active site of a receptor binds with a ligand both spatially and chemically, just as only a specific key fits a given lock. The design of pharmaceuticals molecules based on this principle is usually referred to as structure-based drug design [1]. Modern methods for structure-based drug design can be divided into two categories. The first category, which is usually referred as Database Searching, searches ligands for a given receptor, through large-scale screens in databases of known 3D molecules. The goal of the search is to retrieve molecules, with shape and chemical complementarity to a given active site of a receptor. The second category of structurebased drug design, which is usually referred as de novo design, builds gradually the proper ligands [2]. In this case, ligands, are built up within the constraints (both geometrical and chemical) of the binding receptor, by assembling atoms or small fragments in a stepwise manner. A number of such approaches have already been reported. Several ligands designed in this manner are now in clinical trials. The success of structure-based drug design has encouraged the development of various computational methods that can combine structural information and modern molecular graphics techniques to suggest novel structures, which may prove to be useful lead compounds. Ideally, these methods should be fast, objective and produce a set of diverse yet chemically reasonable structures. In this paper we will present and analyze the current methods used and the problems encountered in the field of Molecular Graphics and especially in the field of structure-based drug design.
Fig. 1: A 3D molecule represented with ball and sticks 2. Molecular Representations Molecular Graphics deals with the representation and manipulation of biological molecules in a computer. It is a newly established scientific field encompassing the theoretical and application areas of computer science that deal with geometry and visualization. Among these areas are computer graphics, computer animation, computer vision, computational geometry and computer-aided geometric design. In this section we will try to introduce some definitions for the concept of molecular surface and molecular volume. The three-dimensional geometric structure of a molecule is often represented as a set of atoms (Figure 1). In this representation each atom is modeled as a hard sphere with a certain atomic radius. The three-dimensional placement of the spheres is expressed according to the internal distances between the centers of each pair of atoms (interatomic distances) and their relative angles (Table 1). The equilibrium configuration of this representation is defined as the low energy conformation. In many representations, the spheres are allowed to interpenetrate one another. In structure-based drug design, one wishes to manipulate a molecule in the presence of another molecule to see whether they fit together. During such manipulation the molecules should have only limited interpenetration. Table 1: Table with internal coordinates of C 2 H 6 1 C 2 C 1.54 1 3 Η 1,0 1 109,5 2 4 Η 1,0 2 109,5 1 180,0 3 5 Η 1,0 1 109,5 2 60,0 4 6 Η 1,0 2 109,5 1-60,0 5 7 Η 1,0 1 109,5 2 180,0 6 8 Η 1,0 2 109,5 1 60,0 7
2.1 Molecular Surfaces The determination of the molecular surface of a collection of atom spheres of a molecule is of great importance in various applications in molecular biology for the interpretation of molecular properties, interactions and processes. Molecular surfaces are classified as contact surfaces and re-entrant surfaces. Various definitions have been proposed in the relative bibliography. The van der Waals surface (Figure 2), Lee and Richards solvent accessible surface [4], and Richards smooth molecular surface. In Lee and Richard s approach, a sphere is rolled on a reference surface, to obtain a new surface described by the center of the rolling sphere, which is assumed to be fairly small. In a simplified approach the molecular surface of a molecule can be described as the accessible parts of the modeled spheres (assuming that the molecules are modeled by the hard sphere model), or as the boundary of the union of the balls in the hard sphere model. 2.2 Molecular Volumes A critical issue in most 3D molecular graphics packages is the choice of the model for representing, the three dimensional structure of a molecule. Obviously the molecular geometry is affected from the selected model. Generally the use of any model means that a significant level of detail is lost as molecules are visualized to have surfaces and volumes similar to our perception of surfaces and volumes of macroscopic objects. Although it is clear that the more accurate the model used, the better are the chances in predicting molecular interactions certain abstractions need to be made in order to calculate molecular interactions efficiently. The sphere model is the most popular model for representing approximately the volume of a molecule. A sphere is drawn around the centre of every atom of the molecule. The radius of each sphere reflects the space requirements of the corresponding atom and consequently the total volume of the molecule (the used radii have been determined by a combination of experimental observations and quantum mechanical calculations). Sticks models are also used to represent the bonds between two consecutive atoms and the angles they form in 2 dimensions. The bond lengths and bond angles are degrees of freedom (DOF) for each molecule. This way a conformation of a molecule is obtained by assigning values to all the DOF of the molecule. Fig 2: a) Representation of Van der Waals surfaces, b) Representation of the VDW surface of the Myelin Basic Protein 72-85
2.3 Color Coding and Texture Mapping Techniques The molecular surface concept is not only useful for a representation of the shape of molecules. These surfaces can be used as screens for the visualization of arbitrary properties using color coding techniques. Color coding is a popular means of displaying scalar information on a surface [3]. In interactive molecular graphics, high contrast color code variation can be realized by using texture mapping techniques which are available on graphical workstations and PC s to represent a color ramp as a 1D texture. Texture mapping is a technique that applies an image to an object s surface. 2.4 Conformational Analysis Having represented the surface and volume of a molecule the next step in molecular modeling is to find the global energy minimum conformation of a molecule, regardless of the size of the molecule under consideration. This procedure is known as conformational analysis (Figure 3). A variety of conformational analysis methods exist and fall into two classes: deterministic and stochastic. Deterministic methods search all the rotatable bonds in a molecule. The time required for this type of search increases exponentially to the number of rotatable bonds. The success of a deterministic method relies on the selected granularity, which is difficult to define. For example it may be possible to perform a complete search if one limits the number of rotamers per bond to a small number. On the other hand the same search may become computationally infeasible if the number of scanned rotamers increases. Stochastic methods are based on algorithms that limit the conformational search space to the lowest energy conformations of a molecule. Some of the stochastic methods use molecular dynamics, Monte Carlo, genetic algorithms, random perturbations to the coordinates, or a combination in order to find and optimize local minima [9]. Unfortunately stochastic methods are not guaranteed to converge to the same set of low energy conformations as produced with a deterministic method. A slight deviation in a torsion angle may miss the lowest energy state, although we are very close to structurally to the global minimum. Fig 3: a)representation of the Myelin Basic Protein (MBP 72-85 ) b) Representation of the MBP 72-85 after a conformational analysis for an optimization process using steepest descent (SD) and adopted-basis Newton Raphon algorithms performed on a Silicon Graphics workstation using the QUANTA program of Molecular Simulations
2.5 Quantitative structure-activity relationships (QSAR) As we stated earlier one of the most important goals of structure-based drug design is to support the industrial pharmaceutical research in the development of new active compounds, with the aim to bring effective and safe drugs to the market. More active or higher specific products have to be developed, which show no adverse reactions or even dangerous side effects. Years of laboratory research are required to discover a new chemical compound while bringing the new drug to the market involves additional years of planning, research and analysis. Starting from compounds which have a very similar structure to existing active synthetic or natural products a large number of derivatives have to be synthesized for a systematic screening. In average from these potential candidates for new therapeutic agents only one of ten thousand synthesized compounds can be used commercially. To improve this very unfavourable ratio between synthesized compounds and drugs on the market, various theoretical methods were introduced in the last years, which are widely used nowadays in pharmaceutical research. Molecular calculations with sophisticated computer program packages allow the prediction of molecular properties in such a way, that accurate proposals for a target oriented synthesis can be performed. The broad spectrum of Molecular Graphics packages allows to consider the steric behaviour of molecules, dynamical molecular properties, molecular similarity and drugreceptor interactions. 3D receptor modelling, Comparative Molecular Field Analysis (CoMFA) and Active Analog Approach (AAA) are well established procedures for the investigation of structure activity relationships. Structure activity relationships are mathematical relationships linking chemical structure and pharmacological activity in a quantitative manner for a series of compounds. Methods which can be used in QSAR include various regression and pattern recognition techniques.computer assisted drug design (CADD) and computer assisted molecular design (CAMD) are widespread methods, which support classical quantitative structure activity relationship (QSAR) techniques and make drug screening and pharmacophor identification more efficient. Many successful applications confirm the importance of calculations on three dimensional quantitative structure activity relationship (3D QSAR). 3. Open Challenges and Problems in Molecular Graphics Studying a hard sphere model from a computational geometry point of view, we have to take into consideration several properties such as: (i) the radii ranges in a fairly restricted area, and (ii) sphere centers cannot get too close to one another. One of the tasks of molecular modeling packages is to display molecules. This should preferably be done so fast that the user can interact with the model by turning it around to look at it from different directions or by moving different molecules with respect to each other, to see for example whether they fit together. In this paragraph we will refer to problems and challenges that arise in the field of molecular graphics and are crucial for application in computer-aided structure-based drug design. 3.1 Reconstructing a 3D Model The reconstruction of a three-dimensional set of points using information about its inter-point distances is a task of great importance in determining molecular structure.
In particular, spectroscopic methods such as two-dimensional NMR provide a mean for determining a labeled subset of the distances between atoms in large structures such as proteins and RNA, and such methods have therefore proved extremely valuable in conformational studies of such molecules. Determining the structure of n labeled points in R 3 is easy if the exact value of every inter-point distance is given. However, this is far from what one obtains using NMR and its variants. In particular, not all distances can be measured, and the distance values that one does obtain are susceptible to a wide variety of sources of error. NMR experiments are frequently based on measuring proton resonances, since hydrogen atoms are easily detected by these techniques; this is in general sufficient for conformational studies, since hydrogen atoms are abundant in organic molecules. To determine spatial information, one typically requires two NMR experiments: one based on spin-spin coupling to determine protons that are close in the covalent geometry of the molecule (COSY, Correlated Spectroscopy), and another to determine distance information for pairs of protons that are close in space but not necessarily close in the covalent structure (NOESY, Nuclear Overhauser Enhancement Spectroscopy). Combining information from these two types of experiments, one obtains labeled interatomic distance data- more precisely which distance corresponds to which pair of atoms. The nuclear Overhauser Effect (NOE), on which the latter technique is based, is manifested in crosspeaks in an NMR spectrum that arise from dipole- dipole coupling of one proton with nearby protons. NOE intensity is given by a formula with a term for d ij -6, where d ij is the Euclidean distance between two protons. If one has approximate knowledge of the other variables involved, then it is feasible to recover a value for d ij. Distance calculations based on NOESY are subject to numerous sources of systematic error. Some basic examples are the following: i) Spin-diffusion is a phenomenon whereby NOE effects between protons i and j are in some sense transferred through a third proton k, resulting in spurious cross-peaks. These effects often result in a compression of the apparent distances derived from transferred NOE experiments. ii) For reasons of experimental efficiency, the relaxation delays between successive NMR scans are typically too short to allow for full recovery of the magnetization effects between nearby protons This is particularly a problem in RNA structure determination, where proton relaxation times are quite long and vary considerably [8]. iii) As the NMR spectrum becomes more and more dense, it is possible for two peaks to essentially coincide, and thus for one or more of these peaks to escape detection when the data is interpreted. This is a problem that becomes increasingly significant precisely when more of the inter-point distances are measurable. iv) As noted above, in order to produce labeled distance data from a NOESY experiment in other words to determine that a given distance d is in fact associated with protons i and j one must solve a complex assignment problem using information from an accompanying COSY experiment. Several error-correcting algorithms have been proposed in order to deal with arbitrary errors in distance measurements and reconstruct an accurate 3D model. Most of them belong in the area of distance geometry and use general model-building tool. 3.2 Hidden Surface Removal Hidden surface raises a difficult problem among sets of intersecting spheres. In practice most 3D graphics workstations available nowadays, use a Z-buffer algorithm either in software or in their graphics hardware. Unfortunately, such an implementation can only handle polyhedral objects, which means that triangular meshes can approximate spheres. To get a reasonable approximation for a sphere one needs more than 100 triangles and these implementations lead to a large number of needed triangles in 3 dimensional space in order to model a molecule.
Another common approach to hidden surface removal used in computer graphics is the painter s algorithm. Here one tries to define a depth order on the objects, sorting them from back to front. Next one draws the objects in this order on top of each other where each new object hides the parts of other objects that lie below it. Such an approach does not require special graphics hardware. The problem with this approach is that it requires a valid depth order on the objects. Such an order does not always exist, as in the case of intersecting objects. In such applications the combinatorial representation of visible pieces, can be seen in the visibility map. The visibility map can be defined as the subdivision of the viewing plane into maximal connected regions in each of which a single object is seen, or no object is seen. 3.3 Modelling Water What is important in a software package is the ability to simulate the presence of water molecules, in order to understand the effects of water on the shapes of biological molecules. Taking into consideration when building a cell that it contains billions of water molecules and the space not occupied by the atoms of biological molecules is filled with water, it is clear why water plays an important role in molecular interactions. A single water molecule (H 2 O) has a tetrahedral geometry, which gives water a loosely packed structure compared with that of most other liquids, such as oils. To construct a computer aided model of water, we need to take into account two different types of forces: intramolecular and intermolecular. Moreover there are two types of waters that must be considered when building a computer model of a biological molecule in aqueous solution: the ordered waters that surround and strongly interact with the molecule and the bulk waters that may be buried within the molecule [6]. Subsequent simulations of DNA in water have revealed that water molecules are able to interact with nearly every part of DNA double helix, including the base pairs that constitute the genetic code. In other words water help us to reveal the structure of several biomolecules. In contrast, water is not able to penetrate deeply into the structures of proteins. Experimental results have shown that water profoundly influences the interactions of proteins and DNA. The polarity and hydrogen-bonding capability of water make it a highly interacting molecule. Water can greatly weaken the electrostatic forces and H-bonds between polar molecules by competing for their attraction, play both roles of an H-bond donor and, by lending electron pairs of the oxygen atom, an acceptor, and form a variety of bridges between molecular donors and acceptors [7]. 3.4 Molecular Docking One important application in the field of computer aided molecular modeling is related to the molecular docking process. An effective and interactive handling of the molecular complementarity is needed in order to investigate possible docking partners. Hidden surface problem is highly related to the protein docking process. Although the mechanisms of docking reactions are not well understood, two complementarity principles seem to be important for the recognition and binding of docking partners. The first principle is the shape complementarity principle: the shapes of the molecules that build a docking complex are complementary, that is, there is a large fit between the surfaces of the docking partners. The second complementarity principle is the chemistry principle: there is a strong chemical complementarity (with respect to hydrogen bonds, electrostatic interactions, hydrophobicity and so on), between the sites of docking partners. Although the second principle is the more important one, it is possibly to identify many docking sites solely with the help of the shape complementarity principle.
In order to find these sites for two molecules A, B with n and m atoms, the following 3D matching problem has to be solved: determine all transformations of B such that there is a large fit between the surface of A and the surface of B, and no penetration of B into the interior of A. For all the candidates with a good geometric fit the potential energy function of the docking conformation and the molecules A and B has to be computed. In the above description of the geometric 3D matching problem two strong assumptions were made: 1) the molecules are considered rigid and 2) there is no penetration. Of course not all molecules are rigid and certain parts of the molecules are very flexible influencing their chemical activity. More over local changes of the shape of the docking sites happen during the docking reactions. So with the structure of even one target protein, and the knowledge of function of its receptor or active site, it is now possible to use computer tools to build and dock a ligand or inhibitor ( new leads ) prior to investing time and resources for synthesis and testing. Thus molecular modeling is essential for understanding and exploring the structure-function relationship and evaluating novel compounds before being synthesized. Based on Fischer s lock and key principle, the mechanical view of molecular interactions can be easily understood and applied to biomolecules. However, even rigid molecules have local flexibility, as previously mentioned, and water molecules as previously described are usually a structural appendage of both the lock and the key, which means the in vivo structure may differ significantly from that on the display screen. 4. Conclusion Researchers in the areas of computer graphics, molecular biology and computational chemistry have long studied representations of molecular models as well as other computational problems related to the geometry of molecules. Current trends incorporate tools from computational geometry in order to provide algorithms that are provably efficient and at the same moment fast in practice. Taking as an input the information determined by X-ray crystallography and NMR structure determination techniques, scientists can move from sequence level to structure level, which is an important step for understanding the molecular details of biological processes. Various fields start to develop like genome modeling and annotation; comparative protein modeling and fold assignment; in silica drug design and modeling of cellular processes. Acknowledgments We would like to express our thanks to Prof. J. Matsoukas and T. Mavromoustakos for their support and useful guidelines. This work was partially supported by the General Secretariat of Research and Technology of Greece during the research project EPET II/ EKBAN 115. References [1] D. Kuntz, Structure-Based Strategies for Drug Design and Discovery, Science 257 (1992) 1078-1082. [2] R. Wang, Y. Gao, L. Lai, LigBuilder: A Multi-Purpose Program for Structure-Based Drug Design, J. Mol. Modeling 6 (2000) 498-516.
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