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A computational kinematics and evolutionary approach to model molecular flexibility for bionanotechnology

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Title:
A computational kinematics and evolutionary approach to model molecular flexibility for bionanotechnology
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Book
Language:
English
Creator:
Brintaki, Athina
Publisher:
University of South Florida
Place of Publication:
Tampa, Fla
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Subjects

Subjects / Keywords:
Collision detection
Molecular conformational search
Flexible molecules
Molecular stability
Computational geometry
Differential evolution
Dissertations, Academic -- Industrial & Management Systems Engineering -- Doctoral -- USF   ( lcsh )
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non-fiction   ( marcgt )

Notes

Abstract:
ABSTRACT: Modeling molecular structures is critical for understanding the principles that govern the behavior of molecules and for facilitating the exploration of potential pharmaceutical drugs and nanoscale designs. Biological molecules are flexible bodies that can adopt many different shapes (or conformations) until they reach a stable molecular state that is usually described by the minimum internal energy. A major challenge in modeling flexible molecules is the exponential explosion in computational complexity as the molecular size increases and many degrees of freedom are considered to represent the molecules' flexibility. This research work proposes a novel generic computational geometric approach called enhanced BioGeoFilter (g.eBGF) that geometrically interprets inter-atomic interactions to impose geometric constraints during molecular conformational search to reduce the time for identifying chemically-feasible conformations. Two new methods called Kinematics-Based Differential Evolution (kDE) and Biological Differential Evolution (BioDE) are also introduced to direct the molecular conformational search towards low energy (stable) conformations. The proposed kDE method kinematically describes a molecule's deformation mechanism while it uses differential evolution to minimize the inta-molecular energy. On the other hand, the proposed BioDE utilizes our developed g.eBGF data structure as a surrogate approximation model to reduce the number of exact evaluations and to speed the molecular conformational search. This research work will be extremely useful in enabling the modeling of flexible molecules and in facilitating the exploration of nanoscale designs through the virtual assembly of molecules. Our research work can also be used in areas such as molecular docking, protein folding, and nanoscale computer-aided design where rapid collision detection scheme for highly deformable objects is essential.
Thesis:
Dissertation (Ph.D.)--University of South Florida, 2010.
Bibliography:
Includes bibliographical references.
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Mode of access: World Wide Web.
System Details:
System requirements: World Wide Web browser and PDF reader.
Statement of Responsibility:
by Athina Brintaki.
General Note:
Title from PDF of title page.
General Note:
Document formatted into pages; contains X pages.
General Note:
Includes vita.

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ABSTRACT: Modeling molecular structures is critical for understanding the principles that govern the behavior of molecules and for facilitating the exploration of potential pharmaceutical drugs and nanoscale designs. Biological molecules are flexible bodies that can adopt many different shapes (or conformations) until they reach a stable molecular state that is usually described by the minimum internal energy. A major challenge in modeling flexible molecules is the exponential explosion in computational complexity as the molecular size increases and many degrees of freedom are considered to represent the molecules' flexibility. This research work proposes a novel generic computational geometric approach called enhanced BioGeoFilter (g.eBGF) that geometrically interprets inter-atomic interactions to impose geometric constraints during molecular conformational search to reduce the time for identifying chemically-feasible conformations. Two new methods called Kinematics-Based Differential Evolution (kDE) and Biological Differential Evolution (BioDE) are also introduced to direct the molecular conformational search towards low energy (stable) conformations. The proposed kDE method kinematically describes a molecule's deformation mechanism while it uses differential evolution to minimize the inta-molecular energy. On the other hand, the proposed BioDE utilizes our developed g.eBGF data structure as a surrogate approximation model to reduce the number of exact evaluations and to speed the molecular conformational search. This research work will be extremely useful in enabling the modeling of flexible molecules and in facilitating the exploration of nanoscale designs through the virtual assembly of molecules. Our research work can also be used in areas such as molecular docking, protein folding, and nanoscale computer-aided design where rapid collision detection scheme for highly deformable objects is essential.
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A Computational Kinematics a nd Evolutionary Approach t o Model Molecular Flexibility f or Bionanotechnology by Athina N. Brintaki A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Industrial and Management Systems Engineering College of Engineering University of South Florida Major Professor: Susana K. Lai Yuen Ph.D. Les Piegl, Ph.D. Alfredo Cardenas, Ph.D. Tapas Das, Ph.D. Kimon Valavanis, Ph.D. Ali Yalcin, Ph.D. Date of Approval: November 3, 2009 Keywords: collision detection, molecular conformational search, flexible molecules, molecular stability, computational geometry, differential evolution Copyright 2010 Athina N. Brintaki

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Dedication To the living memory of my father Nikolao E. Brintaki for his exceptional strength, boundless love and support as well as for his unforgettable spirit! And to my mom ... my love harbor!

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i Table of Contents List of Tables iv List of Figures v Abstract i x Preface x i Chapter 1: Introduction 1 1.1 Motivation 1 1.2 Dissertation Objectives and Contributions 2 1.3 Dissertation Outline 4 Chapter 2: Literature Review 5 2.1 Molecular Mechanics Model 5 2.2 Evolutionary Algorithms (EAs) 6 2.2.1 EAs in Molecular Docking 6 2.2.2 EAs in Molecular Conformational Search 7 2.3 Haptic Rendering Approaches 8 2.4 Computational Geometry 9 2.4.1 Collision Detection in Molecular Conformati onal Search 9 2.4.2 Geometric Based Molecular Docking 11 2.5 Current Literature Limitations 12 Chapter 3: A Geometric Interpretation of Molecular M echanics 14 3.1 Background on Molecules 14 3.2 Molecular Energy 17 3.3 A Geometric Molecular Methodology from Molecular Mechanics 18 Chapter 4: BioGeoFilter (BGF) M ethodology 23 4.1 Overview of the P roposed BGF Model 23 4.2 BGF: Lower Level Hierarchy 24

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ii 4.3 B GF: Upper Level Hierarchy 27 4.3.1 Constructing the Hierarchy 27 4.3.2 Molecular Geometric Constraints 28 4.3.3 Updating the Hierarchy and Self Collision Detection 29 4.4 Computer Implementation and Results 29 4.5 Conclusions 33 Chapter 5: Enhanced BioGeoFi lter (eBGF) Molecular Model 34 5.1 Differences Be tween eBGF and BGF Models 34 5.2 Proposed eBGF Overview 35 5.3 eBGF: Lower Layer Hierarchy 36 5.4 eBGF: Upper Layer Hierarchy 38 5.4.1 Constructing the BVH 39 5.4.2 Randomization 40 5.4.3 Updating the Hierarchy 40 5.4.4 Self Collision Detection 41 5.5 Computer I mplementation and Results 44 5.6 Conclusions 49 Chapter 6: Generic Enhanced BioGeoFilter ( g .eBGF) Model 50 6.1 Differences Between eBGF and g eBGF Models 50 6.2 Ligand Modeling 51 6.3 Protein Modeling 51 6.4 Proposed g eBGF Methodology 53 6.4.1 Overview of the Proposed g eBGF Model 53 6.4.2 C hemically Artificial Bonds for the g .eBGF Method 54 6.4.3 Description of the g.eBGF Algorithm 56 6.5 Computer Implementation and Results 58 6.6 Conclusions 65 Chapter 7: Identifying the Molecular Stability 67 7.1 Fundamentals of Evolutionary Algorithms (EAs) 67 7.2 EAs Advantages, Limitations and How to Compensate 69 7.3 Differential Evolution 71 7.4 Proposed kDE Model 74 7.4.1 Overview of the K inematics Based Differential Evolution (kDE) Model 74 7.4.2 Pre C omputation Module 75 7.4.3 DE L oop Module 75 7.4.4 Computer Implementation and Results 77 7.4.5 Conclusions 82 7.5 Proposed BioDE Approach 83 7.5.1 BioDE Overview 83

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iii 7.5.2 Input Files 85 7.5.3 Pre C omputation Module 86 7.5.4 DE L oop Module 87 7.5.5 Computer Implementation and Results 88 7.5.6 Conclusions 93 7.6 Comparison Between the kDE and BioDE Approaches 94 Chapter 8 : Conclusions Discussion and Future Work 97 8.1 R esearch Summary 97 8.2 Future Research Work 99 References 102 About the Author E nd Page

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iv List of Tables Table 4.1 Statistical data for four different ligand molecules 32 Table 5.1 Performance analysis of the proposed eBGF algorithm for two proteins 44 Table 5.2 Performance analysis of current approaches 48 Table 6.1 Performance analysis of the proposed g .eBGF methodology 59 Table 6.2 Computati onal complexity comparison 65 Table 7 .1 Performa nce analysis of the k DE algorithm on ligands 78 Table 7. 2 Performa nce analysis of the k DE algorithm on proteins 79 Table 7.3 RMSD performance of the kD E algorithm 82 Table 7. 4 Performance analysis of the BioDE algorithm on ligand s 90 Table 7. 5 Performance analysis of the BioDE algorithm on proteins 90 Table 7.6 RMSD performance of the BioDE algorithm 93 Table 7 .7 Comparison between kDE and BioDE a pproaches 95

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v List of Figures Figure 1.1 Receptor and ligand molecules used in drug design 2 Figure 2.1 Molecular manipulatio n and assembly with haptics 8 Figure 3.1 Graphi cal representation of three different molecular structures 15 Figure 3.2 Graphical representation of amino acids topology and link procedure through a covalent bond 16 Figure 3.3 Pattern of a proteins backbone chain 16 F igure 3.4 Mechanical molecular model 19 Figure 3.5 Example of a druglike molecule as an articulated body 19 Figure 3.6 Three geometric molecular models deve loped in this research work 22 Figure 4.1 Overall structure of the proposed BioGeoFilter methodology 24 Figure 4.2 1STP ligand molecule divided into AtomGroups based on the location of the torsion bonds 25 Figure 4.3 AtomGroups for a hypothetical small molecule 25 Figure 4.4 Local Cartesian coordinate frame assigned to iGroup and 1 iGroup 26

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vi Figure 4.5 Sc hematic representation of the smallest enclosing sphere of spheres 27 Figure 4.6 Proposed hierarchical structure for 1STP ligand molecule 27 Figure 4.7 Computational time comparison for four different ligand molecules 30 Figure 4.8 Examples of random conformations for three ligand molecules 31 Figure 5.1 Overview of the proposed eBGF approach 36 Figure 5.2 Graphical representation of the degrees of freedom of a protein 37 F igure 5.3 Graphical representation of the AtomGroup concept along with the proposed splitting procedure for a hypothetical protein segment 37 Figure 5.4 Schematics representation of the rigid and flexible AtomGroups within a hypothetical protein segment and the accordance BVH 39 Figure 5.5 Graphical representation of the proposed collision detection algorithm 42 Figure 5.6 Two example macromol ecules tested in t his work 43 Figure 5.7 Comparison of the average collision time by the proposed eBGF vs. the average energy calculation time for different sets of pre selected flexible residues/dof 45 Figure 5.8 Average total time comparison between the proposed eBGF algorithm and the energy calculation approach to output feasibility for 1STP and 1DO3 proteins in a logarithmic scale 46 Figure 5.9 Schematic demonstration of the accu racy of the proposed eBGF methodology 47 Figure 6.1 Example s of ligand molecules 51

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vii Figure 6.2 VDW representation of two different protein molecule examples 52 Figure 6.3 Graphical representation of the degrees o f freedom of a protein 52 Figure 6.4 Overview of the proposed g .eBGF methodology 54 Figure 6.5 Structure of the protein with PDB ID: 1NS1 55 Figure 6.6 Closest resi due pair between the first helic es of the two 1NS1 pro teins chains 55 Figure 6.7 Graphical representation of the AtomGroup concept along with the proposed splitting procedure for a hypothetical protein segment with two chains 57 Figure 6.8 Time comparison bet ween the traditional energy calculation approach and the proposed g eBGF methodology for ligand molecules 60 Figure 6.9 Time comparison between the traditional energy calculation approach and the proposed g eBGF methodology for protein molecules 61 Figure 6.10 Computational time performance of the proposed g .eBGF approach for molecules of different size and dof 62 Figure 6.11 Splitting threshold impact on the g .eBGF results for protein modeling 63 Figure 6.12 Accuracy comparison between the traditional energy calculation approach and the proposed g eBGF method 63 Figure 7.1 Onepoi nt crossover (recombination) operator 68 Figure 7.2 Uniform mutation operator 68

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viii Figure 7. 3 Overview of the proposed kDE model 75 Figure 7.4 Schematic representation of the chromosome structure used in this work 76 Figure 7. 5 Ligand molecules tested with the kDE model 77 Figure 7. 6 Protein molecules tested with the kDE model 78 Figure 7. 7 kDEs convergence performance for ligands 80 Figure 7. 8 kDE s convergence performance for proteins 81 Figure 7. 9 Overview of the proposed BioDE approach 84 Figure 7. 10 Ligand molecules tested with the BioDE model 89 Figure 7. 11 Protein molecules tested with the BioDE model 89 Figure 7. 12 Convergence performance of the BioDE method for ligands 91 Figure 7. 13 Convergence performance of the BioDE method for proteins 92

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ix A Computational K inematics and Evolutionary Approach to Model Molecul ar Flexibility for Bionanotechnology Athina N. Brintaki ABSTRACT Modeling molecul ar structures is critical for understanding the principles that govern the behavior of molecules and for facilitating the exploration of potential pharmaceutical drugs and nanoscale designs. Biological molecules are flexible bodies that can adopt many different shapes (or conformations) until they reach a stable molecular state that is usually described by the minimum internal energy. A major challenge in modeling flexible molecules is the exponential explosion in computational complexity as the molecular size increases and many degrees of freedom are considered to represent the molecules flexibility. This research work proposes a novel generic computational geometric approach called enhanced BioGeoFilter ( g .eBGF ) that geometrically interprets inter atomic interactions to impose geometric constraints during molecular conformational search to reduce the time for identifying chemically feasible conformations Two new met hods called Kinematics Based Differential Evolution ( kDE ) and Biological Differential Evolution ( BioDE ) are also introduced to direct the molecular conformational search towards low energy (stable) conformations. The proposed kDE method kinematically desc ribes a molecules deformation mechanism while it uses differential evolution to minimize the inta molecular energy On the other hand, the pro posed BioDE utilizes our developed g eBGF data structure as a surrogate

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x approximation model to reduce the number of exact evaluations and to speed the molecular conformational search. This research work will be extremely useful in enabling the modeling of flexible molecules and in facilitating the exploration of nanoscale designs through the virtual assembly of mole cules. Our research work can also be used in areas such as molecular docking, protein folding, and nanoscale computer aided design where rapid collision detection scheme for highly deformable objects is essential.

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xi Preface Four y ears of learning filled with contradictory experiences enforced facing myself, realizing my strengths and limitations while maturing me as a person and as a scientist. This has been a savory adventure within one s emotions : balancing between hope and despa ir, happiness and sadness, solitude and networking. For this reason, I have to express my sincere gratitude to those who made this journey possible and all people that helped and supported me during this adventure. First and foremost, I would like to thank my advisor Dr. Lai Yuen for providing the opportunity to work towards enhancing computer aided molecular design and engineering which really captured my imagination and inspired me as both a student and a researcher. Her continuing support, guidance and encouragement throughout the challenging years we have worked together has been invaluable She allowed me to find my own research path, always listening to my ideas and offering sound advice. I have learned much from her on how to be a scientist, conduct research and, I hope, an inspiring teacher. I would also like to thank Dr. Cardenas and his research group from the Department of Chemistry at USF, who taught me much of the biology I know, and for all their helpful discussions and suggestions. Their feedback and commen t s were very helpful in establishing the biological relevance of my work. I owe special thanks to Dr. Piegl from the Department of Computer Science & Engineering at USF for letting me into his CAD & Graphics research group, supporting and g uiding me throughout this research journey. Numerous times I have been challenged by Dr. Piegl on how to justify and support my research. I have learned much from him on how to present and project my work as a critical component of some of the emerging areas of engineering research in 21st century. I would also like to thank my CAD &

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xii Graphics colleagues Olya and Khairan, for being always attentive and encouraging as well as for our exceptional collaboration. My sincere thanks also to my colleagues Konstant ino Dalamakidi and Soumayaroop Roy from the Department of Computer Science & Engineering at USF for helping me with C++ programming, be ing patient and accessible in answering my numerous questions. Very genuine thanks to my former advisor Dr. Nikolos from the Production Engineering & Management Department at Technical University of Greece for initiating my research and teaching experiences. I am grateful for all his guidance, discussions and support that allowed my preparation for accomplishing this challe nging adventure. I would also like to thank him for his collaboration in one of the projects that made up this dissertation as well as for allowing me to have at my disposal his lab equipment to conduct some of my final experiments. I also owe very special thanks to my former advisor Dr. Valavanis from the Electrical & Computer Engineering Department at University of Denver for believing i n me and inviting me to continue my Ph.D. studies at USF. I am truthfully grateful for the offered opportunity as well as for inspiring me to conduct robotics research and transfer this knowledge at the bionanoscale. I am also extremely thankful towards my Committee Members Dr. Das and Dr. Yalcin and my committee Chair Dr. Tsokos for their insightful feedback, support and attention to my research work I would like to give thanks to my colleagues Wilkistar, Chaitra, Vishnu, Patricio, Laila, Alfredo, Diana, Dayna, Andres, Alcides, Ozan, Sinan, Fet hul lah all my IMSE Professors Chair Dr. Zayas and of course Jackie and Glori a for our excellent collaboration, their support and attention to my research work and academic development. I truthfully would like to thank my family and friends for their unconditional and continued support to this challenging and exciting adventure. I am extremely grateful for all the unreserved support and motivation received from my true friends Kaliopi, Katerina, Despoina, Nikoleta, Maria and of course my very good friend A hn a for always being there for me to strengthen my confidence during all the difficulties I faced. But

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xiii above all, my sincere thanks to my parents Nikolao and Euagelia, my sister Evi, my cousin Gianni my aunt Vagia and my grandparents Athanasio and Pinelopi for strongly believing i n me, supporting me throughout this long journey and all the years preceding it. I am deeply grateful to them for reinforcing me to pursue my dreams and accomplish this goal while teaching me to be a sincere person. They are the source of my strength, joy and love. My only regret however, is for not finishing my dissertation earlier, before my father passed away. I miss watching my fathers eyes filled with sincere love and pride. I am confident though, that in his eternal peace he is proud of me as he was through all my life and as I have been proud of hi m My sincere love love without ego, for my father reinforced my efforts for accomplishing this work while he was fighting for his life. This is my gift to my father for all his courage, strength a nd for his unforgettable spirit!

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1 Chapter 1 Introduction The scope of this chapter is to introd uce the motivation underneath this research work as well as the current molecular m odeling challenges. The proposed research objectives and contributions are also discussed followed by the dissertation outline. 1.1 Motivation Bionanotechnology is the new frontier in research and technology and is vital for the realization of biomedical and nanoscale products. It consists of manipulating biological molecules to create structures or devices with new molecular arrangements. The control, manipul ation, and assembly of molecu les will enable the design of innovative materials, new pharmaceutical drugs enhanced textiles, and precise nanoscale devices with new capabilities for diagnosis and tr eatment of diseases. It is estimated that within the next 10 years, at least half of the newly designed advanced materials and manufacturing processes will be build at the nanoscale [NIST]. To achieve bionanotechnology, it is crucial to enable real-tim e visualization of interactions between biologica l molecules during the design stage so that fully functional nanoscale products can be designed and evaluated prior to actual fabrication. A main key for enabling the visualization of biological components is the understanding and effective modeling of molecules behavior Molecules are very flexible in nature and can adopt many molecular conformations (or shapes) while searching for a stable or low-energy molecular state. The major challenge in modeling flexible molecules (or molecular conformations) lies on the exponential explos ion in computational complexity as the molecular size increases and a large number of degrees of freedom (dof) are considered

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to represent the molecules flexibility. For example, Figure 1.1 shows a small drug molecule (called a ligand) that can dock or assemble into a larger molecule (called a receptor) leading to the identification of pharmaceutical drugs and new molecular arrangements with specific capabilities. Receptor molecules can consist of hundreds or thousands of atoms with hundreds or even thous ands of degrees of freedom. Therefore, modeling molecular conformations is a highly intensive computational task and remains the main challenge in molecular design. Figure 1.1: Receptor and ligand molecules used in drug design. 1.2 Dissertation Objectiv es and Contributions The proposed research aims to address the main molecular modeling challenge and current literature limitations for modeli ng flexible molecules and identifying stable conformations. This research work pres ents a novel computational geometric and evolutionary inspired approach for the eff ective identification of chemically-feasible, low-energy molecular structures of any size, shape and topology. The main expected research outcome is the design of novel algorithms to minimize molecular conformational search and to speed collision detection queries that will enable the visualization and virtual manipulation of flexible molecules for interactive molecular design. The major objectiv es of this dissertation are: 2

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3 1. to develop a novel bounding vol ume data structure called BioGeoFilter ( BGF) for the effective and real-time identifi cation of feasible conformations for flexible drug-like or ligand molecules 2. to develop a generic biologically -inspired data structure called generic enhanced BioGeoFilter ( g.eBGF ) methodology for simplifying the molecular representation regardless of type, size and shape. This methodology considers certain chemical factors that influence the molecular flexibility to effectively provide more realistic and chemically -feasible molecular conformations. 3. to investigate and design a kinematics and evolutionary based direct search technique called kinematics differential evolution or kDE model that effectively searches for stable or lo w-energy molecular conformations with a good convergence performance. 4. to design a novel direct search method called biological differential evolution ( BioDE ) that will utilize our proposed g.eBGF approach as a surrogate approximation model to speed the sear ch towards alternative low-energy molecular conformations and to ach ieve a good convergence performance. The proposed computational geometric a nd evolutionary based research work contributes to the molecular modeling and differential evolution literature through the design of a new geometric-based model for si mplifying the molecular representation and two innovative evolutionary-based algorithms for directing the search towards lowenergy molecular conformations. This hybrid approach will imp act nanoscale design by speeding the modeling of flexible molecu les and enabling the development of an indispensable computer-aided design tool for bionanotechnology. The proposed research can be applied in areas such as molecular do cking/ assembly and protein folding where a rapid collision detection scheme for hi ghly deformable objects is essential. This research has resulted in two jour nal papers [Brintaki and Lai-Yuen, 2008a, 2009a], two submitted journal papers [Brintaki et.al. 2010a,d], five conference proceedings [Brintaki and Lai-Yuen 2008a ,b, 2009b, 2010b,c], three papers in progress and several poster presentations. The research work has been partially supported by NSF, SME and USF grants.

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4 1.3 Dissertation Outline Chapter 2 discusses current research work in molecular modeling, geometric techniques and evolutionary approaches in mo lecular applications. Chapter 3 describes our computational geometric interpretation of molecular inter-atomic interactions for addressing the molecular conformational search problem for highly deformable objects. Chapters 4, 5, and 6 present the development of three computational geometric models for the effective identification of feasible molecular conformations. The first model called BioGeoFilter or BGF, effectively identifies feasible conformations for small molecules in real-time as discussed in Ch apter 4. The second model called enhanced BioGeoFilter or eBGF analyzes the structure of much larger mo lecules such as proteins to model them more effectively as discussed in Chapter 5. Chapter 6 introduces the generic eBGF or g.eBGF model that incorporates chemically-based constraints that result in more realistic molecular conformations for molecules of different type, size, shape and topology. Chapter 7 proposes two new en ergy minimization algorithms: the kinematics differential evolution or kDE and the biological differential evolution or BioDE methods. Both kDE and BioDE models utilize our previously developed differential evolution (DE) algorithm to direct the search towa rds low-energy molecular conformations. The main algorithmic difference between the kDE and BioDE models is that the latest utilizes the g.eBGF data structure as a surrogate approximation model to speed convergence. Chapter 8 provides a summary of the rese arch methodologies presented and future research work.

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5 Chapter 2 Literature Review This chapter provides the background on prev ious work in the areas of molecular modeling, evolutionary algorithms and computa tional geometry techniques in molecular applications. Previous work is analyzed and their limitations identified. 2.1 Molecular Mechanics Models The molecular mechanics or force-field method uses Newtonian procedure to describe a molecular structur e and its properties energeti cally as a function of its conformation. Molecular mechanics approaches are widely used in molecular structure refinement, molecular dynamics (MD), Monte Carlo (MC), or molecular docking simulations. The molecular mechanics model considers atoms as spheres and bonds as springs that have the ability to move along different directions. The mathematics of spring deformation is used to measure the ab ility of the bond to stre tch, bend and twist. Dynamic-based simulation models such as molecular dynamics (MD) simulations [Leech 1996, Branner 2000, Renambot 2001, Tanfer 2004, Phillips 2005, Adckock 2006,] and Monte Carlo (MC) methods [Liu 1999, Kima 2002] are commonly used to obtain information related with the time e volution of molecular conformations. These methods aim to determine molecular feasib ility by calculating atom s position and hence their internal energy in small time steps. This results in a more accurate but slow progress towards the search of a feasible molecular conformation. As the number of atoms within a molecular structure increases, the time to calculate the intra-molecular energy for determining a molecules feasibility (sta bility) increases signi ficantly, making these methods unsuitable for interactive molecular design and assembly.

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6 2.2 Evolutionary Algorithms (EAs) The choice of an appropriate optimizati on method is essential for directing the conformational search to identify the desi red solution or the be st potential molecular conformation. The optimization of molecular geometry was one of the very first applications of evolutiona ry algorithms (EAs) in chemistry. EAs have shown good results in problems where other methods ha ve struggled. In addition, their governing principles are clearly understood, intuitively ap pealing and relatively easy to implement. In this section, we focus on EAs applicat ions in chemical problems that require optimization such as molecular docking and molecular conformational search. Detailed information on EAs is provided in Chapter 7. 2.2.1 EAs in Molecular Docking Current literature in mol ecular docking demonstrates the effectiveness of Evolutionary Algorithms (EAs) for desc ribing complex systems [Thomsen 2003, 2006] and for solving problems involving large sear ch spaces, where traditional optimization techniques are less efficient [Yang 2001]. Ge netic Algorithms (GAs) are presented as an effective local search method that behave s really well for median energy solutions [Westhead 1997, Jones 1997]. Additionally, Morr is et al. compared the efficiency of Monte Carlo (MC) simulated annealing met hod against a classic GA and Lamarckian GA (LGA) for predicting the bound association of flex ible ligands to macromolecule targets. Results showed that both LGA and GA are the most reliable, efficient and successful methods. However, many modifications have been proposed to improve the solution quality and to speed convergence. One of the best EAs for solving real-v alued energy functions is Differential Evolution (DE) initially proposed by [Storn & Price 1995, 2005]. DE is a population based stochastic function minimizer that adds the weighted difference between two individual vectors to a third vector (donor ). Currently DE has been implemented by [Yang 2001, Thomsen 2003, 2006] for investigati ng the docking of a flexible ligand to a

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7 rigid receptor where their numerical results indicate the algorit hms robustness and remarkable performance in terms of convergence speed. 2.2.2 EAs in Molecular Conformational Search Wehrens presented a survey focused on the differences, strengths and weaknesses between EAs and other structure optimization methods such as distance geometry, eigen value decomposition, simulated anneali ng, Monte Carlo or molecular dynamics simulations [Wehrens 2000]. The main conc lusion was that EAs are consistently among the best performing general search algorithms On the other hand, GAs are particularly useful for rapidly producing a family of low energy conformations but are less successful in fine-tuning these conformations towards the exact global optimum. Various evolutionary-based studies have been performed to study flexible ligand, flexible protein or polypeptide molecules conformational search. Wawer et al. [Wawer 2004] presented a real-coded (a s opposed with the binary coding of the classic GAs) genetic algorithm to analyze the conforma tional behavior of Vitamin E (a small molecule). Wang and Ersoy [Wang and Er soy 2005] presented a Mixture Gaussian Optimization (MGO) algorithm as a continuous stochastic approach for flexible ligand conformational search. The MGO method wa s compared against a systematic and a stochastic conformational search algorithm and it was concluded that the MGO algorithm can locate the global minimum faster as the molecular size increases. On the other hand, as the molecular size increases, the system atic search method became non-applicable whereas the stochastic was trapped in local minima. However, the MGO was tested for small molecular structures only and was not a pplied to large molecules such as proteins. Chong and Tremayne [Chong and Tremayne 2006] presented a new DE algorithm based on Cultural Evolution concepts called CD E to study the structure search for ligand molecules. The CDE algorithm was compared against a classic DE method and it was concluded that both methods succeeded to find the global minimum and the convergence performance of the CDE algorithm was 54% faster.

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Damsbo et al. [Damsbo 2004] presented the FOLDAWAY system, an evolutionary based approach for finding th e low-energy conformations of polypeptides. The proposed model found large groups of lo w-energy structures within the expected low-energy globule that were not identified in previously developed MD simulations. Bitello and Lopes [Bitello and Lopes 2004] used a DE algorithm to solve the protein folding problem. Their approach was consistent in finding the global minimum for structures consisted up to 64 amino acids (relatively small protein size) and performed better compared with a classic GA. Figure 2.1: Molecular manipulation and assembly with haptics. 2.3 Haptic Rendering Approaches In recent years, new methods have been investigated to facilitate molecular design and nanoscale engineering by providing realtime force feedback using haptic devices [Sherill, Baxter 1998, Nagata 2002, Gray son 2003, Lee 2004, Lai-Yuen 2006a,b, Morin 2007]. Haptic devices are electromechanical devices that exert forces on users giving them the illusion of touching something in the virtual world. These devices have been used to manipulate virtual molecules and to feel the forces as the molecules interact with each other providing an essential design and vi sualization tool as shown in Figure 2.1. However, current methods using haptics either model molecules as rigid bodies or are 8

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9 limited to local molecular motions and short periods of simulation time. Modeling molecules as rigid bodies can simplify the calc ulation of forces but does not represent the molecular interactions realistically. To achieve a realis tic molecular representation, it is necessary to model molecules as flexible bodie s that attain different conformations while searching for a stable molecular state. Incorp orating real-time haptic force feedback into molecular design requires rapid update and modeling of molecular conformations for providing realistic and continuous visualization and sense of touch to the users. 2.4 Computational Geometry Recently, computational geometry has been successfully used in molecular design since important constraints influencing molecular behavi or can have geometrical interpretation. The representation of intra-molecu lar interactions through a computational geometric approach can allow the approximation of molecules behavior rapidly and efficiently for real-time molecula r design. From a geometric point of view, a molecular conformation can be considered f easible when there are no overlapping atoms or all possible atomic interactions are collision-free. Collision detection (CD) is an essential problem in robotics, computational geometry, and computer graphics and is a major bottleneck in any inter active simulation. A wide range of techniques have been proposed to deal with collisi on detection such as hierarch ical representations, spatial partitioning, analytical methods, and geometric reasoning. The algorithmic design depends on the representation of the mode l, the query types, and the simulated environment [Lin 1998]. 2.4.1 Collision Detection in Molecular Conformational Search Bounding volume hierarchies (BVH) ar e the most popular methods for capturing self-collision and coll isions between objects [Teschner 2005]. The key idea is to use a hierarchical structure to describe the shape of an object at successive levels of detail. The

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object of interest is enclos ed by bounding volumes that can ha ve various shapes such as spheres, axis-aligned bounding boxes (AABBs), and oriented bounding boxes (OBBs) [Lin 1998]. These bounding volumes become the tree leaves of the hierarchy that are enclosed by subsequent bounding volumes form ing a hierarchical data structure. For molecular structures, collision detection is a computationally expensive problem given the many degrees of freedom th at a molecule can have. Lotan et al. [Lotan 2002] used a kinematics chain model to re present proteins flexibility. In respect to the chain topology the authors built a B VH using object-oriented bounding boxes to detect overlapping atoms. They tested various proteins of different size and concluded an updated and testing computational time ra nging in hundreds milliseconds. Their proposed approach requires performance for building the BVH and computational complexity for the collision detection queries. )N(O) N(O3/4 Agarwal et al. [Agarwal 2004] used a BVH with the objects being modeled as spheres to detect collisions for defo rming and moving necklaces (sequence of balls/beads). The authors built a balanced bi nary tree with spheres as bounding volumes to assist in the search for overlapping atom s within flexible protein molecules. They proposed two methods for computing the sphe res: wrapped and layered hierarchy that provides an upper bound of in 2D space or in d-dimensional space for the collision detection plan. )NlogN(O) N(Od/32 Angulo et al. [Angulo 2005] proposed the BioCD algorithm for efficient selfcollision search and distance computations The algorithm maintains two levels of bounding volume hierarchies (BVH). In the low leve l, it identifies the rigid groups of the articulated model and builds a hierarchy for eac h of them. In the upper level, it arranges the roots of the low level hierarchies. The au thors tested various pr oteins with different size and they reached a collis ion detection time measured in tens of milliseconds with performance and a complexity for building the BVH. )N(O )NlogN(O Redon et al. [Redon 2005] proposed an adaptive dynamic algorithm (ADA) for articulated bodies built upon the divide-and-conquer algorithm (DCA). An articulated body is the recursive link pair of articulated parts. The series of the assembly actions is 10

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represented by a binary tree. Each node in the tree represents a sub-assembly motion. Morin and Redon [Morin 2007] utilized the ADA algorithm and proposed a forcefeedback algorithm for adaptive dynamic simulation of proteins. The authors used a multithreaded structure to couple the ad aptive dynamic simulation loop from the computation of the force applied to the us er (through the haptics) requiring a force feedback of complexity. )N(log O 2.4.2 Geometric-Based Molecular Docking Current computational docking methods co me from the areas of surface matching, object recognition and motion planning. Moti on planning is a funda mental problem in robotics that consists of finding a valid seque nce of configurations that moves an object from an initial position to a target point. Automatic motion planning is applicable not only to robotics, but also to virtual reality systems, computer-aided design and computational biology. Recently, researchers realized that both automatic motion planning and molecular docking problem relies upon the same basic principles. A drug molecule can be considered like a robot with many degrees of freedom (dof) whose motion can be predicted by an automatic planner determining its ability to bind with a protein. The binding configuration should satisfy all the geometric, electrostatic and chemical constraints of the problem. A good binding site should also be reachable to the ligand from an outside location. Hence, the path to the binding site is highly important and motivates the use of motion planning in th e molecular docking problem. These methods are known as probabilistic roadmap methods (P RMs) and are widely used in robotics, intelligent CAD systems and lately in computational biology. PRMs randomly construct a graph in C-space (configuration space), a roadma p, as it is called. The motion planning is then solved by connecting the start and goal configurations in the roadmap and searching for the feasible path on it [Bay azit 2003, Cortes 2003, 2005, 2007]. A modification of the PRM framework is the rapidlyexploring random trees (RRT) for solving single-query problems w ithout preprocessing the complete roadmap. 11

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12 These algorithms are well fitted for highl y constrained problems [Cortes 2002, 2003, 2005, 2007, La Valle 1999, 2000a,b]. A recently developed PRM variation to study molecular motions is the Stochastic Road map Simulation (SRS) [Bayazit 2000, Apaydin 2002a, b, 2003, Chiang 2006]. A stochastic ro admap contains many Monte Carlo (MC) simulation paths at the same time. The SRS studi ed all the paths together in a closed form and resulted in significant co mputational time reduction. Zhang and Kavraki [Zhang and Kavraki 2002] compared their proposed atomgroup-local-frame method with the simple rotations and Denavit-Hartenburg model [Hartenburg and Denavit 1955]. It was conc luded that the atom-group-local-frame method not only eliminates all th e disadvantages of the other two but also resulted in a lazy evaluation of atom positions and in co mputational time reduction. This technique appears extremely useful in cases that deal with many conformations. Zhang et al. [Zhang 2005] extended the atom-group-local-frame work by adding a geometric screening phase for identifying feasible molecular conformations. 2.5 Current Literature Limitations Although remarkable advances in comput ational biology have been performed over the years, modeling molecular flexibility remains the main challenge in molecular design. Most of the above discussed methods do not address the modeling of molecules for real-time rendering or only allow a limited nu mber of degrees of freedom to change. In addition, a more generic methodology is required that: 1. is not limited to the topology of the mol ecules for self-collisions or collisions between them, 2. evaluates arbitrary conformations in dependently of the previous query, 3. is adaptive to the molecular structure by exploiting the fact that when limited degrees of freedom change some of th e atomic distances remain constant, 4. identifies molecular feasibility rapidly, efficiently and is evaluated in terms of both computational time and accuracy,

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13 5. incorporates the chemical information that controls molecules flexibility into the molecular design to simplify and realistically represent the molecular interactions 6. effectively directs the search towards low energy and chemically-feasible molecular conformations. To address current literature limitati ons, this research work presents a new biologically-inspired geometric method for simpli fying the representation of molecules of different type, size, shape and topology while considering certain ch emical factors that influence molecules flexibility. To direct the search towards low-energy molecular conformations, we propose the use of a new evol utionary based algorithm that will utilize the developed geometric method as a surr ogate approximation model for reducing the algorithms convergence rate and finding the global minimum. The proposed work can facilitate interactive molecular modeling and nanoscale design.

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14 Chapter 3 A Geometric Interpretation of Molecular Mechanics This chapter introduces the basic molecular concepts and presents our proposed geometric interpretation of molecular mechanics. A brief background on the various types of molecules and their basic functions is provided where molecules are categorized into ligands and receptors. The central con cepts on the internal molecular energy and our geometric interpretation of the molecular conformation mechanics are also explained in this chapter to provide the ba sis for our developed algorithms presented in Chapters 4, 5 and 6. 3.1 Background on Molecules A molecule is a sufficiently stable electrically neutral group of at least two atoms, in a definite arrangement, he ld together by very strong chem ical bonds or covalent bonds. A covalent bond is a chemical bond where el ectrons are shared between atoms. As shown in Figure 3.1, the size, shape and topology of a molecular structure varies according to its chemical characteristics a nd function. These molecules are displayed using the VMD software [Humphrey 1999] as shown in Figure 3.1. Geometrically, a molecule can be considered as a collection of atoms and bonds between each atom pair. Each atom can be represented as a sphere with van der Waals radius while chemical bonds can be represented as springs.

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(a) 1A5Z ligand (b) 1HVR ligand (c) 1DO3 protein molecules, displayed with VMD software Figure 3.1: Graphical repres entation of three different molecular structures. Molecules are essential to a variety of biological processe s and activities of fundamental importance to life. There are f our basic types of mo lecules that are the major players in biological systems: carbohydr ates, lipids, nucleic acids and proteins. Both carbohydrates and lipids are small molecu les that are less complex compared with the nucleic acids and proteins. Carbohydrates tend to be the least complicated molecular structures used as energy sources for cell proce sses. Lipids are also fairly simple organic molecules that have several us es in living organisms such as acting as water barriers in cell membranes. Lipids are also used for extra waterproofing or as insulation around nerves for long-term storage of energy in the fo rm of fat, or used as heat insulation, as cushions, and as messenger molecules. Nucl eic acids and proteins are typically much larger complex molecules. The primary role of the nucleic acids or DNA (one type of nucleic acid) is to store proteins main in formation. Proteins are very important macromolecules in living organi sms and perform many distinct functions. The functions of a protein depend on its 3-dimensional shape, which can be virt ually infinite in variety. Most proteins are enzymes performing biochemical functions such as bond-making and bond-breaking reactions. Other proteins ac t as molecular motors or structural components by performing biophysical functions. 15

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The understanding and modeling of the molecu lar functions and behaviors is very important in nanoscale design since many probl ems associated with the development of bionanotechnology require specif ically-designed molecules. For example, drug design and discovery relies increasin gly on structured-based methods for improving efficiency. The main objective in drug design is to find or build molecules (ligands) that target proteins (receptors) crucial to the proliferation of microbes, cancer cells or viruses. This is a very long and expensive process calle d molecular docking that typically requires years of research, experimentation, and resources. A ligand or drug-like molecule is a small molecular structure that usually consists of at most 50 atoms as shown in Figure 3.1( a) and Figure 3.1(b). A ligand molecule has a tendency to bind to large molecules called receptors that can lead to the identification of new pharmaceutical drugs and the creation of new molecular structures with specific capabilities for diagnosis a nd treatment of diseases. Figure 3.2: Graphical repr esentation of amino acids topology and link procedure through a covalent bond. Figure 3.3: Pattern of a proteins backbone chain. As shown in Figure 3.1(c), a protein or a receptor molecule is a much larger molecular structure that consists of hundreds or even thousands of atoms. Proteins are chains of smaller molecular entities called amino acids. The amino acids consist of a central carbon atom, denoted as connected to an amino group a carboxyl aC2NH16

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groupCOOH, a single hydrogen atom H and a side chain specific for each amino acid, as shown in Figure 3.2(a). There are 20 basic amino acids that serve as building blocks of proteins. Amino acids differ from each other by their side chains, which also determine their chemical characteristics. The amino acids may be linked to each other by the peptide bond (a covalent bond) between an amino group of one amino acid and a carboxyl group of another amino acid releasi ng a water molecule, as shown in Figure 3.2(b). These peptide bonds lead to a linear sequence of amino acids forming a polypeptide chain. The backbone of the chain is formed by a pep tide sequential pattern schematically shown in Figure 3.3. Theref ore, any protein can be considered as a polypeptide chain characterized by the amino acid sequence along the chain in order. R 3.2 Molecular Energy Molecules are very flexible in nature and can attain different conformations. A feasible molecular conformati on indicates a stable molecular state that is usually described by the minimum intra-molecular energy. This energy is a function composed of different energy factors that depict the interactions be tween bonded and non-bonded atoms. The major energy contributors are the non-bonded van der Waals (VDW) potential and electrostatic forces. A ma thematical representation of the non-bonded molecular energy is given by Eqn. 3.1: nbE 11 11 6 ij ij12-n i n j ij ji n i n j ij ij nbr qq k r A r B E (3.1) The first term in Eqn. 3.1 represents the VDW interaction that models the pairwise potential over all pairs of non-bonded atoms i, j and are the VDW repulsion and attraction parameters, respectively; and is the distance between every exclusive non-bonded atom pair i and j The second term in Eqn. 3. 1 represents the electrostatic forces between any non-bonded atom-pair. Th e electrostatic contribution is modeled ijBijAijr17

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through a Coulomb potential where represent the atomic charges, the interatomic distance, and k describes a molecular dielectric constant. jiqq ,ijrAs the number of atoms within a molecu lar structure increases, the time to calculate the intra-molecular energy for determining a molecules feasibility (stability) increases significantly. This makes the energy calculation method unsuitable for realtime molecular design and assembly. For this reason, alternative approaches for identifying feasible molecular conformati ons are needed. Recently, computational geometry has been successfully used in molecular design since important constraints influencing molecular behavior can have geometrical interp retation. The representation of intra-molecular interactions through a computational geometric approach can allow the approximation of molecules beha vior rapidly and efficiently. Hence, this research work focuses on developing a new com putational geometry approach to effectively identify feasible molecular conformations fo r molecular design and assembly. 3.3 A Geometric Molecular Met hodology from Molecular Mechanics The molecular mechanics or force-field method uses Newtonian procedure to describe a molecular structur e and its properties energeti cally as a function of its conformation. The internal forces experienced in the model structure are described using simple mathematics functions. For example, Hookes law is commonly used to describe bonded interactions, whereas the unbounded atom s might be treated as inelastic hard spheres that interact according to the Lennard-Jones potential. Based on these mathematical models, molecular dynamics simulations numerically solve Newtons equation of motion to observe the structural motions with respect of time. These simulations consider atoms as spheres and bonds as springs that have the ability to move along different directions. The mathematics of spring deformation is used to measure the ability of the bond to stretch, bend an d twist as shown in Figure 3.4. 18

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Figure 3.4: Mechanical molecular model. As shown in Eqn. 3.1, the internal no n-bonded energy is calculated based on the VDW potential and the electrostatic forces for every non-bonded atom pair within the molecular structure. Both VDW and electros tatic forces are usually computed for atoms connected by no less than two atoms (non-bonded atoms in a 1, 4 relationship or further apart). In the mechanical model, non-bonde d atoms are those atoms linked by three or more chemical bonds as indicated by the blue-colored spheres in Figure 3.4. Figure 3.5: Example of a drug-like molecule as an articulated body. From a geometric point of view, a molecu le can be modeled as an articulated body with at least six degrees of freedom (dof): three translat ional and three rotational. In addition, each chemical bond within a molecular structure carries information related to the van der Waals radius This information is linked to the bond length; the bond angle (the angle between bond and ) and the set of torsion anglesibirib1 ib ) 2,0[ i. A torsion bond is the bonds cap ability to rotate along its ow n axis. In most molecular studies, the bond length and angl es are kept fixed since they do not contribute 19

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significantly to the molecular shape. Therefore, a molecular conformation is defined in this work as the changes in the angles of the torsion bondsi as shown in Figure 3.5. From molecular mechanics, the VDW repulsion force between two non-bonded atoms increases exponentially as the distance between the atoms decreases. The VDW attraction occurs at shor t range until the non-bonded at oms relative distance d is equal to their equilibrium distance jirrd 0 and fades away as the interacting atoms move apart. A geometric interpretation of the VD W atomic interactions is given by Eqn. 3.2 under which an overlapping atom-pair exists: 1 0 )(, ji atomsrr dji (3.2) Where represents the distance between the non-bonded atoms i and j ; are the VDW radii for the non-bonded atoms i and j respectively; and jiatomsd,jirr is a constant parameter that controls the impact of th e VDW equilibrium distance on each non-bonded atomic interaction. The electrostatic poten tial provides a smooth transition between the attraction and repulsion regimes. The overall impact of the non-bonded atomic interactions can be geometrical ly interpreted by Eqn. 3.3: 1 <0 +)+(<, rrr dijji atomsji (3.3) A stable molecular state can be represen ted by a feasible molecular conformation with low internal energy E In a force field, the VDW forces are the dominant energy contributors while the electrostatic interactions dominate the computational time [Sherrill]. Thus, identifying a molecular conformation with E lower or equal than the VDW interactions guarantees that E will be less than or e qual to the total non-bonded energy : nbE20

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on conformati molecular stable EE that guarantees r A r B Efinding then r qq k r A r B given r qq k r A r B Enb n i n j ij ij ij ij ij ji ij ij ij ij n i n j ij ji n i n j ij ij ij ij nb += 1 =1 = 612 612 1 =1 = 1 =1 = 612 (3.4) Similarly, as shown in Eqn. 3.3, a molecular conformation is considered infeasible when overlapping atoms exist within the molecular structure. Finding a pairwise atomic distance d that satisfies Eqn. 3.2 ensures that a self-collision occurs as it is demonstrated below: occurs collision self ddthat guarantees rrdfindingthenrrrgiven where rrr dji jiatoms ji ijji ijji atoms )( )( 1 0 )(, (3.5) Given that the VDW potential dominates th e molecular interactions chemically and geometrically as demonstrated in Eqn. 3.6 and Eqn. 3.2, respectively, the intramolecular energy can be approximated by the VDW interactions only as follows: 11 612n i n j ij ij ij ij nbr A r B E (3.6) Given that the number of possible molecu lar conformations grows in proportion to the power of the number of torsio n bonds, identifying feasible molecular conformations remains the main challenge in molecular design. This research work presents a biologically-inspired geometri c method that incorporates the above assumptions on atoms connectivity and chemi cal factors to rapidly identify chemicallyfeasible molecular conformations. Our approa ch aims to geometrically approximate the behavior of molecules of any size, shap e, and topology efficiently while minimizing molecular conformational search an d collision detection queries. 21

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In the following three chapters, the devel opment of three com putational geometric molecular models for identifying molecules feasibility is discussed as shown in Figure 3.6. In Chapter 4, a biologically-inspire d geometric method called BioGeoFilter (BGF) methodology is presented for modeling the behavi or of drug-like molecules in real-time. The enhanced BioGeoFilter algorithm (eBGF) is presented in Chapter 5 to model the behavior of macromolecular structures such as protein molecules. Chapter 6 presents a generic computational geometric molecular approach (generic eBGF) for modeling the behavior of molecular struct ures of any size, shape and topology for real-time molecular design and assembly. BGF algorithm for drug-like molecules eBGF algorithm for protein molecules generic eBGF algorithm for any molecular structure Figure 3.6: Three geometric molecular models developed in this research work. 22

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23 Chapter 4 BioGeoFilter (BGF) Methodology In this chapter, a new methodology called BioGeoFilter (BGF) is introduced to approximate drug-like molecule s behavior in real-time s ubject to both chemical and geometric constraints. The BGF approach consists of a two-layer hierarchical data structure that simplifies the molecular repr esentation to effectively identify molecular self-collisions. Experimental results show that the BGF approach significantly decreases the computational time for identifying feasible conformations. This can facilitate the real-time modeling of molecula r components to enable inte ractive molecular design and assembly. 4.1 Overview of the Proposed BGF Model The proposed BGF model consists of a hier archical structure that comprises two layers: a lower level and an upper level as s hown in Figure 4.1. At the lower level, the molecule is modeled as an articulated body with the internal degrees of freedom representing the number of torsion bond a ngles. At the upper level, a bounding volume hierarchy (BVH) is introduced to identify atoms within the molecule that are in collision. A new updating scheme for the BVH is presen ted to identify self-c ollisions during the update phase of the algorithm. This signi ficantly speeds the com putational time so the proposed BGF methodology can be used for both real-time molecular modeling and for reducing the energy minimization time. The follo wing sections describe the two levels of the proposed BGF algorithm.

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Figure 4.1: Overall structure of the proposed BioGeoFilter methodology. 4.2 BGF: Lower Level Hierarchy As shown in Figure 4.2, a drug-like or ligand molecule is modeled as an articulated body, where an arbitrarily-selec ted atom acts as the base of the body. A flexible molecule has at least six degrees of freedom (dof): three translational and three rotational. In addition, each torsion bond angle ) 2,0[ iaccounts for an additional 24

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dof. Hence, a molecular conformation is defi ned as the changes in the angles of the torsion bonds. 1 2 3 Figure 4.2: 1STP ligand molecule divided into AtomGroups based on the location of the torsion bonds. To reduce the computational complexity of a molecular structure, atoms of a molecule are clustered into AtomGroups ba sed on the approach by [Zhang and Kavraki 2004]. Based on the location of the torsion bond s, atoms are clustered into AtomGroups. In other words, all the atoms within an AtomGroup are connected by rigid bonds while AtomGroups are connected by to rsion bonds, as shown in Figure 4.2. Therefore, the number of the AtomGroups required to represen t molecules flexibility is equal to the number of the torsion angles plus one. 25 1 2 3 4 5 6 Figure 4.3: AtomGroups for a hypothetical small molecule.

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Once the AtomGroups are defined, one group is chosen as the root Atomgroup. The root Atomgroup is important since it represents the base of the molecular structure where the molecular motions will be projecte d. The lower hierarchical layer of the proposed BGF is a tree where each vertex represents an AtomGroup and each edge denotes a torsion bond as shown in Figure 4.3. Figure 4.4: Local Cartesian coordinate frame assigned to and iGroup1iGroup To speed the update of molecular conformations, each is assigned a local Cartesian coordinate frame and a relationship is generated between all the AtomGroups. Since each AtomGroup contai ns atoms whose distance will not change when torsion changes occur, the distance be tween atoms in the same AtomGroup do not need to be checked for collision. Only non-bonded atoms that correspond to different AtomGroups will be checked thus reducing the time to identify geometrically feasible conformations. This significantly redu ces the computational time and decreases calculation inaccuracies when updating atom pos itions during conformational changes. The clustering of atoms in groups will be used to form the upper level hierarchy of the proposed BioGeoFilter approach as described in following section. iAtomGroupiF 26

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4.3 BGF: Upper Level Hierarchy 4.3.1 Constructing the Hierarchy Once the different AtomGroups of the molecu le have been built at the lower level hierarchy, the smallest enclosing sphere that contains all the atoms within each AtomGroup is calculated as shown in Figur e 4.5. The spheres (each containing an AtomGroup) are organized into a binary tree-lik e data structure that will serve to detect molecular self-collisions subject to both chemical and geometric constraints during conformational search. Figure 4.5: Schematic representation of th e smallest enclosing sphere of spheres. Level 0 Level 1 Level 2 1S2S3S4S5S6S7S 1S3S123 1S2S3S4S Figure 4.6: Proposed hierarchical st ructure for 1STP ligand molecule. 27

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Figure 4.6 shows the proposed bounding volume hierarchy for the molecule previously shown in Figure 4.2. At the botto m of the tree there are four spheres (called leaf nodes) representing the f our AtomGroups for the 1STP ligand molecule. For each pair of nodes, an intermediate node is create d that encloses the two nodes. This process continues in a bottom-up manner until all the s pheres result into one si ngle root sphere as shown by the number S7 in Figure 4.6. 4.3.2 Molecular Geometric Constraints The VDW interactions are converted into geometric constraints to decrease the time to identify infeasible molecular confor mations. As discussed in Sections 3.2 and 3.3, the VDW repulsion between two non-bonde d atoms increases exponentially as the distance between the atoms decreases. The VDW attraction occurs at short range until the non-bonded atoms relative distance d is equal to their equilibrium distance : Hence, based on these interactions, we introduce the first geometric constraint that no neighboring atoms or atoms within neighboring AtomGroups should be checked for self-collision. 0d0= dd The distance between nonbonded atoms within a molecule can often become very short leading to large va lues in the non-bonded energy and forces. For this reason, the VDW interaction for non-bonded atoms is modeled as a pair-wise potential over all pairs of atoms except 1-2 and 1-3 bonded atom s pairs based on the concept of [Dendzik 2005]. Thus, the second geometric constraint consists of considering as non-bonded atoms the atoms linked by four or more chem ical bonds. Moreover, the detection of an actual self-collision between a non-bonded atom pair along with the algorithms selectivity mechanism depends on the constant of the equilibrium distance where Therefore, the third geometric constr aint is the constr aint given by Eqn. 3.2 that detects atoms self-collisi ons. If Eqn. 3.2 is satisfied, then an actual self collision occurs between the non-bonded atoms i and j 0d)+(=0 jirrd28

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By decreasing the value, the output set of feas ible solutions obtained by the BGF algorithm increases as it is further analy zed in Section 4.4. Therefore, based on the above geometric constraints, the BGF me thodology rejects any mo lecular conformation that does not satisfy the geometric filtering as described in the following section. 4.3.3 Updating the Hierarchy and Self-Collision Detection As new conformations are being searched through changes in the torsion bonds, the new position of the atoms needs to be calcu lated. The new atom positions affect the location and radius of the spheres in the hierarchy so they need to be updated accordingly. In this work, the spheres in the hierarchy are updated in a bottom-up manner and one level at a time. Therefore, the tree nodes are updated from the leaf nodes to their parents and this pr ocess continues until the root node is reached and updated. During the update phase, a new updating algor ithm is introduced so that if a selfcollision is detected, the algorithm will imme diately stop and reject the conformation due to overlapping atoms (self-collision) as show n in Figure 4.1. The algorithm first updates the leaf nodes (e.g., S1, S2, S3, and S4 in Figure 4.6). One leve l at a time, the algorithm updates the spheres radius and centers base d on the new atom locations. Then, the parent nodes of the leaf nodes are tested for update. If ther e is a collision between the children nodes, the algorithm returns that the conformation is infeasible and stops. If no collision is detected, the process continue s until the root node is reached and updated. 4.4 Computer Implementation and Results The presented method and algorithms have been implemented on Intel Pentium 4 with 2.7 GHz personal computers using Vi sual C++ programming language, the OpenGL and CGAL libraries [CGAL]. Four different molecules with different number of atoms and number of degrees of freedom were tested using the proposed BioGeoFilter methodology. The molecules were obtained from the Protein Data Bank (PDB) [Berman 29

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2000] with PDB IDs as follows: 1HVR, 1HTB 1A5Z, and 1JBO. Their corresponding number of atoms and degrees of freedom are indicated in Table 4.1 presented below. 0 0.2 0.4 0.6 0.8 1 1.2 1HVR1HTB1A5Z1JBO (4 DOF)(10 DOF)(11 DOF)(13 DOF)time in ms T_energy T_BGF Figure 4.7: Computational ti me comparison for four different ligand molecules. Figure 4.7 compares the performance of the proposed BGF algorithm and the energy calculation for the four different mol ecules with different pre-selected dof. For each example molecule, random conformations we re generated and test ed for feasibility using both methods. As shown in Figure 4.7, the proposed BG F methodology greatly reduces the computational time needed to identify feasible molecular conformations compared to the energy calculation approach. This reduction in time is significant as multiple flexible molecules will need to be modeled in real-time at the same time for nanoscale assembly. It can also be observed that as the dof increases, the time reduction percentage also increases. The computationa l times for all the tested molecules satisfy the real-time haptic constraint and scale well as the number of degrees of freedom increases. 30 Various feasible conformations obtained from the BGF approach are shown in Figure 4.8(c) for the example molecule 1A5Z. These conformations satisfy the geometric constraints of the BGF met hodology and have been validated using the energy values obtained from the energy calculation method.

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Conformation A Conformation B Conformation C Figure 4.8: Examples of random conf ormations for three ligand molecules. Table 4.1 shows the results in terms of computationa l time (milliseconds) and accuracy (number of feasible conformations identified) between the energy calculation approach ( T_en and F_en columns) and BGF approach ( T_BGF and F_BGF columns). In the proposed methodology, the same molecu lar conformation is used for comparing the two methods. As shown in the percentage time reduction ( T_red) in Table 4.1, the proposed algorithm can identify feasible conf ormations at least twice faster than the energy calculation method and with similar accur acy. It can also be observed that as the dof increases, the time reduction percentage obtained from th e BGF methodology also increases. 31

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Table 4.1: Statistical data for four different ligand molecules. # of Ligand atoms DOFpT_en.T_BGFF_enF_BGFT_red. 0.80.60.113/1000/10083% 0.70.50.214/1006/10060% 0.60.720.15817/10014/10078% 0.80.5860.21818/10018/10063% 0.70.60.343/10039/10050% 1HVR 46410.70.30/1002/10057% 0.80.70.469/10043/10043% 0.71.10.277/10067/10082% 0.60.60.185/10078/10083% 1A5Z 44 1HTB 44 1JBO 43 11 10 13 Table 4.1 also shows the sensitivity analysis performed to study the impact of the different values for of the equilibrium distance on th e results. The entire range of values, where was tested for each molecule but only the most significant 1<0 < values are shown in the table for explanati on purposes. As shown in Table 4.1, it was found that by varying and depending on the size of the molecule, the selectivity of the BGF methodology can be adjusted. As decreases, BGF accepts more molecular conformations as feasible, which leads to a relaxed filtering. The main objective of the BGF methodology is to identify infeasible conformations while not rejecting any feasible conformations. Hence, the se lection of an appropriate value for each molecule depends on the molecule and the desired level of selectivity. In Table 4.1, the grey colored rows denote the best values for each molecule in terms of selectivity and accuracy. An analysis on the relationship between the value and the molecules size is addressed in Chapter 5. The sensitivity analysis was performed by relaxing the third geometric constraint ( value) only. The relaxation of the other geometric constraints was shown to increase the acceptance of unfeasible molecular conformations and computational time. For this reason, the sensitivity analysis only focu sed on relaxing the third constraint while keeping other constraints fixed. 32

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33 4.5 Conclusions This chapter presents a new met hod called BioGeoFilter (BGF) for modeling and approximating the molecular behavior subject to geometric constraints in real-time. BGF consists of a novel two-layer hierarchical stru cture that identifies self-collisions during the hierarchys updating phase. The proposed methodology is presented as a filtering tool based on chemical and geometric concepts for effectively identifying feasible molecular conformations. Computer implemen tation and results demonstrate that the proposed BGF approach significantly decreases the computational time for identifying feasible ligand conformations to satisfy r eal-time update requirements. The proposed BGF methodology can facilitate the real-time modeling and visualization of molecular components and enable the development of an essential interactiv e nanoscale computeraided design tool for bionanotechnology. The following chapter presents the extended BGF algorithm for macromolecular structures.

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34 Chapter 5 Enhanced BioGeoFilter (eBGF) Molecular Model This chapter analyzes the structure of much larger molecules such as proteins to model them more effectively using an e nhanced BGF (eBGF) model. The proposed eBGF approach addresses current limitations in protein modeling through a biologicallyinspired geometric filter for speeding self-collision detection queries. The presented eBGF methodology can facilita te the modeling of flexible macromolecules for applications such as molecular docking, na noscale assembly, and protein folding. 5.1 Differences Between eBGF and BGF Models The proposed enhanced BioGeoFilter (e BGF) algorithm is similar to the BGF approach presented in Chapter 4 in that they both build a hierarchical data structure that consists of two layers: a lower level and an upper level. Both algorithms geometrically interpret the inter-atomic interactions to im pose the geometric constraints that define a feasible molecular conformation. However, gi ven that a protein molecule can consist of hundreds or thousands of atoms with hundreds or even thousands degrees of freedom, the modeling of proteins requires: 1. a fu rther AtomGroup subdivisi on of the proteins backbone structure, 2. an independent upda ting of the BVH and collision detection functions, and 3. an additional geometric constraint for the collision detection queries. To effectively model protein molecules, the eBGF algorithm incorporates new algorithmic concepts. The core differences between the eBGF and BGF models are: 1. Given the particular structure of pr oteins, a new algorithm to divide the protein backbone into smaller groups is incorporated into the eBGF model to handle protein updating and collision detection more effectively.

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35 2. The eBGF algorithm updates the BVH independently from the collision detection query compared to the comb ined updating and collision detection approach in the BGF model. This resu lted in a significantly faster model that is more suitable for large mol ecules such as proteins. 3. To compensate with the not so tight fitt ing that results from the selection of spheres as bounding volumes, the eBGF al gorithm incorporates an additional geometric constraint for the collision dete ction query. 5.2 Proposed eBGF Overview Figure 5.1 shows the overview of the enhanced BioGeoFilter methodology that consists of two layers: the lower and upper hi erarchical layers as indicated by the white colored boxes. At the lower la yer of the hierarchy, the eB GF algorithm starts with any molecular conformation. The dof of the mol ecular structure are defined to form atom groups following the concept presented in Chapter 4. A further simplification in molecular representation is proposed by split ting the backbone atom cluster into smaller groups of atoms as it is discussed in Secti on 5.3. At the upper layer of the proposed approach, we build a BVH for the initial molecular conformation as it is described in Section 5.4. New random molecular confor mations are obtained by arbitrary changing the values for each degree of freedom. For each candidate molecular conformation, the BVH is updated to incorporate the corresponding changes in the dof as it is presented in Section 5.4.3. A collision detection scheme is then performed to identify the feasibility of each random molecular conformation as it is described in Section 5.4.4. At the end of the eBGF algorithm, the intramolecular en ergy value for each random conformation is calculated for evaluating the proposed approach as it is discussed in Section 5.5. The following sections describe in details each hierarchical layer of the proposed eBGF methodology.

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Lower Level Hierarchy Input Any Protein Conformation Read DOF Histidine Lysine Arginine ... ... A segment of a proteins backbone atoms Split BackBone AtomGroup Upper Level Hierarchy Energy Calculation Build BVH Update BVH & Collision Detection Randomize Protein Create AtomGroups . . . eBGF Figure 5.1: Overview of the proposed eBGF approach. 5.3 eBGF: Lower Layer Hierarchy Torsion changes can occur anywhere with in a proteins topology. However, considering random torsions within a protei ns backbone can break its structure making it extremely difficult to evaluate whether the generated molecular conformation is chemically feasible. Therefore, in this paper, torsions are a ssumed only between the central carbon atom of a proteins backbone (C A) and a side chain at om (CB) or within the side chain atoms as shown in Figure 5.2. Furthermore, given the increasing complexity by a proteins size, torsions at th e end of each side chain are neglected (i.e. the bond between CD and OE1 atoms in Figure 5.2(a) since they do not contribute significantly to the molecular conformation. 36

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Figure 5.2: Graphical repres entation of the degrees of freedom of a protein. Once the torsion bonds of a protein are id entified, the atoms are clustered into AtomGroups based on the approach propos ed by [Zhang and Kavraki 2004] and analytically shown in Chapter 4. However, the application of the AtomGroup concept in a protein molecule results in the generation of two different sized atom clusters: clusters of side chain atoms and a cluster of bac kbone atoms as shown in Figure 5.3(a). The cluster of backbone atoms contains hundreds of atoms whereas the clusters of side chain atoms contain tens of atoms. This large size difference in the atom clusters increases the time needed to determine if an actua l molecular self-collision occurs. Figure 5.3: Graphical repres entation of the AtomGroup concept along with the proposed splitting procedure for a hypot hetical protein segment. 37

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38 To address this challenge, this research work proposes to spli t the backbone atom cluster into smaller AtomGroups based on a threshold defined by the maximum number of atoms allowed within each atom cluster. By splitting the backbone cluster, a flexible AtomGroup (i.e. the green and pink sphere in Figure 5.3(b) is obt ained along with a number of rigid AtomGroups (i.e. the six pink/dashed-li ne spheres shown in Figure 5.3(b)). This splitting procedure furthe r simplifies the molecular representation by reducing the collision queries while eliminati ng the collision searches between the rigid groups. Hence, the collision detection is now performed between similar sized flexible groups of atoms significantly reducing th e computational time for identifying a molecules feasibility. The splitting of the backbone cluster into smaller groups of atoms significantly reduces the computational time for updati ng the atoms positions by eliminating the calculation of the relatio n matrices for the rigid groups of atoms. As shown in Figure 5.3(b), the relation matrix for the big green sphere (initial flexible AtomGroup) is the same as the relation matrix of the green and pink sphere (modified flexible AtomGroup) and as the relation matrices of the pink/dashed-line spheres (rigid AtomGroups). Therefore, instead of calculating relation matr ices for all the seven new groups of atoms, we just calculate a single re lation matrix for the modified flexible group as depicted by the specific protein segment shown in Figure 5.3. The clustering of atoms into both rigid and flexible groups will be used to form th e upper layer of the hierarchy of the proposed eBGF methodology as described in the following section. 5.4 eBGF: Upper Layer Hierarchy At the upper level of the eBGF me thod, a bounding volume hierarchy (BVH) depicted as a balanced binary tree similar to the BGF model is introduced to identify atoms self-collisions. The main difference between the bounding volume hierarchies of the BGF and eBGF models is that the leaves in the BGF model only represent flexible group of atoms whereas the leaves of the eB GF model can represent both flexible and

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rigid atom clusters. This impacts both the updating and collisi on detection time. Furthermore, an additional geometric constraint is proposed to compensate the not so tight fitting that results from the selec tion of spheres as bounding volumes. 5.4.1 Constructing the BVH Once the different AtomGroups (both flexible and rigid) have been defined at the lower layer of the hierarchy, the smallest en closing sphere that contains all the atoms within each AtomGroup is calc ulated as in [Fischer and Gartner 2003]. The spheres (each containing an AtomGroup) are organized in to a binary tree-like data structure that will serve to detect molecular self-collisions subject to both chemical and geometric constraints during conformational search as it will be discussed in Section 5.4.4. Figure 5.4: Schematic representation of the rigid and flexible AtomGroups within a hypothetical protein segment and the accordance BVH. Figure 5.4(a) shows a hypothetical pr otein segment with its corresponding bounding volume hierarchy shown in Figure 5.4(b) At the bottom of the tree are the 14 spheres (called leaf nodes) representing the 14 AtomGroups of the molecule that includes flexible (green colored) and rigid (pink colore d or dashed-lines). For each pair of nodes, an intermediate node is created that encloses the two nodes. This process continues in a bottom-up manner until all the spheres result in to one single root sphere as shown by the 39

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purple colored sphere in Figure 5.4, which is the sphere that encloses the whole protein segment. The BVH is built only once at the beginning of the algorithm allowing a total construction time of where )N(O w/o,1N2 oddNifN2 rOfNodes TotalNumbe srigidGroup oups flexibleGr srigidGroup 1DOFN (5.1) 5.4.2 Randomization As soon as the pre-selected dof have been defined for the specific protein molecule, a uniform generator is used to create random values for the torsion anglesi where When random torsion changes occur, the new atom positions are updated based on the concept presented in Chapter 4 to obtain a new molecular conformation. For each new random molecula r state, the BVH is updated and the new molecular conformation is tested for self-collision. )2,0[i 5.4.3 Updating the Hierarchy Every time the torsion bonds change, a ne w molecular conformation is generated. The new atom positions affect th e location and radius of the spheres in the hierarchy so they need to be updated accordingly. In this work, the spheres in the BVH are updated in a bottom-up manner and one level at a time. Therefore, the tree nodes are updated from the leaf nodes to their parents until the root node of the tree is reached and updated. As shown in Figure 5.4(b), the BVH is fo rmed by both flexible (green colored) and rigid (pink colored) sphere s. It can be observed that the updating of the spheres around the rigid groups (pink colored spheres an d their parents) can be neglected since the atom distances within and between the ri gid groups remain unchanged. This occurs when the atom cluster of the molecules backbone has been defined as the root 40

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AtomGroup or else the base of the molecule s body. Omitting the update of the rigid nodes ( k) results in a reduction of the comput ational time for updating the BVH and for identifying molecular feasibility. This contributes to a total updating time of )( k N O that never exceeds )( NO 5.4.4 Self-Collision Detection The fundamental concept underneath the proposed collision detection algorithm is the geometric interpretation of the chemical information provided by the van der Waals (VDW) interaction as discussed in Chapter 3 an d in Section 4.3.3. The main difference lies in that the collision search presented in the eBGF model is handled independently from the BVH update procedure. In other wo rds, the BVH is updated first and then the tree is traversed down (in a top-bottom mode) to check fo r possible overlapping atoms. The geometric constraints used in this work to handle the collision detection queries are depicted by Eqn. 5.2. As shown by th e first relation in Eqn. 5.2, an additional constraint is considered to search for overlapping spheres. 1 ,<0 ) + (< ) + (<21 2 1 atomRadius atomRadius d us sphereRadi us sphereRadi dj i atoms j i spheresij ij (5.2) Where, denotes the distance between the sphere objects i and j ; represents the distance be tween the non-bonded atoms i and j ; and are constants that control the proposed algorith ms selectivity mechanism. jisphered, jiatomsd,12The first constraint in Eqn. 5.2 (the additional constraint for the eBGF model) embodies a primary filtering while checking fo r possible collisions be tween two spherical objects. If this constraint is satisfied, then a possible collision occurs between the sphere objects i and j The second constraint in Eqn. 5.2 ensures that an act ual self-collision occurs by comparing pair-wise atomic dist ances. The physical interpretation of the parameter is that it controls the not so tight ob ject fitting that result from the selection of 141

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spheres as the type of bounding volumes. The parameter controls the algorithms selectivity or the number of feasible solutions generated. By decreasing the value of selectivity parameter, the proposed eBGF algorithm accepts more solutions (molecular conformations) as feasible. From a biological point of view, handles the impact that the VDW equilibrium distance has on the results. 222 Figure 5.5: Graphical repres entation of the proposed collision detection algorithm. Figure 5.5 shows an intermediate and th e worst case scenario of the proposed collision detection scheme fo r a protein segment. During the tree traversal, each nonconstraint pair of nodes is checked for a possi ble collision using the first constraint in Eqn. 5.2, where the actual self-collision detection is performed between non-bonded atom pairs by using the second constraint in Eqn. 5.2. A constrained node pair embodies any of the following properties: 1. For self-collision queries, collision detection between the ro ot of the tree against itself should be omitted. 2. The collision search between rigid At omGroups should be ignored since the atomic distances within and betw een these groups remain unchanged. 3. Collision queries between bonded nei ghboring AtomGroups should also be eliminated since the atomic distances between these two groups do not change significantly. 42

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4. Given that the impact of the VDW in teraction increases as the pair-wise atomic distances decreases, the collis ion detection between any atom pair linked by three or less chemical bonds should be avoided as discussed in Section 3.3. Therefore, non-bonded at oms are the atoms linked by four or more chemical bonds. Under these assumptions, if the roots ch ild nodes are collision free, then the specific molecular conformation is feasible and is accepted. Otherwise, the tree is traversed down to identify whether any atoms ar e actually in collision to reject the current molecular conformation. The comput ation collision detection time has )(log k N O performance that never exceeds where k is a constant that represents the number of constraint nodes (i.e. rigid AtomGroups) that are neglected in the proposed collision detection scheme. Finally, each random molecular conformation is tested with both the proposed eBGF approach and the traditional energy calculation method using Eqn. 3.4. )(log NO Figure 5.6: Two example macromolecules tested in this work. 43

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5.5 Computer Implementation and Results The presented method and algorithms ha ve been implemented on a dual 3 GHz CPU workstation using Visual C++ programming language, Op enGL and CGAL libraries [CGAL]. Two different protein molecules with different number of atoms, residues, and number of degrees of freedom have been tested using the proposed eBGF methodology as shown in Figure 5.6. The molecules were obtained from the Protein Data Bank (PDB) [Berman 2000] with PDB IDs as follows: 1STP and 1DO3 protein molecules. They are displayed using the VMD [Humphrey 1999]. Table 5.1: Performance analysis of the proposed eBGF algorithm for two proteins. Table 5.1 shows a representative list of the performance analysis for the proposed eBGF method applied to the two example macromolecules. The molecules have been tested for feasibility after random tors ion changes have occurred. The same conformations for both molecules have been examined with both the energy ( TimeEnergy FeasibleEnergy and TH columns) and eBGF ( TimeBVHUpdate TimeCollision TimeRand, FeasibleCollision columns) approaches and compared in terms of computational time (in milliseconds) and accuracy (percentage of feasible conformations identified). Furthermore, different case scenarios regarding the number, arrangement and the location of the pre-select ed dof have been tested for assessing their 44

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impact on the proposed eBGF methodology (FreeResidues and DOF columns). Column FreeResidues indicates the allowed number of completely flexible residues and column DOF represents the total number of dof assume d. For example, in 1DO3 protein section at the bottom of Table 5.1: column-pair 35-36 ( FreeResiduesDOF ) indicates that 35 completely flexible residues have been tested for the 1DO3 protein that results in a total of 36 dof; the pair 35-14 indi cates that 14 dof were tested only between backbone and side chain atoms; and the pair 35-22 corresponds to torsions only with in the side chains of the 35 flexible residues. Further discussion of the impact in molecular behavior by the pre-selected number of flexible residues and dof is performed below. In addition, different values for the algorithms selectivity control parameters ( and columns) have been tested and discussed below. 12 Figure 5.7: Comparison of the average co llision time by the proposed eBGF vs. the average energy calculation time for different se ts of pre-selected flexible-residues/dof. Figure 5.7 compares the performance of the eBGF method against the energy calculation approach in terms of computational time needed to identify molecules feasibility for the two protei n (1STP and 1DO3) examples. As shown in Figure 5.7, the eBGF algorithm significantly reduces the comput ational time needed to identify feasible molecular conformations compared to the ener gy approach. In fact, the time reduction is so enormous that two different scales were needed to schematically display the two methods in the same graph. The left scal e for both figures denotes the time in 45

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milliseconds (ms) required by the energy approach to determine the feasibility of a molecular conformation whereas the right scale denotes the computational time (in ms) for the proposed collision detection algorithm to identify molecular feasibility. For both molecules and in all tested sets of pre-sel ected flexible-residues/ dof, the eBGF requires less than 1ms to output if the tested molecular conformation is feasible. This time reduction is noteworthy as multiple flexible mo lecules will need to be modeled in realtime simultaneously for the molecular assembly or molecular docking problems. 0 0.5 1 1.5 2 2.5 3 3.5 1STP(78) 1DO3(153)log(Time in ms) Energy Method eBGF Method Figure 5.8: Average total time comparison between the proposed eBGF algorithm and the energy calculation approach to output feasib ility for 1STP and 1DO3 proteins in a logarithmic scale. 46 Similarly, Figure 5.8 compares the com putational time performance for the two methods (eBGF vs. energy calculation) while co nsidering the scenario that both protein molecules are completely flexible (the tota l number of residues forming each protein structure assumed to be completely flexible). Analytically, Figure 5.8 displays the total computational time (in ms) for the eBGF me thod (the collision detection time + BVH update time + update atoms position time) agai nst the energy approach in a logarithmic scale. As it is shown in Figure 5.8, th e proposed eBGF metho dology is significantly faster than the energy calculation approach in identifying feasible molecular conformations. In addition, the eBGF algorithm scales very well as the protein size and problems complexity increases.

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0 10 20 30 40 50 60 70 1DO3 (1)1DO3 (5)1DO3 (10)1DO3 (35)1DO3 (153) Energy Method Enhanced BGF Method 0 10 20 30 40 50 60 70 80 90 100 1STP(1)1STP(5)1STP(10)1STP(16)1STP(78) Energy Method Enhanced BGF Method Figure 5.9: Schematic demonstration of the accuracy of the proposed eBGF methodology. Figure 5.9 measures the accuracy (percentage of feasible conformations identified) of our proposed method under different considerations regarding the allowed number of flexible residues within each protein structure. For both protein examples, the two methods demonstrate similar accuracy. In fact, the selectivity of the eBGF algorithm can be adjusted by varying the control parameters ( and ). In other words, by decreasing the values, the proposed algorithm can accept more molecular conformations as feasible lead ing to a relaxed filtering. The physical interpretation of selectivity parameter is that it handles the not so tight object fitting resulted by the selection of spheres as the t ype of bounding volumes whereas the selectivity parameter controls the impact that the VDW equilibrium distance has on the results as it has been analyzed in Section 5.3.4. The ma in objective of the eBGF approach is to identify infeasible molecular conformations ra pidly while not rejecti ng any feasible ones. Hence, the selection of appropriate values for each molecule depends on the molecules size along with the desired level of selectivity by the user. It is also worth mentioning here that the molecules feasibil ity is traditionally measured using the energy calculation shown in Eqn. 3.4. A conformati on might be considered as feasible or not depending on the molecular internal energy value. If a candida te conformation has negative internal energy, then it corresponds to a stable molecular state. However, feasible molecular conformations exist while having positive intramolecular energy. 121247

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Therefore, a threshold ( TH column in Table 5.1) has been selected based on the proteins size to define the maximum energy value for which a molecular conformation is considered to be feasible. Moreover, as it is shown in Table 5.1 and Figure 5.9, ther e is a significant dependency among the pre-selected number of flexible residues a nd dof considered in each protein molecule and the output (percentage of feasible mol ecular conformations) derived by both (eBGF and energy) methods Computer implementation and results demonstrate that as the number of dof consid ered increases, the output set of feasible solutions obtained by the energy approach decreases; whereas the output set by the eBGF algorithm can be adjusted as it has been disc ussed previously. In addition, when many dof are assumed between backbone atom and side chain atoms, the output set of feasible solutions by the energy calculation approach de creases. Therefore, an additional direct search method is essential to identify ar bitrarily low energy molecular conformations after they have been filtered by the proposed eBGF methodology. Table 5.2: Performance analys is of current approaches. Methods Build BVH Update BVH Collision Detection ChainTree [Lotan et.al. 2002] )(NO )(NO )(3/4NO SpatialAdaptiveHierarchy [Angulo et.al. 2005] --)log(NNO )(NO DeformingNecklaces [Aqarwal et.al. 2004] )log(NNO )log(NNO )(3/4NO BGF model [Brintaki & Lai-Yuen 2008] --)( NO )(log NO eBGF model [Brintaki & Lai-Yuen 2009] )( NO )( NO )(log NO 48

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Table 5.2 demonstrates the worst case scenarios in terms of computational complexity for eBGF and current methods in the literature. The proposed eBGF methodology requires performance for building and updating the BVH and never exceeds when searching for overlapping atoms. Hence, the eBGF algorithm succeeds to keep the BVH complexity in the lower level ( ) while significantly reducing collision detection complexity from toO. )( NO)(log NO)( NO )(log N) (NO 5.6 Conclusions This chapter presents the enhan ced BioGeoFilter (eBGF) methodology for modeling the behavior of macromolecules such as proteins. The pr oposed approach is presented as a rapid filtering tool for the identification of molecules feasibility. The eBGF algorithm has been tested against the traditional energy calculation approach in terms of computational time and accur acy under different cases. Computer implementation and results demonstrate that the proposed eBGF algorithm significantly decreases the computational time for identifying feasible molecular conformations without sacrificing accuracy. Therefore, the eBGF me thod facilitates the modeling of flexible macromolecules that can be used in molecular modeling, protein folding, and nanoscale design. 49

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50 Chapter 6 Generic Enhanced BioGeoFilter (g.eBGF) Model This chapter presents the generic enhan ced BioGeoFilter (g.eBGF) algorithm for simplifying the representation of molecules of different type, size, shape and topology by considering certain chemical factors that influence molecules flexibility. The incorporation of chemically-based constrai nts can provide more realistic molecular conformations that will si gnificantly improve molecular modeling. The proposed methodology can be used to enable the intera ctive modeling of mol ecules for molecular docking or assembly, and for prot ein folding applications. 6.1 Differences Between eBGF and g.eBGF Models Both geometric methods rely upon the same basic algorithmic concepts for modeling molecules conformation mechanism during conformational search. The main differences between the two methods are that the g.eBGF approach: 1. incorporates certain chemically-based factors that a. control molecules flexibility for providing more realistic and chemicallyfeasible molecular conformations b. further simplifies the molecular re presentation since they reduce the allowed number of degrees of freedom for the molecule. 2. is a generic model applicab le to molecules of different type, size, shape and topology. The g.eBGF methodology can effectively model molecular structures such as ligands and prot eins with one or multiple chains.

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6.2 Ligand Modeling A ligand or drug-like molecule is a small molecular structure that usually consists of at most 50 atoms. Ligand molecules may contain rings of atoms as graphically shown with [Humphrey 1999] molecular graphics software in Figure 6.1. These rings are considered rigid during modeling as the loca tion of the ring atoms does not change with respect to each other. Therefore, torsion can be assumed everywhere within a ligands topology except within the rings and within doubleand triple-bonded atoms, which correspond to stronger (not eas ily breakable) chemical bonds. The proposed generic eBGF approach considers the above information for defining the chemically-feasible dof within a ligand molecule. Figure 6.1: Examples of ligand molecules. 6.3 Protein Modeling A protein molecule may contain one chain (backbone) as shown by using the molecular graphics software [Humphrey 1999] in Figure 6.2(a) or multiple chains as shown in Figure 6.2(b). Multiple chains are usually not connected by chemical bonds but by electrostatic forces that keep the chains close to each other. In contrast to ligand 51

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modeling, macromolecules such as proteins consist of hundreds or thousands of atoms with hundreds or even thousands dof. A prot ein molecule can also be considered as a highly articulated body where an arbitrarily-sele cted atom or atom-group acts as the base of the body. A protein that contains more than one chain can be viewed as multiple kinematics chains with hundreds or thousands of links and joints. Figure 6.2: VDW representa tion of two different protein molecule examples. N CA C O CB CG CD OE1 OE2 GLU residue/ amino-acid Histidine Lysine Arginine ... ... A segment of a proteins backbone atoms Figure 6.3: Graphical repres entation of the degrees of freedom of a protein. Torsion changes can occur everywhere with in a proteins topology, except within rings and doubleand triple-bonded atoms. However, random torsions within a proteins backbone (chain) can break its structure maki ng it extremely difficult to evaluate whether the generated molecular confor mation is chemically feasible. For this reason, similar with Section 5.3 for the generic eBGF model, torsions are assumed only between the 52

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53 central carbon atom of a proteins backbone (CA) and a side chain atom (CB) or within the side chain atoms as shown in Figure 6.3. Furthermore, torsions at the end of each side chain are neglected (i.e. the bond between CD and OE1 atoms in Figure 6.3(a) since they do not contribute significantly to the molecular conformation. Some regions within a protein structure attain higher flexibility. These higher flexible regions are the remote proteins porti ons or the amino acids (residues) located at the end of the chain. The residues within the turns of a proteins chain should also be considered as highly flexible molecular bodi es since those have a tendency to move more. These highly flexible regions are considered in th e proposed g.eBGF model for determining the chemically-feasible dof th at controls the molecular flexibility. 6.4 Proposed g.eBGF Methodology 6.4.1 Overview of the Proposed g.eBGF Model Figure 6.4 shows the overview of the pr oposed generic enhanced BioGeoFilter (g.eBGF) methodology that aims to effectively identify feasible conformations for molecular structures of diff erent type, size, shape or topology while considering the underlying chemical information. The g.eB GF approach similarly with the BGF and eBGF models consists of two layers as indicated by the two larger boxes in Figure 6.4. At the lower layer of the hierarchy, any mo lecular conformation can be input into the g.eBGF algorithm. The pre-selected dof for a molecular structure are defined based on the concepts presented in S ections 6.2 and 6.3 to form the atom groups. A further simplification in macromolecular representa tion is proposed by splitting the backbone atom cluster (or clusters) into smaller groups of atoms. At the uppe r layer of the g.eBGF algorithm, the corresponding bounding volume hier archy (BVH) of the initial molecular conformation is built. New random molecula r conformations are obtained by arbitrarily changing the values for each degree of freedom. For each candidate molecular conformation, the BVH is updated and a collision detection scheme is performed to

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identify the feasibility of the molecular conformation. At the end of the g.eBGF algorithm, the intramolecular energy value for each random conformation is calculated to evaluate the proposed approach. Figure 6.4: Overview of th e proposed g.eBGF methodology. 6.4.2 Chemically-Artificial Bonds for the g.eBGF Method When a protein molecule contains more than one chain as shown with the VMD molecular graphics software [Humphrey 1999] in (Figure 6.5(a)), an artificial rigid bond is introduced between the closest residue-pair of the chains by the proposed generic eBGF approach. This artificial bond is used to simulate the electrostat ic forces that keep the chains in contact and shoul d be created in the least flex ible region of the protein to avoid the risk of breaking its structure. For example, the least flexible region in the 1NS1 protein shown previously in Fi gure 6.5(a) is the area betwee n the first helixes of the two chains as shown in Figure 6.5(b). Howeve r, an artificial bond should not be placed 54

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arbitrarily between any residue-pair within the 1st helices but between the closest possible residue-pair as shown by the circ le in Figure 6.6. Moreover, the selected residue pair should have the same polarity. In other words, the closest residues that are both either hydrophobic, polar or ionized w ould be good candidates for pl acing an artificial rigid bond. Figure 6.5: Structure of the protein with PDB ID: 1NS1. Figure 6.6: Closest residue-pair between the first helices of the two 1NS1 proteins chains. 55

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56 6.4.3 Description of the g.eBGF Algorithm The proposed g.eBGF methodology for identifying feasible molecular conformations requires two input files: the atomic coordina te information and the atoms within the molecular topology that share a tors ion bond. The first input file is usually the PDB file obtained from the Protein Data Bank (PDB) [Berman 2000] and the second input file is the file that describes the preselected dof considered for each experiment. The VMD software [Humphrey 1999] is used to define atoms connectivity information for proteins and to construct the first input file (coordinate fi le). If the protein contains multiple chains, an artificial bond is added as described in Section 6.4.2. Finally, to create the file that contains the pre-select ed dof, the concepts about the allowed number and location of the pre-selected dof (torsion angles) presented in Sections 6.2 and 6.3 (i.e. torsions are not allowed with in the rings or double-bonded atom s, etc.) are incorporated for studying chemically-feasible ra ndom molecular conformations. Once the two required input files have been defined, groups of atoms are formed following the concept presented in Section 4.2 to create the lower layer of the proposed g.eBGF approach. Based on the location of the torsion bonds, atoms are clustered into AtomGroups where all the atoms within an AtomGroup are connected by rigid bonds while AtomGroups are connected by torsion bonds. Figure 6.7(a), schematically represents the Atomgroups for a hypotheti cal protein segment consisted by two symmetric chains. As it is shown in Figure 6. 7(a), since the two chains are symmetric, the defined Atomgroups for both chains are the same in terms of both number and atom clustering. If the tested molecular stru cture is a protein molecule, then an additional step within the g.eBGF algorithm is performed for splitting the backbone atom cluster (or clusters in the case of multiple chain proteins) into smaller AtomGroups. The purpose of this additional step is to reduce the time needed to identify an actual molecular selfcollision while decreasing the computati onal time for updating the atoms positions as discussed in Section 5.3. The size of thes e AtomGroups is based on a threshold defined by the maximum number of atoms allowed within each atom cluster. By splitting a

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backbone cluster, a flexible AtomGroup (i.e. the green-pink sphere in Figure 6.7(b)) is obtained along with a number of rigid AtomGroups (i.e. th e six pink spheres shown in Figure 6.7(b)) for each chain within a protei n molecule. The splitting of the backbone group eliminates collision dete ction within and between the rigid groups since the atomic distances remain unchanged between and within these groups. Moreover, after splitting the backbone cluster, collision search is performed between si milar sized flexible groups of atoms. Figure 6.7: Graphical repres entation of the AtomGroup concept along with the proposed splitting procedure for a hypothetical protein segment with two chains. At the upper hierarchical layer of the pr oposed g.eBGF method similarly with the eBGF model, a BVH depicted as a balanced binary tree is introduced to identify overlapping atoms. In respect the BVH the di fference between the tw o models lays on the geometric representation of the preselected dof. Based on the loca tion of the torsion bond angles (dof) within the mol ecule, groups of atoms are generated for both models. To construct the hierarchy a sphere is attached around each atom cluster. The spheres around each generated AtomGroup depict the leaves of the BVH for each method. As opposed to eBGF method the g.eBGF model defines the chemically-allowed torsion movements for each molecular structure. These chemically -oriented factors influence both the number and location of the generated AtomGroups a nd hence, the size and location of the 57

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generated spheres for the BVH. Therefore, even if the same protein molecule is tested with both models (eBGF and g.eBGF) the resulted spheres for each method will have different size and location in respect to the world coordinate frame. In view that the chemically-allowed torsion movements (dof) pr ovide a more realistic representation for molecules deformation mechanism, the g.eBGF compared with the eBGF model provides a more realistic geometric representation of molecules flexibility. The BVH is built only once at the beginning of the algorithm allowing a total construction time of where: )( NOtestedare proteins whenonlyexist:srigidGroup w/o1N2 oddNifN2 rOfNodes TotalNumbe srigidGroup oups flexibleGr srigidGroup 1DOFN (6.1) Likewise with the eBGF model presente d in Section 5.4.3, the BVH for the g.eBGF method is updated for each new molecula r conformation as the torsion angles are randomly modified contributi ng to a total updating time of )( k N O that never exceeds Where, k is the number of rigid node s that remain unchanged. )( NOAs soon as the BVH is updated, a collision detection algorithm is performed to search for overlaps between non-bonded atoms within the new molecular conformation. The collision detection queries follow the same concepts of the eBGF model as analyzed in Section 5.4.4, attaining a total )(log k N O performance that never exceeds Where, k is a constant that represents the number of constraint nodes (i.e. rigid AtomGroups). )(log NO 6.5 Computer Implementation and Results The presented method and algorithms have been implemented on a dual 3.0 GHz CPU workstation using Visual C++ programming language, Op enGL and CGAL libraries [CGAL]. Different molecules with different number of atoms, chains, residues and dof 58

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have been tested with the proposed g.eBGF algorithm. The example molecules were obtained from the Protein Data Bank (PBD ) [Berman 2000] with PDB IDs as follows: 1STP, 1A5Z, 1HVR, 1HTB, and 1JBO ligand molecules, along with 1STP, 1DO3, and 1NS1 protein molecules. Table 6.1: Performance analysis of the proposed g.eBGF methodology. MoleculeNumberNumber Time Time TimeTime % FeasibleTH % Feasible FlexFlexFlex TH PDB NameChainsAtoms Energy BVHupdateCollisionRand by EnergyEnergy by Collision HelixesTurnsResidues Split 1STP 160.1830.10.00160.0082100 100 3 1JBO 431.370.230.230.02417 20 13 1HTB 441.380.210.0140.02552 52 10 1A5Z 441.720.230.0160.02456 55 11 1HVR 461.870.170.0040.01100 100 4 1STP 160.180.10.00270.0088100 100 3 1JBO 431.580.240.0180.0249 10 13 1HTB 441.620.220.020.02240 43 0.70.8 10 1A5Z 441.740.230.0020.0250 41 11 1HVR 461.420.170.0040.018100 100 4 233.161.50.210.3285 90 10 220.960.740.730.380 90 15 222.450.640.880.3190 90 20 224.151.380.120.3380 90 10 218.90.670.190.3170 70 15 219.560.570.20.2870 70 20 214.351.330.280.314 5 10 2300.890.710.312 6 15 225.020.760.950.313 4 20 1456.044.470.30.830 25 10 1515.642.761.090.7927 30 15 1461.592.232.610.7415 15 20 1479.414.30.070.836 3 10 15132.910.770.818 10 15 1491.052.422.770.84 2 20 1398.633.690.060.750 0 10 1511.753.340.0650.850 0 15 1469.472.760.130.780 0 20 x 100 x 4218 5 267 41818 3 3058 2nd,3rd 3498 x 4 0.6 x 1 1STP1903 100000 0.5 1NS122342 Rho1Rho2 DOF 0.60.7 x Table 6.1 shows a representative list of the performance analysis for the proposed g.eBGF methodology on different example mole cules. For each molecule, sets of random torsion angles were randomly generated. The same random molecular conformation was tested with both the g.eBGF (TimeBVHUpdate, TimeCollision, TimeRand, %FeasibleByCollision columns) and the energy (TimeEnergy, %FeasibleByEnergy, and TH Energy columns) approaches a nd evaluated in terms of computational time (in milliseconds) and accuracy (percentage of feasible conformations identified). A molecular conformation can be considered as feasible based on the internal molecular energy value. Therefore, a threshold (TH column in Table 6.1) has been 59

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selected based on the molecules size to define the maximum energy value for which a molecular conformation is considered to be feasible with the energy approach. Different scenarios regard ing the number, arrangement and the location of the pre-selected dof were studied for asse ssing their impact on the proposed g.eBGF algorithm. Columns FlexHelixes, FlexTurns and FlexResidues denote the number of flexible helixes, residues and turns respectively, where column dof states the total number of chemically and geometrically feasible dof considered for each experiment. In addition, different values for the algorithms selectivity parameters ( and ) have been tested for evaluating their impact on the g.eB GF results. Table 6.1 shows the results for the ligand molecules using two different pair -values for the selectivity parameters (rho values) and for the protein molecules using one selectivity parameter pair. A detailed analysis on the impact of the rho values on protein modeling can be found in our Chapter 5. Finally, three different values for the splitting threshold have been tested for each protein molecule as shown by the last column (TH Split) in Table 6.1. 12 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 1STP(3)1HVR(4)1HTB(10)1A5Z(11)1JBO(13) li g a n d s Time in millisecondsEnergy vs. Generic eBGF Approach Time_eBGF Time_Energy Figure 6.8: Time comparison between the trad itional energy calculation approach and the proposed g.eBGF methodology for ligand molecules. 60

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 200 400 600 800 1000 1200 1400 1600 1STP(5)1STP(7)1STP(18)1NS1(42)1NS1(58)1NS1(98) proteinsTime in millisec ondsEnergy vs. Generic eBGF Approach Time_Energy Time_eBGF eBGF_axis Energy_axis Figure 6.9: Time comparison between the trad itional energy calculation approach and the proposed g.eBGF methodology for protein molecules. Figure 6.8 and Figure 6.9 compares the performance of the proposed g.eBGF method (update atoms position time + update BVH time + collision detection time) against the energy calculation approach in terms of computational time required to identify the molecules feasibility. Figur e 6.8 and Figure 6.9 s how the computational time performance for both methods as the num ber of dof increases for ligand and protein molecules, respectively. Results show that the proposed approach significantly reduces the computational time compared to the energy approach for identifying the feasibility of a random molecular conformation. It was observed that as the molecular size and problems complexity increases, the time benefit provided by the proposed g.eBGF method increases significantly. Moreover, as shown in Figure 6.9, the time difference between the two methods is so significant th at two different scal es were needed for displaying both methods in the same graph. The left scale in Figure 6.9 denotes the computational time (in ms) required by the en ergy calculation approach whereas the right scale denotes the computational time (i n ms) for the proposed g.eBGF methodology. Figure 6.10 displays the computational ti me performance of the proposed g.eBGF methodology while considering the scenario that all the example molecules are 61

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completely flexible. The tested molecules are assumed to be fully flexible bodies by randomly varying the total allowed number of chemically-feasible dof as presented in Sections 6.2 and 6.3. Results show that the proposed g.eBGF approach scales very well as the molecular size and problems complexity increase. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 1STP(3)1HVR(4)1HTB(10)1A5Z(11)1JBO(13)1STP(18)1NS1(98) ligands proteinsTime in millisecondseBGF Performance Over Molecules of Different Size Figure 6.10: Computational time performan ce of the proposed g.eBGF approach for molecules of different size and dof. The impact of the splitting threshold sele ction value on the g.eBGF algorithm is shown in Figure 6.11. As discussed in Section 6.4.3, the splitting threshold (TH Split column in Table 6.1) defines the maximum allowed number of atoms in each atom cluster. As the splitting threshold value d ecreases, it can be observed from Figure 6.11 that the following occur: the computational time required to update the BVH increases, the computational time for self-col lision detection decreases, and the overall computational time for the g.eBGF algorithm remains approximately the same. 62

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Figure 6.11: Splitting threshold impact on th e g.eBGF results for protein modeling. Figure 6.12: Accuracy comparison between th e traditional energy ca lculation approach and the proposed g.eBGF method. Therefore, incorporating the splitting con cept for the backbone atom clusters into the proposed algorithm speeds the identification of molecular feasibility while it does not affect the overall performance of the algor ithm. The best selection for a splitting threshold depends on the pre-selected set of dof. A good splitting threshold is a value that allows the construction of similar sized groups of atom s while reducing the 63

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computational time for self-collision. In th is work, it was found that a splitting threshold value of 15 atoms for each atom cluster provided similar-sized atom groups and the best results for self-collision detection time. Figure 6.12 demonstrates the accuracy (percentage of feasible molecular conformations identified) between the traditional energy method and the proposed g.eBGF methodology. For the ligand molecules, two different pairs of the selectivity parameters are presented to show their impact on the results, whereas one selectivity pair is shown for the protein molecules. As s hown in Figure 6.12, both methods (g.eBGF and Energy) demonstrate similar accura cy for all tested molecules. In fact, the selectivity of the g.eBGF algorithm can be adjusted by varying the control parameters ( and ). In other words, by decreasing the values, the proposed algorithm can accept more molecular conformations as feasible leading to a relaxed filtering. It is important to select appropriate values based on the molecules size and the desired level of selectivity by the user to avoid rejecting molecular conformations that are feasible. 12 As shown in Table 6.1 and Figure 6.12, there is a significant dependency between the pre-selected number of chemically-feasible dof considered in each molecule and the percentage of feasible molecular conformations from both the g.eBGF and energy methods. Results demonstrate that as the number of dof increases the output set of feasible solutions obtained by the energy approach decreases; whereas the output set by the g.eBGF algorithm can be adjusted as it ha s been discussed previ ously. In addition, when a macromolecule is assumed to be a fu lly flexible body, the output set of feasible solutions by the energy calculation approach decreases and tends to approach zero. Therefore, an additional direct search method is necessary to identify low-energy molecular conformations af ter they have been filte red by the proposed g.eBGF methodology. Table 6.2 demonstrates the worst case scenarios in terms of computational complexity for g.eBGF and current methods in the literature. The proposed g.eBGF methodology requires performance for building and updating the BVH and never exceeds when searching for overlapping atoms. Hence, the g.eBGF algorithm )( NO)(log NO64

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succeeds to keep the BVH complexity in the lower level ( ) while significantly reducing collision detection complexity from toO. )( NO)(log N)( NO Table 6.2: Computational complexity comparison. Methods Build BVH Update BVH Collision Detection ChainTree [Lotan et.al. 2002] ) O 65 )N( O N( )N(O3/4 SpatialAdaptiveHierarchy [Angulo et.al. 2005] --)N logN(O )N(O DeformingNecklaces [Aqarwal et.al. 2004] )Nlog O N( )N O logN( )N(O3/4 BGF model [Brintaki & Lai-Yuen 2008] --) N(O )N(logO eBGF model [Brintaki & Lai-Yuen 2009] )N( O ) N(O )N(logO Generic eBGF model [submitted to CAD journal 2009] )N( O ) )N(logO N(O 6.6 Conclusions This chapter presented a new generic mo lecular modeling tool called enhanced BioGeoFilter (g.eBGF) for effectively identifying chemically-feasible conformations for molecules of different type, size and topology. The proposed g.eBGF methodology incorporates chemical fact ors that control molecules conformation into a bounding volume hierarchy to rapidly identify chemica lly-feasible molecular conformations. The g.eBGF approach is presented as a filt ering tool to rapidly identify molecular

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66 conformations for speeding mo lecular conformational search and collision detection queries. Computer implementation and result s demonstrate that the g.eBGF methodology significantly decreases the computational time for identifying feasible molecular conformations while maintaining accuracy. Th erefore, the g.eBGF method can be used to facilitate the modeling of flexible mol ecular structures for a pplications such as molecular docking and assembly, and protein folding.

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67 Chapter 7 Identifying the Molecular Stability The scope of this chapter is to investig ate the performance of evolutionary-based optimization methods on effectively searchi ng for low-energy molecular conformations. Two novel differential evolutionbased me thods are presented. The first proposed method is a kinematics-based DE algorith m called kDE model that kinematically represents and simplifies the molecular repres entation to direct the conformational search towards stable solutions. The second approach, called Biologi cal Differential Evolution (BioDE) is based on our previously develope d differential evolution algorithm and our developed biologically-inspired geometric representation of molecules conformation mechanism. The BioDE algorithm utilizes the g.eBGF model as a surrogate approximation model to reduce the number of exact evaluations and to reduce the algorithms convergence rate. Both proposed methodologies will be extremely useful in speeding the search for low-energy molecular conformations while enabling the modeling of flexible molecules for molecular design. 7.1 Fundamentals of Evolutionary Algorithms (EAs) Evolutionary Algorithms (EA) simulates the natural selection process using a number (population) of indivi duals (candidate solutions to the problem) to evolve through certain procedures. Similar to natu re, each individual is represented as a chromosome a string of numbers (bit strings, intege rs or floating point numbers) which contain the design variables for the optimizatio n problem. Each individuals quality is represented by a fitness function tailored to the problem under consideration.

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Classic Genetic Algorithms (GAs) use bina ry coding for the representation of the genotype. However, floating point coding move s EAs closer to the problem space. This allows the operators to be more problem specific while providi ng a better physical representation of the space constraints. In general, EA starts by generating, randomly, the initial chromosome population with their genes (the design variables in the case of floating point coding) taking values inside the desired constrained space of each design variable. The lower and higher constraints of each gene may be chosen in a way that specific undesirable solutions may be avoided. Although the shortening of the search space reduces the computation time, it may also lead to sub-optimal solutions due to the lower variability between the potential solutions. Figure 7.1: One-point crossove r (recombination) operator. After the evaluation of each individuals fi tness function, operators are applied to the population, simulating the natural processes. Applied operators include various forms of recombination, mutation and selection, wh ich are used to provide the next generation of chromosomes. The first classic operator applied to the selected chromosomes is the one-point crossover scheme. In this operato r, two randomly selected chromosomes are divided in the same (random) position while the first part of the first one is connected to the second part of the second one and vice-vers a as shown in Figure 7.1. The crossover operator is used to provide information excha nge between different potential solutions to the problem. Figure 7.2: Uniform mutation operator. 68

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69 The second classic operator applied to th e selected chromosomes is the uniform mutation scheme. This asexual operator alters a randomly selected gene of a chromosome as shown in Figure 7.2. The new gene takes its random value from the constrained space that is determined at th e beginning of the process. The mutation operator is used to introduce some extra variability into the population. The resulting intermediate population is evaluated a nd a fitness function is assigned to each member of the population. Using a selection procedure (different for each type of EA), the best individuals of the intermediate population (or the best individuals of the intermediate and the previous population) will form the next generation. The process of a new genera tion evaluation and crea tion is successively repeated, resulting in individuals with higher values of fitness function. 7.2 EAs Advantages, Limitations and How to Compensate Evolutionary algorithms are a class of s earch methods with remarkable balance between exploitation of the best solutions and exploration of the search space. They combine elements of directed and stochastic search and consequently, are more robust than directed search methods. The EAs are algorithms parallel by nature and may be easily tailored to the specific application of interest taking into account the special characteristics of the problem under consider ation. In addition, the EAs are easy to implement in problems with a relatively high nu mber of constraints and design variables, as well as with many and contradictory objectives [Michalewicz 1999, Goldberg 1989, Holland 1992]. However, EAs main limitations are the c onvergence uncertainty and trapping into local minima. To compensate for the algorithm s failure, the first step is to adjust the algorithms selective pressure, which is defined as the predominance of exploitation versus exploration. By increasing the se lective pressure, the algorithms convergence rate and the probability of trapping into loca l minima are enhanced. Therefore, a balance between exploration and exploi tation is essential. The cro ssover and mutation operators

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70 are responsible for exploring the solution space (exploration) wh ile leading to an increasing variation of the population. On the other hand, the selection process pushes the search into the region with the best f itness function values (exploitation) aiming to decrease the population variation. Thus, the balance between exploration and exploitation is given by the specific type of selection operator for the problem under consideration. Typically, high selective pressure requires high variation in the population to avoid any local minima traps. Another way to compensate for algorithms potential failure is to define suitable values for the control parameters of the EA. The control parameters for an evolutionary algorithm are th e crossover and mutation proba bilities. These probabilities remain constant during the search process while affecting the conve rgence behavior and algorithms robustness. The values for the crossover and mutation probabilities are strongly dependent on the objective function, the characteristics of the problem, and the population size. Usually, when an algorithm is mostly based on a crossover operator, it requires low selective pressure to avoid trapping into local minima solutions. Additionally, when a high mutation probability is applied, the algorithm entails a high selective pressure to compensate for failure Based on these concep ts, a trial and error testing for the EAs control parameters will tune the algorithms robustness and convergence rate. An alternative approach is the use of a differential evolution (DE) algorithm since it has shown a better convergence performa nce compared with other EAs [Storn and Price 1995, 2005, Nikolos and Brintaki 2005a,b,c, 2007, Thomsen 2003, 2006]. Incorporating the scheme presented by [Hui-Yuan 2003] for determining the donor scheme for the mutation operator accelerates the algorithms convergence rate, without sacrificing accuracy or the algorithms robustnes s. In this scheme, the donor is randomly selected (with uniform distribution) from the region within the hyper triangle formed by the three members of the triplet. With this scheme, the donor comprises the local information of all members of the triplet. This provides a better starting-point for the mutation operation and results in a bette r distribution of th e trial-vectors.

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71 Besides using special operators, a substitute solution for compensating for the algorithms convergence failure and for d ecreasing the computat ional time is the utilization of surrogating m odels and approximations. Surrogate models are auxiliary simulations that are less accurate, but also le ss computationally costly than the expensive (exact model) simulations. Surrogate approxi mations are algebraic summaries obtained from previous runs of the expensive simulation [Torczon 1998, Giannakoglou 2002]. Such approximations are the various type s of Artificial Neural Networks (ANN) [Giannakoglou 2002, Nikolos and Brintaki 2005b 2007]. The basic concept of using an approximation method is to replace the costly exact evaluations with fast inexact approximations while maintaining the algor ithms robustness. The surrogate model predictions replace exact and co stly evaluations only for the less-promising individuals, while the more-promising ones are always exactly evaluated. Our developed biologically-inspired geometric filter (generic eBGF method) presented in Chapter 6 can also set the base for a surrogate approximati on model as it is analyzed in Section 7.4. 7.3 Differential Evolution In this work, a Differential Evolution (DE) algorithm based on the concepts by [Storn and Price 1995, 2005], improved by [H ui-Yuan 2003] and presented in [Nikolos and Brintaki 2005a,b,c, 2007] is used to dir ect the search towards low energy molecular conformations. The DE algorithm is a simple evolutionary algorithm to implement and demonstrates better convergence performan ce compared with other EAs. Differential Evolution embodies a type of evolutionary stra tegy (ES) especially formed to deal with continuous optimization problems often en countered in engineering design. The classic DE algorithm e volves a fixed population si ze consisted by candidate problem solutions (population members or else chromosomes), randomly initialized. After initializing the population, an iterative process starts to direct the search towards better fitted population member s. At each iteration (gener ation), a new population of candidate solutions is produced until a stopping condition is satisfied. At each generation,

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each element (member) of the population can be replaced with a new generated one. The new element is a linear combination betw een a randomly selected population member and a difference between two other randomly se lected members. Below is the analytical description of the algorithms structure. Given an objective function as shown by Eqn. 7.1: R RXFparamn objective :)( (7.1) the optimization goal is to minimize the objec tive function value by optimizing the values of its parameters (design variables) as shown as follows: Rx xxxXj nparam ),,...,,(21 (7.2) where X denotes the vector composed of objective function parameters (design variables). The design variables take valu es between the specific upper and lower bounds: paramnparam U jj L jnjxxx,...,1, (7.3) The DE algorithm implements real-numbe r encoding for the design variables. Often, the only information available is the boundaries of the parameters. Hence, to obtain a starting point for the algorithm, we initialize the pop ulation by randomly assigning values to the design variables with in their boundaries as given by Eqn. 7.4: param pop L j L j U jnjnixxxr,...,1,,...,1,)(x0 ji, (7.4) where r is a uniformly distributed random value within the range [0, 1]. DEs mutation operator is based on a triplet of randomly selected individuals (different from each other). A new parameter vector is generated by adding the weighted difference vector between the two members of the triplet to the third one (the donor). In this way, a perturbed individual is generated. The perturbed individual and the initial population member are then subject to a cr ossover operation for generating the final candidate solution as shown as follow: wo x njkjCrifxxFx xG ji param r G jB G jA G jc G jii i i/ ,...,1 ) (), ()( )( )( )( )1(' (7.5) 72

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]1,0[ ,]1,0[ ],1,0[ ],...,1[ ,],...,1[ ],,...,1[ ,...,1,,...,1 r F C nCnBnA njnir pop i pop i pop i param pop (7.6) Where is called the donor, G is the current generation, and k a random integer within [1, ], chosen once for all members of the population. The random number r is seeded for every gene of each chromosome. F and Cr are DE control parameters, which remain constant during the search proc ess and affect the conv ergence behavior and robustness of the algorithm. Their values also depend on the objective function, the characteristics of the problem, and the population size. )( ,G jcixnparamThe population for the next generation is selected between the current population and the final candidates. If each candidate vect or is better fitted than the corresponding current one, the new vector replaces the v ector with which it was compared. The DE selection scheme for a minimization problem is described as follow: wo X XF XFif X XG i G i obj G i obj G i G i/ )()( ,)( )( )1(' )1(' )1( (7.7) In this research work, the new improved scheme by [Hui-Yuan 2003] for determining the donor for the mutation operation is used to accelerate the convergence rate. In this scheme, the donor is randomly selected (with uniform distribution) from the region within the hyper triangle, formed by the three members of the triplet. With this scheme, the donor comprises the local inform ation of all the members of the triplet, providing a better starting-point for the mutation operation that result in a better distribution of the trial-vect ors. As it is reported in [H ui-Yuan 2003], the modified donor scheme accelerated the DE convergence rate, wi thout sacrificing the solution precision or robustness of the DE algorithm. The random number generation (with uniform probability) is based on the algorithm presen ted in [Hui-Yuan 2003], which computes the remainder of divisions involving integers th at are longer than 32 bits, using 32-bit (including the sign bit) words. The corres ponding algorithm, using an initial seed, produces a new seed and a random number. In each different oper ation inside the DE 73 algorithm that requires a random number ge neration, a different sequence of random

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74 e 7.4 roposed kDE Model In this section, a novel kinematics and evolutionary inspired approach called kinema l .4.1 Overview of the Kinematics-Ba sed Differential Evolution (kDE) Model Figure 7.3 shows the overview of the proposed kinematics-based differential evoluti e es a number numbers is produced, by using a different initial seed for e ach operation and a separat storage of the corresponding produced seeds. By using specific initial seeds for each operation, it is ensured that the differe nt sequences differ by 100,000 numbers. P tics-based differential evolution (kDE) is proposed to model flexible biologica molecules and to rapidly identify low-ener gy molecular conformations. The proposed kDE model consists of two modules: the pre-computation and the DE-loop. The kDE model provides the global minimum region fo r molecular structures of different type, size, shape and topology. This region consists of a number of alternative stable molecular conformations that attain the same low-energy value. 7 on (kDE) model that consists of tw o modules: the pre-computation and the DE loops. During the pre-computation, a molecu le is represented as a highly articulated body that can adopt different conformations. As shown in Figure 7.3, the kDE model starts with any random molecular conformati on where the dof of the molecular structur are defined to form groups of atoms. During the DE-loop, our previously developed DE algorithm is incorporated to direct the search towards low-energy molecular conformations and to provide the global mini mum region. This region includ of alternative stable molecular conformations that attain the same low-energy value.

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Figure 7.3: Overview of the proposed kDE model. 7.4.2 Pre-Computation Module During the pre-computation module, a ge ometric interpretation of the underlying chemical information is performed to repres ent the molecules flexibility mechanism as discussed in Section 3.3. As a result, each molecular structure is represented as a highly articulated body able to deform and adopt different molecular conformations. A further simplification in the molecular representation is performed by applying the atom clustering approach presented in Section 4.2 to form groups of atoms based on the number and location of the torsion bonds. 7.4.3 DE-Loop Module 75 Once the various atom clusters within each molecular structure are formed during the pre-calculation stage, the DE algorithm presented in Section 7.3 is incorporated into the kDE model. The DE algorithm is used to direct the search towards low-energy molecular conformations. As shown in Figure 7. 3, two steps are required in the DE loop:

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formulate the chromosome structure and define the fitness function. Each chromosome structure through the defined genotype repres ents a candidate solution to the problem under consideration, whereas the chromosome genes represent the design variables. Hence, to direct the search towards low-energy molecular conformations, the chromosome for the proposed kDE model should represent a candidate molecular conformation. The simplest possible chromo some structure for describing a molecular conformation is to consider each gene to be a degree of freedom or in our case, a torsion bond angle as shown in Figure 7.4. i total 21 Figure 7.4: Schematic representation of th e chromosome structure used in this work. The fitness function (ff) plays the role of the evaluation criterion for each candidate solution. Choosing a good mathematical representation for the ff is very important and challenging since it directs the search towards the optimal solution or in our case, towards stable (l ow-energy) molecular conformations. A good mathematical representation for the fitness function is the use of the total intra-molecular energy. As discussed in Section 3.2, the internal ener gy of a molecule is a function composed of different energy factors that depict the interactions between bonded and non-bonded atoms. However, the major energy contri butors are the non-b onded van der Waals (VDW) potential and electrostatic forces. Given that the VDW potential dominates the molecular interactions chemically and geometrically at short-range and the electrostatic forces dominate the computational time, the internal molecular energy can be approximated by the VDW interactions measurement as demonstrated in Section 3.3. Therefore, to evaluate the fitness of each candidate chromosome (molecular conformation), we propose the use of the VDW non-bonded atoms potential as shown by Eqn. 7.8: 11 612-n i n j ij ij ij ijr A r B ff (7.8) 76

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77 is Where and are the VDW repulsion and attraction parameters, respectively; and the distance between every ex clusive non-bonded atom pair i and j. ijBijAijr 7.4.4 Computer Implementation and Results The presented method and algorithms have been implemented on a dual 3.0 GHz CPU workstation using Visual C++, Visual Basic programming languages and OpenGL. Different molecules with different number of atoms, chains, residues and dof have been tested with the proposed kDE approach. The example molecules were obtained from the Protein Data Bank (PDB) [Berman 2000] with PDB IDs as follows: BTN, CYC, NAD, XK2 for ligand molecules and 1STP, 1DO3, 1N S1 for protein molecules. Figure 7.5 and Figure 7.6 show some of the tested molecu les that are graphically displayed using the VMD package [Humphrey 1999]. Figure 7.5: Ligand molecules te sted with the kDE model. The kDE algorithms termination criterion for each experiment but the 1NS1 protein was set to maxgen = 500 generations performed and popsize = 100 candidate molecular conformations (population members) considered in each generation. For the 1NS1 protein, given the large number of dof considered in each experimental scenario, the maxgen was set to 600 generations and 300 population members used as popsize. Finally, the DEs control parameters used in all experiments were F = 0.6 for the mutation parameter, and Cr = 0.45 for the crossover probability.

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Figure 7.6: Protein molecules tested with the kDE model. Tables 7 alysis for the roposed kDE approach on different ligand and protein molecules, respectively. As shown ach The analysis of the kDE algorithm on ligands. .1 and 7.2, show a representative list of the performance an p in Tables 7.1 and 7.2, the first co lumns indicate the PDB IDs for the tested molecules. The second column in both tables specifies the number of atoms within e molecular structure and the third column show s the dof considered in each scenario preselected dof for each experiment are the chemically-allowed dof to study chemicallyfeasible molecular conformations. Table 7.1: Performance LigandsNumber AtomsDOFE_crystalE_kDEConv.Gener.T_kDE (kcal/mol)(kcal/mol) (ms) BTN163 3.25 3.5985/5002.57 CYC4313 16.75 17.93209/5002.1 NAD4411 12.42 13.68125/5002.18 XK2464 10.91 11.35117/5002.34 78

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79 Table 7.2: Performance analysis of the kDE algorithm on proteins. ProteinsNumber AtomsDOFE_crys talE_kDEConv.Gener.T_kDE (s) 1STP9037 554.78 576.16118/5000.35 9 876.37189/5002.55 22 881.53191/5002.62 36 879.38211/5002.58 42 1127.49170/6002.3 58 1128.061172/6002.27 98 1126.076138/6002.25 1DO32466 876.33 1NS12342 1126.35 (kcal/mol)(kcal/mol) To evaluate our proposed kDE molecula r model, we compared our obtained results with the crystal stru ctures published in the Prot ein Data Bank [Berman 2000]. Analyt re onding tion generation is the generation where t r, t ically, we calculated the van der Waals (VDW) energy for each crystal structu (E_crystal) and compared it against the obta ined VDW energy value by the kDE algorithm (E_kDE) accordingly. As shown in Tables 7.1 and 7.2, the kDE algorithm succeeded to converge in a smaller VDW energy value compared with the corresp VDW energy of the crystal structure for all the performed experiments. This phenomenon occurs since all the incorporated energy terms (i.e., VDW potential or electrostatic forces) in a molecules internal energy are in fact an approximation of the potential energy and not the molecules free energy, which requires entropy calculations, among others. Therefore, the proposed kDE model managed to output a stable molecular conformation for all the tested molecules. This is very important given that most evolutionary algorithms suffer from local minima traps. The sixth column (Conv.Gener.) in Tables 7.1 and 7.2 indicates the genera when the BioDE algorithm converged. This convergence he fitness function of the worst p opulation member equals with the fitness function of the best one, which is also the same as the obtained E_kDE value. Moreove Figure 7.7 shows the convergence performa nce of the kDE algorithm over differen tested ligand molecules. Similarly, Figure 7.8 demonstrates the convergence performance of the kDE model for 1STP, 1NS1 and 1DO3 protein molecules over a selection of

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experimental scenarios based on the dof considered. As shown in these figures, the kDE algorithm converges really fast. This is ve ry important given that one of the main drawbacks in an evolutionary-based al gorithm is the convergence uncertainty. -4 -3.8 -3.6 -3.4 -3.2 -3 -2.8 -2.6 -2.4 -2.2 -2 0100200300400500Internal Energy (kcal/mol)Number of Generations Performed Convergence Performance, BTN Ligand80 Best ff value Worst ff value -30 -20 -10 0 10 20 30 40 50 0100200300400500Internal Energy (kcal/mol)Number of Generations Performed nd Convergence Performance, CYC Liga Best ff value Worst ff value -30 -20 -10 0 10 20 30 40 50 0100200300400500Internal Energy (kcal/mol)Number of Generations Performed Convergence Performance, NAD Ligand Best ff value Worst ff value -13 -12 -11 -10 -9 -8 -7 -6 -5 0100200300400500Internal Energy (kcal/mol)Number of Generations Performed Convergence Performance, XK2 Ligand Best ff value Worst ff value Figure 7.7: kDEs convergen ce performance for ligands.

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81 -700 -600 -500 -400 -300 -200 -100 0 100 0100200300400500Internal Energy (kcal/mol)Number of Generations Performed Convergence Performance, 1STP, 7 dof Best ff value Worst ff value -1000 -900 -800 -700 -600 -500 -400 -300 -200 -100 0 0100200300400500Internal Energy (kcal/mol)Number of Generations Performed Convergence Performance, 1DO3, 36 dof Best ff value Worst ff value -1300 -1100 -900 -700 -500 -300 -100 0100200300400500Internal Energy (kcal/mol)Number of Generations Performed Convergence Performance, 1NS1, 58 dof Best ff value Worst ff value -1300 -1100 -900 -700 -500 -300 -100 0100200300400500Internal Energy (kcal/mol)Number of Generations Performed Convergence Performance, 1NS1, 98 dof Best ff value Worst ff value Figure 7.8: kDEs convergence performance for proteins. The last column in Tables 7.1 and 7.2 indicate the average computational time (T_kDE) required by the kDE algorithm to evaluate each generation. T_kDE is given in milliseconds for the ligands and in seconds for the proteins. Considering the slowest experimental scenario depicted by the largest tested 1DO3 protein molecule, the required total computational time to output a stable conformation is T_kDE x Converge Generation x popsize = 2.68 s/generation x 200 gene rations x 100 members = 53600 s = 893.333 min = 13.89 hours. Therefore, in less than a day, a stable molecular conformation for a large enough protein mol ecule can be obtained using our proposed methodology.

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Table 7.3: RMSD performance for the kDE algorithm. min_RMSD2.092871.179231.43770.104940.150970.064550.289080.491950.94288 max_RMSD2.408819.947542.174940.359870.233430.244660.311830.692521.25286 average_RMSD2.264756.118041.825150.237220.19420.16980.300670.611111.09315 molecules CYCNADXK21STP dof131147922365898 1DO3kDE1NS1Proteins Ligands At the end, the kDE algorithm outputs the final population of stable solutions or the obtained global minimum region for any tested ligands and protein molecules. This final population contains a large number of different molecular conformations for each tested molecule that attains the same lo w-energy value. However, to evaluate the structural feasibility of each obtained molecu lar solution, we have calculated their Root Mean Square Deviation (RMSD) in Angstroms from their corresponding crystal structure. Generally, lower RMSD values in dicate closer resemblance between observed and predicted structures with RM SD values below or near 2.0 usually considered being sufficiently close. AAs shown in Table 7.3, the kDE algorithm succeeded to identify stable structures with RMSD values in the range of 0.10 to 0.94 for any tested protein and in the range of 1.18 to 2.09 for the tested ligands. This m eans that the results of the kDE method lie within the acceptable structural range of [0, 3) for all tested molecules. Therefore, the kDE approach outputs a large nu mber of alternative stable molecular conformations that can be clustered based on their structures to identify those closest to their crystal structure. 7.4.5 Conclusions This section presented a new kinematicsbased differential evolution (kDE) model for effectively searching for low-energy mo lecular conformations. The proposed model consists of two modules: the pre-computation and the DE-loop. At the pre-computation module, a molecule is represented as a highl y articulated body able to adopt different 82

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molecular conformations. At the DE-loop, a di fferential evolution algorithm is used as a direct search technique towards low-en ergy molecular conformations. Computer implementation and results demonstrate that the proposed kDE approach rapidly and accurately finds low-energy (stable) molecula r conformations for molecular structures of different size, shape and topol ogy. Results also show that the kDE algorithm attains a very good convergence performance while it outputs the global minimum region for any tested molecule. This region provides a number of alternative stable molecular solutions that have the same low internal energy value. As demonstrated the proposed kDE model outputs sufficient molecular conformations with RMSD values below or near 2.0 The predicted molecular confor mations can then be clustered based on their structure similarity to identify those clos est to their crystal structure. A 7.5 Proposed BioDE Approach This section presents a new algorithmic scheme called Biol ogical Differential Evolution (BioDE) to minimize the molecular energy based on the differential evolution algorithm presented in Section 7.3 and the hierarchical data structure presented in Chapter 6. The proposed BioDE utilizes our previously developed data structure called g.eBGF, as a surrogate approximation model to reduce the number of exact evaluations, speed molecular conformational search, and reduce the algorithms convergence rate as discussed in this section. 7.5.1 BioDE Overview 83 The proposed BioDE methodology is a novel, generic evolutionary and geometric-based direct search technique to id entify molecules minimum internal energy. The BioDE algorithm aims to effectively iden tify stable conformations for any molecular structure regardless of type, size and shape while considering the underlying chemical information. The main algorithmic difference between the kDE and BioDE approaches is

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that the latest utilizes our previously develo ped g.eBGF data structure as a primary filter of molecules feasibility to speed convergence. Figure 7.9: Overview of th e proposed BioDE approach. Figure 7.9 illustrates the overview of th e proposed BioDE approach. As shown in Figure 7.9, the BioDE approach employs tw o modules: the pre-computation and the DEloop modules. At the pre-computation stage, a geometric interpretation of the interatomic interactions is performed to set the constraints under which a molecular conformation is considered as feasible as disc ussed in Section 3.3. During this stage, our previously developed g.eBGF data structure is utilized as a primary filter for feasible molecular conformations. As shown in Figure 7.9, a molecular conformation (coordinate file) is input into the BioDE algorithm. The pre-selected dof (TorsionAtoms file) for a molecular structure are defined to form the atom groups. To further simplify the macromolecular representation, the backbone atom cluster(s) is split into smaller groups of atoms based on the concepts discussed in Section 5.3. A bounding volume hierarchy (BVH) denoted as a balanced binary tree is constructed for the initial molecular conformation to capture the molecular shape at successive level of details and to assist in 84

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85 the collision detection queries. The type of bounding volumes used in this work is spheres since spheres are invariant to rotations and simpler to implement. At the DE-loop module, the DE algorithm presented in Section 7.3 is used to direct the search towards low-energy minima or stable molecular conformations. For each candidate molecular conformation (population member) in each generation, the BVH is updated and a collision detection scheme is performed to determine the feasibility of the molecular conformation. The fundamental c oncept underneath the proposed collision detection algorithm is the geometric interpretation of the chemical information provided by the intra-molecular energy. If the collis ion detection queries output a feasible molecular conformation, then the fitness function for that conformation measures the van der Waals energy value; otherwise, a penalty function is computed to reject the specific unfit solution. This is an iterative process to provide better fitted i ndividuals. As soon as the termination criterion denoted by the maximum allowed number of generations performed is satisfied, the BioDE algorith m outputs the final population of stable molecular conformations. This final populati on describes the global minimum region that provides a number of different molecular conf ormations that attain the same low-energy value and hence, provides alternative stable molecular solutions. 7.5.2 Input Files The proposed BioDE methodology consists of two modules: the pre-computation and the DE-loop modules. Our proposed algo rithm requires two input files for the precomputation module: the atomic coordinate information (coordinate file) and the atoms within the molecular topology that share a torsion bond (torsionAtoms file). In addition, the BioDE algorithm requires one input file for the DE-loop module, the chromosome file. The coordinate input file is usually the PDB file obtained from the Protein Data Bank (PDB) [Berman 2000]. The VMD software [Humphrey 1999] is used to define the atoms connectivity information for proteins and to construct the coordinate input file.

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The torsionAtoms input file is the file that describes the atoms that share a torsion bond and hence, depicts the number and the locati on of the pre-selected dof considered for each experiment. To create this file, the c oncepts about the allowed number and location of the pre-selected dof (torsion bond angles) presented in Section 6.2 and Section 6.3 are incorporated for studying chemically -feasible molecular conformations. Finally, the chromosome file denotes the required dof or design variables for representing a candidate molecular conforma tion. Similar to nature, each chromosome structure through the defined genotype emb odies a candidate solution to the problem under consideration whereas the chromosome genes represent the de sign variables that take values within their constrained space. Hence, to direct the search towards lowenergy molecular conformations, the ch romosome for the BioDE algorithm should represent a candidate molecular conformation with the genes accounting for the dof or the torsion bond angles in the [0, 360) range. i 7.5.3 Pre-Computation Module During the pre-computation module, once the two input files (coordinate and torsionAtoms files) have been defined, groups of atoms are formed following the concept presented in Section 4.2 to simplify the molecu lar representation. If the tested molecular structure is a protein molecule, then an a dditional step within the BioDE algorithm is performed for splitting the backbone atom cl uster (or clusters in the case of multiple chain proteins) into smaller AtomGroups as discussed in Section 5.3. As soon as the atom clusters both rigid and flexib le have been defined, a bounding volume hierarchy (BVH) depicted as a balanced binary tree is introduced to capture the shape of the molecule at successive levels of detail and to facilitate the collision de tection search for overlapping atoms. The BVH is built only once at th e beginning of the algorithm during the precomputation stage. 86

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7.5.4 DE-Loop Module At the beginning of the DE-loop module, similarly to the kDE model, the chromosome file is input to the BioDE algorit hm to define a candidate molecular conformation as a function of the torsion bond angles i as shown in Figure 7.4. The DE algorithm evolves a fixed population size (popsize) composed of candidate problem solutions (population members or chromoso mes), randomly initialized. Consequently, each population member corresponds to a candidate molecular conformation. After initializing the population, an iterative proce ss starts to direct the search towards better fitted population members or stable molecular solutions. At each generation (iteration), a new population of candidate solutions (c onformations) is produced until a stopping criterion is satisfied. In this work, the termination criterion is the maximum allowed number of generations performed (maxgen). At each generation, each population member (candidate molecular conformation) can be replaced with a new generated one. The new member is a linear combination between a randomly selected member (the donor) and a difference between two other randomly selected members. Genetic operators (mutation, crossover, and selection) are applied to prov ide the next generation of better fitted candidate problem solutions (molecular conformations). For each candidate molecular confor mation (population member) in each generation, the BVH is updated and a collision detection scheme is performed to determine the feasibility of the molecular conformation. The BVH is updated for each new molecular conformation as the torsion angl es are randomly modified. As soon as the BVH is updated, the collision detection algo rithm presented in Chapter 5 and Chapter 6, is performed to search for potential overlaps between non-bonded atoms within the tested molecular conformation. One of the major components in an evoluti onary-based algorithm is to define an appropriate fitness function ff. The ff plays the role of the evaluation criterion for each candidate problem solution. Choosing a go od mathematical representation for the ff is very important and challenging since it directs the search towards the optimal solution or in our case, towards stable molecular confor mations. A good mathematical representation 87

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for the ff is the use of the total intra-molecular energy. However, as discussed in Section 3.2, this would result in a ve ry slow progress towards the se arch for low-energy solutions. Therefore, we propose an al ternative fitness function definition utilizing our developed g.eBGF data structure as an approximation model of molecules feasibility. In other words, if the collision detection algorithm outputs a feasible molecular conformation, then the fitness function for that confor mation measures the van der Waals (VDW) energy value, else a penalty function is computed to rej ect the specific bad solution as shown by Eqn. 7.9: wo m mpenalty feasible on conformati if r A r B al VDWpotenti ffm k kjki m k kjik n i n j ij ij ij ijrr dd/ )1( -1 ,, 2 2 1 ,,,0 2 11 612)( )( (7.9) Where, and are the equilibrium distance and th e relative inter-atomic distance for the kth colliding non-bonded pair of atoms i and j, and m is the total number of colliding atom pairs within th e tested molecular conformation. As shown in Eqn. 7.9, the proposed penalty function is a distance function betw een the equilibrium and the relative inter-atomic distances for each overlapping atom pair within the molecular topology. The purpose of using this penalty function is to train the algorithm to avoid searching space regions mostly occupied by infeasible solutions. This training is defined as a function of how much infeasible these solutions are or alternatively how many overlapping atom pairs exist within the examin ed molecular conformation. kd,0 kjid,, 7.5.5 Computer Implementation and Results The presented method and algorithms have been implemented on a dual 3.0 GHz CPU workstation using Visual C++, Visual Basic programming languages, OpenGL and CGAL libraries [CGAL]. As shown in Figure 7.10 and Figure 7.11, different molecular structures with different number of atoms, chains, residues and dof are tested with the proposed BioDE approach. The example molecu les were obtained from the Protein Data 88

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Bank (PDB) [Berman 2000] with PDB IDs as follows: BTN, CYC, NAD, XK2 ligand molecules and 1STP, 1DO3, 1NS1 protein mol ecules. These molecules are the same ones used to evaluate the kDE method presented in Section 7.4. Using these molecules as the test-bed for the BioDE algorithm, a perf ormance comparison between BioDE and kDE models is provided in the following Section 7.6. Figure 7.10: Ligand molecules te sted with the BioDE model. Figure 7.11: Protein molecules tested with the BioDE model. The termination criterion for the BioDE algorithm was set to maxgen = 500 generations performed and popsize = 100 candidate molecular conformations (population members) considered in each generation for each experiment except for the 1NS1 protein. For the 1NS1 protein, given the large number of dof considered in each experimental scenario, the maxgen was set to 600 generations and 300 population 89

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members used as popsize. Finally, the DEs control parameters used in all experiments were F = 0.6 for the mutation parameter, and Cr = 0.45 for the crossover probability. Tables 7.4 and 7.5, show a representative list of the performance analysis for the proposed BioDE approach on different ligand and protein molecules, respectively. As shown in Tables 7.4 and 7.5, the first co lumns indicate the PDB IDs for the tested molecules. The second column on each table specifies the number of atoms within each molecular structure, whereas the third co lumn denotes the dof considered in each scenario. The preselected dof for each experime nt are the chemically-allowed dof for the molecules. Table 7.4: Performance analysis of th e proposed BioDE algorithm on ligands. LigandsNumber AtomsDOFE_crystalE_BioDEConv.Gener.T_BioDE (kcal/mol)(kcal/mol) (ms) BTN163 3.25 3.5987/5001.86 CYC4313 16.75 17.93132/5002.07 NAD4411 12.42 13.68162/5002.31 XK2464 10.91 11.35117/5002.36 Table 7.5: Performance analysis of the BioDE algorithm on proteins. ProteinsNumber AtomsDOFE_crystal E_BioDEConv.Gener.T_BioDE (kcal/mol)(kcal/mol) (s) 1STP9037 554.78 576.16187/5000.34 9 876.37180/5002.49 22 881.53188/5002.49 36 879.38175/5002.46 42 1127.49150/6002.28 58 1128.061127/6002.2 98 1126.076150/6002.21 2342 876.33 1126.35 1DO3 1NS1 2466 To evaluate our proposed BioDE molecular model, we compared our results with the results for crystal structures published in the Protein Data Bank [Berman 2000]. Analytically, we calculated the van der Waals (VDW) energy for each crystal structure (E_crystal) and compared it against the VDW en ergy value obtained from the BioDE 90

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algorithm (E_BioDE). As shown in Tables 7.4 and 7.5, the BioDE algorithm succeeded to converge in a smaller VDW energy valu e compared with the corresponding VDW energy for the crystal structure on all th e performed experiments. This phenomenon occurs since when measuring a molecules internal energy all the incorporated energy terms (i.e. VDW or electrostatic forces) are in fact an approximation of the potential energy and not the molecules free ener gy, which among other requires entropy calculations. Therefore, the proposed BioDE algorithm succeeded to output a stable molecular conformation for all the tested mol ecules. This is very important given that most evolutionary algorithms suffer from local minima traps. 91 -4 -3.8 -3.6 -3.4 -3.2 -3 -2.8 -2.6 -2.4 -2.2 -2 0100200300400500Internal Energy (kcal/mol)Number of Generations Performed Convergence Performance, BTN Ligand Best ff value Worst ff value -40 -30 -20 -10 0 10 20 30 40 0100200300400500Internal Energy (kcal/mol)Number of Generations Performed Convergence Performance, CYC Ligand Best ff value Worst ff value -20 -10 0 10 20 30 40 0100200300400500Internal Energy (kcal/mol)Number of Generations Performed Convergence Performance, NAD Ligand Best ff value Worst ff value -13 -12 -11 -10 -9 -8 -7 -6 -5 0100200300400500Internal Energy (kcal/mol)Number of Generations Performed Convergence Performance, XK2 Ligand Best ff value Worst ff value Figure 7.12: Convergence performan ce of the BioDE method for ligands.

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The sixth column (Conv.Gener.) in Tables 7.4 and 7.5 indicates the BioDE algorithms convergence generation. This c onvergence generation is the generation where the fitness function of the worst population member equals with the fitness function of the best individual, which is also the same as the E_BioDE value. Figure 7.12 shows the convergence performance of the BioDE algo rithm over the different tested ligand molecules. Figure 7.13 demonstrates BioD Es convergence performance for 1NS1, 1DO3 and 1STP proteins, over diffe rent number of dof. As shown in these figures, the BioDE algorithm attains a very good c onvergence performance. This is very important given that one of the main drawbacks in an evolu tionary-based algorithm is the convergence uncertainty. 92 -800 -700 -600 -500 -400 -300 -200 -100 0 100 0100200300400500Internal Energy (kcal/mol)Number of Generations Performed Convergence Performance, 1STP, 7 dof Best ff value Worst ff value -1000 -800 -600 -400 -200 0 200 0100200300400500Internal Energy (kcal/mol)Number of Generations Performed Convergence Performance, 1DO3, 36 dof Best ff value Worst ff value -1300 -1100 -900 -700 -500 -300 -100 100 0100200300400500Internal Energy (kcal/mol)Number of Generations Performed Convergence Performance, 1NS1, 58 dof Best ff value Worst ff value -1300 -1100 -900 -700 -500 -300 -100 100 300 0100200300400500Internal Energy (kcal/mol)Number of Generations Performed Convergence Performance, 1NS1, 98 dof Best ff value Worst ff value Figure 7.13: Convergence performance of the BioDE method for proteins.

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The last column in Tables 7.4 and 7.5 indicates the average computational time (T_BioDE) required by the BioDE algorithm to evaluate each generation. The T_BioDE is given in milliseconds for the ligands and in seconds for the proteins. Similarly to the kDE model, at the end of the BioDE algorithm the final population of stable solutions is obtained. This final population contains a large number of different molecular conformations for each tested molecule that attains the same lowenergy value. To evaluate the structural f easibility of each obtained molecular solution, we have calculated their Root Mean Square Deviation (RMSD) in Angstroms from their corresponding crystal structure. Generall y, lower RMSD values indicate closer resemblance between observed and predicted st ructures with RMSD values below or near 2.0 usually considered being sufficiently close. A Table 7.6: RMSD performance of the BioDE algorithm. min_RMSD3.131681.179231.43770.082780.003350.294710.445380.488780.88915 max_RMSD12.616410.02172.173940.502260.358590.512790.772540.69371.2621 average_RMSD8.708626.214251.823670.332430.273510.400330.646550.604181.08408 molecules CYCNADXK21STP dof131147922365898Ligands1DO3 1NS1BioDE Proteins As shown in Table 7.6, the BioDE al gorithm succeeded to identify stable structures with RMSD values in the range of 0.003 to 0.889 for any tested protein and with RMSD values of 1.17 and 1.43 for NAD and XK2 ligands, respectively. However, the result for CYC ligand is relatively highe r. This means that the BioDE method lies within the acceptable structural range of [0 3) for all tested molecules except for the CYC ligand. 7.5.6 Conclusions Section 7.5 presented a novel generic co mputational geometric and evolutionarybased molecular methodology called biologically -inspired differential evolution (BioDE) 93

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94 approach for effectively identifying chemically-feasible and low-energy conformations for molecules of different type, size, shape and topology. The proposed BioDE approach employs a differential evolution algorithm to direct the search towards stable molecular conformations. It incorporates the underlying geometric interpretation of the inter-atomic interactions as a primary filter for feasible molecular conformations to reduce the number of exact evaluations performed and to speed the molecular conformational search. Computer implementation and results demo nstrate that the proposed BioDE algorithm accurately and rapidly identif ies low-energy molecular conformations for different molecular structures. The proposed BioDE a pproach attains a very good convergence performance while it outputs the global minimu m region for the tested molecule. It also provides a set of alternative low-energy molecular conformati ons for researchers to test during molecular design. As demonstrated the BioDE algorithm outputs sufficient molecular conformations for all the tested prot eins and most of the tested ligands. These predicted molecular conformations can then be clustered based on their structures to identify those closest to their crystal structure. 7.6 Comparison Between the kDE and BioDE Approaches The main algorithmic difference between the BioDE and kDE approaches is that the BioDE algorithm utilizes the g.eBGF technique presented in Chapter 6, as a surrogate approximation model to speed convergence and to reject any unfeasible generated solution. This is possible through a penalty function that is assigne d to those individuals (population members or else conformations) that attain overlapping atoms within their generated topology. Therefore, the g.eBGF algorithm is used by the BioDE model as a primary filter of molecules feasibility. If a candidate molecular conformation passes this first filtering step (no overlapping atoms: penalty = 0), then the fitness function (ff) measures its VDW energy to define the individuals feasibility level. Different to this algorithmic scheme, the kDE model is a much simpler methodology that calculates the VDW energy for all candidate molecular conformations to determine their fitness level.

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As demonstrated in Section 7.4.4 an d Section 7.5.5, both kDE and BioDE methods are effectively pushing the molecula r conformational search towards the global minimum region occupied by a large number of alternative stable conformations. Both methods succeeded in identifying molecules stability for any type of molecular structure tested while attaining a very good conver gence performance. However, given the algorithmic difference of the two approaches, the main question lies in which from the proposed direct search algorithms perform better. Table 7.7: Comparison between kDE and BioDE approaches. ProteinsNumber AtomsDOFBioDE_Conv.Gener.kDE_Conv.Gener.T_BioDE (s)T_kDE (s) 1STP9037187/500 118/5000.340.35 9180/500 189/5002.492.55 22188/500 191/5002.492.62 36175/500 211/5002.462.58 42150/600 170/6002.282.3 58127/600 172/6002.22.27 98150/600 138/6002.212.25 1DO32466 1NS12342 To evaluate and validate the kDE and BioDE models, we compared the obtained molecular structures (kDE and BioDE outp uts) against their corresponding crystal structures published in PDB [Berman 2000] Two different performance assessments were performed: an energy-oriented (l owest obtained VDW energy values) and a structural-based (RMSD) performed. To co mpare the BioDE algorithm against the kDE method, we have tested both methods us ing the same molecular structures and workstations. Computer implementation and results dem onstrate that for ligand molecules both methods perform approximately the same. For pr otein molecules, there is no significant computational time improvement of usin g the BioDE approach for identifying the molecular feasibility as shown in Table 7. 7. However, the BioDE provides a convergence enhancement over the kDE method for proteins. As shown in Table 7.7, the BioDE algorithm converged about 25% faster on over 70% of the experiments performed. 95

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96 Therefore, the BioDE algorithm provides convergence enhancement while identifying alternative low-energy molecular conformations. In regards to the identified molecular st ructure, both the kDE and BioDE methods succeeded to converge in conformations close to the crystal structures for any tested flexible protein and for most of the examined flexible ligands. As shown in Tables 7.3 and 7.6, the RMSD values obtained by the BioDE approach are much smaller compared with those obtained by the kDE model for all the tested proteins bu t 1DO3 protein with 36 dof. Regarding the predicted ligand struct ures, it appears that both models had the same performance for all the tested ligands but CYC ligand where the kDE model calculated a smaller RMSD value.

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97 Chapter 8 Conclusions, Discussion and Future Work The scope of this chapter is to provide a summary of the research methodologies presented to study the molecular flexibil ity and stability mechanism. The general conclusions, including encountered challenges and limitations, are also discussed here, followed by a description of future research work. 8.1 Research Summary This research work presented three diffe rent molecular models: the g.eBGF, kDE and BioDE algorithms. Two computational geom etric models (BGF and eBGF) were also implemented to assist in the developmen t of the g.eBGF approach. The g.eBGF algorithm is responsible for rapidly and accura tely identifying the molecular feasibility whereas the kDE and BioDE algorithms direct the conformational search towards stable molecular conformations. All proposed algor ithms rely upon the basic algorithmic concepts for kinematically representing the molecular structure. They also integrate concepts from robotics, evolutionary-oriente d optimization, computational geometry and computational biology. The core algorithmic architecture of th e BGF, eBGF and g.eBGF methods is a two layer hierarchical data structure that kinematically represents the molecular flexibility using a bounding volume hierarchy to assist in the collision detection. The BGF or BioGeoFilter approach effectively identi fies the molecular feasibility for ligands (drug-like) molecules and performs really we ll in identifying the molecular feasibility rapidly and accurately. In addition, the BGF algorithm satisfies the haptic-rate requirement, enabling re al-time ligand design.

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98 The eBGF or enhanced BioGeoFilter al gorithm presented in Chapter 5 is a significant enhancement of the BGF approach as it can model macromolecules such as proteins. The proposed eBGF method effec tively studies the flexibility mechanism of proteins while addressesing current limitations in protein modeling. Therefore, the eBGF algorithm is presented as a rapid and accurate filtering tool of proteins feasibility, significantly facilitating protein modeling and design. The generic eBGF or g.eBGF approach discussed in Chapter 6 is a generic molecular modeling tool able to represent the flexibility mechanism of any molecule independently of type, size, shape and to pology. The proposed g.eBGF model is the generic enhancement of the eBGF algorith m. The g.eBGF model considers some chemically-based constraints to provide more realistic and chemically-feasible molecular conformations compared with those of the eBGF. This is a significant improvement in computational-aided molecular design. The kDE and BioDE models presented in Chapter 7 direct the molecular conformational search towards low-energy (s table) solutions. To achieve this, both models employ our previously developed DE algorithm. The main algorithmic difference is that the BioDE algorithm utilizes the g. eBGF method as a surrogate approximation model to speed convergence. Both approach es effectively identify stable molecular conformations for any molecular structure independently of type, size, shape and topology. Both models also succeed in providing the global minimum region for any tested molecule while attaining a very good convergence performance. However, the BioDE algorithm slightly speeds the computational time for identifying stable protein solutions while significantly speeding th e algorithms convergence rate in protein conformational search. To evaluate and validate our proposed re search work, we have tested the BGF, eBGF and g.eBGF algorithms against the tr aditional energy calculation approach. All methods succeeded in significantly decreasing the computational time for identifying feasible molecular conformations without sacrificing accuracy. To evaluate and validate the kDE and BioDE models, we compared the obtained molecular structures (kDE and BioDE ou tputs) against the corresponding crystal

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99 structures published in [Berman 2000]. Tw o different performance assessments were performed: an energy-oriented and a stru ctural-based. Both methods succeeded to converge in a smaller VDW energy value compared with the VDW energy for the corresponding crystal structure, thus identi fying the molecules stability state. The structures of the obtained molecular confor mations were compared to the corresponding crystal structure using the Root Mean Square Deviation (RMSD). Both methods managed to output molecular conformations clos e to the crystal structure for all the tested proteins and for most of th e tested ligands while atta ining a very good convergence performance. 8.2 Future Research Work Studying the atomic-scale processes is an open research problem that requires many disciplines to collaborate for providing reliable results and enabling bionanotechnology applications. Our proposed research work facilitates the modeling of flexible molecules and the identification of stable or low-energy conformations. This work can be used in molecular docking, na noscale assembly problems and towards the development of an indispensable computer-a ided design tool for bionanotechnology. To build a fully functional molecular system, ma ny challenges need to be addressed and many different research pathways can be pursued. One of the fundamental principles of Indu strial Systems Engineering is that the first step in a product/system development is the idea itself, followed by the design and production stages. It is also well known th at any candidate produc t/system modifications are better performed during the design stage for the product/system to be cost-effective. Under these assumptions, future research work lies within the real-t ime visualization of the molecular interactions during the design stage so that fully functional bionanoscale products can be designed and evaluated prior to actual fabrication. In this research work, new methods have been investig ated that provide real-time force feedback using haptic devices. These devices are currently used to manipulate virtual molecules and to feel the forces as the molecules interact with each other providing an essential design and

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100 visualization tool. However, to achieve a realistic molecular representation, continuous visualization as well as sense of touch to the us ers, a rapid molecular tool is essential that satisfies the haptic-rate requirement. To achieve haptic-rate performance, the rapid update and modeling of molecular conformatio ns are the main prerequisites. Focusing towards this objective, our research work presented the BGF algorithm for real-time ligand modeling that satisfie s the haptic-rate requirement. Although the proposed eBGF and g.eBGF are fast molecula r feasibility tools, they do not satisfy the haptic constraint. Further research work is required for speeding the identification of a proteins flexibility mechanism to allow hap tic interaction between macromolecules. As shown in Table 6.1, the bottleneck function of the proposed g.eBGF algorithm is the update of the bounding volume hierarchy. The GPU-oriented modeling technique seems a promising approach for speeding the BVH update and hence, to enable haptic interactions between flexible proteins or between a flexible ligand and a flexible protein. As soon as a molecular modeling system that satisfies the haptic constraint is developed, haptic devices can be utilized to study the r eal-time molecular docki ng and/or assembly problems. In addition, the main limitation of the proposed kDE and BioDE methods is that both methods require approximately one to three days outputting a stable molecular solution. Although these results are significantl y faster than current literature, they are unsuitable for real-time haptic design. To allo w haptic molecular interactions in a virtual environment, further research work is requ ired to speed the molecular conformational search. Parallel computing is a very promisin g approach towards this direction. In other words, instead of using the DE algorithm pr esented in Section 7.3, we could utilize parallel computing such as a parallel DE al gorithm to direct the molecular conformational search. This is expected to enable the ha ptic manipulation of molecules in a virtual environment and facilitate bionanoscale design and engineering. An off-line improvement of the current rese arch work is to target the molecular docking/assembly problem through a molecula r path planning approach. Under this perspective, the BioDE or kDE algorithm can be enhanced by using a modified chromosome structure, which represents a ca ndidate problem solution. Currently in the

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101 kDE and BioDE models, the genes of the ch romosome measure the torsion bond angles representing a candidate molecular conforma tion. The chromosome for the molecular path planning operation will depict both the torsion bond angles and the way-points along the optimum trajectory. This trajectory is the optimum path of a totally flexible ligand towards the cavity site of a total flexible protein. The location of the cavity site will be assumed to be known and will define the last chromosome gene, signifying the molecular path planning target. Alternatively, an on-line molecular path -planning improvement can be envisioned by utilizing the aforementioned concepts abou t GPUs and/or parall el computing as well as haptic-rendering approaches. Using hap tics to control a flexible ligand around a flexible protein with unknown binding site may lead to the identification of the proteins binding site. This may also provide strong insights for identifying feasible land-marks (coordinate points) along the molecules optimum trajectories. These land-marks may be used as path planning targets through the corresponding chromosomes genes to speed the algorithm convergence. As mentioned above, studying the molecula r flexibility is only a step closer to fully understand and model the molecular behavior. Molecules are very flexible bodies in nature that usually exist in a solvent envi ronment. Further research is required to holistically model the molecular interactions in a solvent environment. Additionally, future research work lies in studying the pr otein folding problem fo r providing structural insights for artificial macromolecular design. To conclude, bionanoscale research is still at the early stages and all the possible rese arch ideas, concepts and pathways can be limited only by the researchers vision and imagination.

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About the Author Athina N. Brintaki was born on April 26, 1978, in Athens, Greece. She received her B.S. and M.S. degrees from the Department of Production Engineering & Management at Technical University of Crete, Greece, in 2004 and 2006, respectively. In 2005, she received a graduate fellowship from th e University of South Florida (USF) to continue her graduate studies and completed her Ph.D. in Industrial Engineering in 2010. From 2005-2010, she worked as a research assi stant at USF, where she received the 2008 College of Engineering Research Poster Awa rd and the 2010 Graduate and Professional Student Council Award from USF. She publis hed 3 journal and 6 conference papers and currently has 2 journal papers submitted, 1 journal paper to be submitted, 3 working papers in different stages of completi on and a number of conference and poster presentations. She continues conducting research to understand the bionanoscale processes.