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1 Useful Concepts in Molecular Modelling 1

1.1 Introduction 1

1.2 Coordinate Systems 2

1.3 Potential Energy Surfaces 4

1.4 Molecular Graphics 5

1.5 Surfaces 6

1.6 Computer Hardware and Software 8

1.7 Units of Length and Energy 9

1.8 The Molecular Modelling Literature 9

1.9 The Internet 9

1.10 Mathematical Concepts 10

Further Reading 24

References 24

2 An Introduction to Computational Quantum Mechanics 26

2.1 Introduction 26

2.2 One-electron Atoms 30

2.3 Polyelectronic Atoms and Molecules 34

2.4 Molecular Orbital Calculations 41

2.5 The Hartree-Fock Equations 51

2.6 Basis Sets 65

2.7 Calculating Molecular Properties Using ab initio Quantum Mechanics 74

2.8 Approximate Molecular Orbital Theories 86

2.9 Semi-empirical Methods 86

2.10 Hückel Theory 99

2.11 Performance of Semi-empirical Methods 102

Appendix 2.1 Some Common Acronyms Used in Computational Quantum Chemistry 104

Further Reading 105

References 105

3 Advanced ab initio Methods， Density Functional Theory and Solid-state Quantum Mechanics 108

3.1 Introduction 108

3.2 Open-shell Systems 108

3.3 Electron Correlation 110

3.4 Practical Considerations When Performing ab initio Calculations 117

3.5 Energy Component Analysis 122

3.6 Valence Bond Theories 124

3.7 Density Functional Theory 126

3.8 Quantum Mechanical Methods for Studying the Solid State 138

3.9 The Future Role of Quantum Mechanics： Theory and Experiment Working Together 160

Appendix 3.1 Alternative Expression for a Wavefunction Satisfying Bloch’s Function 161

Further Reading 161

References 162

4 Empirical Force Field Models： Molecular Mechanics 165

4.1 Introduction 165

4.2 Some General Features of Molecular Mechanics Force Fields 168

4.3 Bond Stretching 170

4.4 Angle Bending 173

4.5 Torsional Terms 173

4.6 Improper Torsions and Out-of-plane Bending Motions 176

4.7 Cross Terms： Class 1， 2 and 3 Force Fields 178

4.8 Introduction to Non-bonded Interactions 181

4.9 Electrostatic Interactions 181

4.10 Van der Waals Interactions 204

4.11 Many-body Effects in Empirical Potentials 212

4.12 Effective Pair Potentials 214

4.13 Hydrogen Bonding in Molecular Mechanics 215

4.14 Force Field Models for the Simulation of Liquid Water 216

4.15 United Atom Force Fields and Reduced Representations 221

4.16 Derivatives of the Molecular Mechanics Energy Function 225

4.17 Calculating Thermodynamic Properties Using a Force Field 226

4.18 Force Field Parametrisation 228

4.19 Transferability of Force Field Parameters 231

4.20 The Treatment of Delocalised π Systems 233

4.21 Force Fields for Inorganic Molecules 234

4.22 Force Fields for Solid-state Systems 236

4.23 Empirical Potentials for Metals and Semiconductors 240

Appendix 4.1 The Interaction Between Two Drude Molecules 246

Further Reading 247

References 247

5 Energy Minimisation and Related Methods for Exploring the Energy Surface 253

5.1 Introduction 253

5.2 Non-derivative Minimisation Methods 258

5.3 Introduction to Derivative Minimisation Methods 261

5.4 First-order Minimisation Methods 262

5.5 Second Derivative Methods： The Newton-Raphson Method 267

5.6 Quasi-Newton Methods 268

5.7 Which Minimisation Method Should I Use？ 270

5.8 Applications of Energy Minimisation 273

5.9 Determination of Transition Structures and Reaction Pathways 279

5.10 Solid-state Systems： Lattice Statics and Lattice Dynamics 295

Further Reading 300

References 301

6 Computer Simulation Methods 303

6.1 Introduction 303

6.2 Calculation of Simple Thermodynamic Properties 307

6.3 Phase Space 312

6.4 Practical Aspects of Computer Simulation 315

6.5 Boundaries 317

6.6 Monitoring the Equilibration 321

6.7 Truncating the Potential and the Minimum Image Convention 324

6.8 Long-range Forces 334

6.9 Analysing the Results of a Simulation and Estimating Errors 343

Appendix 6.1 Basic Statistical Mechanics 347

Appendix 6.2 Heat Capacity and Energy Fluctuations 348

Appendix 6.3 The Real Gas Contribution to the Virial 349

Appendix 6.4 Translating Particle Back into Central Box for Three Box Shapes 350

Further Reading 351

References 351

7 Molecular Dynamics Simulation Methods 353

7.1 Introduction 353

7.2 Molecular Dynamics Using Simple Models 353

7.3 Molecular Dynamics with Continuous Potentials 355

7.4 Setting up and Running a Molecular Dynamics Simulation 364

7.5 Constraint Dynamics 368

7.6 Time-dependent Properties 374

7.7 Molecular Dynamics at Constant Temperature and Pressure 382

7.8 Incorporating Solvent Effects into Molecular Dynamics： Potentials of Mean Force and Stochastic Dynamics 387

7.9 Conformational Changes from Molecular Dynamics Simulations 392

7.10 Molecular Dynamics Simulations of Chain Amphiphiles 394

Appendix 7.1 Energy Conservation in Molecular Dynamics 405

Further Reading 406

References 406

8 Monte Carlo Simulation Methods 410

8.1 Introduction 410

8.2 Calculating Properties by Integration 412

8.3 Some Theoretical Background to the Metropolis Method 414

8.4 Implementation of the Metropolis Monte Carlo Method 417

8.5 Monte Carlo Simulation of Molecules 420

8.6 Models Used in Monte Carlo Simulations of Polymers 423

8.7 ’Biased’ Monte Carlo Methods 432

8.8 Tackling the Problem of Quasi-ergodiciry： J-walking and Multicanonical Monte Carlo 433

8.9 Monte Carlo Sampling from Different Ensembles 438

8.10 Calculating the Chemical Potential 442

8.11 The Configurational Bias Monte Carlo Method 443

8.12 Simulating Phase Equilibria by the Gibbs Ensemble Monte Carlo Method 450

8.13 Monte Carlo or Molecular Dynamics？ 452

Appendix 8.1 The Marsaglia Random Number Generator 453

Further Reading 454

References 454

9 Conformational Analysis 457

9.1 Introduction 457

9.2 Systematic Methods for Exploring Conformational Space 458

9.3 Model-building Approaches 464

9.4 Random Search Methods 465

9.5 Distance Geometry 467

9.6 Exploring Conformational Space Using Simulation Methods 475

9.7 Which Conformational Search Method Should I Use？ A Comparison of Different Approaches 476

9.8 Variations on the Standard Methods 477

9.9 Finding the Global Energy Minimum： Evolutionary Algorithms and Simulated Annealing 479

9.10 Solving Protein Structures Using Restrained Molecular Dynamics and Simulated Annealing 483

9.11 Structural Databases 489

9.12 Molecular Fitting 490

9.13 Clustering Algorithms and Pattern Recognition Techniques 491

9.14 Reducing the Dimensionality of a Data Set 497

9.15 Covering Conformational Space： Poling 499

9.16 A ’Classic’ Optimisation Problem： Predicting Crystal Structures 501

Further Reading 505

References 506

10 Protein Structure Prediction， Sequence Analysis and Protein Folding 509

10.1 Introduction 509

10.2 Some Basic Principles of Protein Structure 513

10.3 First-principles Methods for Predicting Protein Structure 517

10.4 Introduction to Comparative Modelling 522

10.5 Sequence Alignment 522

10.6 Constructing and Evaluating a Comparative Model 539

10.7 Predicting Protein Structures by ’Threading’ 545

10.8 A Comparison of Protein Structure Prediction Methods： CASP 547

10.9 Protein Folding and Unfolding 549

Appendix 10.1 Some Common Abbreviations and Acronyms Used in Bioinformatics 553

Appendix 10.2 Some of the Most Common Sequence and Structural Databases Used in Bioinformatics 555

Appendix 10.3 Mutation Probability Matrix for 1 PAM 556

Appendix 10.4 Mutation Probability Matrix for 250 PAM 557

Further Reading 557

References 558

11 Four Challenges in Molecular Modelling： Free Energies， Solvation， Reactions and Solid-state Defects 563

11.1 Free Energy Calculations 563

11.2 The Calculation of Free Energy Differences 564

11.3 Applications of Methods for Calculating Free Energy Differences 569

11.4 The Calculation of Enthalpy and Entropy Differences 574

11.5 Partitioning the Free Energy 574

11.6 Potential Pitfalls with Free Energy Calculations 577

11.7 Potentials of Mean Force 580

11.8 Approximate/’Rapid’ Free Energy Methods 585

11.9 Continuum Representations of the Solvent 592

11.10 The Electrostatic Contribution to the Free Energy of Solvation：The Born and Onsager Models 593

11.11 Non-electrostatic Contributions to the Solvation Free Energy 608

11.12 Very Simple Solvation Models 609

11.13 Modelling Chemical Reactions 610

11.14 Modelling Solid-state Defects 622

Appendix 11.1 Calculating Free Energy Differences Using Thermodynamic Integration 630

Appendix 11.2 Using the Slow Growth Method for Calculating Free Energy Differences 631

Appendix 11.3 Expansion of Zwanzig Expression for the Free Energy Difference for the Linear Response Method 631

Further Reading 632

References 633

12 The Use of Molecular Modelling and Chemoinformatics to Discover and Design New Molecules 640

12.1 Molecular Modelling in Drug Discovery 640

12.2 Computer Representations of Molecules， Chemical Databases and 2D Substructure Searching 642

12.3 3D Database Searching 647

12.4 Deriving and Using Three-dimensional Pharmacophores 648

12.5 Sources of Data for 3D Databases 659

12.6 Molecular Docking 661

12.7 Applications of 3D Database Searching and Docking 667

12.8 Molecular Similarity and Similarity Searching 668

12.9 Molecular Descriptors 668

12.10 Selecting ’Diverse’ Sets of Compounds 680

12.11 Structure-based De Novo Ligand Design 687

12.12 Quantitative Structure-Activity Relationships 695

12.13 Partial Least Squares 706

12.14 Combinatorial Libraries 711

Further Reading 719

References 720

Index 727