Schlick T. Molecular Modeling and Simulation: An Interdisciplinary Guide
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Contents xxiii
3.2 The Amino Acid Building Blocks ............... 82
3.2.1 Basic C
α
Unit ..................... 82
3.2.2 Essential and Nonessential Amino Acids ....... 83
3.2.3 Linking Amino Acids ................. 85
3.2.4 The Amino Acid Repertoire: From Flexible
GlycinetoRigidProline ............... 85
3.3 Sequence Variations in Proteins ................ 89
3.3.1 Globular Proteins ................... 90
3.3.2 Membrane and Fibrous Proteins ........... 90
3.3.3 Emerging Patterns from Genome Databases ..... 92
3.3.4 Sequence Similarity .................. 92
3.4 Protein Conformation Framework ............... 97
3.4.1 The Flexible φ and ψ and Rigid ω Dihedral Angles . 97
3.4.2 Rotameric Structures ................. 99
3.4.3 Ramachandran Plots .................. 99
3.4.4 Conformational Hierarchy .............. 103
4 Protein Structure Hierarchy 105
4.1 Structure Hierarchy ...................... 106
4.2 Helices: A Common Secondary Structural Element ..... 106
4.2.1 Classic α-Helix .................... 106
4.2.2 3
10
and π Helices ................... 107
4.2.3 Left-Handed α-Helix ................. 109
4.2.4 Collagen Helix .................... 110
4.3 β-Sheets: A Common Secondary Structural Element ..... 110
4.4 Turns and Loops ....................... 110
4.5 Formation of Supersecondary and Tertiary Structure ..... 113
4.5.1 Complex 3D Networks ................ 113
4.5.2 Classes in Protein Architecture ............ 113
4.5.3 Classes are Further Divided into Folds ........ 114
4.6 α-Class Folds ......................... 114
4.6.1 Bundles ........................ 114
4.6.2 Folded Leafs ...................... 115
4.6.3 Hairpin Arrays .................... 115
4.7 β-Class Folds ......................... 115
4.7.1 Anti-Parallel β Domains ............... 116
4.7.2 Parallel and Antiparallel Combinations ........ 116
4.8 α/β and α + β-Class Folds .................. 117
4.8.1 α/β Barrels ...................... 117
4.8.2 Open Twisted α/β Folds ............... 118
4.8.3 Leucine-Rich α/β Folds ................ 118
4.8.4 α+β Folds ....................... 118
4.8.5 Other Folds ......................
118
4.9 Number of Folds ........................ 118
4.9.1 Finite Number? .................... 119
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4.10 Quaternary Structure ...................... 119
4.10.1 Viruses ......................... 119
4.10.2 From Ribosomes to Dynamic Networks ....... 123
4.11 Protein Structure Classification ................ 126
5 Nucleic Acids Structure Minitutorial 129
5.1 DNA, Life’s Blueprint .................... 130
5.1.1 The Kindled Field of Molecular Biology ....... 130
5.1.2 Fundamental DNA Processes ............. 132
5.1.3 Challenges in Nucleic Acid Structure ......... 133
5.1.4 Chapter Overview ................... 134
5.2 The Basic Building Blocks of Nucleic Acids ......... 135
5.2.1 Nitrogenous Bases .................. 135
5.2.2 Hydrogen Bonds ................... 136
5.2.3 Nucleotides ...................... 137
5.2.4 Polynucleotides .................... 137
5.2.5 Stabilizing Polynucleotide Interactions ........ 140
5.2.6 Chain Notation .................... 140
5.2.7 Atomic Labeling ................... 141
5.2.8 Torsion Angle Labeling ............... 142
5.3 Nucleic Acid Conformational Flexibility ........... 142
5.3.1 The Furanose Ring .................. 143
5.3.2 Backbone Torsional Flexibility ............ 145
5.3.3 The Glycosyl Rotation ................ 148
5.3.4 Sugar/Glycosyl Combinations ............ 148
5.3.5 Basic Helical Descriptors ............... 150
5.3.6 Base-Pair Parameters ................. 151
5.4 Canonical DNA Forms .................... 155
5.4.1 B-DNA ........................ 156
5.4.2 A-DNA ........................ 157
5.4.3 Z-DNA ........................ 160
5.4.4 Comparative Features ................. 161
6 Topics in Nucleic Acids Structure: DNA Interactions
and Folding 163
6.1 Introduction .......................... 164
6.2 DNA Sequence Effects .................... 165
6.2.1 Local Deformations
.................. 165
6.2.2 Orientation Preferences in Dinucleotide Steps .... 166
6.2.3 Orientation Preferences in Dinucleotide Steps
With Flanking Sequence Context: Tetranucleotide
Studies......................... 169
6.2.4 Intrinsic DNA Bending in A-Tracts .......... 169
6.2.5 Sequence Deformability Analysis Continues ..... 173
Contents xxv
6.3 DNA Hydration and Ion Interactions ............. 174
6.3.1 Resolution Difficulties ................ 175
6.3.2 Basic Patterns ..................... 176
6.4 DNA/Protein Interactions ................... 180
6.5 Cellular Organization of DNA ................. 182
6.5.1 Compaction of Genomic DNA ............ 182
6.5.2 Coiling of the DNA Helix Itself ............ 184
6.5.3 Chromosomal Packaging of Coiled DNA ...... 185
6.6 Mathematical Characterization of DNA Supercoiling ..... 195
6.6.1 DNA Topology and Geometry ............ 195
6.7 Computational Treatments of DNA Supercoiling ...... 197
6.7.1 DNA as a Flexible Polymer .............. 198
6.7.2 Elasticity Theory Framework ............. 199
6.7.3 Simulations of DNA Supercoiling .......... 200
7 Topics in Nucleic Acids Structure: Noncanonical Helices
and RNA Structure 205
7.1 Introduction .......................... 205
7.2 Variations on a Theme ..................... 206
7.2.1 Hydrogen Bonding Patterns in Polynucleotides ... 206
7.2.2 Hybrid Helical/Nonhelical Forms ........... 210
7.2.3 Unusual Forms: Overstretched and Understretched
DNA.......................... 214
7.3 RNA Structure and Function ................. 216
7.3.1 DNA’s Cousin Shines ................. 216
7.3.2 RNA Chains Fold Upon Themselves ......... 216
7.3.3 RNA’s Diversity .................... 217
7.3.4 Non-Coding and Micro-RNAs ............ 221
7.3.5 RNA at Atomic Resolution .............. 222
7.4 Current Challenges in RNA Modeling ............ 225
7.4.1 RNA Folding ..................... 225
7.4.2 RNA Motifs ...................... 225
7.4.3 RNA Structure Prediction ............... 226
7.5 Application of Graph Theory to Studies of RNA Structure
andFunction.......................... 229
7.5.1 Graph Theory ..................... 229
7.5.2 RNA-As-Graphs (RAG) Resource .......... 230
8 Theoretical and Computational Approaches to Biomolecular
Structure 237
8.1 The Merging of Theory and Experiment
........... 238
8.1.1 Exciting Times for Computationalists! ........ 238
8.1.2 The Future of Biocomputations ............ 240
8.1.3 Chapter Overview ................... 240
xxvi Contents
8.2 Quantum Mechanics (QM) Foundations of Molecular Mechan-
ics(MM)............................ 241
8.2.1 The Schr¨odinger Wave Equation ........... 241
8.2.2 The Born-Oppenheimer Approximation ....... 242
8.2.3 Ab Initio QM ..................... 242
8.2.4 Semi-Empirical QM .................. 244
8.2.5 Recent Advances in Quantum Mechanics ...... 244
8.2.6 From Quantum to Molecular Mechanics ....... 247
8.3 Molecular Mechanics: Underlying Principles ......... 251
8.3.1 The Thermodynamic Hypothesis ........... 251
8.3.2 Additivity ....................... 252
8.3.3 Transferability ..................... 254
8.4 Molecular Mechanics: Model and Energy Formulation .... 256
8.4.1 Configuration Space .................. 258
8.4.2 Functional Form .................... 259
8.4.3 Some Current Limitations ............... 262
9 Force Fields 265
9.1 Formulation of the Model and Energy ............. 266
9.2 Normal Modes ......................... 267
9.2.1 Quantifying Characteristic Motions .......... 267
9.2.2 Complex Biomolecular Spectra ............ 269
9.2.3 Spectra As Force Constant Sources .......... 269
9.2.4 In-Plane and Out-of-Plane Bending .......... 271
9.3 Bond Length Potentials .................... 272
9.3.1 Harmonic Term .................... 273
9.3.2 Morse Term ...................... 274
9.3.3 Cubic and Quartic Terms ............... 275
9.4 Bond Angle Potentials ..................... 276
9.4.1 Harmonic and Trigonometric Terms ......... 277
9.4.2 Cross Bond Stretch / Angle Bend Terms ....... 278
9.5 Torsional Potentials ...................... 281
9.5.1 Origin of Rotational Barriers ............. 281
9.5.2 Fourier Terms ..................... 281
9.5.3 Torsional Parameter Assignment ........... 282
9.5.4 Improper Torsion
................... 286
9.5.5 Cross Dihedral/Bond Angle and Improper/Improper
DihedralTerms .................... 287
9.6 The van der Waals Potential .................. 288
9.6.1 Rapidly Decaying Potential .............. 288
9.6.2 Parameter Fitting From Experiment ......... 289
9.6.3 Two Parameter Calculation Protocols ......... 289
9.7 The Coulomb Potential .................... 291
9.7.1 Coulomb’s Law: Slowly Decaying Potential ..... 291
9.7.2 Dielectric Function .................. 292
9.7.3 Partial Charges .................... 294
Contents xxvii
9.8 Parameterization ........................ 295
9.8.1 A Package Deal .................... 295
9.8.2 Force Field Comparisons ............... 295
9.8.3 Force Field Performance ............... 297
10 Nonbonded Computations 299
10.1 A Computational Bottleneck ................. 301
10.2 Approaches for Reducing Computational Cost ........ 302
10.2.1 Simple Cutoff Schemes ................ 302
10.2.2 Ewald and Multipole Schemes ............ 303
10.3 Spherical Cutoff Techniques .................. 304
10.3.1 Technique Categories ................. 304
10.3.2 Guidelines for Cutoff Functions ........... 305
10.3.3 General Cutoff Formulations ............. 306
10.3.4 Potential Switch .................... 307
10.3.5 Force Switch ..................... 308
10.3.6 Shift Functions .................... 309
10.4 The Ewald Method ...................... 311
10.4.1 Periodic Boundary Conditions ............ 311
10.4.2 Ewald Sum and Crystallography ........... 314
10.4.3 Mathematical Morphing of a Conditionally
ConvergentSum.................... 316
10.4.4 Finite-Dielectric Correction .............. 320
10.4.5 Ewald Sum Complexity ................ 320
10.4.6 Resulting Ewald Summation ............. 321
10.4.7 Practical Implementation: Parameters, Accuracy,
andOptimization ................... 322
10.5 The Multipole Method ..................... 324
10.5.1 Basic Hierarchical Strategy .............. 324
10.5.2 Historical Perspective ................. 329
10.5.3 Expansion in Spherical Coordinates ......... 330
10.5.4 Biomolecular Implementations ............ 332
10.5.5 Other Variants ..................... 333
10.6 Continuum Solvation ..................... 333
10.6.1 Need for Simplification! ............... 333
10.6.2 Potential of Mean Force ................ 334
10.6.3 Stochastic Dynamics ................. 335
10.6.4 Continuum Electrostatics ............... 338
11 Multivariate Minimization in Computational Chemistry 345
11.1 Ubiquitous Optimization: From Enzymes to Weather to Eco-
nomics............................. 347
11.1.1 Algorithmic Sophistication Demands Basic
Understanding..................... 347
11.1.2 Chapter Overview ................... 347
xxviii Contents
11.2 Optimization Fundamentals .................. 348
11.2.1 Problem Formulation ................. 348
11.2.2 Independent Variables ................. 349
11.2.3 Function Characteristics ................ 349
11.2.4 Local and Global Minima ............... 351
11.2.5 Derivatives of Multivariate Functions ......... 353
11.2.6 The Hessian of Potential Energy Functions ...... 353
11.3 Basic Algorithmic Components ................ 356
11.3.1 Greedy Descent .................... 356
11.3.2 Line-Search-Based Descent Algorithm ........ 359
11.3.3 Trust-Region-Based Descent Algorithm ....... 361
11.3.4 Convergence Criteria ................. 362
11.4 The Newton-Raphson-Simpson-Fourier Method ....... 364
11.4.1 The One-Dimensional Version of Newton’s Method . 364
11.4.2 Newton’s Method for Minimization ......... 367
11.4.3 The Multivariate Version of Newton’s Method .... 368
11.5 Effective Large-Scale Minimization Algorithms ....... 369
11.5.1 Quasi-Newton (QN) .................. 370
11.5.2 Conjugate Gradient (CG) ............... 372
11.5.3 Truncated-Newton (TN) ................ 374
11.5.4 Simple Example .................... 376
11.6 Available Software ....................... 378
11.6.1 Popular Newton and CG ............... 378
11.6.2 CHARMM’s ABNR ................. 379
11.6.3 CHARMM’s TN ................... 379
11.6.4 Comparative Performance on Molecular Systems .. 379
11.7 Practical Recommendations .................. 380
11.8 Future Outlook ......................... 383
12 Monte Carlo Techniques 385
12.1 MC Popularity ........................ 386
12.1.1 A Winning Combination ............... 386
12.1.2 From Needles to Bombs ............... 387
12.1.3 Chapter Overview ................... 387
12.1.4 Importance of Error Bars ............... 388
12.2 Random Number Generators ................. 388
12.2.1 What is Random? ................... 388
12.2.2 Properties of Generators ............... 389
12.2.3 Linear Congruential Generators (LCG) ........ 392
12.2.4 Other Generators ................... 396
12.2.5 Artifacts ........................ 400
12.2.6 Recommendations ................... 401
12.3 Gaussian Random Variates .................. 403
12.3.1 Manipulation of Uniform Random Variables ..... 403
12.3.2 Normal Variates in Molecular Simulations ...... 403
Contents xxix
12.3.3 Odeh/Evans Method .................. 404
12.3.4 Box/Muller/Marsaglia Method ............ 405
12.4 Means for Monte Carlo Sampling ............... 406
12.4.1 Expected Values .................... 406
12.4.2 Error Bars ....................... 409
12.4.3 Batch Means ...................... 410
12.5 Monte Carlo Sampling ..................... 411
12.5.1 Density Function ................... 411
12.5.2 Dynamic and Equilibrium MC: Ergodicity,
DetailedBalance ................... 411
12.5.3 Statistical Ensembles ................. 412
12.5.4 Importance Sampling: Metropolis Algorithm
andMarkovChains.................. 413
12.6 Monte Carlo Applications to Molecular Systems ....... 418
12.6.1 Ease of Application .................. 418
12.6.2 Biased MC ...................... 419
12.6.3 Hybrid MC ...................... 420
12.6.4 Parallel Tempering and Other MC Variants ...... 421
13 Molecular Dynamics: Basics 425
13.1 Introduction: Statistical Mechanics by Numbers ....... 426
13.1.1 Why Molecular Dynamics? .............. 426
13.1.2 Background ...................... 427
13.1.3 Outline of MD Chapters ................ 428
13.2 Laplace’s Vision of Newtonian Mechanics .......... 429
13.2.1 The Dream Becomes Reality ............. 429
13.2.2 Deterministic Mechanics ............... 432
13.2.3 Neglect of Electronic Motion ............. 432
13.2.4 Critical Frequencies .................. 433
13.2.5 Hybrid Quantum/Classical Mechanics Treatments .. 435
13.3 The Basics: An Overview ................... 435
13.3.1 Following the Equations of Motion .......... 435
13.3.2 Perspective on MD Trajectories ............ 436
13.3.3 Initial System Settings ................ 437
13.3.4 Sensitivity to Initial Conditions and
OtherComputationalChoices............. 440
13.3.5 Simulation Protocol .................. 442
13.3.6 High-Speed Implementations ............. 443
13.3.7 Analysis and Visualization .............. 445
13.3.8 Reliable Numerical Integration ............ 445
13.3.9 Computational Complexity .............. 446
13.4 The Verlet Algorithm ..................... 448
13.4.1 Position and Velocity Propagation .......... 449
13.4.2 Leapfrog, Velocity Verlet, and Position Verlet .... 451
13.5 Constrained Dynamics ..................... 453
xxx Contents
13.6 Various MD Ensembles .................... 455
13.6.1 Need for Other Ensembles .............. 455
13.6.2 Simple Algorithms .................. 456
13.6.3 Extended System Methods .............. 459
14 Molecular Dynamics: Further Topics 463
14.1 Introduction .......................... 464
14.2 Symplectic Integrators ..................... 465
14.2.1 Symplectic Transformation .............. 466
14.2.2 Harmonic Oscillator Example ............. 467
14.2.3 Linear Stability .................... 467
14.2.4 Timestep-Dependent Rotation in Phase Space .... 469
14.2.5 Resonance Condition for Periodic Motion ...... 470
14.2.6 Resonance Artifacts .................. 471
14.3 Multiple-Timestep (MTS) Methods .............. 472
14.3.1 Basic Idea ....................... 472
14.3.2 Extrapolation ..................... 473
14.3.3 Impulses ....................... 474
14.3.4 Vulnerability of Impulse Splitting to Resonance
Artifacts........................ 475
14.3.5 Resonance Artifacts in MTS ............. 476
14.3.6 Limitations of Resonance Artifacts on Speedup;
PossibleCures..................... 478
14.4 Langevin Dynamics ...................... 479
14.4.1 Many Uses ...................... 479
14.4.2 Phenomenological Heat Bath ............. 480
14.4.3 The Effect of γ .................... 480
14.4.4 Generalized Verlet for Langevin Dynamics ...... 482
14.4.5 The LN Method .................... 482
14.5 Brownian Dynamics (BD) ................... 487
14.5.1 Brownian Motion ................... 487
14.5.2 Brownian Framework ................. 489
14.5.3 General Propagation Framework ........... 491
14.5.4 Hydrodynamic Interactions .............. 491
14.5.5 BD Propagation Scheme: Cholesky vs. Chebyshev
Approximation .................... 494
14.6 Implicit Integration ...................... 496
14.6.1 Implicit vs. Explicit Euler ..............
497
14.6.2 Intrinsic Damping ................... 498
14.6.3 Computational Time ................. 498
14.6.4 Resonance Artifacts .................. 499
14.7 Enhanced Sampling Methods ................. 503
14.7.1 Overview ....................... 503
14.7.2 Harmonic-Analysis Based Techniques ........ 503
14.7.3 Other Coordinate Transformations .......... 505
Contents xxxi
14.7.4 Coarse Graining Models ............... 507
14.7.5 Biasing Approaches .................. 508
14.7.6 Variations in MD Algorithm and Protocol ...... 509
14.7.7 Other Rigorous Approaches for Deducing
Mechanisms, Free Energies, and Reaction Rates . . . 511
14.8 Future Outlook ......................... 513
14.8.1 Integration Ingenuity ................. 513
14.8.2 Current Challenges .................. 514
15 Similarity and Diversity in Chemical Design 519
15.1 Introduction to Drug Design .................. 520
15.1.1 Chemical Libraries .................. 520
15.1.2 Early Drug Development Work ............ 521
15.1.3 Molecular Modeling in Rational Drug Design .... 523
15.1.4 The Competition: Automated Technology ...... 524
15.1.5 Chapter Overview .................. 526
15.2 Problems in Chemical Libraries ................ 526
15.2.1 Database Analysis ................... 526
15.2.2 Similarity and Diversity Sampling .......... 527
15.2.3 Bioactivity Relationships ............... 529
15.3 General Problem Definitions ................. 532
15.3.1 The Dataset ...................... 532
15.3.2 The Compound Descriptors .............. 534
15.3.3 Characterizing Biological Activity .......... 535
15.3.4 The Target Function .................. 536
15.3.5 Scaling Descriptors .................. 536
15.3.6 The Similarity and Diversity Problems ........ 538
15.4 Data Compression and Cluster Analysis ........... 540
15.4.1 Data Compression Based on Principal Component
Analysis(PCA) .................... 540
15.4.2 Data Compression Based on the Singular Value
Decomposition (SVD) ................ 542
15.4.3 Relation Between PCA and SVD ........... 544
15.4.4 Data Analysis via PCA or SVD and Distance
Refinement ...................... 545
15.4.5 Projection, Refinement, and Clustering Example ... 546
15.5 Future Perspectives ...................... 551
Epilogue 555
Appendices 556
A Molecular Modeling Sample Syllabus
557
B Article Reading List 559
xxxii Contents
C Supplementary Course Texts 563
D Homework Assignments 571
References 623