FREE ENERGY SIMULATIONS OF COMPLEX BIOLOGICAL SYSTEMS AT CONSTANT PH By JASON M. SWAILS A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2013 © 2013 Jason M. Swails 2 I dedicate this dissertation to the late Professor Frederick P. Arnold whose intelligence, excitement, and guidance propelled me into this field. 3 ACKNOWLEDGMENTS I would like to thank my parents, Mark and Mindy Swails, for their love, guidance, and constant encouragement. I thank my siblings, Kerri and Jeffrey Swails, for all the great times and their (vain) attempts to keep me humble. Thank you to my extended family for the incredible support system I’ve always had growing up. The Roitberg Group provided a great deal of support and comaraderie, and Adrian Roitberg provided guidance and instruction during my graduate studies. I also thank Quantum Theory Project for all the good times. And finally, I would like to thank my wife, Roxy Lowry Swails, for everything. 4 TABLE OF CONTENTS page ACKNOWLEDGMENTS .................................. 4 LIST OF TABLES ...................................... 9 LIST OF FIGURES ..................................... 10 LIST OF ABBREVIATIONS ................................ 13 LIST OF CONSTANTS AND OPERATORS ....................... 15 ABSTRACT ......................................... 16 CHAPTER 1 INTRODUCTION ................................... 18 1.1 Origins of Computational Chemistry ..................... 18 1.1.1 Quantum Mechanics .......................... 19 1.1.1.1 Born-Oppenheimer Approximation ............. 20 1.1.1.2 Computational Quantum Mechanics ............ 21 1.1.2 Statistical Mechanics .......................... 21 1.1.2.1 Monte Carlo ......................... 24 1.1.2.2 Molecular Dynamics and the Ergodic Hypothesis ..... 28 1.2 Molecular Mechanics .............................. 30 1.2.1 Force Fields ............................... 31 1.2.1.1 Bonds ............................. 31 1.2.1.2 Angles ............................ 33 1.2.1.3 Torsions ............................ 33 1.2.1.4 Electrostatic Interactions .................. 35 1.2.1.5 van der Waals Interactions ................. 36 1.2.1.6 Other Force Field Terms .................. 37 1.2.2 The Amber Force Field ......................... 39 1.2.2.1 Functional Form ....................... 40 1.2.2.2 Implementation ........................ 41 2 BIOMOLECULAR SIMULATION: SAMPLING AND FREE ENERGY ...... 43 2.1 Simulations in the Condensed Phase ..................... 43 2.1.1 Implicit Solvent ............................. 44 2.1.1.1 Distance-dependent Dielectric ............... 44 2.1.1.2 Poisson-Boltzmann ..................... 46 2.1.1.3 Generalized Born ...................... 48 2.1.1.4 Non-polar Solvation ..................... 53 2.1.2 Explicit Solvent ............................. 54 2.1.2.1 Periodic Boundary Conditions ............... 55 5 2.1.2.2 Cutoff Methods ........................ 55 2.1.2.3 Ewald Summation ...................... 59 2.1.2.4 Other Approaches ...................... 62 2.2 Sampling .................................... 64 2.2.1 Umbrella Sampling ........................... 65 2.2.2 Steered Molecular Dynamics ..................... 68 2.2.3 Expanded Ensemble .......................... 69 2.2.4 Replica Exchange Molecular Dynamics ............... 71 2.3 Free Energy Calculations ........................... 72 2.3.1 Thermodynamic Integration ...................... 73 2.3.2 Free Energy Perturbation ....................... 77 2.3.3 End-state Calculations ......................... 78 2.3.3.1 MM-PBSA .......................... 79 2.3.3.2 LIE .............................. 80 3 CONSTANT pH REPLICA EXCHANGE MOLECULAR DYNAMICS ....... 83 3.1 Constant pH and pKa Calculations ...................... 83 3.2 Theory ...................................... 86 3.2.1 The Semi-Grand Ensemble ...................... 86 3.2.2 CpHMD ................................. 87 3.2.3 pH-REMD ................................ 89 3.3 Methods ..................................... 89 3.3.1 Starting Structure ............................ 89 3.3.2 Molecular Dynamics .......................... 90 3.3.3 Replica Exchange ........................... 91 3.4 Results and Discussion ............................ 91 3.4.1 Simulation Stability ........................... 92 3.4.2 Accuracy of Predicted pKas ...................... 92 3.4.3 Enhancing Protonation State Sampling with pH-REMD ....... 94 3.5 Exchange Attempt Frequency and Protonation State Sampling ...... 98 3.5.1 Enhancing Conformational State Sampling with pH-REMD ..... 101 3.5.2 Scalability With Increasing Exchange Attempt Frequency ..... 109 3.6 Conclusion ................................... 110 4 CONSTANT pH MOLECULAR DYNAMICS IN EXPLICIT SOLVENT ...... 112 4.1 Introduction ................................... 112 4.2 Theory and Methods .............................. 114 4.2.1 Conformational and Protonation State Sampling ........... 114 4.2.2 Explicit Solvent CpHMD Workflow ................... 116 4.2.3 pH-based Replica Exchange ..................... 117 4.3 Calculation Details ............................... 119 4.3.1 Model Compounds ........................... 119 4.3.2 ACFCA ................................. 121 4.3.3 Proteins: HEWL and RNase A ..................... 122 6 4.3.4 Simulation Details ........................... 123 4.4 Results and Discussion ............................ 124 4.4.1 Box Size Effects ............................ 124 4.4.2 τrlx Effects ................................ 125 4.4.3 ACFCA: CpHMD vs. pH-REMD .................... 128 4.4.4 Hen Egg White Lysozyme ....................... 131 4.4.5 Ribonuclease A ............................. 134 4.5 Conclusion ................................... 137 5 REMD: GPU ACCELERATION AND EXCHANGES IN MULTIPLE DIMEN- SIONS ......................................... 142 5.1 Temperature REMD .............................. 142 5.2 Hamiltonian REMD ............................... 145 5.3 Multi-Dimensional REMD ........................... 147 5.4 Implementation ................................. 148 5.4.1 Exchange Attempts ........................... 149 5.4.2 Message Passing: Data Exchange in REMD Simulations ...... 152 6 FLEXIBLE TOOLS FOR AMBER SIMULATIONS ................. 155 6.1 MMPBSA.py .................................. 155 6.1.1 Motivation ................................ 155 6.1.2 Capabilities ............................... 156 6.1.2.1 Stability and Binding Free Energy Calculations ...... 156 6.1.2.2 Free Energy Decomposition ................ 158 6.1.2.3 Entropy Calculations ..................... 159 6.1.3 General Workflow ............................ 160 6.1.4 Running in Parallel ........................... 162 6.1.5 Differences to mm pbsa.pl ....................... 162 6.2 ParmEd ..................................... 164 6.2.1 Motivation ................................ 164 6.2.2 Implementation and Capabilities ................... 165 6.2.2.1 Lennard-Jones Parameter Modifications .......... 167 6.2.2.2 Changing Atomic Properties ................ 168 6.2.2.3 Setting up for H-REMD Simulations ............ 169 6.2.2.4 Changing Parameters .................... 169 APPENDIX A NUMERICAL INTEGRATION IN CLASSICAL MOLECULAR DYNAMICS .... 170 A.1 Lagrangian and Hamiltonian Formulations .................. 170 A.2 Numerical Integration by Finite Difference Methods ............. 171 A.2.1 Predictor-corrector ........................... 171 A.2.2 Verlet Integrators ............................ 173 7 B AMBER PARAMETER-TOPOLOGY FILE FORMAT ............... 176 B.1 Layout ...................................... 176 B.2 List of SECTIONs ................................. 178 B.3 Deprecated Sections .............................. 192 B.4 CHAMBER Topologies ............................. 193 C MESSAGE PASSING INTERFACE ......................... 199 C.1 Parallel Computing ............................... 199 C.1.1 Data Models ............................... 199 C.1.2 Memory Layout ............................. 199 C.1.3 Thread Count .............................. 201 C.2 The Mechanics of MPI ............................. 202 C.2.1 Messages ................................ 202 C.2.2 Communicators ............................. 202 C.2.3 Communications ............................ 202 C.2.3.1 Point-to-point ......................... 203 C.2.3.2 All-to-one and One-to-all .................. 203 C.2.3.3 All-to-all ............................ 204 C.2.4 Blocking vs. Non-blocking Communications ............. 206 REFERENCES ....................................... 207 BIOGRAPHICAL SKETCH ................................ 218 8 LIST OF TABLES Table page 3-1 Reference pKa values for the acidic residues treated in this study. Values are the same as those used in the original Amber CpHMD implementation.[130] .. 88 3-2 pKa and Hill coefficients for each residue taken from each set of simulations. The pKas and Hill coefficients (n) are shown for each EAF. ............ 94 3-3 Value of RSS according to Eq. 3–6 for the 8 residues shown in Figs. 3-2 and 3-3. Larger values represent more deviation from the fitted curve ........ 97 3-4 Standard deviations of pKa (σpKa ) and Hill coefficient (σn) and average Hill co- efficient (n¯) calculated by dividing each simulation into sections of 0.25 ns. ... 99 3-5 Average timings for CpHMD and pH-REMD simulations. ............. 109 4-1 Model compound pKa values and reference energies. .............. 121 4-2 Calculated pKas for acid-range titratable