Molecular Dynamics Simulation Methods Revised H. Bekker CIP-DATA KONINKLIJKE BIBLIOTHEEK, DEN HAAG Bekker, Hendrik Molecular Dynamics Simulation Methods Revised / Hendrik ] Bekker, - [S.l. : s.n. ([S.l.] : Febo Druk). - Ill. Thesis Rijksuniversiteit Groningen. - With ref. ISBN 90-367-0604-1 Subject headings: molecular dynamics / simulation / algorithms. Omslag: Bart ten Hoonte/letter & lijn. RIJKSUNIVERSITEIT GRONINGEN Molecular Dynamics Simulation Methods Revised Proefschrift ter verkrijging van het doctoraat in de Wiskunde en Natuurwetenschappen aan de Rijksuniversiteit Groningen op gezag van de Rector Magni®cus Dr. F. van der Woude in het openbaar te verdedigen op vrijdag 14 juni 1996 des namiddags te 4.00 uur door Hendrik Bekker geboren op 5 juni 1950 te Ee Promotores: Prof.dr. H.J.C. Berendsen Prof.dr. N. Petkov CONTENTS 1 Introduction 1 1.1 M.D. simulation in outline : :: ::: :: :: :: :: ::: :: :: : 1 1.2 The subjects of this thesis : :: ::: :: :: :: :: ::: :: :: : 8 1.3 Research goals : :: :: :: :: ::: :: :: :: :: ::: :: :: : 10 1.4 Discussion ::: :: :: :: :: ::: :: :: :: :: ::: :: :: : 11 2 An Ef®cient Non-Bonded Force Algorithm 13 2.1 Introduction :: :: :: :: :: ::: :: :: :: :: ::: :: :: : 13 2.2 Derivations ::: :: :: :: :: ::: :: :: :: :: ::: :: :: : 15 2.2.1 Notation :: :: :: :: ::: :: :: :: :: ::: :: :: : 15 2.2.2 Force derivations : :: ::: :: :: :: :: ::: :: :: : 16 2.2.3 Virial derivations : :: ::: :: :: :: :: ::: :: :: : 18 2.2.4 Neighbour searching :: ::: :: :: :: :: ::: :: :: : 21 2.3 The Implementation :: :: :: ::: :: :: :: :: ::: :: :: : 22 2.3.1 The algorithm : :: :: ::: :: :: :: :: ::: :: :: : 22 2.3.2 The implementation :: ::: :: :: :: :: ::: :: :: : 25 2.3.3 The machines used : :: ::: :: :: :: :: ::: :: :: : 26 2.3.4 The test M.D. system : ::: :: :: :: :: ::: :: :: : 27 2.3.5 The Test Runs : :: :: ::: :: :: :: :: ::: :: :: : 27 2.3.6 Results : :: :: :: :: ::: :: :: :: :: ::: :: :: : 27 2.4 Extensions ::: :: :: :: :: ::: :: :: :: :: ::: :: :: : 29 2.4.1 Generalisation to other box shapes :: :: :: ::: :: :: : 29 2.4.2 Applicability :: :: :: ::: :: :: :: :: ::: :: :: : 30 2.5 Conclusion ::: :: :: :: :: ::: :: :: :: :: ::: :: :: : 31 3 Uni®cation of Box Shapes in Molecular Simulations 33 3.1 Introduction :: :: :: :: :: ::: :: :: :: :: ::: :: :: : 33 3.2 De®ning primitive cells by a lattice and a metric : :: ::: :: :: : 36 3.3 De®ning boxes by their edges : ::: :: :: :: :: ::: :: :: : 39 3.4 Constructing simple boxes : :: ::: :: :: :: :: ::: :: :: : 42 3.5 Translating particles between primitive cells. :: :: ::: :: :: : 45 v vi 3.6 An example transformation of a simulation : :: :: ::: :: :: : 49 3.7 Related Topics : :: :: :: :: ::: :: :: :: :: ::: :: :: : 50 3.7.1 Pressure scaling :: :: ::: :: :: :: :: ::: :: :: : 50 3.7.2 Lattice reduction :: :: ::: :: :: :: :: ::: :: :: : 51 3.7.3 Long range order : :: ::: :: :: :: :: ::: :: :: : 53 3.7.4 How to set up a simulation :: :: :: :: :: ::: :: :: : 53 3.7.5 Which box to use: the triclinic or the rectangular? :::::: 55 3.8 Conclusion ::: :: :: :: :: ::: :: :: :: :: ::: :: :: : 57 Appendix A :: ::: :: :: :: :: ::: :: :: :: :: ::: :: :: : 58 Appendix B :: ::: :: :: :: :: ::: :: :: :: :: ::: :: :: : 59 4 Constraint Dynamics 63 4.1 Introduction :: :: :: :: :: ::: :: :: :: :: ::: :: :: : 63 4.2 Zeroth order equations of motion :: :: :: :: :: ::: :: :: : 64 4.3 First order equations of motion ::: :: :: :: :: ::: :: :: : 67 4.4 Second order equations of motion :: :: :: :: :: ::: :: :: : 69 4.5 Discussion ::: :: :: :: :: ::: :: :: :: :: ::: :: :: : 71 4.6 Example applications : :: :: ::: :: :: :: :: ::: :: :: : 72 4.7 Conclusion ::: :: :: :: :: ::: :: :: :: :: ::: :: :: : 79 Appendix C :: ::: :: :: :: :: ::: :: :: :: :: ::: :: :: : 79 5 Torsional-angle Potentials 81 5.1 Introduction :: :: :: :: :: ::: :: :: :: :: ::: :: :: : 81 5.2 Dihedral-angle force expressions :: :: :: :: :: ::: :: :: : 84 5.3 The virial of angle-dependent interactions :: :: :: ::: :: :: : 87 Appendix D :: ::: :: :: :: :: ::: :: :: :: :: ::: :: :: : 87 Appendix E :: ::: :: :: :: :: ::: :: :: :: :: ::: :: :: : 89 Appendix F :: ::: :: :: :: :: ::: :: :: :: :: ::: :: :: : 90 6 The Virial of Angle Dependent Potentials 93 6.1 Introduction :: :: :: :: :: ::: :: :: :: :: ::: :: :: : 93 6.2 Theory :: ::: :: :: :: :: ::: :: :: :: :: ::: :: :: : 94 6.2.1 The virial of interactions with angle dependent potentials :: 95 6.3 Simulated system and methods ::: :: :: :: :: ::: :: :: : 97 6.4 Simulation results : :: :: :: ::: :: :: :: :: ::: :: :: : 98 Appendix G :: ::: :: :: :: :: ::: :: :: :: :: ::: :: :: : 98 vii 7 Mapping MD on a Ring Architecture 105 7.1 Introduction :: :: :: :: :: ::: :: :: :: :: ::: :: :: : 105 7.2 M.D. simulation in more detail : ::: :: :: :: :: ::: :: :: : 106 7.2.1 Non Bonded Forces :: ::: :: :: :: :: ::: :: :: : 106 7.2.2 Bonded forces and constraints :: :: :: :: ::: :: :: : 108 7.3 Allocation of the NBF calculations on a ring : :: :: ::: :: :: : 109 7.4 Allocation of constraint- and BF calculations :: :: ::: :: :: : 113 7.4.1 Theory : :: :: :: :: ::: :: :: :: :: ::: :: :: : 113 7.4.2 Triplet and quadruplet allocation :: :: :: ::: :: :: : 116 7.5 Results and discussion : :: :: ::: :: :: :: :: ::: :: :: : 117 7.5.1 Test of Reduce on protein molecules :: :: ::: :: :: : 117 7.5.2 Discussion : :: :: :: ::: :: :: :: :: ::: :: :: : 118 7.6 Conclusion ::: :: :: :: :: ::: :: :: :: :: ::: :: :: : 120 8 Delay Insensitive Synchronisation 123 8.1 Introduction :: :: :: :: :: ::: :: :: :: :: ::: :: :: : 123 8.2 Constraint Molecular Dynamics simulation : :: :: ::: :: :: : 125 8.3 SHAKE on a ring architecture : ::: :: :: :: :: ::: :: :: : 128 8.4 The function ACWT implemented with the open collector bus :: : 131 8.5 Discussion ::: :: :: :: :: ::: :: :: :: :: ::: :: :: : 133 8.6 Conclusion ::: :: :: :: :: ::: :: :: :: :: ::: :: :: : 134 Samenvatting 137 Nawoord 143 Curriculum Vitae 145 Index 147 viii ix This thesis is based on the following papers: Chapter 2: ªAn ef®cient box shape independent non-bonded force algorithm for Molecular Dynamicsº, H. Bekker, E.J. Dijkstra, H.J.C. Berendsen, M.K.R. Renardus, Molecular Simulation, 1995, Vol. 14, pp. 137-151. Chapter 3: Submitted to Journal of Computational Chemistry. Chapter 4: To be submitted. Chapter 5: ªForce and Virial of Torsional-Angle-Dependent Potentialsº, H. Bekker, H.J.C. Berendsen, W.F. van Gunsteren, Journal of Computational Chemistry, Vol. 16, No. 5, 527-533 (1995). Chapter 6: ªThe Virial of Angle Dependent Potentials in Molecular Dynamics Simulationsº, H. Bekker and P. AhlstrÈom, Molecular Simulation, 1994, Vol. 13, pp. 367-374. Chapter 7: ªMapping molecular dynamics simulation calculations on a ring archi- tectureº, H. Bekker, E.J. Dijkstra, H.J.C. Berendsen, In Parallel Computing: From Theory to Sound Practice, ed. W. Joosen and E. Milgrom, pp. 268-279, IOS Press, Amsterdam, 1992. Chapter 8: ªDelay insensitive synchronisation on a message-passing architecture with an open collector busº, H. Bekker, E.J. Dijkstra Proceedings of PDP '96, IEEE Comp. Soc. Press, 1996. Other publications: ªDesign of a transputer network for searching neighbours in M.D. simulationsº, H. Bekker, M.K.R. Renardus, Microprocessing and Microprogramming, Vol. 30, 1-5, August 1990. x ªGROMACS: a Parallel Computer for Molecular Dynamics Simulationº, H. Bekker, et al, Conf. Proc. Physics Computing '92, pp. 252-256, World Scienti®c Publishing Co. Singapore, New York, London, 1993. ªGROMACS: Method of Virial Calculation Using a Single Sumº, H. Bekker, et al, Conf. Proc. Physics Computing '92, pp. 257-261, World Scienti®c Publishing Co. Singapore, New York, London, 1993. ªMolecular Dynamics simulation on an i860 based ring architectureº, H. Bekker, E.J. Dijkstra, H.J.C. Berendsen. Supercomputer 54, X-2, 4±10, 1993. 1 INTRODUCTION Molecular Dynamics (M.D.) Simulation is in principle very simple: the time devel- opment of a many particle system is evaluated by numerically integrating Newton's equations of motion. But, as with most simple principles, many additional concepts and techniques have to be applied to make the main principle operational. The ad- ditional concepts and techniques are required, not because the main principle needs correction or re®nement, but because a bare implementation of the main principle would result in very sluggish software without practical value. For that reason, from the outset in the ®fties, much work has been done to turn a simple principle into a useful tool. In this thesis no new techniques are proposed, but existing techniques are revised. As a result, a number of concepts of M.D. simulation are simpli®ed, and alternative implementations are proposed. In this chapter the main concepts of the M.D. simulation technique are introduced and an overview of this thesis is given. Also the goal of this thesis is formulated and some remarks are made about the way current M.D. implementations are created. 1.1 M.D. simulation in outline Main principle (t ) The main principle of M.D. simulation is as follows: given the system state S 0 , that is, the position r and velocity v of every particle (atom) in the system at time S (t + ∆t) S (t + ∆t) ::: t0, subsequent states 0 , 0 2 , , are calculated by using Newton's m ∆t law F = a. For accurate results small timesteps have to be used. To calculate S (t +(n+ )∆t) S (t + n∆t) i (t + n∆t) 0 1from 0 , ®rst for every particle , Fi 0 is calculated. (t + n∆t) i Fi 0 is the sum of the forces on as exerted by the other particles of the t + n∆t i (t + n∆t) system at time 0 . For every particle the force Fi 0