1. 1 Nonlinear Instability in Multiple Time Stepping Molecular Dynamics Jesús Izaguirre, Qun Ma, Department of Computer Science and Engineering University of Notre Dame and Robert Skeel Department of Computer Science and Beckman Institute University of Illinois, Urbana-Champaign SAC’03 March 10, 2003 Supported by NSF CAREER and BIOCOMPLEXITY grants

2. 2 Overview Background Classical molecular dynamics (MD) Multiple time stepping integrator Linear instability Nonlinear instabilities Analytical approach Numerical approach Concluding remarks Acknowledgements Key references

3. 3 Overview Background Classical molecular dynamics (MD) Multiple time stepping integrator Linear instability Nonlinear instabilities Analytical approach Numerical approach Concluding remarks Acknowledgements Key references

4. 4 Classical molecular dynamics Newton’s equations of motion: Atoms Molecules CHARMM potential (Chemistry at Harvard Molecular Mechanics) Bonds, angles and torsions

5. 5 The CHARMM potential terms BondAngle Dihedral Improper

6. 6 Energy functions

7. 7 Overview Background Classical molecular dynamics (MD) Multiple time stepping integrator Linear instability Nonlinear instabilities Analytical approach Numerical approach Concluding remarks Acknowledgements Key references

8. 8 Multiple time stepping Fast/slow force splitting Bonded: “fast” (small periods) Long range nonbonded: “slow” (large char. time) Evaluate slow forces less frequently Fast forces cheap Slow force evaluation expensive Fast forces,  t Slow forces,  t

9. 9 Verlet-I/r-RESPA/Impulse Grubmüller,Heller, Windemuth and Schulten, 1991 Tuckerman, Berne and Martyna, 1992 The state-of-the-art MTS integrator Fast/slow splitting of nonbonded terms via switching functions 2 nd order accurate, time reversible Algorithm 1. Half step discretization of Impulse integrator

10. 10 Overview Background Classical molecular dynamics (MD) Multiple time stepping integrator Linear instability Nonlinear instabilities Analytical approach Numerical approach Concluding remarks Acknowledgements Key references

11. 11 Linear instability of Impulse Total energy(Kcal/mol) vs. time (fs) Linear instability: energy growth occurs unless longest  t < 1/2 shortest period. Impulse MOLLY - ShortAvg MOLLY - LongAvg

12. 12 Overview Background Classical molecular dynamics (MD) Multiple time stepping integrator Linear instability Nonlinear “instabilities” (overheating) Analytical approach Numerical approach Concluding remarks Acknowledgements Key references

13. 13 Nonlinear instability of Impulse Approach Analytical: Stability conditions for a nonlinear model problem Numerical: Long simulations differing only in outer time steps; correlation between step size and overheating Results: energy growth occurs unless longest  t < 1/3 shortest period. Unconditionally unstable 3 rd order nonlinear resonance Flexible waters: outer time step less than 3~3.3fs Constrained-bond proteins w/ SHAKE: time step less than 4~5fs Ma, Izaguirre and Skeel (SISC, 2003)

14. 14 Overview Background Classical molecular dynamics (MD) Multiple time stepping integrator Linear instability Nonlinear instabilities Analytical approach Numerical approach Concluding remarks Acknowledgements Key references

15. 15 Nonlinear instability: analytical Approach: 1-D nonlinear model problem, in the neighborhood of stable equilibrium MTS splitting of potential: Analyze the reversible symplectic map Express stability condition in terms of problem parameters Result: 3 rd order resonance stable only if “equality” met 4 th order resonance stable only if “inequality” met Impulse unstable at 3 rd order resonance in practice

16. 16 Nonlinear: analytical (cont.) Main result. Let 1. (3 rd order) Map stable at equilibrium if and unstable if Impulse is unstable in practice. 2. (4 th order) Map stable if and unstable if May be stable at the 4 th order resonance.

17. 17 Overview Background Classical molecular dynamics (MD) Multiple time stepping integrator Linear instability Nonlinear instabilities Analytical approach Numerical approach Concluding remarks Acknowledgements Key references

18. 18 Nonlinear resonance: numerical Fig. 1: Left: Flexible water system. Right: Energy drift from 500ps MD simulation of flexible water at room temperature revealing 3:1 and 4:1 nonlinear resonance (3.3363 and 2.4 fs)

19. 19 Nonlinear resonance: numerical Fig. 2. Energy drift from 500ps MD simulation of flexible water at room temperature revealing 3:1 (3.3363)

20. 20 Fig. 3. Left: Flexible melittin protein (PDB entry 2mlt). Right: energy drift from 10ns MD simulation at 300K revealing 3:1 nonlinear resonance (at 3, 3.27 and 3.78 fs). Nonlinear: numerical (cont.)

21. 21 Overview Background Classical molecular dynamics (MD) Multiple time stepping integrator Linear instability Nonlinear instabilities Analytical approach Numerical approach Concluding remarks Acknowledgements Key references

22. 22 Concluding remarks MTS restricted by a 3:1 nonlinear resonance that causes overheating Longest time step < 1/3 fastest normal mode Important for long MD simulations due to: Faster computers enabling longer simulations Long time kinetics and sampling, e.g., protein folding Use stochasticity for long time kinetics For large system size, NVE  NVT

23. 23 Overview Background Classical molecular dynamics (MD) Multiple time stepping integrator Linear instability Nonlinear instabilities Analytical approach Numerical approach Concluding remarks Acknowledgements Key references

24. 24 Acknowledgements People Dr. Thierry Matthey Dr. Ruhong Zhou, Dr. Pierro Procacci Dr. Andrew McCammon hosted JI in May 2001 at UCSD Dept. of Mathematics, UCSD, hosted RS Aug. 2000 – Aug. 2001 Resources Hydra and BOB clusters at ND Norwegian Supercomputing Center, Bergen, Norway Funding NSF CAREER Award ACI-0135195 NSF BIOCOMPLEXITY-IBN-0083653

25. 25 Key references [1] Overcoming instabilities in Verlet-I/r-RESPA with the mollified impulse method. J. A. Izaguirre, Q. Ma, T. Matthey, et al.. In T. Schlick and H. H. Gan, editors, Proceedings of the 3rd International Workshop on Algorithms for Macromolecular Modeling, Vol. 24 of Lecture Notes in Computational Science and Engineering, pages 146-174, Springer-Verlag, Berlin, New York, 2002 [2] Verlet-I/r-RESPA/Impulse is limited by nonlinear instability. Q. Ma, J. A. Izaguirre, and R. D. Skeel. Accepted by the SIAM Journal on Scientific Computing, 2002. Available at http://www.nd.edu/~qma1/publication_h.html. [3] Targeted mollified impulse – a multiscale stochastic integrator for molecular dynamics. Q. Ma and J. A. Izaguirre. Submitted to the SIAM Journal on Multiscale Modeling and Simulation, 2003. [4] Nonlinear instability in multiple time stepping molecular dynamics. Q. Ma, J. A. Izaguirre, and R. D. Skeel. In Proceedings of the 2003 ACM Symposium on Applied Computing (SAC’03), pages 167-171, Melborne, Florida. March 9-12, 2003

26. 26 Key references [5] Long time step molecular dynamics using targeted Langevin Stabilization. Q. Ma and J. A. Izaguirre. In Proceedings of the 2003 ACM Symposium on Applied Computing (SAC’03), pages 178-182, Melborne, Florida. March 9-12, 2003 [6] Dangers of multiple-time-step methods. J. J. Biesiadecki and R. D. Skeel. J. Comp. Phys., 109(2):318–328, Dec. 1993. [7] Difficulties with multiple time stepping and the fast multipole algorithm in molecular dynamics. T. Bishop, R. D. Skeel, and K. Schulten. J. Comp. Chem., 18(14):1785–1791, Nov. 15, 1997. [8] Masking resonance artifacts in force-splitting methods for biomolecular simulations by extrapolative Langevin dynamics. A. Sandu and T. Schlick. J. Comut. Phys, 151(1):74-113, May 1, 1999

27. 27 THE END. THANKS!

28. 28 Nonlinear: numerical (cont.) Fig. 4. Left: Melittin protein and water. Right: Energy drift from 500ps SHAKE- constrained MD simulation at 300K revealing combined 4:1 and 3:1 nonlinear resonance.