1 Targeted Langevin Stabilization of Molecular Dynamics Qun (Marc) Ma and Jesús A. Izaguirre Department of Computer Science and Engineering University of Notre Dame CSE’03 February 9, 2003 Supported by: NSF BIOCOMPLEXITY-IBN , and NSF CAREER Award ACI

2 Overview Background Classical molecular dynamics (MD) Multiple time stepping integrator Nonlinear instabilities Targeted MOLLY Objective statement Targeted Langevin coupling Results Acknowledgements

3 Overview Background Classical molecular dynamics (MD) Multiple time stepping integrator Nonlinear instabilities Targeted MOLLY Objective statement Targeted Langevin coupling Results Acknowledgements

4 Classical molecular dynamics Newton’s equations of motion: Atoms Molecules CHARMM potential (Chemistry at Harvard Molecular Mechanics) Initial value problem Require correct statistics Bonds, angles and torsions

5 Overview Background Classical molecular dynamics (MD) Multiple time stepping integrator Nonlinear instabilities Targeted MOLLY Objective statement Targeted Langevin coupling Results Acknowledgements

6 Multiple time stepping Fast/slow force splitting Bonded: “fast” Small periods Long range nonbonded: “slow” Large characteristic time Evaluate slow forces less frequently Fast forces cheap Slow force evaluation expensive

7 The Impulse integrator Grubmüller,Heller, Windemuth and Schulten, 1991 Tuckerman, Berne and Martyna, 1992 The impulse “train” Time, t Fast impulses,  t Slow impulses,  t How far apart can we stretch the impulse train?

8 Overview Background Classical molecular dynamics (MD) Multiple time stepping integrator Nonlinear instabilities Targeted MOLLY Objective statement Targeted Langevin coupling Results Acknowledgements

9 Stretching slow impulses  t ~ 100 fs if accuracy does not degenerate 1/10 of the characteristic time MaIz, SIAM J. Multiscale Modeling and Simulation, 2003 (submitted) Resonances (let  be the shortest period) Natural:  t = n , n = 1, 2, 3, … Numerical: Linear:  t =  /2 Nonlinear:  t =  /3 MaIS_a, SIAM J. on Sci. Comp. (SISC), 2002 (in press) MaIS_b, 2003 ACM Symp. App. Comp. (SAC’03), 2002 (in press) MTS limited by instabilities, not acuracy!

10 Overview Background Classical molecular dynamics (MD) Multiple time stepping integrator Nonlinear instabilities Targeted MOLLY Objective statement Targeted Langevin coupling Results Acknowledgements

11 Objective statement Design multiscale integrators that are not limited by nonlinear and linear instabilities Allowing longer time steps Better sequential performance Better scaling

12 Overview Background Classical molecular dynamics (MD) Multiple time stepping integrator Nonlinear instabilities Targeted MOLLY Objective statement Targeted Langevin coupling Results Acknowledgements

13 Targeted MOLLY (TM) TM = MOLLY + targeted Langevin coupling Mollified Impulse (MOLLY) to overcome linear instabilities Izaguirre, Reich and Skeel, 1999 Stochasticity to stabilize MOLLY Izaguirre, Catarello, et al, 2001 MaIz, 2003 ACM Symp. App. Comp. (SAC’03), 2002 (in press) MaIz, SIAM J. Multiscale Modeling and Simulation, 2003 (submitted)

14 Mollified Impulse (MOLLY) MOLLY (mollified Impulse) Slow potential at time averaged positions, A(x) Averaging using only fastest forces Mollified slow force = A x (x) F(A(x)) Equilibrium and B-spline B-spline MOLLY Averaging over certain time interval Needs analytical Hessians Step sizes up to 6 fs (50~100% speedup)

15 Introducing stochasticity Langevin stabilization of MOLLY (LM) Izaguirre, Catarello, et al, fs for flexible waters with correct dynamics Dissipative particle dynamics (DPD): Pagonabarraga, Hagen and Frenkel, 1998 Pair-wise Langevin force on “particles” Time reversible if self-consistent V i VjVj F R i, F D i F R j = - F R i F D j = - F D i

16 Targeted Langevin coupling Targeted at “trouble-making” pairs Bonds, angles Hydrogen bonds Stabilizing MOLLY Slow forces evaluated much less frequently Recovering correct dynamics Coupling coefficient small

17 Overview Background Classical molecular dynamics (MD) Multiple time stepping integrator Nonlinear instabilities Targeted MOLLY Objective statement Targeted Langevin coupling Results Acknowledgements

18 TM: main results 16 fs for flexible waters Correct dynamics Self-diffusion coefficient, D. leapfrog w/ 1fs, D = 3.69+/-0.01 TM w/ (16 fs, 2fs), D = 3.68+/-0.01 Correct structure Radial distribution function (r.d.f.)

19 TM: correct r.d.f. Fig. 4. Radial distribution function of O-H (left) and H-H (right) in flexible waters.

20 ProtoMol: the framework for MD Front-end Middle layer back-end libfrontend libintegrators libbase, libtopology libparallel, libforces Modular design of ProtoMol (Prototyping Molecular dynamics). Available at Matthey, et al, ACM Tran. Math. Software (TOMS), submitted

21 Overview Background Classical molecular dynamics (MD) Multiple time stepping integrator Nonlinear instabilities Targeted MOLLY Objective statement Targeted Langevin coupling Results Acknowledgements

22 Acknowledgements People Dr. Jesus Izaguirre Dr. Robert Skeel, Univ. of Illinois at Urbana-Champaign Dr. Thierry Matthey, University of Bergen, Norway Resources Hydra and BOB clusters at ND Norwegian Supercomputing Center, Bergen, Norway Funding NSF BIOCOMPLEXITY-IBN , and NSF CAREER Award ACI

23 THE END. THANKS!

24 Key references [1] J. A. Izaguirre, Q. Ma, T. Matthey, et al. Overcoming instabilities in Verlet-I/r-RESPA with the mollified impulse method. 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 , Springer-Verlag, Berlin, New York, 2002 [2] Q. Ma, J. A. Izaguirre, and R. D. Skeel. Verlet-I/r-RESPA/Impulse is limited by nonlinear instability. Accepted by the SIAM Journal on Scientific Computing, Available at [3] Q. Ma and J. A. Izaguirre. Targeted mollified impulse method for molecular dynamics. Submitted to the SIAM Journal on Multiscale Modeling and Simulation, [4] T. Matthey, T. Cickovski, S. Hampton, A. Ko, Q. Ma, T. Slabach and J. Izaguirre. PROTOMOL, an object-oriented framework for prototyping novel applications of molecular dynamics. Submitted to the ACM Transactions on Mathematical Software (TOMS), [5] Q. Ma, J. A. Izaguirre, and R. D. Skeel. Nonlinear instability in multiple time stepping molecular dynamics. Accepted by the 2003 ACM Symposium on Applied Computing (SAC’03). Melborne, Florida. March 2003 [6] Q. Ma and J. A. Izaguirre. Long time step molecular dynamics using targeted Langevin Stabilization. Accepted by the 2003 ACM Symposium on Applied Computing (SAC’03). Melborne, Florida. March 2003 [7] M. Zhang and R. D. Skeel. Cheap implicit symplectic integrators. Appl. Num. Math., 25: , 1997

25 Other references [8] J. A. Izaguirre, Justin M. Wozniak, Daniel P. Catarello, and Robert D. Skeel. Langevin Stabilization of Molecular Dynamics, J. Chem. Phys., 114(5): , Feb. 1, [9] T. Matthey and J. A. Izaguirre, ProtoMol: A Molecular Dynamics Framework with Incremental Parallelization, in Proc. of the Tenth SIAM Conf. on Parallel Processing for Scientific Computing, [10] H. Grubmuller, H. Heller, A. Windemuth, and K. Schulten, Generalized Verlet algorithm for efficient molecular dynamics simulations with long range interactions, Molecular Simulations 6 (1991), [11] M. Tuckerman, B. J. Berne, and G. J. Martyna, Reversible multiple time scale molecular dynamics, J. Chem. Phys 97 (1992), no. 3, [12] J. A. Izaguirre, S. Reich, and R. D. Skeel. Longer time steps for molecular dynamics. J. Chem. Phys., 110(19):9853–9864, May 15, [13] L. Kale, R. Skeel, M. Bhandarkar, R. Brunner, A. Gursoy, N. Krawetz, J. Phillips, A. Shinozaki, K. Varadarajan, and K. Schulten. NAMD2: Greater scalability for parallel molecular dynamics. J. Comp. Phys., 151:283–312, [14] R. D. Skeel. Integration schemes for molecular dynamics and related applications. In M. Ainsworth, J. Levesley, and M. Marletta, editors, The Graduate Student’s Guide to Numerical Analysis, SSCM, pages Springer-Verlag, Berlin, 1999 [15] R. Zhou,, E. Harder, H. Xu, and B. J. Berne. Efficient multiple time step method for use with ewald and partical mesh ewald for large biomolecular systems. J. Chem. Phys., 115(5):2348– 2358, August

26 Other references (cont.) [16] E. Barth and T. Schlick. Extrapolation versus impulse in multiple-time-stepping schemes: Linear analysis and applications to Newtonian and Langevin dynamics. J. Chem. Phys., [17] I. Pagonabarraga, M. H. J. Hagen and D. Frenkel. Self-consistent dissipative particle dynamics algorithm. Europhys Lett., 42 (4), pp (1998). [18] G. Besold, I. Vattulainen, M. Kartunnen, and J. M. Polson. Towards better integrators for dissipative particle dynamics simulations. Physical Review E, 62(6):R7611–R7614, Dec [19] R. D. Groot and P. B. Warren. Dissipative particle dynamics: Bridging the gap between atomistic and mesoscopic simulation. J. Chem. Phys., 107(11):4423–4435, Sep [20] I. Pagonabarraga and D. Frenkel. Dissipative particle dynamics for interacting systems. J. Chem. Phys., 115(11):5015–5026, September [21] Y. Duan and P. A. Kollman. Pathways to a protein folding intermediate observed in a 1- microsecond simulation in aqueous solution. Science, 282: , [22] M. Levitt. The molecular dynamics of hydrogen bonds in bovine pancreatic tripsin inhibitor protein, Nature, 294, , 1981 [23] M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids, Oxford University Press, New York, [24] E. Hairer, C. Lubich and G. Wanner. Geometric numerical integration: structure-preserving algorithms for ordinary differential equations. Springer, 2002 [25] C. B. Anfinsen. Principles that govern the folding of protein chains. Science, 181, , 1973