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### Targeted Langevin Stabilization of Molecular Dynamics

OverviewOverviewOverviewOverviewOverview

Qun (Marc) Ma and Jesús A. Izaguirre

Department of Computer Science and EngineeringUniversity of Notre Dame

CSE’03

February 9, 2003

Supported by:

NSF BIOCOMPLEXITY-IBN-0083653, and

NSF CAREER Award ACI-0135195

Overview

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

Overview

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

Classical molecular dynamics

- Newton’s equations of motion:
- Atoms
- Molecules
- CHARMM potential(Chemistry at Harvard MolecularMechanics)
- Initial value problem
- Require correct statistics

Bonds, angles and torsions

Overview

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

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

The Impulse integrator

Grubmüller,Heller, Windemuth and Schulten, 1991 Tuckerman, Berne and Martyna, 1992

- The impulse “train”

Fast impulses, t

Time, t

Slow impulses, t

How far apart can we stretch the impulse train?

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

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!

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

Objective statement

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

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

Targeted MOLLY (TM)

MaIz, 2003 ACM Symp. App. Comp. (SAC’03), 2002 (in press)

MaIz, SIAM J. Multiscale Modeling and Simulation, 2003 (submitted)

- 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

Mollified Impulse (MOLLY)

- MOLLY (mollified Impulse)
- Slow potential at time averaged positions, A(x)
- Averaging using only fastest forces
- Mollified slow force = Ax(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)

Vj

FRi, FDi

FRj = - FRiFDj = - FDi

Introducing stochasticity- Langevin stabilization of MOLLY (LM)

Izaguirre, Catarello, et al, 2001

- 12 fs for flexible waters with correct dynamics
- Dissipative particle dynamics (DPD):

Pagonabarraga, Hagen and Frenkel, 1998

- Pair-wiseLangevin force on “particles”
- Time reversible if self-consistent

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

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

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.)

TM: correct r.d.f.

Fig. 4. Radial distribution function of O-H (left) and H-H (right) in flexible waters.

libfrontend

Middle layer

libintegrators

back-end

libbase, libtopology

libparallel, libforces

ProtoMol: the framework for MDMatthey, et al, ACM Tran. Math. Software (TOMS), submitted

Modular design of ProtoMol (Prototyping Molecular dynamics).

Available at http://www.cse.nd.edu/~lcls/protomol

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

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-0083653, and
- NSF CAREER Award ACI-0135195

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 146-174, 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, 2002. Available at http://www.nd.edu/~qma1/publication_h.html.

[3] Q. Ma and J. A. Izaguirre. Targeted mollified impulse method for molecular dynamics. Submitted to the SIAM Journal on Multiscale Modeling and Simulation, 2003.

[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), 2003.

[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., 25:297-302, 1997

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):2090-2098, Feb. 1, 2001.

[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, 2001.

[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), 121-142.

[11] M. Tuckerman, B. J. Berne, and G. J. Martyna, Reversible multiple time scale molecular dynamics, J. Chem. Phys 97 (1992), no. 3, 1990-2001

[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, 1999.

[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, 1999.

[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 119-176. 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 1 2001.

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., 1997.

[17] I. Pagonabarraga, M. H. J. Hagen and D. Frenkel. Self-consistent dissipative particle dynamics algorithm. Europhys Lett., 42 (4), pp. 377-382 (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. 2000.

[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 15 1997.

[20] I. Pagonabarraga and D. Frenkel. Dissipative particle dynamics for interacting systems. J. Chem. Phys., 115(11):5015–5026, September 15 2001.

[21] Y. Duan and P. A. Kollman. Pathways to a protein folding intermediate observed in a 1-microsecond simulation in aqueous solution. Science, 282:740-744, 1998.

[22] M. Levitt. The molecular dynamics of hydrogen bonds in bovine pancreatic tripsin inhibitor protein, Nature, 294, 379-380, 1981

[23] M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids, Oxford University Press, New York, 1987.

[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, 223-230, 1973

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