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Nonlinear Instability in Multiple Time Stepping Molecular Dynamics

Jesús Izaguirre, Qun Ma, Department of Computer Science and Engineering

University of Notre Dameand

Robert SkeelDepartment of Computer Science and Beckman Institute

University of Illinois, Urbana-Champaign

SAC’03March 10, 2003

Supported by NSF CAREER and BIOCOMPLEXITY grants

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Overview Background

Classical molecular dynamics (MD) Multiple time stepping integrator Linear instability

Nonlinear instabilities Analytical approach Numerical approach Concluding remarks

Acknowledgements Key references

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Overview Background

Classical molecular dynamics (MD) Multiple time stepping integrator Linear instability

Nonlinear instabilities Analytical approach Numerical approach Concluding remarks

Acknowledgements Key references

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Classical molecular dynamics Newton’s equations

of motion:

Atoms Molecules CHARMM potential

(Chemistry at Harvard Molecular Mechanics)

'' ( ) ( ). - - - (1)U Mr r F r

Bonds, angles and torsions

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The CHARMM potential terms

Bond Angle

Dihedral Improper

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Energy functions

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Overview Background

Classical molecular dynamics (MD) Multiple time stepping integrator Linear instability

Nonlinear instabilities Analytical approach Numerical approach Concluding remarks

Acknowledgements Key references

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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

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Verlet-I/r-RESPA/ImpulseGrubmüller,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 2nd order accurate, time reversible

slow1/ 2half kick: ( ) / 2

oscillate: update positions and momentum

using Verlet/leapfrog ( /2, much smaller time steps)

n n nv v t f r

t

Algorithm 1. Half step discretization of Impulse integrator

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Overview Background

Classical molecular dynamics (MD) Multiple time stepping integrator Linear instability

Nonlinear instabilities Analytical approach Numerical approach Concluding remarks

Acknowledgements Key references

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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

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Overview Background

Classical molecular dynamics (MD) Multiple time stepping integrator Linear instability

Nonlinear “instabilities” (overheating) Analytical approach Numerical approach Concluding remarks

Acknowledgements Key references

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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 3rd 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)

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Overview Background

Classical molecular dynamics (MD) Multiple time stepping integrator Linear instability

Nonlinear instabilities Analytical approach Numerical approach Concluding remarks

Acknowledgements Key references

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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:

3rd order resonance stable only if “equality” met 4th order resonance stable only if “inequality” met Impulse unstable at 3rd order resonance in practice

2 2 2 3 4 5oscillate kick( ) ( / 2) ( / 2 / 3 / 4) ( ).U q q Aq Bq Cq O q

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Nonlinear: analytical (cont.) Main result. Let

1. (3rd order) Map stable at equilibrium if and unstable if

Impulse is unstable in practice. 2. (4th order) Map stable if

and unstable if

May be stable at the 4th order resonance.

2 , where , where

1 ' /(2 ') if ' 0, 0 if ' 0, and

1 ' /(2 ') if s '/ 0, 0 if s '/ 0, and

' sin( / 2) and c ' cos( / 2).

i

hs A c c c

hc A s

s h h

0, 0,B C 0.B

20 or 2 ' '/ ,C C hB s c 20 2 ' '/ .C hB s c

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Overview Background

Classical molecular dynamics (MD) Multiple time stepping integrator Linear instability

Nonlinear instabilities Analytical approach Numerical approach Concluding remarks

Acknowledgements Key references

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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)

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Nonlinear resonance: numerical

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

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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.)

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Overview Background

Classical molecular dynamics (MD) Multiple time stepping integrator Linear instability

Nonlinear instabilities Analytical approach Numerical approach Concluding remarks

Acknowledgements Key references

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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

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Overview Background

Classical molecular dynamics (MD) Multiple time stepping integrator Linear instability

Nonlinear instabilities Analytical approach Numerical approach Concluding remarks

Acknowledgements Key references

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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

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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

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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

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THE END. THANKS!

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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.