Nonlinear Instability in Multiple Time Stepping Molecular Dynamics

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Nonlinear Instability in Multiple Time Stepping
Molecular Dynamics
Jesús Izaguirre, Qun Ma, Department of Computer
Science and EngineeringUniversity of Notre
Dame and Robert Skeel Department of Computer
Science and Beckman Institute University of
Illinois, Urbana-Champaign SAC03 March 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

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
Classical molecular dynamics

• Newtons equations of motion
• Atoms
• Molecules
• CHARMM potential(Chemistry at Harvard Molecular
  Mechanics)

Bonds, angles and torsions
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The CHARMM potential terms
Bond
Angle
Improper
Dihedral
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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/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
• 2nd order accurate, time reversible

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
Linear instability energy growth occurs unless
longest ?t lt 1/2 shortest period.
Total energy(Kcal/mol) vs. time (fs)
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
Ma, Izaguirre and Skeel (SISC, 2003)

• 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 lt 1/3 shortest period.
• Unconditionally unstable 3rd order nonlinear
  resonance
• Flexible waters outer time step less than
  33.3fs
• Constrained-bond proteins w/ SHAKE time step
  less than 45fs

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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

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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.

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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 31 and 41
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 31
(3.3363)
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Nonlinear numerical (cont.)
Fig. 3. Left Flexible melittin protein (PDB
entry 2mlt). Right energy drift from 10ns MD
simulation at 300K revealing 31 nonlinear
resonance (at 3, 3.27 and 3.78 fs).
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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 31 nonlinear resonance that
  causes overheating
• Longest time step lt 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 (SAC03),
  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 (SAC03), 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)318328, 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)17851791, 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 41 and 31
nonlinear resonance.