Transition Path Sampling

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Transition Path Sampling

• Janne Grunau, Alexander Riemer
• Seminar Fortgeschrittene Methoden in der
  Molekulardynamik
• 8 June 2005

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Table of Contents

• Introduction
• Metropolis Monte Carlo
• Efficient Trial Moves
• Shooting
• Shifting
• Generating an Initial Path
• Determining Reaction Rates
• Example 7-Atom Lennard-Jones cluster
• Conclusion

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Introduction
Intro MC Trial Moves Initial Path
Reaction Rates Example Conclusion

• MD trajectories sample potential energy surface
• Good sampling of local minima ? (meta)stable
  states
• What about transitions between stable states?
• Identify activated transition states
• Describe dynamics of complex systems
• Example protonated water

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Protonated Water
Intro MC Trial Moves Initial Path
Reaction Rates Example Conclusion
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Looking at Rare Events
Intro MC Trial Moves Initial Path
Reaction Rates Example Conclusion

• Important transitions happen slowly or
  infrequently
• Pseudo-rotation in cyclohexane 10µs
• Folding of small to medium-sized proteins
  10-5-10-3s
• Protonated water trimer hours to observe 1
  transition
• Time step 1-10fs ? transitions occur very rarely
• Approach
• Look for saddle points in the energy surface
• Identify activated states that lie on
  transition paths
• Rough energy surface, high dimension
• No analytical solution

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Transition Path Sampling
Intro MC Trial Moves Initial Path
Reaction Rates Example Conclusion

• Solution
• Monte Carlo method over trajectory space
• Look only at trajectories connecting two
  interesting stable states
• Carefully reweight trajectories to preserve
  true dynamics / kinetics
• Throwing Ropes Over Mountains in the Dark

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Transition Path Ensemble
Intro MC Trial Moves Initial Path
Reaction Rates Example Conclusion

• Given ensemble of trajectories, stable states A,
  B
• Trajectory x has statistical weight P(x), e.g.
• Subset of trajectories connecting A B ?
  reactive
• Reweighting yields transition path ensemble

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Defining Initial Final Regions
Intro MC Trial Moves Initial Path
Reaction Rates Example Conclusion

• Defined by low dimensional parameter
• Parameter must distinguish sharply
• Otherwise
• SolutionPhysical intuitiontrial error

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Metropolis Monte Carlo Sampling
Intro MC Trial Moves Initial Path
Reaction Rates Example Conclusion

• Pick a trajectory x(o) from transition path
  ensemble.
• Generate new pathway x(n) with probability
  Pgen(x(o)?x(n)).
• Accept or reject according to Metropolis
  acceptance criterion obeying detailed balance
  w.r.t. transition path ensemble
• Repeat from 1.

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Acceptance Step
Intro MC Trial Moves Initial Path
Reaction Rates Example Conclusion

• Detailed balance criterion
• Application of Metropolis rule
• Acceptance rate (
  )

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Shooting Moves
Intro MC Trial Moves Initial Path
Reaction Rates Example Conclusion

• Modify point ?
• Propagate forward backward in time
• Proposal probability

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Shooting Moves (2)
Intro MC Trial Moves Initial Path
Reaction Rates Example Conclusion

• Acceptance probability
• Blackboard
• For symmetric generation probabilities
• Efficient implementation due to
• First acceptance/rejection decision for
• Then acc./rej. decisions for fwd. bwd.
  trajectory

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Shifting/Reptation Moves
Intro MC Trial Moves Initial Path
Reaction Rates Example Conclusion

• Translate an existing trajectory forward or
  backward in time
• Delete segment of length dt from start or end
• Then extend the other end by dt simulation steps

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Shifting Moves (2)
Intro MC Trial Moves Initial Path
Reaction Rates Example Conclusion

• Proposal probability
• Acceptance probability
• Blackboard
• Under certain restrictions

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Shifting Moves (3)
Intro MC Trial Moves Initial Path
Reaction Rates Example Conclusion

• Average acceptance rate controlled by dt
• Low computation time for short width shifting
• Hard to change behaviour in transition region
• Random walk less random than w/ shooting
• Should be combined with other trial moves

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Generating an Initial Path
Intro MC Trial Moves Initial Path
Reaction Rates Example Conclusion

• Problem Need at least one reactive trajectory
• Continue sampling until found
• Construct arbitrary pathway from A to B
• ? Relaxing through Monte Carlo method
• Conduct high temperature sampling
• Employ assumptions about reaction kinetics

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Determining Reaction Rates
Intro MC Trial Moves Initial Path
Reaction Rates Example Conclusion

• True dynamical pathways ? reflect kinetics
• Most important reaction rates kAB and kBA
• Look at correlation coefficient
• Reversible work is a free energy difference

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Reaction Rates (2)
Intro MC Trial Moves Initial Path
Reaction Rates Example Conclusion

• Correlation coefficient C(t) ? kinetics

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Sampling C(t)
Intro MC Trial Moves Initial Path
Reaction Rates Example Conclusion

• Problem How to calculate C(t) efficiently?
• Recall Low dimensional parameter ? distinguishes
  between A and B

• Idea
• Sample ? within discrete windows

? Not
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Umbrella Sampling
Intro MC Trial Moves Initial Path
Reaction Rates Example Conclusion

• Approach Umbrella sampling
• Introduce bias to force better sampling of
  interesting regions in parameter space
• Bias Modified potential function
• ? sampled within discrete windows
• distribution of ? at time t in
  trajectories started in A

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Efficiently Computing C(t)
Intro MC Trial Moves Initial Path
Reaction Rates Example Conclusion

• Idea
• Define

Umbrella sampling 1x
Calculated efficiently for all t
Reweighted trajectories with statistical weight
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Efficiently Computing C(t) (2)
Intro MC Trial Moves Initial Path
Reaction Rates Example Conclusion

• Use MMC again for PAB-ensemble
• Simply replace hB(xt) with HB(x)
• ? R(t,t) for all t simultaneously
• Recall that R(t,t) C(t)/C(t)
• Calculate C(t) by umbrella sampling
• Get C(t) for all other t from PAB
• kAB is derivative of asymptotic slope of C(t)

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Example
Intro MC Trial Moves Initial Path
Reaction Rates Example Conclusion

• Application of the method to a 7-atom
  Lennard-Jones cluster in 2D
• Potential
• Langevin equation
• modified Verlet integration

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Initial transition path
Intro MC Trial Moves Initial Path
Reaction Rates Example Conclusion
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Generated transition path
Intro MC Trial Moves Initial Path
Reaction Rates Example Conclusion
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Example
Intro MC Trial Moves Initial Path
Reaction Rates Example Conclusion
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Example Path quenching
Intro MC Trial Moves Initial Path
Reaction Rates Example Conclusion

• Gradient search on the transition path ensemble
• Generates least action paths
• Determines meta-stable states and transition
  states

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Example - Observations
Intro MC Trial Moves Initial Path
Reaction Rates Example Conclusion

• Hard to find initial path
• For high energy initial paths, all generated
  paths have similar energy
• Shifting increases acceptance rate for following
  shooting step
• Creates very diverse paths

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Conclusion
Intro MC Trial Moves Initial Path
Reaction Rates Example Conclusion

• TPS performs MMC in trajectory space rather than
  long MD/MC in coordinate space ? reduced
  computational cost
• Reweighting of trajectories preserves true
  dynamics and allows inference of rate const.
• Requires no knowledge of true dynamics and no
  assumptions about transitions
• Applicable to deterministic and stochastic
  dynamics likewise

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References
Intro MC Trial Moves Initial Path
Reaction Rates Example Conclusion

• Understanding Molecular Simulation, Daan
  Frenkel Berend Smit, Academic Press, 2. edition
  2001
• Transition Path Sampling, Christoph Dellago,
  Peter G. Bolhuis Phillip L Geissler, Adv. Chem.
  Phys. 123, 1 (2002)
• Efficient Transition Path Sampling Application
  to Lennard-Jones Cluster Rearrangements,
  Christoph Dellago, Peter G. Bolhuis David
  Chandler, J. Chem. Phys. 108, 9236 (1998)
• Transition Path Sampling Throwing Ropes over
  Mountains in the Dark, Christoph Dellago, Peter
  G. Bolhuis, Phillip L Geissler David Chandler,
  J. Phys. Cond. Matt. 12, A 147 (2000)