A Superlinear Minimization Scheme for the Nudged Elastic Band Method

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A Super-linear Minimization Scheme for the Nudged
Elastic Band Method

• Jhih-Wei Chu1, Bernard R. Brooks2 and Bernhardt
  Trout1
• 1Department of Chemical Engineering, MIT
• 2National Institutes of Health

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Motivation

• Understanding transition processes in complicated
  systems is important for scientists and engineers
• Catalytic reactions
• Enzymatic reactions
• Conformational changes of macromolecules

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Studying Transition Processes in Complicated
Systems

• Reaction mechanism
• Rate constants (transition state theory)
• Minimum energy paths (MEPs) gradient only exist
  along the path direction

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Motivation for Computing MEPs

• Defines a one-dimensional reaction coordinate
• Gas phase reactions
• Surface reactions
• Catalytic reactions
• Most probable Brownian Trajectory
• (R. Olender and R. Elber, Theochem, 1997)
• Inexpensively gain insight into possible reaction
  paths of complicates systems (reactions in
  solution)
• Can be used to define a blue-moon ensemble to
  compute free energy profile

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Outline

• Review of the NEB method
• Minimization issues
• The ABNR (Adopted Basis Newton-Raphson) scheme
• Super-linear minimization method for NEBexamples
• Blue-moon ensemble strategy using MEP
• Conclusions

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The Replica/Path Formulation for Searching
Reaction Paths
(R. Elber and M Karplus, Chemical Physics
Letters, 1987)
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Problems of Mixing Potential Energy and Spring
Penalty
Corner cutting
Sliding down
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The Nudged Elastic Band Method(H. Jónsson, in
Classical and Quantum Dynamics in Condensed Phase
Simulations, 1997)
Minimize the projected force vector gives a MEP
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Minimization Issues of NEB

• Define a Lagrangian is not straightforward
• Applying conjugate gradient minimization method
  may be inhibited
• Quenched molecular dynamics
• Linear convergence

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A Quasi-Newton Minimization Scheme Adopted
Basis Newton-Raphson (ABNR)

• The NR procedure is applied to a subspace of the
  system

• The subspace is updated by a SD step at each step

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Extend ABNR to Minimize a Path With NEB Force
Projections
Nonlinear and under-determined
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Results of NEB Force Projection in ABNR
Minimization

• A set of nonlinear and under-determined NR
  equations
• Self-consistent solutions of the tangent vector
• Use Singular Value Decomposition

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Test Case 1 Alanine Dipeptide

• United atom model
• CHARMM 19 force field
• 25 replicas
• Cyclic path

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Minimum Energy Path of Alanine Dipeptide
Isomerization
1
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Projection of MEP onto ?, ? surface
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Speed-up of Convergence Using ABNR
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Test Case 2 ?-Helix to ?-Helix Transition of an
Alanine Decapeptide

• All-atom model
• 105 atoms
• CHARMM 22 force field
• 51 replicas
• Three initial paths
• Concerted
• Zipup
• Zipdown

?-helix
?-helix
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Speed-up of Convergence Using ABNR
Concerted initial path
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Characteristics of Different Pathways
Concerted
Zipdown
Zipup
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Energy Profiles of Different Pathways
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Test Case 3 Oxidation of Dimethyl Sulfide by H2O2

• B3LYP/6-31G
• 25 atoms
• 20 Replicas
• CHARMM/GAMESS-UK interface

Reactant
Product
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Speed-up of Convergence Using ABNR
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Using NEB to Investigate Different Mechanisms
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Blue-Moon Ensemble Using Projected Order
Parameter from MEP
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Examine the Reaction Coordinate in Condense Phase
Simulations

• Examine the reaction coordinate by calculating
  commitment probability distribution on constraint
  MD trajectories
• Generate initial dynamic pathways
• Transition path sampling

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Example Oxidation of Methionine in Solution
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Conclusions

• Quadratic convergence of optimizing a reaction
  path with NEB force projections is achieved
• Rapidly evaluate different proposals of reaction
  mechanisms and activation barriers
• The method is generalized to the RMS best fit
  space for flexible choice of activate atoms
  (important for biological processes)
• Blue-Moon ensemble sampling procedure for the
  free energy profile using MEP is demonstrated
• Initial input for dynamic method

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Acknowledgement

• Amgen Inc.
• National University of Singapore
• SMA
• NIH Biowulf/Lobos3 System

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Blue-Moon Ensemble Approach for Free Energy
Simulation
M. Sprik and G. Ciccotti, JCP, 1998