Theory and Modeling of Rare Events

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Theory and Modeling of
Rare Events

• Weinan E
• Princeton University
• Joint work with Weiqing Ren and
• Eric Vanden-Eijnden (Courant Inst, New York
  University)
• Supported by ONR

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Problem Conformational change of biomolecules
in explicit solvent, at room temperature
?
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Time history of torsion angles (in vacuum)
Conformational changes are rare events. This is a
multi-time-scale problem.
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Rare events are of general interest

• Conformational changes of biomolecules
• Chemical reactions
• Nucleation events
• ? ? ? ? ? ?
• Transition caused by very small thermal noise.

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What do we want to know?

• Transition pathways, bottlenecks,
• Transition mechanism.
• Transition rates

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Plan for the talk

• Theoretical foundation
• Numerical modeling
• Two different situations
• Classical case Simple energy landscape
• Complex systems Rough energy landscape

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Two different situations
Complex
Simple
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Consider the following (frictional) dynamics
Issues of dynamics Most results can be
extended to Langevin dynamics. Many issues
for Hamiltonian (NVE) dynamics remain open.
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Simple system Transition state theory
(H. Eyring, E. Wigner, H. Kramers, 1930-40)

• Identify transition state (saddle point)
• Compute rates by harmonic approximation

Extended to higher dimensions
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Transition State Theory Extensions

• Dividing surfaces, replace saddle points
• Variational TST optimal dividing surfaces
• Transmission coefficients (re-crossing issue)

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Pathways Minimum energy path (MEP)

• When transition involves several saddle points
  and intermediate states, identify the relevant
  sequence of saddle points.

This is also the most probable path
(consequence of path integral formalism).
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Langevin equation
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Once we know MEP

• Go back to TST with harmonic approximation
• Use MEP as a local reaction coordinate,
• to obtain a one-dimensional reduction
• (compute potential of mean force, etc),
• then use Kramers theory to obtain rates.

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Complex systems Rough energy landscapes

• Saddle points, MEP are not the right object.
• No single most probable path. Many paths
  contribute.
• Notion of transition state needs to be extended.

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Two dimensional rough energy landscapeResults
of finite temperature string method
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How do we describe transition over rough energy
landscapes?
Metastability related to existence of spectral
gap for L
Transition rates
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Reaction coordinate q(x) cst. gives a
foliation of configuration space
Foliation family of non-intersecting surfaces
that fill up
space.
Importance of choosing the right reaction
coordinate (Bolhuis, Chandler, Dellago,
Geissler,
Transition path sampling (TPS)).
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A distinguished reaction coordinate
The committor function q(x) (TPS)

• level sets q(x) cst iso-committor surfaces

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Analytical characterization
Simple mathematical characterization
Backward Kolmogorov equation
Where
This is a very high dimensional problem. Compare
with quantum many-body problem.
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Characterization of iso-committor surfaces
Hitting point distribution for reactive
trajectories Hitting point distribution for
all trajectories Equilibrium distribution
restricted to the surfaces
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Localization assumption
Equilibrium distribution on iso-committor
surface are peaked (localized).
Definition Transition tube -- connecting these
peaked distributions
This is the object of interest!
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Alternative Variational formulation for
committor function q(x)

• We are only interested in the iso-committor
  surfaces around the
• transition tube.
• Locally, we can approximate the iso-committor
  surfaces there by
• hyper-planes.
• The variational problem then simplifies
  dramatically.
• (choosing special test functions)
• Use these to reduce the many-body problem
• to coupled
  one-body problem!

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Finite-temperature string method (FTS)

• 1. parametrizing the family of hyperplanes by
  strings
• (center of mass) or by their normals

2. evolution of the string sampling
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Dynamics of the string
Sampling on the hyper-planes e. g.
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Calculation of the free energy
Value of the committor function
Connection with the variational problem
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Transition state ensemble
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Example Two dimensional rough energy landscape
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Example Isomerization of alanine dipeptide
in vacuum, and in
explicit solvent

• (in collaboration with Paul Maragakis, M.
  Karplus)
• Simple system which exhibits generic features
  common to more complex biomolecules.
• Conformation state described by torsion angles
  but mechanism of transition unknown before string
  method (transition state region? role of solvent?)

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Actual conformation change
Two conformation states
microtime 1 fs
time per transition 1ps
time between transitions 1-10 ns
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Transition path identified by string method
Two conformation states
mean string
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Projection of the transition tube on the torsion
angle map
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Projection of Transition state region
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Free energy and committor function

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Center of the transition tube (black
curve) Transition state region (red dots) Color
lines actual trajectories (stay confined in
tube)
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Checking the result Computing the
committor distribution (operational
definition of committor)
Committer distribution (peaked at 1/2 for the
transition region)
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Alanine dipeptide with explicit solvent
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Alanine dipeptide with explicit solvent
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Committor distribution for the transition state
region
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Martensitic phase transformation
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Two stable configurations
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Dynamics suggested by mean string
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Phase transformed by Twinn boundary propagation
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Propagation of twin dislocation inside the twinn
boundary
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Mean potential energy along the path
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Special case T0, Zero-temperature
string method (ZTS)
Transition tube ? Minimum energy path
?
Connection with nudged elastic band
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Example Mueller potential
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Comparison with nudged elastic band
(Jonsson et al) LEPS potential
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Summary 1. Conceptually

• Saddle points (isolated transition states)
• ? dividing surfaces
• ? foliation
• Minimum energy path (isolated transition path)
• ? transition tube A distribution in the
  state
• space connecting the reactant and
  product
• states, and tells us where the reactive
  
• trajectories are.

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Committor functions and
iso-committor surfaces

• Operational viewpoint (as in TPS)
• launch trajectories and count them
• Analytical viewpoint (as in FTS)
• Backward Kolmogorov equation or
  variational
• problem, in very high dimensional space
• Localization assumption allows us to reduce the
  many-body problem to coupled one-body
  problem.

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2. String method for computing transition
pathways and rates

• Zero-T string method (ZTS) for simple systems
• Finite-T string method (FTS) for complex
  systems, and when entropic effects are important.

A bridge between NEB and TPS
Adaptive blue moon sampling
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3. Advertising
Tutorial, references, sample codes available at
www.math.princeton.edu/string
Mathematics and Chemistry A year-long special
program at IMA, Univ. of Minnesota,
2008-2009. Organizers Don Truhlar, Anne Chaka,
Weinan E, Bill Hase, Claude Le Bris and Tamar
Schlick.
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Example

• Rotation of aromatic ring of Tyr 35 in BPTI
• (Bovine Pancreatic Trypsin Inhibitor)
• computed by Paul Maragakis (Harvard)
• Ring flipping time scale 0.1 1 s
• (compare with time scale for atomic
  vibration
• 1 fs 10-15 s)