Milestoning and the R to T transition in Scapharca Hemoglobin

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Milestoning and the R to T transition in
Scapharca Hemoglobin

• Ron Elber
• and Anthony West
• Department of Computer Science,
• Cornell University
• NIH

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Time Scale Problems in Atomically Detailed
Molecular Dynamics Simulation
seconds
Channel Gating. Fast folding
Protein Activation
Slow folding
Mol. Dyn.
Non equilibrium processes, averages over many
trajectories... Allosteric transition From local
atomic change to global change
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Long time processes By long range diffusion or
activation
Activated processes rare fast
Milestoning slow diffusive
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Divide reaction coordinate into Milestones
R
P

• For each milestone
• Prepare statistical ensemble in orthogonal plane
• Fundamental assumption gives fast equilibration
  in plane, slow between planes (
  )
• Integrate each trajectory until first passage to
  neighboring planes
• Histogram first passage times to obtain first
  passage time (FPT) distribution
• Construct system distn from FPT distn
  (below).

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First passage time distribution K2
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Resulting model Continuous-time random walk

• Complicated reaction now modeled as discrete
  non-Markovian K-dependent 1-D hopping
• t reinterpreted as incubation time (waiting
  time) between hops. FPT distn is now
  waiting time distn
• Process is generally non-Markovian
• NB are the only input, capture all
  relevant details of microscopic dynamics and
  connect them to this abstract model.

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How to solve for

• Define prob density of transition to
  s at t
• Then hopping dynamics are defined by
• Only input is , from microscopic
  calculations.
• Solve integral equation numerically to get what
  we want

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Equivalent to Generalized Master Equation

• The generalized Markov equation has time
  dependent rates
• K in the QK formulation is easier to compute than
  R and the Laplace transforms are related by

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Calculations of rates / time moments (Shalloway
Vanden Eijnden)

• The first passage time (and all its moments) with
  absorbing boundary condition at the product state
  N

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Time moments -- continuation
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Rate constant Inverse of first passage time
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When and why it works

• memory loss system equilibrates in plane much
  faster than between planes.
• Permits fragment then glue approach to
  trajectories along RC...produces effectively
  long-time trajectories
• Fragmentation gives diffusive speedup on flat
  energy surfaces
• and exponential speed-up for systems with
  substantial barriers
• Independence gives parallelizability speedup

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Alanine dipeptide in water
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Sampling Milestones
Absorbing boundary condition
Reflecting boundary conditions
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A sample of first passage time distribution
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Overall time course for alpha to beta transition
in alanine dipeptide
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Total first passage time as a function of the log
of number of milestones
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Table of first passage times
For 19 milestones speed up of 9 compared to exact
trajectories
Rate computed from the smallest eigenvalue of
the Markov Matrix
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Velocity memory is kept below 300fs
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Free energy estimates
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References for Milestoning

• Anton K. Faradjian and Ron Elber, "Computing time
  scales from reaction coordinates by milestoning",
  J. Chem. Phys. 12010880-10889(2004)
• Anthony M.A. West, Ron Elber and David Shalloway,
  Extending Molecular Dynamics Time Scales with
  MilestoningL Example of complex kinetics in a
  solvated peptide, J. Chem. Phys.
  126,145104(2007)
• Ron Elber, "A milestoning study of the kinetics
  of an allosteric transition Atomically detailed
  simulations of deoxy Scapharca hemoglobin",
  Biophysical J. ,2007 92 L85-L87

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The classical problem of the R to T transition
in hemoglobin A

• Four chains two alpha, two beta,
• Large quaternary structural change
• Allosteric transition
• Complex kinetics
• Dependence on pH, other factors
• Tens of microsecond time scale
• Noble to Perutz

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Hemoglobin A indirect heme-heme interactions
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Animation of the transition
Wikipedia
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Scapharca hemoglobin

• Homodimer
• Allosteric
• No large quaternary change
• Hemes in close contact
• Phenylalanine conformational transition
• Changes in water structure
• No pH effect
• Simpler kinetics
• Microsecond time scale

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Scaphraca Hemoglobin
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Reaction path for the R to T transition in
Scapharca Hemoglobin
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References for reaction path algorithms

• Pratt L., JCP. 85, 5045 1986 (global reaction
  path algorithm)
• Elber Karplus CPL, 139, 375 (1987).
  (equi-distance spring)
• A. Ulitsky and R. Elber, JCP, 96, 1510 (1990).
  (exact SDP by updating planes)
• R. Olender and R. Elber, J. Mol. Struct.
  Theochem, 398-399, 63-72 (1997).

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Molecular dynamics simulations in explicit water
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First passage time of milestone 7 along the
reaction coordinate (computed with SPW algorithm)
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Following distance distribution between
allosteric phenylalanines
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Solving the milestoning equation

• The overall rate is 10 microsecond, in accord
  with spectroscopy measurements.
• Identifying late barrier with global motions that
  follow the side chain transitions
• In progress Myosin transition (Tony West)

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Summary

• Milestoning divides RC into fragments whose
  kinetics can be computed independently
• Provides factor of improvement for
  diffusive processes, exponential for activated
  processes. Uses FPTs from microscopic dynamics
• System distribution given by simple
  integral equations that can be easily solved
  numerically
• Some care is needed in
• Choice of reaction coordinate
• Application of equilibrium assumption

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Comparison to TPS

• TPS provide a rate constant for a high energy
  barrier
• Rate constant exists
• System in equilibrium
• Trajectory computed sequentially (must be short)
• No need for a reaction coordinate

• Milestoning computes arbitrary kinetics on rough
  energy landscapes
• Requires a reaction coordinate
• Local equilibrium is assumed but non equilibrium
  along order parameter is possible
• Non-exponential kinetics is fine.
• Trajectory computed in parallel and can be made
  very long (Systems in progress Hemoglobin,
  Myosin)

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Comparison to Bolhuis et al.

• PPTIS and Milestoning use order parameter
• PPTIS assumes loss of memory in time (Markovian
  process) and a single exponential relaxation.
• Milestoning assumes spatial memory loss in the
  direction perpendicular to the order parameter
• PPTIS computes trajectory sequentially
• Milestoning allows for more efficient
  parallelization
• The two approaches can be combined!

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Example FPT distn
2D model, T 0.2, M 8
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Memory loss
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Reaction coordinate/Order parameter

• Fundamental assumption of MLST relaxation to
  equilibrium between milestones is fast. Proper
  choice of

Eric Vanden Eijnden For a system in equilibrium,
Brownian dynamics, and milestones equi-committer
surfaces, milestoning is exact.
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Toy model 1D box simulation

• Microscopic dynamics are Brownian
• Simulations run at various temperatures and for
  4, 8, and 16 milestones

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1D reaction curves (5000 trajs/MLST)
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Extracting the rate
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1D power law (not Arrhenius)
Equally good results with 4, 8, or 16 milestones
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Rate constants
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2D simulation
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Time course of transition
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2D power law
Now clear that 4 milestones is best
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2D simulation
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Memory loss demonstration
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FREE ENERGY is a by-product (from equilibrium
vector)