Enhanced conformational sampling via very large time-step molecular dynamics, novel variable transformations and adiabatic dynamics

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Enhanced conformational sampling via very large
time-step molecular dynamics, novel variable
transformations and adiabatic dynamics

• Mark E. Tuckerman
• Dept. of Chemistry
• and Courant Institute of Mathematical Sciences
• New York University, 100 Washington Sq. East
• New York, NY 10003

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Acknowledgments
Students past and present
Postdocs
Collaborators

• Zhongwei Zhu
• Peter Minary
• Lula Rosso
• Jerry Abrams

• Glenn Martyna
• Christopher Mundy

• Dawn Yarne
• Radu Iftimie

Funding

• NSF - CAREER
• NYU Whitehead Award
• NSF Chemistry, ITR
• Camille and Henry Dreyfus Foundation

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Talk Outline

• Very large time-step multiple time scale
  integration that avoids resonance phenomena.
• Novel variable transformations in the partition
  function for enhancing conformational sampling.
• Adiabatic decoupling along directions with high
  barriers for direct computation of free energies.

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Multiple time scale (r-RESPA) integration
MET, G. J. Martyna and B. J. Berne, J. Chem.
Phys. 97, 1990 (1992)
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Resonance Phenomena

• Large time step still limited by frequency of the
  fast force due to numerical artifacts called
  resonances.
• Problematic whenever there is high frequency
  weakly coupled to low frequency motion

• Biological Force Fields
• Path integrals
• Car-Parrinello molecular dynamics

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Illustration of resonance
A. Sandu and T. S. Schlick, J. Comput. Phys. 151,
74 (1999)
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Illustration of resonance (contd)
Note det(A) 1
Depending on ?t, eigenvalues of A are either
complex conjugate pairs
or eigenvalues are both real
Leads to resonances (Tr(A) ? 2) at ?t np/?
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Resonant free multiple time-scale MD

• Resonance means time steps are limited to 5-10 fs
  for most problems.
• Assign time steps to each force component based
  on intrinsic time scale.
• Prevent any mode from becoming resonant via a
  kinetic energy constraint.
• Ensure ergodicity through Nosé-Hoover chain
  thermostatting techniques.

P. Minary, G. J. Martyna and MET, Phys. Rev.
Lett. 93, 150201 (2004).
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Review of isokinetic dynamics
Constraint the kinetic energy of a system
Introduce constraint via a Lagrange multiplier
Derivative of constraint yields multiplier
Partition function generated
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Review of Nosé-Hoover Equations
For each degree of freeom with coordinate q and
velocity v,

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New equations of motion (Iso-NHC-RESPA)
Couple each degree of freedom to the first
element of L NHCs of length M
Ensures the constraint
is satisfied.
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Classical non-Hamiltonian statistical mechanics
Tuckerman, Mundy, Martyna, Europhys. Lett. 45,
149 (1999) Tuckerman, et al. JCP 115, 1678
(2001).
General equations of motion
If
the equations are non-Hamiltonian. ?(x) called
the compressibility of the equations.
Consider a solution
In order to generalize Liouvilles theorem, we
need to determine
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Classical non-Hamiltonian statistical mechanics
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Classical non-Hamiltonian statistical mechanics
Solution
Note that for Hamiltonian systems, ?(x)0 and
J(xt,x0)1.
Define
Then
Whence
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Classical non-Hamiltonian statistical mechanics
Define a metric factor
In addition, suppose the dynamical equations have
Nc conservation laws of the form
Also, assume equilibrium conditions, i.e. that
and the phase space distribution has no explicit
time dependence.
Then, the dynamical system, assuming ergodicity,
will generate a microcanonical ensemble whose
partition function is
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Phase space distribution
For the Iso-NHC-RESPA method
Metric Factor
For the present system
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Integration of the equations
Liouville operator decomposition
Factorized propagator
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Numerical illustration of resonance
Harmonic oscillator with quartic perturbation
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Flexible TIP3P water
Long-range forces 10 Å Ewald
Short-range forces cutoff 5Å
Intramolecular forces
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HIV-1 Protease in vacuo
g(r)
1.0
1.1
1.2
1.5
2.5
3.5
4.5
0.9
rCH (A)
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Conformational sampling in Biophysics

• Ab initio protein/nucleic acid structure
  prediction Sequence ? Folded/active structure.
• Enzyme catalysis.
• Drug docking/Binding free energy.
• Tracking motion water, protons, other ions.

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Native State
Unfolded State
Misfolded State
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The conformational sampling problem

• Find low free energy structures of complex
  molecules
• Sampling conformations described by a potential
  function
  V(r1,,rN)
• Protein with 100 residues has 1050
  conformations.
• Rough free energy landscape in Cartesian space.
• Solution Find a smoother space in which to
  work.

Z. Zhu, et al. Phys. Rev. Lett. 88, art. No.
100201 (2002) P. Minary, et al. (in preparation)
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REPSWA (Reference Potential Spatial Warping
Algorithm)
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No Transformation
Transformation
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Barrier Crossing Transformations (contd)




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Vref(F)
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A 400-mer alkane chain
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No Transformation
Transformation
RIS Model value 10
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No Transformation
Parallel Tempering
Dynamic transformation
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No Transformations
Transformations
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L. Rosso, P. Minary, Z. Zhu and MET, J. Chem.
Phys. 116, 4389 (2000)
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Conformational sampling of the solvated alanine
dipeptide
L Rosso, J. B. Abrams and MET (in preparation)
?
f
AFED Tf,? 5T, Mf,? 50MC 4.7 ns Umbrella
Sampling 50 ns
CHARM22
aR
ß
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Conformational sampling of the gas-phase alanine
dipeptide
?
f
AFED Tf,? 5T, Mf,? 50MC 3.5 ns Umbrella
Sampling 35 ns
CHARM22
ß
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Conformational sampling of the gas-phase alanine
tripeptide
f1
AFED Tf,? 5T, Mf,? 50MC 4.7 ns Umbrella
Sampling 50 ns
?2
?1
f2
Cax7
ß
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Conformational sampling of the solvated alanine
tripeptide
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Closed 5Å
Open 15Å
R

• Protonation state of the active site important in
  drug binding

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No Transformation
Transformation
RIS Model value 14
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Protease alone
Water Number Density (Å )
-3
Protease alone 0.024
Protease drug 0.015
Bulk water 0.033
Protease drug
Avg. cavity dimensions (Å)
Height
Width
PR alone 20.7 12.3 PR drug
19.2 17.3 PR Saq. 20.2
15.1
Z. Zhu, D. I. Schuster and MET, Biochemistry 42,
1326 (2003)
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Conclusions

• Isokinetic-NHC-RESPA method allows time steps as
  large as 100 fs to be used in typical biophysical
  problems.
• Variable transformations lead to efficient MD
  scheme and exactly preserve partition function.
• Speedups of over 106 possible in systems with
  many backbone dihedral angles.
• Trapped states are largely avoided.
• Future Combine variable transformations with
  Iso-NHC-RESPA
• Future Develop variable transformations for ab
  initio molecular dynamics, where potential
  surface is unknown.