1. 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

2. 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

3. 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.

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

5. 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

6. Illustration of resonance
A. Sandu and T. S. Schlick, J. Comput. Phys. 140, 1 (1998)

7. Illustration of resonance (cont’d)
Note: det(A) = 1
Depending on Δt, eigenvalues of A are either complex conjugate pairs
or eigenvalues are both real
Leads to resonances (Tr(A) → ∞) at Δt = nπ/ω

8. 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. (submitted)

9. Review of Nosé-Hoover Equations
For each degree of freeom with coordinate q and velocity v,

10. New equations of motion (Iso-NHC-RESPA)
For each degree of freeom with coordinate q and velocity v,
Ensures the constraint:
is satisfied.

11. Phase space distribution
MET, C. J. Mundy, G. J. Martyna, Europhys. Lett. (2000)
General non-dissipative non-Hamiltonian dynamics:
General “microcanonical” partition function:
Phase space metric:
For the present system:

12. Integration of the equations
Liouville operator decomposition:
Factorized propagator:

13. Numerical illustration of resonance
Harmonic oscillator with quartic perturbation

14. Flexible TIP3P water

15. HIV-1 Protease in vacuo
g(r)
1.5
2.5
3.5
4.5
0.9
1.0
1.1
1.2
rCH (A)

16. 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.

17. Native State
Unfolded State
Misfolded State

18. 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 (2002)
P. Minary, et al. (in preparation)

22. REPSWA (Reference Potential Spatial Warping Algorithm)

23. No Transformation
Transformation

24. Barrier Crossing Transformations (cont’d)‘ ‘ ‘ ‘

25. Vref(Φ)

29. A 400-mer alkane chain

30. No Transformation
Transformation
RIS Model
value: 10

32. Model sheet protein No Transformation Parallel Tempering
Dynamic transformation

33. Transformations
No Transformations

35. L. Rosso, P. Minary, Z. Zhu and MET, J. Chem. Phys. 116, 4389 (2000)

37. Conformational sampling of the solvated alanine dipeptide
[J. Abrams, L. Rosso and MET (in preparation)] φ ψ
AFED Tφ,ψ = 5T, Mφ,ψ = 50MC ns
Umbrella Sampling ns αR
CHARM22 β

38. Conformational sampling of the gas-phase alanine dipeptideφ ψ
AFED Tφ,ψ = 5T, Mφ,ψ = 50MC ns
Umbrella Sampling ns
CHARM22 β

39. Conformational sampling of the gas-phase alanine tripeptideφ1
AFED Tφ,ψ = 5T, Mφ,ψ = 50MC ns
Umbrella Sampling ns ψ2 ψ1 φ2
Cax7 β

40. Conformational sampling of the solvated alanine tripeptide

41. 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.