Mark E. Tuckerman Dept. of Chemistry and Courant Institute of Mathematical Sciences

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. Zhongwei Zhu Peter Minary Lula Rosso Jerry Abrams NSF - CAREER NYU Whitehead Award NSF – Chemistry, ITR Camille and Henry Dreyfus Foundation Acknowledgments Students past and present Postdocs Collaborators • Glenn Martyna • Christopher Mundy • Dawn Yarne • Radu Iftimie Funding

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) 1.0 1.1 1.2 1.5 2.5 3.5 4.5 0.9 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 ~1050conformations. • “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)

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 Dynamictransformation

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 4.7 ns Umbrella Sampling 50 ns CHARM22 αR β

38. Conformational sampling of the gas-phase alanine dipeptide φ ψ AFED Tφ,ψ = 5T, Mφ,ψ = 50MC 3.5 ns Umbrella Sampling 35 ns CHARM22 β

39. Conformational sampling of the gas-phase alanine tripeptide φ1 AFED Tφ,ψ = 5T, Mφ,ψ = 50MC 4.7 ns Umbrella Sampling 50 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.