Biophysical Chemistry G4170: Introduction to Molecular Dynamics

1. Biophysical Chemistry G4170: Introduction to Molecular Dynamics Ruhong Zhou IBM Thomas Watson Research Center Yorktown Heights, NY 10598

2. Polarizable Force Fields

3. m = 1.85 D m ~ 2.5 D H H H H DU = -9.9 kcal/mol O O Gas phase Aqueous phase Solvent-Induced Dipole It takes energy to change from gas-phase to liquid-phase charges: for H2O, DUpol = 2-5 kcal/mol

4. C terminus -0.42 +0.42 C -0.20 + N O H +0.20 N terminus Amide bond: m = 3.5 D a-helix: m = 5.0 D Conformational Effects Wada, Adv. Biophys., 9, 1 (1976); Duijnen and Thole, Biopolymers, 21, 1749 (1982)

5. Polarizable Force Fields • Atomic charges adjust to different chemical environments • Electrostatic interactions are long-ranged interactions, accurate models needed • Needed to calculate many-body interactions • Hopefully a better transferability A C B EAB (C): Energy between A & B depends on C’s position

6. Polarizable Force Fields • Friesner & Berne: Polarizable OPLSAA Fluctuating charges and fluctuating dipoles • Kollman & Case: AMBER2002 Dipole polarizability • Ponder: TINKER Force Field Dipole polarizability, and higher multipoles such as quadruples

7. Fluctuating Charge Model • Dqi: change in partial charge on atom i • Vi: applied electrostatic potential at atom i • Jij: coefficient representing interaction between partial charges at sites i,j – depends on nuclear configuration • Minimize Upolarization to find Dqi, yielding a set of linear equations. • Alternatively, treat Dqi as dynamical variables and propagate them along with the coordinates {xi,yi,zi,qi} S. Rick, S. Stuart, and B. Berne, J. Chem. Phys. 1994

8. Dipolar Polarizability Models • ai: polarizability of atom i • mi: induced dipole on atom i • Ei: applied electric field at atom i • Jij: dipole interaction tensor representing interaction between dipoles at sites i,j After defining ai = 1/Jii, we can rewrite it into Thole 1981; Rullmann & van Duijnen 1988; Cieplak & Kollman 1990; Bernardo, Ding, Krogh-Jerpersen, & Levy 1994

9. Combined Fluctuating Charge & Dipole Model Each atom can have both a partial charge and a dipole, so it might have up to four variables: one charge and three dipole moments Charges on atoms A and dipoles on atoms B. All models may be written succinctly in matrix form: where vectors f={V, Ex, Ey, Ez} and q={q, mx, my, mz} Minimize to determine charges and/or dipole Moments on each atom J. Bank, G. Kaminski, R. Zhou, D. Mainz, B. Berne, R. Friesner, J. Chem. Phys. 110, 741, 1999 H. Stern, G. Kaminski, J. Banks, R. Zhou, B. Berne, R. Friesner, J. Phys. Chem. B103, 4730, 1999 G Kaminski, H. Stern, B. Berne, R. Friesner, Y. Cao, R. Murphy, R. Zhou, J. Comput. Chem. 23, 1515, 2002 G. Kaminski, R. Friesner, R. Zhou, J. Comput. Chem. 24, 267, 2003

10. Polarizable FF Fitting Philosophy • Polarization: • Treat long-range interactions by Coulomb’s law. Scale short-range interactions by adjustable parameters • Apply a series of electrostatic perturbations to a molecule • For each perturbation, compute the change in the electrostatic potential at a series of grid points from ab initio calculations on the unperturbed and perturbed molecules • Fit the parameters of the model so as to best reproduce these changes when the same perturbation are applied • Gas-phase electrostatics: choose fixed charges so that the total electrostatic potential of the model best reproduces high-level ab initio gas-phase calculations • Intramolecular, Lennard-Jones, and torsional terms: take from OPLSAA. Refit key torsions to ab initio relative conformational energies

11. Three-body Energies for molecules with two probes • E(3) = E123 – E12 –E23 –E13 + E1 + E2 +E3 • 3-body energies are all zero in standard force fields • RMS errors are from comparisons to high level QM calculations

12. Cases where fluctuating charge model fails • Two cases that point-charge-only model fails for three-body energies • Bifurcated hydrogen bond • Probes above or below aromatic rings, out-of-plane polarization

13. Relative Conformational Energy

14. Summary on Polarizable OPLSAA • Force fields incorporating explicit polarization have been developed that accurately predict many-body effects • Polarizable FF dramatically improves the prediction of relative conformational energies for small peptides • Dipolar model can correct errors in fluctuating charge model alone for cases with out-of-plane polarization (aromatic rings) or bifurcated hydrogen bonds (O, S atoms) • Parameterization was systematic and transferable

15. II. Solvation Models

16. Solvation Models • Explicit solvent models • Fixed charge models: SPC, SPC/E, TIP3P, TIP4P, TIP5P, ST2,… • Polarizable water models: TIP4P/FQ, POL5, MCDHO,… • Implicit Solvent models • Poisson-Boltzman solver (Delphi, Honig) • Generalized Born Model (Still) • Karplus’ EEF1 model • Benoit Roux’s Spherical Solvent Boundary Potential (SSBP)

17. Explicit Water modelsSPC, SPC/E, TIPnP, POL5

18. Water Model Geometries

19. Water Model Parameters • SPC, SPC/E (Berendsen) • TIP3P, TIP4P, TIP5P (Jorgensen) • TIP4P/FQ, POL5 (Berne)

20. Properties of Water Models

21. Water density maximum

22. Water structure comparison M. Mahoney and W. L. Jorgensen, J. Chem. Phys. 112, 8910, 2000

23. POL5 Model

24. Gas-phase electrostatic properties

25. r f q Water dimer properties

26. Trimer

27. Tetramer

28. Pentamer

29. Hexamers

30. Book hexamer

31. Prism hexamer

32. Liquid-state properties

33. Water density revisited

34. Implicit Solvent ModelsPBF, GB

35. Continuum Solvent Model continuum solvent e=80 e=1-4 protein

36. Molecular Surfaces • Dotted line: Solvent Accessible Surface (SAS) • Solid line: molecular surface (MS) • Shaded grey area: van der Waals surface

43. R. Levy, JCC 2002

45. Molecular Surface Colored by Potential The molecular surface of acetyl choline esterase molecule color coded by electrostatic potential. the view is directly into the active site and acetyl choline is present in a bond representation. note the depth of the pocket, its negative nature corresponding to the positive charge on the acetyl choline.

46. Trp-cage Folding: Kinetics • OPLS united atom Force Field • Continuum Solvent GBSA • Langevin dynamics • Water viscosity g=91/ps B: MD simulation C: NMR structure 2.1 A Ca RMSD Folding time 1.5ms (3.0 A cutoff) to 8.7 ms (2.5 A cutoff) M. Snow, B. Zagrovic, V. Pande, JACS 124, 14548, 2002

47. Trp-cage Folding: Structure Blue: MD simulation Grey: NMR structure 0.97 A Ca RMSD 1.4 A RMSD heavy atoms • AMBER99 Force Field • Continuum Solvent GBSA • NVT ensemble C. Simmerling, B. Strockbine, A. Roitberg, JACS 124, 11258, 2002

48. Protein (un)Folding Example: a b-hairpin Res. 41-56 Protein G (2gb1) GEWTYDDATKTFTVTE V. Munoz, P. Thompson, J. Hofrichter, W. Eaton, Nature, 390, 196, 1997 R. Zhou, B. Berne and R. Germain, PNAS, 98, 14931, 2001

50. b-hairpin Folding in Various Models • OPLSAA/SPC (explicit) • OPLSAA/SGB • OPLSAA/PB R. Zhou, B. J. Berne, PNAS 99, 2002 R. Zhou, G. Krilov, B. J. Berne, JPC, 2004 R. Zhou, et al, PNAS 98, 2001 R. Zhou, and B. Berne, PNAS 99, 2002