Hybrid Quantum-Classical Molecular Dynamics of Hydrogen Transfer Reactions in Enzymes

1. Hybrid Quantum-Classical Molecular Dynamics of Hydrogen Transfer Reactions in Enzymes Sharon Hammes-Schiffer Penn State University

2. Catalyze chemical reactions: make them faster Enzymes cofactor enzyme substrate chemical reaction

3. Issues to be Explored • Fundamental nature of H nuclear quantum effects • Zero point energy • H tunneling • Nonadiabatic effects • Rates and kinetic isotope effects • Comparison to experiment • Prediction • Role of structure and motion of enzyme and solvent • Impact of enzyme mutations

4. Impact of Enzyme Motion • Activation free energy barrier • equilibrium between transition state and reactant • Dynamical re-crossings of free energy barrier • nonequilibrium dynamical effect

5. Billeter, Webb, Iordanov, Agarwal, SHS, JCP 114, 6925 (2001) Hybrid Approach Real-time mixed quantum/classical molecular dynamics simulations including nuclear quantum effects and motion of complete solvated enzyme • Elucidates relation between specific enzyme motions • and enzyme activity • Distinguishes between activation free energy and • dynamical barrier recrossing effects

6. Two Levels of Quantum Mechanics • Electrons • Breaking and forming bonds • Empirical valence bond (EVB) potential • Warshel and coworkers • Nuclei • Zero point motion and hydrogen tunneling • H nucleus represented by 3D vibrational wavefunction • Mixed quantum/classical molecular dynamics • MDQT surface hopping method

7. D A H D A H Empirical Valence Bond Potential EVB State 1 EVB State 2 Diagonalize • GROMOS forcefield • Morse potential for D-H and A-H bond • 2 parameters fit to reproduce experimental free • energies of activation and reaction

8. Mixed quantum/classical nuclei • r: H nucleus, quantum • R: all other nuclei, classical • Calculate 3D H vibrational wavefunctions on grid Treat H Nucleus QM Fourier grid Hamiltonian multiconfigurational self-consistent-field (FGH-MCSCF) Webb and SHS, JCP 113, 5214 (2000) Partial multidimensional grid generation method Iordanov et al., CPL 338, 389 (2001)

9. Calculation of Rates and KIEs • Equilibrium TST rate • Calculated from activation free energy • Generate adiabatic quantum free energy profiles • Nonequilibrium transmission coefficient • Accounts for dynamical re-crossings of barrier • Reactive flux scheme including nonadiabatic effects

10. Collective reaction coordinate • Mapping potential to drive • reaction over barrier • Thermodynamic integration to connect free energy curves • Perturbation formula to include adiabatic H quantum effects Calculation of Free Energy Profile

11. Reactive flux approach for infrequent events • Initiate ensemble of trajectories at dividing surface • Propagate backward and forward in time Calculation of Transmission Coefficient  = 1/a for trajectories with a forward and a-1 backward crossings = 0 otherwise • MDQT surface hopping method to include vibrationally • nonadiabatic effects (excited vibrational states) • Tully, 1990; SHS and Tully, 1994

12. Mixed Quantum/Classical MD • Classical molecular dynamics • Calculate adiabatic H quantum states • Expand time-dependent wavefunction • quantum probability for state n at time t • Solve time-dependent Schrödinger equation Hynes,Warshel,Borgis,Ciccotti,Kapral,Laria,McCammon,van Gunsteren,Cukier

13. Tully, 1990; SHS and Tully, 1994 MDQT • System remains in single adiabatic quantum state k • except for instantaneous nonadiabatic transitions • Probabilistic surface hopping algorithm: for large number • of trajectories, fraction in state n at time t is • Incorporates zero point energy and H tunneling • Valid in adiabatic, nonadiabatic, and intermediate regimes

14. MDQT Reactive Flux • Reactive flux approach for infrequent events • Initiate ensemble of trajectories at dividing surface • Propagate backward and forward in time • Extension for MDQT [Hammes-Schiffer and Tully, 1995] • Propagate backward with fictitious surface hopping • algorithm independent of quantum amplitudes • Re-trace trajectory in forward direction to determine • weighting to reproduce results of MDQT

15. LADH Alcohol Aldehyde/Ketone NAD+ NADH + H+ Liver Alcohol Dehydrogenase • Critical for key steps in metabolism • Relevant to medical complications of alcoholism • Experiments: Klinman (KIE, mutagenesis) • Other theory • electronic structure: Houk, Bruice, Gready • molecular dynamics: Bruice • VTST-QM/MM: Truhlar, Gao, Hillier, Cui, Karplus

16. Crystal structure: Ramaswamy, Eklund, Plapp, 1994 LADH Simulation System • 75140 atoms in rectangular periodic box • Two protein chains, co-enzymes, benzyl alcohol substrates • 22682 solvent (water molecules)

17. Proton transfer occurs prior to hydride transfer • Experimental data • Electronic structure/classical forcefield calculations • Agarwal, Webb, SHS, JACS 122, 4803 (2000) Active Site of LADH

18. LADH Reaction

19. Two EVB parameters fit to experimental free energies • Plapp and coworkers, Biochemistry 32, 11186 (1993) • Nuclear quantum effects decrease free energy barrier Free Energy Profile for LADH

20. Ground state Excited state Hydrogen Vibrational Wavefunctions Reactant TS Product

21. Isotope Effects of H Wavefunctions at TS Hydrogen Deuterium Tritium

22. KIE from Activation Free Energy TST Calculations Experiment1 kH/kD 5.0 ± 1.8 3.78 ± 0.07 kD/kT 2.4 ± 0.8 1.89 ± 0.01 1Bahnson and Klinman, 1995

23. The Reactive Center

24. Equilibrium Averages of Properties

25. Real-Time Dynamical Trajectories

26. LADH Productive Trajectory

27. LADH Unproductive Trajectory

28. LADH Recrossing Trajectory

29. Transmission Coefficient kH = 0.95 kD = 0.98 • Values nearly unity dynamical effects not dominant • Inverse KIE for k Calculations: kH/kD = 4.8 ± 1.8 Experiment: kH/kD = 3.78 ± 0.07

30. Normalized weighted correlation between geometrical property and barrier re-crossing () Correlation Functions Property Correlation CD-CA distance17.8% Zn-O distance 0.5% CD-O distance 5.0% VAL-203 Cg1-CA distance 5.6% VAL-203 Cg1-NH4 distance 5.2% VAL-203 Cg1-CD distance 0.2% C NAD+/NADH angle - 1.7% N NAD+/NADH angle10.4% Standard deviation for random sample: 6.0%

31. DHFR DHF THF NADPH + H+ NADP+ Dihydrofolate Reductase • Maintains levels of THF required for biosynthesis of • purines, pyrimidines, and amino acids • Pharmacological applications • Experiments: • Benkovic (kinetics, mutagenesis), Wright (NMR) • Previous theory • electronic structure: Houk • QM/MM: Gready and coworkers • molecular dynamics: Radkiewicz and Brooks

32. DHFR Simulation System Crystal structure: 1rx2, Sawaya and Kraut, Biochemistry 1997 • 14063 atoms in octahedral periodic box • NADPH co-enzyme, DHF substrate • 4122 solvent (water molecules)

33. DHFR Reaction

34. Free Energy Profile for DHFR Agarwal, Billeter, Hammes-Schiffer, JPC 106, 3283 (2002) • Two EVB parameters fit to experimental free energies • Fierke, Johnson and Benkovic, Biochemistry 1987 • kH/kD TST: 3.4 ± 0.8, experiment: 3.0 ± 0.4

35. Transmission Coefficient for DHFR kH = 0.80 kD = 0.85 • Values less than unity • dynamical barrier recrossings significant • Physical basis • friction from environment • not due to nonadiabatic transitions

36. DHFR Productive Trajectory

37. Motion in DHFR Agarwal, Billeter, Rajagopalan, Benkovic, Hammes-Schiffer, PNAS 2002 • Conserved residues • (genomic analysis across 36 • species, E. coli to human) • Effects of mutations on • hydride transfer rate: • large effects far from active site, • non-additive double mutants • NMR: dynamic regions • Wright and coworkers • MD: correlated regions • Radkiewicz and Brooks

38. Hybrid Quantum-Classical Simulations • Systematic study of conserved residues • Calculated two quantities per distance • thermally averaged change from reactant to TS (ms timescale of H─ transfer) • correlation to degree of barrier recrossing (fs-ps timescale of dynamics near TS)

39. DHF/NADPH Motion

40. Motions Near DHF/NADPH

41. Loop Motion

42. Network of Coupled Promoting Motions • Located in active site and exterior of enzyme • Contribute to collective reaction coordinate • Occur on millisecond timescale of H- transfer reaction

43. G121V Mutant Free Energy Profile Gly Val Simulations: G121V has higher free energy barrier than WT Experiment: G121V rate 163 times smaller than WT

44. G121V Mutant Motions WT G121V

45. Summary of Hybrid Approach • Generate free energy profiles and dynamical trajectories • Nuclear quantum effects included • Motion of complete solvated enzyme included • Wealth of information • Rates and KIEs • Fundamental nature of nuclear quantum effects • Relation between specific enzyme motions and activity • (activation free energy and barrier re-crossings) • Impact of mutations • Network of coupled promoting motions

46. Acknowledgements Pratul Agarwal Salomon Billeter Tzvetelin Iordanov James Watney Simon Webb DHFR: Ravi Rajagopalan, Stephen Benkovic Funding: NSF, NIH, Sloan, Dreyfus