Hybrid Quantum-Classical Molecular Dynamics of Enzyme Reactions

1. Hybrid Quantum-Classical Molecular Dynamics of Enzyme Reactions Sharon Hammes-Schiffer Penn State University

2. 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

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

4. 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

5. 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

6. 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)

7. 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

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

9. 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 Keck, Bennett, Chandler, Anderson • MDQT surface hopping method to include vibrationally • nonadiabatic effects (excited vibrational states) • Tully, 1990; SHS and Tully, 1994

10. 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,Kapral, Laria,McCammon,van Gunsteren,Cukier,Tully

11. Tully, 1990; SHS and Tully, 1994 • 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 • Combine MDQT and reactive flux • [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 MDQT

12. LADH Alcohol Aldehyde/Ketone DHFR DHF THF NAD+ NADH + H+ NADPH + H+ NADP+ Systems Studied • Liver alcohol dehydrogenase • Dihydrofolate reductase

13. Dihydrofolate Reductase Simulation system > 14,000 atoms • Maintains levels of THF required for biosynthesis of • purines, pyrimidines, and amino acids • Hydride transfer from NADPH cofactor to DHF substrate • Calculated KIE (kH/kD) is consistent with experimental value of 3 • Calculated rate decrease for G121V mutant consistent with • experimental value of 160 • k = 0.88 (dynamical recrossings occur but not significant)

14. DHFR Productive Trajectory

15. DHFR Recrossing Trajectory

16. Network of Coupled Motions • Located in active site and exterior of enzyme • Equilibrium, thermally averaged motions • Conformational changes along collective reaction coordinate • Reorganization of environment to facilitate H- transfer • Occur on millisecond timescale of H- transfer reaction

17. Strengths of Hybrid Approach • Electronic and nuclear quantum effects included • Motion of complete solvated enzyme included • Enables calculation of rates and KIEs • Elucidates fundamental nature of nuclear quantum effects • Provides thermally averaged, equilibrium information • Provides real-time dynamical information • Elucidates impact of mutations

18. Limitations and Weaknesses • System size • LADH (~75,000 atoms), DHFR (~14,000 atoms) • Sampling • DHFR: 4.5 ns per window, 90 ns total • Potential energy surface (EVB) • not ab initio, requires fitting, only qualitatively accurate • Bottleneck: grid calculation of H wavefunctions • - must calculate energies/forces on grid for each MD time step • - scales as • - computationally expensive to include more quantum nuclei • Future US/UK and biomolecules/materials collaborations • Future requirements for HPC hardware and software

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