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Hybrid Quantum-Classical Molecular Dynamics of Enzyme Reactions Sharon Hammes-Schiffer Penn State University 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 Hybrid Quantum/Classical Approach Billeter, Webb, Iordanov, Agarwal, SHS, JCP 114, 6925 (2001) 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 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 Empirical Valence Bond Potential EVB State 1 EVB State 2 D D H A H A V12 V1 (R nuc ) H EVB (R nuc ) V V ( R ) 12 2 nuc 12 Diagonalize H EVB (R nuc ) Vg (R nuc ) • GROMOS forcefield • Morse potential for D-H and A-H bond • 2 parameters fit to reproduce experimental free energies of activation and reaction Treat H Nucleus QM • Mixed quantum/classical nuclei r: H nucleus, quantum R: all other nuclei, classical • Calculate 3D H vibrational wavefunctions on grid TH Vg (r, R ) n (r; R ) n ( R ) n (r; R) 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) Calculation of Rates and KIEs k kTST • kTST k BT h e -G † / k BT – Equilibrium TST rate – Calculated from activation free energy – Generate adiabatic quantum free energy profiles • 0 1 – Nonequilibrium transmission coefficient – Accounts for dynamical re-crossings of barrier – Reactive flux scheme including nonadiabatic effects Calculation of Free Energy Profile • Collective reaction coordinate (R) V11 (r, R) - V22 (r, R) o • Mapping potential to drive reaction over barrier Vmap (r, R; m ) (1 - m )V11 (r, R ) mV22 (r, R ) • Thermodynamic integration to connect free energy curves • Peturbation formula to include adiabatic H quantum effects e - F0 ( n ;m ) e -Vintmap ( R ;m ) e - Fmap ( n ;m ) Cr dre e - [ o ( R ) -Vintmap ( R ;m )] - Vmap ( r , R ;m ) m , n Calculation of Transmission Coefficient • Reactive flux approach for infrequent events – Initiate ensemble of trajectories at dividing surface – Propagate backward and forward in time w = 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 Mixed Quantum/Classical MD Nc H tot PI2 TH Vg (r, R ) I 1 2 M I • Classical molecular dynamics FIeff M I R I -R I V eff (R) • Calculate adiabatic H quantum states TH Vg (r, R ) n (r; R ) ( R ) n (r; R) n • Expand time-dependent wavefunction (r, R, t ) Cn (t ) n (r; R) 2 n C n (t ) : quantum probability for state n at time t • Solve time-dependent Schrödinger equation i Ck Ck k - i C R d j kj d kj k R j j Hynes,Warshel,Borgis,Kapral, Laria,McCammon,van Gunsteren,Cukier,Tully MDQT 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 2 of trajectories, fraction in state n at time t is C n ( t ) • 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 Systems Studied • Liver alcohol dehydrogenase LADH Alcohol Aldehyde/Ketone NAD+ NADH + H+ • Dihydrofolate reductase DHFR DHF NADPH + H+ THF NADP+ 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 • = 0.88 (dynamical recrossings occur but not significant) DHFR Productive Trajectory DHFR Recrossing Trajectory 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 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 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 Ndim - scales as Ngrid pts per dim - computationally expensive to include more quantum nuclei Future US/UK and biomolecules/materials collaborations Future requirements for HPC hardware and software Acknowledgements Pratul Agarwal Salomon Billeter Tzvetelin Iordanov James Watney Simon Webb Kim Wong DHFR: Ravi Rajagopalan, Stephen Benkovic Funding: NIH, NSF, Sloan, Dreyfus