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

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

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

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

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

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 GROMOS forcefield Morse potential for D  H and A  H bond 2 parameters fit to reproduce experimental free energies of activation and reaction EVB State 1EVB State 2 DA H DA H Diagonalize

Treat H Nucleus QM Mixed quantum/classical nuclei r: H nucleus, quantum R: all other nuclei, classical Calculate 3D H vibrational wavefunctions on grid 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 – 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

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

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

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 Tully, 1990; SHS and Tully, 1994

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

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 AlcoholAldehyde/Ketone NAD + NADH + H + LADH

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

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

LADH Reaction

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

Hydrogen Vibrational Wavefunctions Reactant TS Product Ground stateExcited state

Isotope Effects of H Wavefunctions at TS Hydrogen Deuterium Tritium

KIE from Activation Free Energy TST Calculations Experiment 1 k H /k D 5.0 ± ± 0.07 k D /k T 2.4 ± ± Bahnson and Klinman, 1995

The Reactive Center

Equilibrium Averages of Properties

Real-Time Dynamical Trajectories

LADH Productive Trajectory

LADH Unproductive Trajectory

LADH Recrossing Trajectory

Transmission Coefficient  H = 0.95  D = 0.98 Values nearly unity dynamical effects not dominant Inverse KIE for  Calculations: k H /k D = 4.8 ± 1.8 Experiment: k H /k D = 3.78 ± 0.07

Correlation Functions Normalized weighted correlation between geometrical property and barrier re-crossing (  ) Property Correlation C D -C A distance17.8% Zn-O distance 0.5% C D -O distance 5.0% VAL-203 C  1 -C A distance 5.6% VAL-203 C  1 -NH4 distance 5.2% VAL-203 C  1 -C D distance 0.2% C NAD + /NADH angle- 1.7% N NAD + /NADH angle 10.4% Standard deviation for random sample: 6.0%

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 DHFTHF NADPH + H + NADP + DHFR

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

DHFR Reaction

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

Transmission Coefficient for DHFR  H = 0.80  D = 0.85 Values less than unity dynamical barrier recrossings significant Physical basis − friction from environment − not due to nonadiabatic transitions

DHFR Productive Trajectory

Motion in DHFR 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 Agarwal, Billeter, Rajagopalan, Benkovic, Hammes-Schiffer, PNAS 2002

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)

DHF/NADPH Motion

Motions Near DHF/NADPH

Loop Motion

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

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

G121V Mutant Motions WTG121V

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

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