Hybrid Quantum-Classical Molecular Dynamics of Hydrogen

Transfer Reactions in Enzymes

Sharon Hammes-Schiffer Penn State University

Enzymes• Catalyze chemical reactions: make them faster

enzymecofactor

substrate

chemicalreaction

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 dynamicssimulations including nuclear quantum effects andmotion 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 DH and AH bond• 2 parameters fit to reproduce experimental free

energies of activation and reaction

EVB State 1 EVB State 2

D AH D AH

1 nuc 12EVB nuc

12 2 nuc 12

( )( )

( )

V V

V V

RH R

R

EVB nuc g nuc( ) ( )VH R RDiagonalize

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 multiconfigurationalself-consistent-field (FGH-MCSCF)Webb and SHS, JCP 113, 5214 (2000)

Partial multidimensional grid generation methodIordanov et al., CPL 338, 389 (2001)

( , ) ( ; ) ( ) ( ; )nH g n nT V r R r R R r R

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

† /

TSTBG k TBk T

kh e

dyn TSTk k

0 1

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

11 22 o( ) ( , ) ( , )V V R r R r R

map 11 22( , ; ) (1 ) ( , ) ( , )m m mV V V r R r R r R

map intmap0 ( ; ) [ ( ) ( ; )]( ; )

,

n m o mn m

m n

F VFe e e

R R

intmap map( ; ) ( , ; )m mV Ve C d e R r Rr r

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/ 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 MD2

tot1

( , )2

cNI

H gI I

PH T V

M

r R

• 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

eff eff ( )II I IM V RF R R

( , ) ( ; ) ( ) ( ; )nH g n nT V r R r R R r R

( , , ) ( ) ( ; )n nn

t C t r R r R2

( ) :nC t

k k k j kjj

i C C i C R d kj k j Rd

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

2( )nC t

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

Alcohol Aldehyde/Ketone

NAD+ NADH + H+

LADH

LADH Simulation System

• 75140 atoms in rectangular periodic box• Two protein chains, co-enzymes, benzyl alcohol substrates• 22682 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, 11186 (1993)• Nuclear quantum effects decrease free energy barrier

Hydrogen Vibrational Wavefunctions

Reactant

TS

Product

Ground state Excited state

Isotope Effects of H Wavefunctions at TS

Hydrogen

Deuterium

Tritium

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

The Reactive Center

Equilibrium Averages of Properties

Real-Time Dynamical Trajectories

LADH Productive Trajectory

LADH Unproductive Trajectory

LADH Recrossing Trajectory

Transmission Coefficient

H = 0.95D = 0.98

• Values nearly unity dynamical effects not dominant

• Inverse KIE for

Calculations: kH/kD = 4.8 ± 1.8

Experiment: kH/kD = 3.78 ± 0.07

Correlation FunctionsNormalized weighted correlation between geometrical property and barrier re-crossing ()

Property CorrelationCD-CA distance 17.8%Zn-O distance 0.5%CD-O distance 5.0%VAL-203 C1-CA distance 5.6%VAL-203 C1-NH4 distance 5.2%VAL-203 C1-CD 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

DHF THF

NADPH + H+ NADP+

DHFR

DHFR Simulation System

• 14063 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

• kH/kD TST: 3.4 ± 0.8, experiment: 3.0 ± 0.4

Agarwal, Billeter, Hammes-Schiffer, JPC 106, 3283 (2002)

Transmission Coefficient for DHFR

H = 0.80D = 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 WTExperiment: G121V rate 163 times smaller than WT

Gly

Val

G121V Mutant MotionsWT G121V

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 AgarwalSalomon BilleterTzvetelin IordanovJames WatneySimon Webb

DHFR: Ravi Rajagopalan, Stephen Benkovic

Funding: NSF, NIH, Sloan, Dreyfus