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Hybrid Quantum-Classical
Molecular Dynamics of Hydrogen
Transfer Reactions in Enzymes
Sharon Hammes-Schiffer
Penn State University
Enzymes
• Catalyze chemical reactions: make them faster
cofactor
enzyme
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
Billeter, Webb, Iordanov, Agarwal, SHS, JCP 114, 6925 (2001)
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
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
kdyn   kTST
k BT

h
-G † / k BT
e
• kTST
– 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
• Perturbation formula to include adiabatic H quantum effects
e
e
-  F0 (  n ;m )
-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
• 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,Ciccotti,Kapral,Laria,McCammon,van Gunsteren,Cukier
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
of trajectories, fraction in state n at time t is C n ( t ) 2
• Incorporates zero point energy and H tunneling
• Valid in adiabatic, nonadiabatic, and intermediate regimes
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
LADH
Alcohol
NAD+
Aldehyde/Ketone
NADH + H+
• 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
LADH Simulation System
Crystal structure: Ramaswamy, Eklund, Plapp, 1994
• 75140 atoms in rectangular periodic box
• Two protein chains, co-enzymes, benzyl alcohol substrates
• 22682 solvent (water molecules)
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
Ground state
Reactant
TS
Product
Excited state
Isotope Effects of H Wavefunctions at TS
Hydrogen
Deuterium
Tritium
KIE from Activation Free Energy
kH/kD
kD/kT
TST Calculations
5.0 ± 1.8
2.4 ± 0.8
1Bahnson and Klinman,
1995
Experiment1
3.78 ± 0.07
1.89 ± 0.01
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: kH/kD = 4.8 ± 1.8
Experiment: kH/kD = 3.78 ± 0.07
Correlation Functions
Normalized weighted correlation between
geometrical property and barrier re-crossing ()
Property
CD-CA distance
Zn-O distance
CD-O distance
VAL-203 Cg1-CA distance
VAL-203 Cg1-NH4 distance
VAL-203 Cg1-CD distance
C NAD+/NADH angle
N NAD+/NADH angle
Correlation
17.8%
0.5%
5.0%
5.6%
5.2%
0.2%
- 1.7%
10.4%
Standard deviation for random sample: 6.0%
Dihydrofolate Reductase
DHFR
DHF
NADPH + H+
THF
NADP+
• 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
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)
DHFR Reaction
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
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
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
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
Gly
Val
Simulations: G121V has higher free energy barrier than WT
Experiment: G121V rate 163 times smaller than WT
G121V Mutant Motions
WT
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 Agarwal
Salomon Billeter
Tzvetelin Iordanov
James Watney
Simon Webb
DHFR: Ravi Rajagopalan, Stephen Benkovic
Funding: NSF, NIH, Sloan, Dreyfus