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![Page 1: Hybrid Quantum-Classical Molecular Dynamics of Hydrogen Transfer Reactions in Enzymes Sharon Hammes-Schiffer Penn State University.](https://reader035.fdocuments.net/reader035/viewer/2022062714/56649d215503460f949f6d18/html5/thumbnails/1.jpg)
Hybrid Quantum-Classical Molecular Dynamics of Hydrogen
Transfer Reactions in Enzymes
Sharon Hammes-Schiffer Penn State University
![Page 2: Hybrid Quantum-Classical Molecular Dynamics of Hydrogen Transfer Reactions in Enzymes Sharon Hammes-Schiffer Penn State University.](https://reader035.fdocuments.net/reader035/viewer/2022062714/56649d215503460f949f6d18/html5/thumbnails/2.jpg)
Enzymes• Catalyze chemical reactions: make them faster
enzymecofactor
substrate
chemicalreaction
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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
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Impact of Enzyme Motion
• Activation free energy barrier– equilibrium between transition state and reactant
• Dynamical re-crossings of free energy barrier– nonequilibrium dynamical effect
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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)
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LADH Reaction
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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
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Hydrogen Vibrational Wavefunctions
Reactant
TS
Product
Ground state Excited state
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Isotope Effects of H Wavefunctions at TS
Hydrogen
Deuterium
Tritium
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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
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The Reactive Center
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Equilibrium Averages of Properties
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Real-Time Dynamical Trajectories
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LADH Productive Trajectory
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LADH Unproductive Trajectory
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LADH Recrossing Trajectory
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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
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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%
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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
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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
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DHFR Reaction
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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)
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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
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DHFR Productive Trajectory
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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
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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)
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DHF/NADPH Motion
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Motions Near DHF/NADPH
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Loop Motion
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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
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G121V Mutant Free Energy Profile
Simulations: G121V has higher free energy barrier than WTExperiment: G121V rate 163 times smaller than WT
Gly
Val
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G121V Mutant MotionsWT G121V
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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
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Acknowledgements
Pratul AgarwalSalomon BilleterTzvetelin IordanovJames WatneySimon Webb
DHFR: Ravi Rajagopalan, Stephen Benkovic
Funding: NSF, NIH, Sloan, Dreyfus