Hybrid Quantum-Classical Molecular Dynamics of Enzyme Reactions
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Transcript of Hybrid Quantum-Classical Molecular Dynamics of Enzyme Reactions
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Hybrid Quantum-Classical Molecular Dynamics of Enzyme
ReactionsSharon Hammes-Schiffer Penn State University
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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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Hybrid Quantum/Classical Approach
Real-time mixed quantum/classical molecular dynamicssimulations including electronic/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• Identifies effects of motion on both activation free energy and dynamical barrier recrossings
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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
( )( )
( )
RH R
R
V V
V V
EVB nuc g nuc( ) ( )H R RVDiagonalize
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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)
( , ) ( ; ) ( ) ( ; ) r R r R R r RnH g n nT V
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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
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
• Peturbation 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 otherwiseKeck, Bennett, Chandler, Anderson
• 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
r RcN
IH g
I I
PH T V
M
• 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 ( ) RF R RII I IM V
( , ) ( ; ) ( ) ( ; ) r R r R R r RnH g n nT V
( , , ) ( ) ( ; ) r R r Rn nn
t C t2
( ) :nC t
R d k k k j kjj
i C C i C Rdkj k j
Hynes,Warshel,Borgis,Kapral, Laria,McCammon,van Gunsteren,Cukier,Tully
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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 • 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
Tully, 1990; SHS and Tully, 1994
2( )nC t
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Systems Studied
• Liver alcohol dehydrogenase
Alcohol Aldehyde/Ketone
NAD+ NADH + H+
LADH
• Dihydrofolate reductase
DHF THF
NADPH + H+ NADP+
DHFR
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Dihydrofolate Reductase
• 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)
Simulation system> 14,000 atoms
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DHFR Productive Trajectory
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DHFR Recrossing Trajectory
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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
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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
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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
scales as
computationally expensive to include more quantum nuclei
dim
grid pts per dim
NN
Future US/UK and biomolecules/materials collaborationsFuture requirements for HPC hardware and software
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Acknowledgements
Pratul AgarwalSalomon BilleterTzvetelin IordanovJames WatneySimon WebbKim Wong
DHFR: Ravi Rajagopalan, Stephen Benkovic
Funding: NIH, NSF, Sloan, Dreyfus