Transcript Hybrid_Quantu_Classic_Dynamics!!

Hybrid Quantum-Classical
Molecular Dynamics of Enzyme
Reactions
Sharon Hammes-Schiffer
Penn State University
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
Hybrid Quantum/Classical Approach
Billeter, Webb, Iordanov, Agarwal, SHS, JCP 114, 6925 (2001)
Real-time mixed quantum/classical molecular dynamics
simulations including electronic/nuclear quantum effects and
motion of complete solvated enzyme
• Elucidates relation between specific enzyme motions
and enzyme activity
• Identifies effects of motion on both activation free energy and
dynamical barrier recrossings
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
k   kTST
• kTST
k BT

h
e
-G † / k BT
– 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
• Peturbation formula to include adiabatic
H quantum effects
e
-  F0 (  n ;m )
e
-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
Keck, Bennett, Chandler, Anderson
• 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  -RI 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,Kapral, Laria,McCammon,van Gunsteren,Cukier,Tully
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
2
of trajectories, fraction in state n at time t is C n ( t )
• 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
Systems Studied
• Liver alcohol dehydrogenase
LADH
Alcohol
Aldehyde/Ketone
NAD+
NADH + H+
• Dihydrofolate reductase
DHFR
DHF
NADPH + H+
THF
NADP+
Dihydrofolate Reductase
Simulation system
> 14,000 atoms
• 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)
DHFR Productive Trajectory
DHFR Recrossing Trajectory
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
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
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
Ndim
- scales as  Ngrid pts per dim 
- computationally expensive to include more quantum nuclei
Future US/UK and biomolecules/materials collaborations
Future requirements for HPC hardware and software
Acknowledgements
Pratul Agarwal
Salomon Billeter
Tzvetelin Iordanov
James Watney
Simon Webb
Kim Wong
DHFR: Ravi Rajagopalan, Stephen Benkovic
Funding: NIH, NSF, Sloan, Dreyfus