Hybrid QuantumClassical Molecular Dynamics of Hydrogen Transfer Reactions in Enzymes

1
Hybrid Quantum-Classical Molecular Dynamics of
Hydrogen Transfer Reactions in Enzymes
Sharon Hammes-Schiffer Penn State University
2
Enzymes

• Catalyze chemical reactions make them faster

cofactor
enzyme
substrate
chemical reaction
3
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

4
Impact of Enzyme Motion

• Activation free energy barrier
• equilibrium between transition state and
  reactant
• Dynamical re-crossings of free energy barrier
• nonequilibrium dynamical effect

5
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

6
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

7
Empirical Valence Bond Potential
EVB State 1
EVB State 2
Diagonalize

• GROMOS forcefield
• Morse potential for D-H and A-H bond
• 2 parameters fit to reproduce experimental free
• energies of activation and reaction

8
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 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)
9
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

10
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
Calculation of Transmission Coefficient

• Reactive flux approach for infrequent events
• Initiate ensemble of trajectories at dividing
  surface
• Propagate backward and forward in time

? 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

12
Mixed Quantum/Classical MD

• 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

Hynes,Warshel,Borgis,Ciccotti,Kapral,Laria,McCammo
n,van Gunsteren,Cukier
13
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
• Incorporates zero point energy and H tunneling
• Valid in adiabatic, nonadiabatic, and
  intermediate regimes

14
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

15
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

16
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)

17
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)

18
LADH Reaction
19
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

20
Hydrogen Vibrational Wavefunctions
Ground state
Excited state
Reactant
TS
Product
21
Isotope Effects of H Wavefunctions at TS
Hydrogen
Deuterium
Tritium
22
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
23
The Reactive Center
24
Equilibrium Averages of Properties
25
Real-Time Dynamical Trajectories
26
LADH Productive Trajectory
27
LADH Unproductive Trajectory
28
LADH Recrossing Trajectory
29
Transmission Coefficient
kH 0.95 kD 0.98

• Values nearly unity dynamical effects
  not dominant
• Inverse KIE for k

Calculations kH/kD 4.8 1.8 Experiment
kH/kD 3.78 0.07
30
Correlation Functions
Normalized weighted correlation between
geometrical property and barrier re-crossing (?)
Property
Correlation CD-CA distance 17.8 Zn-O
distance 0.5 CD-O distance
5.0 VAL-203 Cg1-CA distance 5.6 VAL-203
Cg1-NH4 distance 5.2 VAL-203 Cg1-CD
distance 0.2 C NAD/NADH angle - 1.7 N
NAD/NADH angle 10.4
Standard deviation for random sample 6.0
31
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

32
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)

33
DHFR Reaction
34
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

35
Transmission Coefficient for DHFR
kH 0.80 kD 0.85

• Values less than unity
• dynamical barrier recrossings significant
• Physical basis
• friction from environment
• not due to nonadiabatic transitions

36
DHFR Productive Trajectory
37
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

38
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)

39
DHF/NADPH Motion
40
Motions Near DHF/NADPH
41
Loop Motion
42
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

43
G121V Mutant Free Energy Profile
Gly
Val
Simulations G121V has higher free energy barrier
than WT Experiment G121V rate 163 times smaller
than WT
44
G121V Mutant Motions
WT
G121V
45
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

46
Acknowledgements
Pratul Agarwal Salomon Billeter Tzvetelin
Iordanov James Watney Simon Webb DHFR Ravi
Rajagopalan, Stephen Benkovic Funding NSF,
NIH, Sloan, Dreyfus