Hybrid Quantum-Classical Molecular Dynamics of Enzyme

ReactionsSharon 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

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

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

• 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

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

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

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

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

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

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

Systems Studied

• Liver alcohol dehydrogenase

Alcohol Aldehyde/Ketone

NAD+ NADH + H+

LADH

• Dihydrofolate reductase

DHF THF

NADPH + H+ NADP+

DHFR

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

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

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

Acknowledgements

Pratul AgarwalSalomon BilleterTzvetelin IordanovJames WatneySimon WebbKim Wong

DHFR: Ravi Rajagopalan, Stephen Benkovic

Funding: NIH, NSF, Sloan, Dreyfus