Download Combined quantum mechanics/molecular mechanics (QM/MM

Combined quantum
mechanics/molecular
mechanics (QM/MM) Methods
Lubomír Rulíšek
Institute of Organic Chemistry and
Biochemistry, Prague, Czech Republic
A Course in Molecular Modelling, Prague, June 9, 2006
Outline
Introduction: few notes on the physical chemistry of enzyme
structure and catalysis
QM/MM Methodology
• Why?
• QM methods: usage, accuracy, limitations
• MM methods
• QM/MM coupling schemes
• Other coupling schemes (QM-MM/X-ray, QM-MM/EXAFS)
QM/MM → (E0, ∆G, ∆G#, pKa, KI, k1,…) Route: towards
experimental data – comparison with in vacuo studies
Proteins: Essential Elements in Biocatalysis
Enzyme Catalysis:
1/ Recognition of the Substrate
2/ Chemical Reaction: Local Dielectrics is (Electrostatic
effects)
Metalloproteins
‘Since 30-50% of proteins contain metal ions, one might be interested in
understanding their role and function’
The role of metal ions is
structural and functional:
Ad functional
1/ redox reactions (electron
transfer reactions)
2/ ‘difficult’ reactions (N2, O2
bond breaking)
3/ spin-forbidden reactions
(spin-orbit coupling)
c.f. metal-catalysts vs.
organocatalysts in organic
synthesis
Rate constant:
k = kBT/h exp(-∆G#/RT)
RDS … rate determining step
Equilibrium constant:
K = exp(-∆G0/RT)
Fig: (adapted from Villa, Warshel)
Reactions occur as the consequence of the system dynamics. The atomic
fluctuations result in finding a path from the reactants to products within a
given time scale.
Can we accurately calculate (model) free energy?
QM/MM (quantum mechanics/molecular
mechanics) method
‘To start with, let us have to look at these two separately’
QM(QC) methods:
1/ ‘classical’ wave function based methods (variational, perturbational)
ĤeΨe (r;R) = Ee Ψe(r;R)
(HF, CASSCF, CI, MP2, CASPT2, CCSD(T), MR-CI, …)
150
→
100
→
30 atoms
2/ QMC – quantum Monte Carlo methods (limited to small systems, benchmark
purposes)
numerical sampling of N-electron wave function
∫Ψ ĤΨdr1…drn = < ĤΨ/Ψ> + a<((ĤΨ/Ψ) - <(ĤΨ/Ψ)>)2>1/2 /N1/2
3/ DFT (density functional theory) methods
In 80’s: the first set of ‘chemically accurate’ exchange-correlation functionals
developed (Becke, Perdew, Parr), later on hybrid functionals followed:
=> B-P86, PBE, B3LYP, BPW91, TPSS
Especially with using RI (resolution-of-the-identity approximation) systems up to
~200 atoms can be studiedIn practice (metalloproteins, DFT is used)
Shortcomings: 1/ ‘empirical’ method
2/ does not properly describe dispersion (DFT(+D) method of
Hobza, Jurecka, Cerny??)
MM (molecular mechanics, force field) methods
QM: nuclei + electrons
MM: atoms
Potential Energy of Molecular System
(adapted from AMBER8 Manual)
MM (force field) methods: molecular simulations
Few notes:
Terminology: optimization (MM) vs dynamics (MD), or Monte Carlo
(MC) sampling
(Dis)advantages: universal parametrization
Systems:
~50.000 atoms can be studied conveniently
Limitations:
not able to described chemical reactions
heterocompounds poorly described
=> QM/MM: Why not to couple these two?
History: A. Warshel and M. Levitt, Theoretical Studies of Enzymic Reactions: Dielectric Electrostatic and
Steric stabilisation of the carbonium ion in the reaction of Lysozyme. J. Mol. Biol. 103, 227 (1976).
System is divided into 2 parts:
1/ Reaction region
described by QM methods
2/ The rest (spectator, bulk)
described by MM methods
Fig: (adapted from Villa, Warshel)
The energy of the total system is given as :
Etot = EQM + EMM + Ecoupling
QM/MM (Example: ONIOM-like approach)
System 1 (S1) is treated at the QM level. It is truncated using
link atoms. The positions of the link atoms are linearly
dependent on the cooresponding heayy atoms (hydrogen
link approach)
System 2 (S2) contains all amino acids and solvent molecules
within a radius R1 of any atom in S1. It is treated with MM
methods.
System 3 (S3) contains all amino acids and solvent molecules
within a radius R2 of any atom in S2. It is also treated with
MM methods.
The energy of the total system is given as :
In this study, Turbomole 5.7 (QM part)
and Amber 8/sander (MM part)
programs, have been used. Using RI
approx. in DFT, QM systems up to 150
atoms can be conveniently calculated
E(QM/MM)=E(QM,S1) + E(MM,S3) - E(MM,S1),
where
E(QM,S1) … QM energy of S1 in the field of point charges
E(MM,S3) … MM energy of S3 with charges of S1 set to zero
E(MM,S1) … MM energy of S1 with charges of S1 set to zero
U. Ryde and M. H. M. Olsson, Int. J. Quantum Chem., 81, 335-347, 2001
QM/MM simulation protocol
(general)
Start with a crystal (NMR) structure
Add hydrogens, missing atoms, loops
Run MM minimization for the system in vacuo with atoms resolved in c.s. as
fixed or constrained
Add solvation shell or solvation box (PBC) to the whole system
Run minimization of the whole solvated system (c.s. atoms fixed)
Run 300-ps of simulated annealing protocol (c.s. atoms fixed) (equilibration)
After the system is equilibrated, remove all ‘solvent’ waters except ~10 Å
solvation shell
Run the final MM minimization
Define QM region, surrounding region (S2) that is relaxed in MM
Start QM/MM production calculations
QM/MM simulation protocol
(technical details)
Evaluate the wave function of System 1 in the field of the pointcharges of System 2 and 3 (QC).
Evaluate the forces of system1 including the electrostatics of system2 and 3 (QC).
Evaluate the forces of system1 and 2 with any electrostatic interactions (MM).
Add the QC and MM forces to obtain the QC/MM forces on the atoms of System 1 (FixForce).
Relax the atoms of System 1 using the QC/MM forces (QC).
Use the relaxed coordinates of System 1 to construct the new coordinate representation of System 1
(FixCoord1).
Insert the charges (mulliken, ESP,...) of the atoms of System 1 obtained from the QC calculation into
the MM representation (FixCharge).
Relax the atoms of System 2 with System 1 fixed (MM).
Insert the new coordinates of the pointcharges of the atoms in System 2 and 3 to be used in the
next QC calculation. (FixCoord2).
Calculate the QC energy of System 1 (QC).
Calculate the MM energy of Systems 1 and 2 (MM).
Add the energies appropriately (FixEnergy)
Check for convergence. If not converged, then go to step 2 above.
Other QM/MM coupling schemes
Quantum Refinement (QM/MM with structure refinement)
improving locally crystal structures
EQM/MM/X-ray = EQM/MM + waEX-ray
QM/MM-EXAFS
more accurate information about the local structure
EQM/MM/EXAFS = wQM/MMEQM/MM + wEXAFSEEXAFS
QM/MM-NMR
(group of Prof. Ryde, Lund University)
QM/MM (ComQum) +/- : few comments and
summary
+ QM/MM method is the natural way to include the environment (electrostatics,
protein restraints on the active site of interest) into the ‘accurate’ QM
calculations (at minimum cost) => ‘good’ structures
+ QM (~100 atoms) vs MM (~1000 atoms) system size: full MM minimization of
S2 for each QM step
- Poor representation of electrostatics in the boundary region (polarizable force
fields?, charge-fitting procedures for S2 (< 5 Å) ?, multipole expansions)
- Need for QM/MM sampling to get a satisfactory accurate energetics,
approaching ‘biochemical accuracy’ (~ kTlog10) => QTCP method
(T. H. Rod & U. Ryde (2005) Phys. Rev. Lett., 94, 138302)
Outline
Introduction: few notes on the physical chemistry of enzyme
structure and catalysis
QM/MM Methodology
• Why?
• QM methods: usage, accuracy, limitations
• MM methods
• QM/MM coupling schemes
• Other coupling schemes (QM-MM/X-ray, QM-MM/EXAFS)
QM/MM → (E0, ∆G, ∆G#, pKa, KI, k1,…) Route: towards
experimental data – comparison with in vacuo studies
QM/MM → (E0, ∆G, ∆G#, pKa, KI, k1,…) Route:
(towards experimental data – comparison with in vacuo studies)
Rate constant:
k = kBT/h exp(-∆G#/RT)
Equilibrium constants:
K = exp(-∆G0/RT)
KI = exp(-∆GB/RT)
KA = exp(-∆GA/RT)
Fig: (adapted from Villa, Warshel)
Accuracy of the calculations vs. Adequate Model vs. ‘the quality of the
experimental data’ ☺
In vacuo Calculations of Free Energy
Thermodynamic cycle used to calculate ∆G(ε=x)
R(l)
∆Gr(ε=x)
∆Gsolv(x,R)
R(g)
P(l)
∆Gsolv(x,P)
∆Gr(ε=1)
P(g)
∆Gr(ε=x) = ∆Gr(ε=1) + ∆Gsolv(x,P) - ∆Gsolv(x,R)
where ∆Gr(ε=1) = ∆Er + ZPE + (∆n)RT -T∆Sr
solvation effects calculated by polarized continuum models (PCM,
COSMO - Conductor-like Screening Model)
QM/MM → (E0, ∆G, ∆G#, pKa, KI, k1,…) Route:
Statistical Thermodynamics
Ensembles
An ensemble is the assembly of all possible configurations that are consistent with the constraints that we
impose on the system. A number of different ensemble averages are possible depending on the conditions of
measurement (or simulation).
The microcanonical ensemble (NVE) is the assembly of all states for a system with fixed total energy, E,
number of molecules, N, and volume, V.
In the canonical ensemble (NVT) the energy can fluctuate. This ensemble describes a closed system in
contact with a heat bath. Calculations done with these conditions yield the Helmholtz free energy.
The isothermal-isobaric ensemble (NPT), with a constant number of particles, pressure, and temperature, is
used to obtain the Gibbs free energy.
The grand canonical ensemble (µVT) is obtained under conditions of constant chemical potential, volume,
and temperature.
QM/MM → (E0, ∆G, ∆G#, pKa, KI, k1,…) Route
=> Need for Configurational Sampling
Thermodynamic Integration (Free Energy Perturbation) Methods
Simply substituting E with EQM/MM gives the analogous results (though we face a
problem with sampling, QM/MM is still too demanding) => sometimes called
QTCP method (Ryde, Rod)
QM/MM → (E0, ∆G, ∆G#, pKa, KI, k1,…) Route
As in In Vacuo calculations, FEP (TI) are used in conjunction with
Thermodynamic Cycles (e.g., ligand binding nergies => KI
QM/MM → (E0, ∆G, ∆G#, pKa, KI, k1,…) Route
Solvation Methods for Protein Energetics (Implicit Water model, such
as MM-PBSA method)
where φi(r) in solution (dielectric) is obtained by solving Poisson-Boltzmann Equation
Practical Hint: Run MD in Explicit Water Model (PBC, droplet, etc.), remove
water molecules and calculate solvation energies by PBSA
Summary
Introduction: few notes on the physical chemistry of enzyme
structure and catalysis
QM/MM Methodology
• Why?
• QM methods: usage, accuracy, limitations
• MM methods
• QM/MM coupling schemes
• Other coupling schemes (QM-MM/X-ray, QM-MM/EXAFS)
QM/MM → (E0, ∆G, ∆G#, pKa, KI, k1,…) Route: towards
experimental data – comparison with in vacuo studies