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Unformatted text preview: QM/MM Calculations and Applications to
QM/MM Calculations and Applications to
Biophysics
Biophysics Marcus Elstner
Physical and Theoretical Chemistry,
Technical University of Braunschweig Proteins, DNA, lipids
Proteins, DNA, lipids N Computational challenge
Computational challenge
~ 1.00010.000 atoms in protein
~ ns molecular dynamics simulation
(MD, umbrella sampling) QM
 chemical reactions: proton transfer
 treatment of excited states Computational problem I: number of atoms
Computational problem I: number of atoms chemical reaction which needs QM treatment immediate environment: electrostatic and steric
interactions solution, membrane: polarization and structural effects
on protein and reaction! 10.000...  several 100.000 atoms Computational problem II: sampling with MD
Computational problem II: sampling with MD flexibility: not one global minimum conformational entropy solvent relaxation ps – ns timescale (timestep ~ 1fs) (folding anyway out of reach!) Optimal setup
Optimal setup Water: = 80 = 20
Membrane: = 10 Protein active = 20
Water: = 80 Membrane: = 10 Combined QM/MM
Combined QM/MM
ε=80 Quantum mechanical (QM)
•
• Bond breaking/formation
Computationally demanding
– DFT, AI: ~ 50 atoms
– SemiEmpirical: ~1023 atoms Molecular mechanical (MM)
•
• Computationally efficient
– ~1035 atoms
Generally for structural properties Combined QM/MM
• Chemical Rx in macromolecules ˆ QM + H QM / MM Ψ + E QM / MM + E MM • DFT (AI) /MM: Reaction path
ˆ
E= Ψ H
van
el
•
•No polarization of MM region!
•No charge transfer between QM and MM SemiEmpirical/MM: Potential of
mean force, rate constants Combined QM/MM
Combined QM/MM 1976 Warshel and Levitt 1986 Singh and Kollman 1990 Field, Bash and Karplus
QM
• Semiempirical
• quantum chemistry packages: DFT, HF, MP2, LMP2
• DFT plane wave codes: CPMD
MM
• CHARMM, AMBER, GROMOS, SIGMA,TINKER, ... Hierarchy of methods
Hierarchy of methods
Continuum electrostatics
Continuum electrostatics ns Molecular Mechanics
Molecular Mechanics ps SEQM
SEQM
approxDFT
approxDFT time fs HF, DFT
HF, DFT
CI, MP
CI, MP
CASPT2
CASPT2 nm Length scale Empirical Force Fields: Molecular Mechanics
Empirical Force Fields: Molecular Mechanics
MM
MM
V = ∑ k ( b − b ) + ∑ k (θ − θ
2 bonds b 0 angles θ )+
2 0 σ
+ ∑ 4ε r i,j ∑ ∑ kφ [1 + cos( ( nφ − δ ) ] + ∑ kω ( ω − ω
N (n) dihedrals n =1 impropers σ qq − + ∑ r Dr 12 i,j i,j i,j 6 i,j i j i,j i, j ij models protein + DNA structures quite well
Problem:
 polarization
 charge transfer
 not reactiv in general kb
k k 0 ) 2 QM/MM Methods
QM/MM Methods Mechanical embedding: only steric effects Electrostatic embedding: polarization of QM due to MM Electrostatic embedding + polarizable MM Larger environment:  box + Ewald summ.
 continuum electrostatics ? ?
QM  coarse graining MM Ho to study reactions and (rare) dynamical events
Ho to study reactions and (rare) dynamical events direct MD accelerated MD
 hyperdynamics (Voter)  chemical flooding (Grubmüller)
 metadynamics (Parinello) reaction path methods
 NEB (nudged elastic band, Jonsson)
 CPR (conjugate peak refinement, Fischer, Karplus)
 dimer method (Jonsson) free energy sampling techniques
 umbrella sampling
 free energy perturbation
 transition path sampling Ho to study reactions and (rare) dynamical events
Ho to study reactions and (rare) dynamical events accelerated MD
 metadynamics reaction path methods
 CPR free energy sampling techniques
 umbrella sampling QM/MM Methods
QM/MM Methods Subtractive vs. additive models
Subtractive vs. additive models  subtractive: several layers: QMMM
doublecounting on the regions is subtracted
 additive: different methods in different regions +
interaction between the regions QM
MM Additive QM/MM
Additive QM/MM total energy = + QM MM + interaction QM MM Subtractive QM/MM:: ONIOM
Subtractive QM/MM ONIOM
Morokuma and co.: GAUSSIAN
Morokuma and co.: GAUSSIAN total energy = MM + QM  MM The ONIOM Method (an ONIONlike method)
Bondformat ion/breaking t akes place.
Use the "High level" (H) method. Sm a ll Mod e l
Sys t e m ( SM) I nt e r m e d i a t e
Mod e l Sys t e m( IM) R e a l Sys t e m( R ) First Layer Second Layer
Elect ronic effect on t he first layer.
Use the "Medium level" (M) method. T hird Layer
Environment al effect s on t he first layer.
Use the "Low level" (L) method. Example: The binding energy of φ3CC φ3 ( HPE)
ONIOM: 16.4 kcal/mol
C C Hexaphenylethane
(HPE) 2
C Triphenylmethyl radical
(T PMR) from S. Irle Link Atoms
Real system
H H H LAYER 1 Model system
H C C RL RLAH
Link atom
Link atom host C LAYER 2 H H H F
F RL = g x RLAH F g: constant from S. Irle ONIOM Energy: The additivity assumption
Level Effect and Size Effect assumed uncoupled
E(HIGH ,REAL) ≅ E(ONIOM) =
= E(LOW,MODEL +
) SIZE (Svalue) + LEVEL = E(LOW,MODEL) + [E(LOW ,REAL)E(LOW,MODEL) ] + [E(HIGH,MODEL)E(LOW,MODEL)]
H
H H H
C
C + F HIGH
F
F LEVEL HIGH H
H H
C H
H LOW H
C
H Approximation  MODEL E(ONIOM ) = E(LOW,REAL )
+ E(HIGH ,MODEL)  E(LOW,MODEL ) +
SIZE H
H H
C
C F LOW
F
F REAL from S. Irle S(ubstituent)Value test: Does the low level work?
Choice of combination of levels is critical
Combinations can be investigated using the SValue test
S(LEVEL) = E(LEVEL,REAL)  E( LEVEL, MODEL) Several E(HIGH,REAL) calculations necessary
HIGH If S(HIGH) = S(LOW)
E(ONIOM) = E(HIGH, REAL) LEV EL S(HIGH) Low level describes subst it uent
effect as good as high level does! S(HIGH)  S(LOW)
L OW S(LOW)
MODEL REAL SIZE Must be as close t o zero as possible from S. Irle ONIOM Potential Energy Surface and Properties
ONIOM energy
E(ONIOM, Real) = E(Low,Real) + E(High,Model)  E(Low,Model) Potential energy surface well defined, and also derivatives are available. ONIOM gradient
G(ONIOM, Real) = G(Low,Real) + G(High,Model) x J  E(Low,Model) x J J = ∂ (Real coord.)/ ∂ (Model coord.) is the Jacobian that converts the model
system coordinate to the real system coordinate ONIOM Hessian
H(ONIOM,Real) = H(Low,Real) + JT x H(High,Model) x J  JT x H(Low,Model) x J
Scale each Hessian by s(Low)**2 or s(High)**2 to get scaled H(ONIOM) ONIOM density
ρ(ONIOM, Real) = ρ(Low,Real) + ρ(High,Model)  ρ(Low,Model) ONIOM properties
< o (ONIOM, Real)> = < o (Low,Real) > + < o (High,Model) >  < o (Low,Model) > from S. Irle Threelayer ONIOM (ONIOM3)
E(ONIOM)= E(LOW,REAL)
HIGH +  E (LOW,INT ERMEDIAT E
) Target + E (MEDIUM,INT ERMEDIATE
) LE V E L  E (MEDIUM,MODEL)
+ E (HIGH,MODEL)
MEDIUM  LOW
MODEL + INT ERMEDIAT E +
REAL MO:MO:MO
MO:MO:MM SIZE
from S. Irle Additive QM/MM: linking
Additive QM/MM: linking Additive QM/MM
Additive QM/MM total energy = + QM MM + interaction QM MM Additive QM/MM:
Additive QM/MM: ˆ
ˆ
ˆ
ˆ
H = H QM + H MM + H QM / MM A
Zq
B
q
ˆ
H QM / MM = −∑ M + ∑ α M + ∑ αM − αM
12
6
RαM
i , M riM
α , M RαM
α , M RαM Elecrostatic ˆ int .coor + H QM / MM mechanical embedding Combined QM/MM
Combined QM/MM
Bonds: Amaro , Field , Chem Acc. 2003 a) take force field terms
b)  link atom
 pseudo atoms
 frontier bonds
Nonbonding:
 VdW  electrostatics
A
Zq
B
q
ˆ
H QM / MM = −∑ M + ∑ α M + ∑ αM − αM
12
6
riM α , M RαM α , M RαM RαM
i,M ˆ int .coor + H QM / MM Combined QM/MM
Combined QM/MM Bonds:
a)
from force field Reuter et al, JPCA 2000 Combined QM/MM: link atom
Combined QM/MM: link atom
a) constrain or not?
(artificial forces)
relevant for MD
b) Electrostatics
 LA included – excluded
(include!)  QMMM:
exclude MMhost
exclude MMhostgroup  Amaro & Field , T. Chem Acc. 2003 DFT, HF: gaussian broadening of MM point
charges, pseudopotentails (e spill out) Combined QM/MM: ffrozen orbitals
Combined QM/MM: rozen orbitals
Reuter et al, JPCA 2000 Warshel, Levitt 1976
Rivail + co. 19962002
Gao et al 1998 A
Zq
B
q
ˆ
H QM / MM = −∑ M + ∑ α M + ∑ αM − αM
12
6
riM α , M RαM α , M RαM RαM
i,M ˆ int .coor + H QM / MM Combined QM/MM: Pseudoatoms
Combined QM/MM: Pseudoatoms
Amaro & Field ,T Chem Acc. 2003 Pseudobond connection atom
Zhang, Lee, Yang, JCP 110, 46
Antes&Thiel, JPCA 103 9290 X No link atom: parametrize Cβ H2 as pseudoatom A
Zq
B
q
ˆ
H QM / MM = −∑ M + ∑ α M + ∑ αM − αM
12
6
riM α , M RαM α , M RαM RαM
i,M ˆ int .coor + H QM / MM Combined QM/MM
Combined QM/MM
Nonbonding terms: Amaro & Field ,T Chem Acc. 2003 VdW
 take from force field
 reoptimize for QM level Coulomb:
which charges?
A
Zq
B
q
ˆ
H QM / MM = −∑ M + ∑ α M + ∑ αM − αM
12
6
RαM
i , M riM
α , M RαM
α , M RαM ˆ int .coor + H QM / MM Combined QM/MM
Combined QM/MM Tests:
 CC bond lengths, vib. frequencies
 CC torsional barrier
 Hbonding complexes
 proton affinities, deprotonation
energies Subtractive vs. additive QM/MM
Subtractive vs. additive QM/MM
 parametrization of methods for all regions required
e.g. MM for Ligands
SE for metals
+ QM/QM/MM conceptionally simple and applicable Local Orbital vs. plane wave approaches:
Local Orbital vs. plane wave approaches:
PW implementations
(most implementations in LCAO)
 periodic boundary conditions and large box! lots of empty space in unit cell
 hybride functionals have better accuracy: B3LYP, PBE0 etc.
+ no BSSE + parallelization (e.g. DNA with ~1000 Atoms) Problems
Problems • QM and MM accuracy
• QM/MM coupling
• model setup: solvent, restraints
• PES vs. FES: importance of sampling All these factors CAN introduce errors in similar
magnitude Modelling Stratgies
Modelling Stratgies How much can we treat ? =
How much can we treat ? =
How much can we afford
How much can we afford Water: = 80 = 20
Membrane: = 4 Protein active = 20
Water: = 80 Membrane: = 4 How to model the environment
How to model the environment 1) Only QM (implicit solvent)
2) QM/MM w/wo MM polarization
3) Truncated systems and charge scaling
System in water with periodic boundary
conditions: pbc and Ewald summation
Truncated system and implicit solvent models How much can we treat ? =
How much can we treat ? =
How much can we afford
How much can we afford Don‘t have or don‘t trust QM/MM or too
complicated Only active site models active = ?? How much can we treat ? =
How much can we treat ? =
How much can we afford
How much can we afford
Small protein Simple QM/MM:  fix most of the protein
 neglect polarization of environment Protein active First approximations:
First approximations:
• solvation charge scaling
• freezing vs. stochastic boundary
• size of movable MM?
• size of QM? How much can we treat ? =
How much can we treat ? =
How much can we afford
How much can we afford
Small protein Simple QM/MM:  fix most of the protein
 include polarization from environment Protein: polarizable active Absolute excitation energies
S1 excitation energy (eV)
exp
vacuum
2.1
8
Hayashi[2000] bR (QM:RET)
Vreven[2003] 2 1 TDB3LYP1 TDDFTB OM2/
CIS CASSCF2 OM2/
MRCI SORCI 2.42 2.14 2.34 2.86 2.13 1.86 2.53 2.21 2.54 3.94 2.53 2.34 0.2 1.0 0.1 • TDDFT nearly zero
• CIS shifts still too small ~50%
• SORCI, CASPT2
• OM2/MRCI compares very well 0.5 Polarizable force field for environment
• MM charges
• MM polarization RESP charges for residues in gas phase atomic polarizabilities: µ = α E Polarization red shift of about 0.1 eV: How much can we treat ? =
How much can we treat ? =
How much can we afford
How much can we afford pbc Protein active
Explicit Watermolecules How much can we treat ? =
How much can we treat ? =
How much can we afford
How much can we afford Water: = 80 = 20
Membrane: = 4 Protein active = 20
Water: = 80 Membrane: = 4 Ion channels
Ion channels Water: = 80
Explicit water Membrane: = 4 Membrane: = 4 Explicit water Water: = 80 Implicit solvent: Generalized Solvent Boundary
Implicit solvent: Generalized Solvent Boundary
Potential (GSBP, B. Roux)
Potential (GSBP, B. Roux)
•Drawback of conventional implicit solvation:
e.g. specific water molecules important
•Compromise: 2 layers, one explicit solvent
layer before implicit solvation model.
• inner region: MD, geomopt
• outer region: fixed QM/MM
explicit MM
implicit GSBP
GSBP φrf = φs − φv Solvation free
energy of point
charges GSBP
GSBP Depends on inner
coordinates! Basis set expansion
of inner density
calculate reaction
field for basis set
QM/MM DFTB implementation by Cui group
(Madison) Water structure in Aquaporin
Water structure in Aquaporin
Water structure only in
agreement with full
solvent simulations when
GSBP is used! Problems with the PES: CPR, NEB etc.
Problems with the PES: CPR, NEB etc.
differences in protein
conformations Zhang et al JPCB 107 (2003) 44459 Problems with the PES: complex energy
Problems with the PES: complex energy
llandscape
andscape
 differences in protein conformations
(starting the reaction path calculation)
 problems along the reaction pathway
* flipping of water molecules
* size of movable MM region
different Hbonding pattern average over these effects:
potential of mean force/free energy Ion channels
Ion channels ...
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 Summer '06
 DuaneJohnson
 Lipids, pH

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