QM/MM Calculations and Applications to Biophysics

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QM/MM Calculations and Applications to Biophysics

Marcus Elstner Physical and Theoretical
Chemistry, Technical University of Braunschweig
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Proteins, DNA, lipids
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Computational challenge

• 1.000-10.000 atoms in protein
• ns molecular dynamics simulation
• (MD, umbrella sampling)
• chemical reactions proton transfer
• treatment of excited states

QM
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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

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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!)

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Optimal setup
Water ? 80
? 20
Protein
Membrane ? 10
Membrane ? 10
active
? 20
Water ? 80
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Combined QM/MM
e80
Quantum mechanical (QM)

• Bond breaking/formation
• Computationally demanding
• DFT, AI 50 atoms
• Semi-Empirical 102-3 atoms

Molecular mechanical (MM)

• Computationally efficient
• 103-5 atoms
• Generally for structural properties

Combined QM/MM

• Chemical Rx in macromolecules
• DFT (AI) /MM Reaction path
• Semi-Empirical/MM Potential of mean force, rate
  constants
  

• No polarization of MM region!
• No charge transfer between QM and MM

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Combined QM/MM
1976 Warshel and Levitt 1986 Singh and
Kollman 1990 Field, Bash and Karplus

• QM
• Semi-empirical
• quantum chemistry packages DFT, HF, MP2, LMP2
• DFT plane wave codes CPMD
• MM
• CHARMM, AMBER, GROMOS, SIGMA,TINKER, ...

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Hierarchy of methods
fs ps
ns time
CI, MP CASPT2
Length scale
nm
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Empirical Force Fields Molecular Mechanics
MM

• models protein DNA structures quite
  well
• Problem
• polarization
• charge transfer
• not reactiv in general

k?
kb
k?
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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
• - coarse
  graining

?
?
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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

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Ho to study reactions and (rare) dynamical events
accelerated MD - metadynamics reaction path
methods - CPR free energy sampling
techniques - umbrella sampling

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QM/MM Methods

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Subtractive vs. additive models

• subtractive several layers QM-MM
• doublecounting on the regions is subtracted
• additive different methods in different regions
  
• interaction between the regions

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Additive QM/MM
total energy
QM


interaction
QM

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Subtractive QM/MM ONIOM Morokuma and co.
GAUSSIAN
total energy
MM

MM
QM
-

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The ONIOM Method (an ONION-like method)
Example The binding energy of f3C-C f3 (HPE)
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Link Atoms
Real system
Model system
H
H
H
H
H
H
LAYER 1
C
C
H
Link atom
C
Link atom host
LAYER 2
F
F
F
g constant
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ONIOM Energy The additivity assumption
Level Effect and Size Effect assumed uncoupled
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
C
C
C
C
C
C
C
HIGH
HIGH

H
H
H
H
H
H
C
F
F
F
Approximation
LEVEL
-

H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
C
C
C
C
C
C
C
LOW
LOW
SIZE
C
H
H
H
H
H
H
F
F
F
REAL
MODEL
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(No Transcript)
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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 r(ONIOM, Real)
r(Low,Real) r(High,Model) - r(Low,Model) ONIOM
properties lt o (ONIOM, Real)gt lt o (Low,Real)
gt lt o (High,Model) gt - lt o (Low,Model) gt
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Three-layer ONIOM (ONIOM3)
Target
MOMOMO MOMOMM
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Additive QM/MM linking
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Additive QM/MM
total energy
QM


interaction
QM

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Additive QM/MM
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Combined QM/MM

• Bonds
• take force field terms
• - link atom
• - pseudo atoms
• - frontier bonds
• Nonbonding
• - VdW
• electrostatics

Amaro , Field , Chem Acc. 2003
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Combined QM/MM
Bonds a) from force field
Reuter et al, JPCA 2000
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Combined QM/MM link atom

• constrain or not?
• (artificial forces)
• relevant for MD
• b) Electrostatics
• LA included excluded
• (include!)
• QM-MM
• exclude MM-host
• exclude MM-hostgroup
• DFT, HF gaussian broadening of MM point
  charges, pseudopotentails (e spill out)

Amaro Field , T. Chem Acc. 2003
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Combined QM/MM frozen orbitals
Reuter et al, JPCA 2000
Warshel, Levitt 1976 Rivail co. 1996-2002 Gao
et al 1998
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Combined QM/MM Pseudoatoms
Amaro Field ,T Chem Acc. 2003
Pseudobond- connection atom Zhang, Lee, Yang,
JCP 110, 46 AntesThiel, JPCA 103 9290 No link
atom parametrize C? H2 as pseudoatom
X
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Combined QM/MM

• Nonbonding terms
• VdW
• - take from force field
• reoptimize for QM level
• Coulomb
• which charges?

Amaro Field ,T Chem Acc. 2003
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Combined QM/MM

• Tests
• C-C bond lengths, vib. frequencies
• C-C torsional barrier
• H-bonding complexes
• proton affinities, deprotonation
• energies

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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

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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)

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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

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Modelling Stratgies

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How much can we treat ? How much can we afford
Water ? 80
? 20
Protein
Membrane ? 4
Membrane ? 4
active
Water ? 80
? 20
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How to model the environment

• Only QM (implicit solvent)
• QM/MM w/wo MM polarization
• Truncated systems and charge scaling
• System in water with periodic boundary
  conditions pbc and Ewald summation
• Truncated system and implicit solvent models

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How much can we treat ? How much can we afford
Dont have or dont trust QM/MM or too
complicated ?Only active site models
? ??
active
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How much can we treat ? How much can we afford
Small protein ? Simple QM/MM - fix most of the
protein - neglect polarization of environment
Protein
active
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First approximations

• solvation? charge scaling
• freezing vs. stochastic boundary
• size of movable MM?
• size of QM?

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How much can we treat ? How much can we afford
Small protein ? Simple QM/MM - fix most of the
protein - include polarization from environment
Protein polarizable
active
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Absolute excitation energies
S1 excitation energy (eV) S1 excitation energy (eV) S1 excitation energy (eV) S1 excitation energy (eV) S1 excitation energy (eV) S1 excitation energy (eV) S1 excitation energy (eV)
exp TD-B3LYP1 TD-DFTB OM2/ CIS CASSCF2 OM2/ MRCI SORCI
vacuum 2.42 2.14 2.34 2.86 2.13 1.86
bR (QMRET) 2.18 2.53 2.21 2.54 3.94 2.53 2.34
0.1 0.2 1.0
0.5
1Vreven2003 2 Hayashi2000

• TDDFT nearly zero
• CIS shifts still too small 50
• SORCI, CASPT2
• OM2/MRCI compares very well

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Polarizable force field for environment

• MM charges
• MM polarization
• ? RESP charges for residues in gas phase
• ? atomic polarizabilities ? ? E
• Polarization red shift of about 0.1 eV

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How much can we treat ? How much can we afford
pbc
Explicit Watermolecules
Protein
active
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How much can we treat ? How much can we afford
Water ? 80
? 20
Protein
Membrane ? 4
Membrane ? 4
active
? 20
Water ? 80
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Ion channels
Water ? 80
Explicit water
Membrane ? 4
Membrane ? 4
Water ? 80
Explicit water
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Implicit solvent Generalized Solvent Boundary
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
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GSBP
Solvation free energy of point charges
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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)
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Water structure in Aquaporin
Water structure only in agreement with full
solvent simulations when GSBP is used!
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Problems with the PES CPR, NEB etc.

• differences in protein
• conformations

Zhang et al JPCB 107 (2003) 44459
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Problems with the PES complex energy landscape

• differences in protein conformations
• (starting the reaction path calculation)
• problems along the reaction pathway
• flipping of water
  molecules
• size of movable MM
  region
• different H-bonding
  pattern
• average over these effects
• potential of mean force/free energy

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Ion channels