QM/MM Calculations and Applications to Biophysics Marcus Elstner Physical and Theoretical Chemistry, Technical University of Braunschweig

Proteins, DNA, lipids N

Computational challenge ~ atoms in protein ~ ns molecular dynamics simulation (MD, umbrella sampling) - chemical reactions: proton transfer - treatment of excited states QM

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!  several atoms

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 Protein Membrane:  = 10 active Water:  = 80  = 20

Combined QM/MM  =80 Computationally efficient –~ atoms Generally for structural properties Bond breaking/formation Computationally demanding –DFT, AI: ~ 50 atoms –Semi-Empirical: ~ atoms Quantum mechanical (QM) Molecular mechanical (MM) 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

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,...

Hierarchy of methods CI, MP CASPT2 CI, MP CASPT2 Length scale Continuum electrostatics Molecular Mechanics fs ps ns time SE-QM approx-DFT SE-QM approx-DFT HF, DFT nm

Empirical Force Fields: Molecular Mechanics MM models protein + DNA structures quite well Problem: - polarization - charge transfer - not reactiv in general kbkb kk kk

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 MM QM ? ?

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 accelerated MD - metadynamics reaction path methods - CPR free energy sampling techniques - umbrella sampling

QM/MM Methods

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 MM QM

Additive QM/MM total energy QM = + + MM interaction

Subtractive QM/MM: ONIOM Morokuma and co.: GAUSSIAN total energy QM MM = - +

The ONIOM Method (an ONION-like method) Example: The binding energy of  3 C-C  3  HPE) from S. Irle

Link Atoms C C F F F H H H C H H H H Link atom Link atom host R L = g x R LAH R LAYER 1 LAYER 2 Real systemModel system R L g: constant from S. Irle

E(ONIOM) = E(LOW,REAL) +E(HIGH,MODEL) - E(LOW,MODEL) C C F F F H H H C C F F F H H H C H H H H C H H H H C H H H H C H H H H C H H H H C H H H H C H H H H C H H H H C H H H H C H H H H C H H H H C H H HIGH H H LOW SIZE LEVEL E(HIGH,REAL)  E(ONIOM) = = E(LOW,MODEL) + SIZE (S-value) + LEVEL Level Effect and Size Effect assumed uncoupled Approximation MODEL REAL = E(LOW,MODEL) + [E(LOW,REAL)-E(LOW,MODEL) ] + [E(HIGH,MODEL)-E(LOW,MODEL)] ONIOM Energy: The additivity assumption 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) + J T x H(High,Model) x J - J T x H(Low,Model) x J Scale each Hessian by s(Low)**2 or s(High)**2 to get scaled H(ONIOM) ONIOM density  (ONIOM, Real) =  (Low,Real) +  (High,Model) -  (Low,Model) ONIOM properties = + - from S. Irle

Three-layer ONIOM (ONIOM3) MO:MO:MO MO:MO:MM Target from S. Irle

Additive QM/MM: linking

Additive QM/MM total energy QM = + + MM interaction

Additive QM/MM: Elecrostatic mechanical embedding

Combined QM/MM Amaro, Field, Chem Acc Bonds: a)take force field terms b) - link atom - pseudo atoms - frontier bonds Nonbonding: - VdW - electrostatics

Combined QM/MM Reuter et al, JPCA 2000 Bonds: a) from force field

Combined QM/MM: link atom Amaro & Field, T. Chem Acc a)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)

Combined QM/MM: frozen orbitals Warshel, Levitt 1976 Rivail + co Gao et al 1998 Reuter et al, JPCA 2000

Combined QM/MM: Pseudoatoms Amaro & Field,T Chem Acc Pseudobond- connection atom Zhang, Lee, Yang, JCP 110, 46 Antes&Thiel, JPCA No link atom: parametrize C  H 2 as pseudoatom X

Nonbonding terms: VdW - take from force field - reoptimize for QM level Coulomb: which charges? Combined QM/MM Amaro & Field,T Chem Acc. 2003

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

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 Problems

Modelling Stratgies

How much can we treat ? = How much can we afford Protein Membrane:  = 4 active Water:  = 80  = 20

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 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 afford Protein active Small protein  Simple QM/MM: - fix most of the protein - neglect polarization of environment

solvation  charge scaling freezing vs. stochastic boundary size of movable MM? size of QM? First approximations:

How much can we treat ? = How much can we afford Protein: polarizable active Small protein  Simple QM/MM: - fix most of the protein - include polarization from environment

Absolute excitation energies S 1 excitation energy (eV) expTD- B3LYP 1 TD- DFTB OM2/ CIS CASSCF 2 OM2/ MRCI SORCI vacuum bR (QM:RET) Vreven[2003] 2 Hayashi[2000] TDDFT nearly zero CIS shifts still too small ~50% SORCI, CASPT2 OM2/MRCI compares very well

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:

How much can we treat ? = How much can we afford Explicit Watermolecules pbc Protein active

How much can we treat ? = How much can we afford Protein Membrane:  = 4 active Water:  = 80  = 20

Ion channels Membrane:  = 4 Water:  = 80 Explicit water

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

GSBP Solvation free energy of point charges

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 only in agreement with full solvent simulations when GSBP is used!

-differences in protein conformations Problems with the PES: CPR, NEB etc. Zhang et al JPCB 107 (2003) 44459

- 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 Problems with the PES: complex energy landscape

Ion channels