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QM /MM C alculations and A pplications to Biophysics

QM /MM C alculations and A pplications to Biophysics. Marcus Elstner Physical and Theoretical Chemistry, Technical Universi ty of Braunschweig. N. Proteins, DNA, lipids. Computational challenge. ~ 1.000-10.000 atoms in protein ~ ns molecular dynamics simulation

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QM /MM C alculations and A pplications to Biophysics

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  1. QM/MM Calculations and Applications to Biophysics Marcus Elstner Physical and Theoretical Chemistry, Technical Universityof Braunschweig

  2. N Proteins, DNA, lipids

  3. 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

  4. 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

  5. Computational problem II: sampling with MD • flexibility: not one global minimum  conformational entropy • solventrelaxation • ps – ns timescale (timestep ~ 1fs) (folding anyway out of reach!)

  6. Optimal setup Water:  = 80  = 20 Membrane:  = 10 Protein Membrane:  = 10 active  = 20 Water:  = 80

  7. Computationally efficient ~103-5 atoms Generally for structural properties Bond breaking/formation Computationally demanding DFT, AI: ~ 50 atoms Semi-Empirical: ~102-3 atoms Combined QM/MM e=80 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

  8. 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, ...

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

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

  11. MM QM 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 ? ?

  12. 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

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

  14. QM/MM Methods

  15. MM QM 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

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

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

  18. from S. Irle The ONIOM Method(an ONION-like method) Example: The binding energy of f3C-C f3 (HPE)

  19. R R L LAH R = g x R L LAH from S. Irle 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

  20. @ E( HIGH , REAL ) E( ONIOM ) = = E( LOW , MODEL ) + SIZE (S-value) + LEVEL = E( LOW , MODEL ) + [E( LOW , REAL )-E( LOW , MODEL ) ] + [E( HIGH , MODEL )-E( LOW , MODEL )] E( ONIOM ) = E( LOW , REAL ) + E( HIGH , MODEL ) - E( LOW , MODEL ) from S. Irle 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 H H H H H H C F F F REAL MODEL

  21. from S. Irle

  22. 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 r(ONIOM, Real) = r(Low,Real) + r(High,Model) - r(Low,Model) ONIOM properties < o (ONIOM, Real)> = < o (Low,Real) > + < o (High,Model) > - < o (Low,Model) >

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

  24. Additive QM/MM: linking

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

  26. Elecrostatic mechanical embedding Additive QM/MM:

  27. Combined QM/MM • Bonds: • take force field terms • - link atom • - pseudo atoms • - frontier bonds • Nonbonding: • - VdW • electrostatics Amaro , Field , Chem Acc. 2003

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

  29. 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

  30. Combined QM/MM: frozen orbitals Reuter et al, JPCA 2000 Warshel, Levitt 1976 Rivail + co. 1996-2002 Gao et al 1998

  31. Combined QM/MM: Pseudoatoms Amaro & Field ,T Chem Acc. 2003 Pseudobond- connection atom Zhang, Lee, Yang, JCP 110, 46 Antes&Thiel, JPCA 103 9290 No link atom: parametrize C H2 as pseudoatom X

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

  33. Combined QM/MM • Tests: • C-C bond lengths, vib. frequencies • C-C torsional barrier • H-bonding complexes • proton affinities, deprotonation • energies

  34. 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

  35. 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)

  36. 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

  37. Modelling Stratgies

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

  39. 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

  40. 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

  41. 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

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

  43. 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

  44. Absolute excitation energies 0.1 0.2 1.0 0.5 1Vreven[2003] 2 Hayashi[2000] • TDDFT nearly zero • CIS shifts still too small ~50% • SORCI, CASPT2 • OM2/MRCI compares very well

  45. 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:

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

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

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

  49. 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

  50. GSBP Solvation free energy of point charges

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