mixed quantum-classical molecular dynamics simulations of biomolecular systems

1. mixed quantum-classical molecular dynamics simulations of biomolecular systems concepts, machinery & applications Gerrit Groenhof dept. of biophysical chemistry University of Groningen Nijenborg 4, 9747 AG Groningen The Netherlands

2. biomolecular simulation • biomolecules - proteins, DNA, lipid membranes, … - biochemistry, biology, farmacy, medicine, … • physical composition of biomolecules - molecules are composed of atoms - atoms are composed of electrons and nuclei • laws of physics - interaction - motion • computing properties of biomolecules - static: energies, structures, spectra, … - dynamic: trajectories, …

3. molecular simulation • standard molecular dynamics - forcefield - single overall connectivity: no chemical reactions - single electronic state: no photo-chemical reactions • example - aquaporin-1 mechanism B. de Groot & H. Grubmüller  Science294: 2353-2357 (2001)

4. molecular simulation • QM/MM molecular dynamics - combination of quantum mechanics and forcefield - connectivity varies: chemical reactions • electronic state varies: photo-chemical reactions • examples - Diels-Alder reaction cycloaddition of ethene and butadiene in cyclo-hexane (not shown)

5. molecular simulation • QM/MM molecular dynamics - combination of quantum mechanics and forcefield - connectivity varies: chemical reactions • electronic state varies: photo-chemical reactions • examples • photo-isomerization QM/MM simulation of Photo-active yellow protein J. Amer. Chem. Soc.126: 4228-4233 (2004)

6. molecular simulation • concepts & machinery - molecular dynamics (MD) (5m) (5m) - molecular mechanics forcefield (MM) - molecular quantum mechanics (QM) (60m) - mixed quantum/classical mechanics (QM/MM) (30m) - geometry optimization (10m) • applications - Photoactive Yellow Protein (45m) - Diels-Alderase enzyme (you!) (3h)

7. molecular dynamics • nuclei are classical particles - Newton’s equation of motion - numerically integrate equations of motion • potential energy and forces - molecular mechanics - quantum mechanics

8. molecular dynamics • numerically integrate eoms of atoms

9. molecular mechanics forcefield • approximation for energy V - analytical lower dimensional functions (n << N) bonded interactions - empirical parameters (pk) thermodynamic data & QM calculations

10. molecular mechanics forcefield • approximation for energy V - analytical lower dimensional functions (n << N) non-bonded interactions - empirical parameters (pk) thermodynamic data & QM calculations

11. molecular mechanics forcefield • bonded interactions: bonds , angles & torsions

12. molecular mechanics forcefield • non-bonded interactions: Lennard-Jones & Coulomb

13. molecular mechanics forcefield • popular forcefields - CHARMM, OPLS, GROMOS, AMBER, … • advantages - fast large systems: proteins, DNA, membranes, vesicles • disadvantages - limited validity only valid inside harmonic regime no bond breaking/formation - limited transferrability new molecules need new parametrization

14. fundamental quantum mechanics • subatomic particles Louis de Broglie Erwin Schrödinger Werner Heisenberg Paul Dirac Max Born Albert Einstein many others - wave character electron diffraction - energy quantization • wavefunction - Schrödinger wave equation time-dependent time-independent - Hamilton operator kinetic potential

15. molecular quantum mechanics • solving electronic Schrödinger equation - Born-Oppenheimer approximation electronic and nuclear motion decoupled - electrons move in field of fixed nuclei • electronic hamiltonian kinetic elec-nucl elec-elec nucl-nucl • forces on classical nuclei

16. molecular quantum mechanics • applications for molecular modeling - electron density (charge distribution)

17. molecular quantum mechanics • applications for molecular modeling - reaction pathways Diels-Alder cyclo-addition mechanism

18. molecular quantum mechanics • Hartree approximation to wavefunction - product of one electron functions - hamiltonian without electron-electron term - hamiltonian with electron-electron term kinetic elec-nucl elec-elec - mean field approximation electron i in average static field of other electrons - iterative solution (self consistent field)

19. molecular quantum mechanics • Hartree approximation - illustration of mean-field approach electronic structure of O2; atom conf.: (1s22s22px2)2py12pz1

20. molecular quantum mechanics • Hartree approximation - illustration of mean-field approach electronic structure of O2; atom conf.: (1s22s22px2)2py12pz1

21. molecular quantum mechanics • Hartree approximation - illustration of mean-field approach electronic structure of O2; atom conf.: (1s22s22px2)2py12pz1

22. molecular quantum mechanics • Pauli principle - electrons are fermions (spin ½ particles) - electron wavefunction is anti-symmetric - no two electrons can occupy same state • Hartree approximation - product of one electron functions: - not anti-symmetric: • Hartree-Fock approximation - anti-symmetric combination of Hartee products

23. molecular quantum mechanics • anti-symmetric sum of Hartree products - e.g. product of two one electron functions Hartree approximation: Fock (Slater) correction: - anti-symmetric - no effect on wavefunction’s properties energy, density, …

24. molecular quantum mechanics • Hartree-Fock approximation - anti-symm. product of one electron wavefunctions - Slater determinant

25. molecular quantum mechanics • one electron wavefunctions - spatial & spin part - Ĥ does not operate on s, only on x,y,z - s(s) is a spinlabel - spatial part (x,y,z) is a molecular orbital max. two electrons (Pauli principle) - Slater determinant with molecular orbitals

26. molecular quantum mechanics • molecular orbitals - linear combination of atomic orbitals - e.g. H2 ;

27. molecular quantum mechanics • atomic orbitals - combination of simple spatial functions Slater-type orbitals: gaussian-type orbitals: - mimic atomic s,p,d,… orbitals - basisset: STO-3G, 3-21G, …, 6-31G*, …

28. molecular quantum mechanics • restricted Hartree-Fock wavefunction - Slater determinant - molecular orbitals - atomic orbitals (basisset) • optimization of MO coefficents cji - variation principle - find cji that minimize the energy (just 3 slides)

29. molecular quantum mechanics • Hartree-Fock equations - minimization problem - HF equation for single moleclar orbital (meanfield) - nonlinear set of equations coulomb operator exchange operator - total electronic energy (1/3)

30. molecular quantum mechanics • Roothaan-Hall equations - HF equation for molecular orbitals - expressed in atomic orbitals - multiply by atomic orbital ci* and integrate - matrix equation - solution ({cja} and {ea}) if (2/3)

31. molecular quantum mechanics • self consistent field procedure - iterate until energy no longer changes (converged) e.g. Gaussian SCF output: Closed shell SCF: Cycle 1 Pass 1 IDiag 1: E= -2929.02815281902 Cycle 2 Pass 1 IDiag 1: E= -2929.07991917607 Delta-E= -0.051766357053 Rises=F Damp=T Cycle 3 Pass 1 IDiag 1: E= -2929.13887276782 Delta-E= -0.058953591741 Rises=F Damp=F ...skipping... Cycle 12 Pass 1 IDiag 1: E= -2929.14125348456 Delta-E= -0.000000000195 Rises=F Damp=F Cycle 13 Pass 1 IDiag 1: E= -2929.14125348457 Delta-E= -0.000000000008 Rises=F Damp=F Cycle 14 Pass 1 IDiag 1: E= -2929.14125348456 Delta-E= 0.000000000012 Rises=F Damp=F SCF Done: E(RHF) = -2929.14125348 A.U. after 14 cycles Convg = 0.9587D-08 -V/T = 1.9993 S**2 = 0.0000 (3/3)

32. molecular quantum mechanics • Hartree-Fock based methods • Hartree Fock wavefunction as starting point no electron correlation • MCSCF (CI, CASSCF) • perturbation theory (MP2, MP4, CASPT2) • high demand on computational resources small to medium-size molecules in gas phase • alternative methods • semi-empirical methods • density functional theory methods

33. molecular quantum mechanics • limitations of HF wavefunction • no electron correlation dynamic: electronic motion is correlated static: electrons avoid each other • improving HF wavefunction • multi-configuration self-consistent field (mcscf) single, double, triple, quadruple, quintuple, … excitations resolves (part of) static correlation

34. molecular quantum mechanics • multi-configuration self-consisitent field • size of sum

35. molecular quantum mechanics • limitations of HF wavefunction • no electron correlation dynamic: electronic motion is correlated static: electrons avoid each other • improving HF wavefunction • perturbation theory • Møller-Plesset (MP): MP2, MP4, CASPT2, …

36. molecular quantum mechanics • semi-empirical methods • Roothaan-Hall equations • zero differential overlap • empirical parameters in Fij fitted to thermochemical data CNDO, INDO, NDDO, MINDO, MNDO, AM1, PM3

37. molecular quantum mechanics • density functional theory • Hohenberg-Kohn Theorem (1964) electron density defines all ground-state properties • Kohn-Sham equation (1965) • Kohn-Sham orbitals • exchange-correlation functional Exc[re(r)] - find cji that minimize the energy functional E[re(r)] • self-consistent Roothaan-Hall equations

38. molecular quantum mechanics • summary • solving electronic Schrödinger equation • computational techniques Hartree-Fock and beyond (RHF, UHF, CASSCF, MP2,…) semi-empirical methods (INDO, AM1, PM3, …) density functional theory (Becke, BP87, B3LYP, …) • forces on nuclei • more accurate than any forcefield bond breaking/formation excited states, transitions between electronic states

39. molecular quantum mechanics • high demand on computational recources small to medium sized gas-phase systems

40. mixed quantum/classical methods • reaction in condensed phase - reactions in solution - enzymatic conversions • subdivision of the total system (QM) - reactive center - environment (MM) • QM/MM hybrid model - compromise between speed and accuracy - realistic chemistry in realistic system

41. QM/MM hybrid model • QM subsystem embedded in MM system A. Warshel & M. Levitt. J. Mol. Biol.103: 227-249 (1976)

42. QM/MM hybrid model • application for molecular modeling - catalytic Diels-Alderase antibody J. Xu et al. Science286: 2345-2348 (1999) (experimental) http://md.chem.rug.nl/~groenhof/EMBO2004/html/tutorial.html

43. QM/MM hybrid model • interactions in QM subsystem - QM hamiltonian • interactions in the MM subsystem - forcefield • interactions between QM and MM subsystems - QM/MM interface - forcefield bonded and dispersion interactions - QM hamiltonian electrostatic interactions

44. QM/MM hybrid model • QM/MM bonded interactions bonds , angles & torsions

45. QM/MM hybrid model • QM/MM dispersion interactions Lennard-Jones

46. QM/MM hybrid model • QM/MM boundary link atom , frozen orbital

47. QM/MM hybrid model • QM/MM electrostatic interactions point charges:

48. QM/MM hybrid model • Roothaan-Hall equations - HF equation for molecular orbitals - QM subsystem in cloud of pointcharges - polarization of QM subsystem • forcefield terms - QM/MM (bonds, angles, torsions & LJ) - MM

49. QM/MM hybrid model

50. QM/MM hybrid model • electrostatic QM/MM interaction - QM subsystem in cloud of pointcharges core nucl-MMatom elec-MMatom - polarization of QM subsystem • problems & inconsistencies - no polarization of MM subsystem implicitly incorporated in LJ and atomic charges - pointcharges of MM atoms forcefield dependent