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

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, …

molecular simulation standard molecular dynamics example - forcefield - single overall connectivity: no chemical reactions - single electronic state: no photo-chemical reactions example - aquaporin-1 mechanism B. de Groot & H. Grubmüller  Science 294: 2353-2357 (2001)

molecular simulation QM/MM molecular dynamics examples - 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)

molecular simulation QM/MM molecular dynamics examples - 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)

molecular simulation concepts & machinery applications - molecular dynamics (MD) (5m) - molecular mechanics forcefield (MM) (5m) - 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)

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

molecular dynamics numerically integrate eoms of atoms

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

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

molecular mechanics forcefield bonded interactions: bonds , angles & torsions

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

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

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 Intro on quantum mechanics time-dependent time-independent - Hamilton operator kinetic potential

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

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

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

molecular quantum mechanics Hartree approximation to wavefunction - product of one electron functions - hamiltonian with electron-electron term - hamiltonian without 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)

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

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

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

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

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, …

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

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

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

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*, …

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)

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)

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)

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)

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

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

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

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, …

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

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

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

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

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

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

QM/MM hybrid model application for molecular modeling - catalytic Diels-Alderase antibody J. Xu et al. Science 286: 2345-2348 (1999) (experimental) http://md.chem.rug.nl/~groenhof/EMBO2004/html/tutorial.html

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

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

QM/MM hybrid model QM/MM dispersion interactions Lennard-Jones

QM/MM hybrid model QM/MM boundary link atom , frozen orbital

QM/MM hybrid model QM/MM electrostatic interactions point charges:

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

QM/MM hybrid model

QM/MM hybrid model electrostatic QM/MM interaction - QM subsystem in cloud of pointcharges core elec-MMatom nucl-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

alternative QM/MM interface ONIOM F. Maseras & K. Morokuma, J. Comp. Chem. 16, 1170 (1995) two layer ONIOM energy

alternative QM/MM interface multilayer ONIOM QM/QM/.../…/MM

geometry optimization potential energy surface energy & forces MM (forcefield) QM (HF, DFT, …) QM/MM …

geometry optimization stationary points reactants → products minima on PES reactants products saddle-points transition states reaction mechanism

geometry optimization stationary points Hess matrix (Hessian) - matrix of second derivatives minima on potential energy surface Hessian has only positive eigenvalues saddle-points on potential energy surface Hessian has one negative eigenvalue

geometry optimization locating minima - general procedure follow the gradient downhill locating saddle points - optimization with constraints one eigenvalue of Hessian is negative good guess TS geometry (intuition & experience) linear transit reaction coordinate interpolation between reactant and product geometries always check the eigenvalues of Hessian!!

geometry optimization linear transit calculation - reaction coordinate (experience & intuition) e.g. Diels-Alder cycloaddition constrain/restrain reaction coordinate minimize/sample all other degrees of freedom Dvreact or even DGreact (potential of mean force)

geometry optimization linear transit calculation - result for the Diels-Alder cycloaddition

end of part I coming up part II QM/MM concepts & machinery QM/MM applications