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Melissa Gajewski & Jonathan Mane Melissa Gajewski & Jonathan Mane Molecular Mechanics (MM) Methods Force fields & potential energy calculations Example using noscapine Molecular Dynamics (MD) Methods Ensembles & trajectories Example using 18-crown-6 Quantum Mechanics (QM) Methods Schrödinger’s equation Semi-empirical (SE) Wave Functional Theory (WFT) Density Functional Theory (DFT) Hybrid QM/MM & MD Methods Comparison of hybrid methods 2 3 Molecular Mechanics (MM) Methods Force fields & potential energy calculations Example using noscapine Molecular Dynamics (MD) Methods Ensembles & trajectories Example using 18-crown-6 Quantum Mechanics (QM) Methods Schrödinger’s equation Semi-empirical (SE) Wave Functional Theory (WFT) Density Functional Theory (DFT) Hybrid QM/MM & MD Methods Comparison of hybrid methods 4 Useful for all system size ◦ Small molecules, proteins, material assemblies, surface science, etc … ◦ Based on Newtonian mechanics (classical mechanics) d F = (mv) dt ◦ The potential energy of the system is calculated using a force field € 5 An atom is considered as a single particle Example: H atom Particle variables: ◦ Radius (typically van der Waals radius) ◦ Polarizability ◦ Net charge Obtained from experiment or QM calculations ◦ Bond interactions Equilibrium bond lengths & angles from experiment or QM calculations 6 All atom approach ◦ Provides parameters for every atom in the system (including hydrogen) Ex: In –CH3 each atom is assigned a set of data MOLDEN MOLDEN (Radius, polarizability, netMOLDENMOLDENMOLDENMOLDENMOLDEN charge, & bond interactions)MOLDENMOLDENMOLDENMOLDENMOLDEN MOLDENMOLDENMOLDENMOLDENMOLDEN MOLDENMOLDENMOLDENMOLDENMOLDEN defaults used defaults used single point single point 7 United atom approach ◦ Represents group of atoms as a single particle All hydrogens not participating in hydrogen- bonding are lumped into heavier atoms ◦ Methyl group in large molecules ◦ Amino Acids represented by 2 to 4 particles MOLDEN Ex: -CH is replaced withMOLDENMOLDENMOLDENMOLDENMOLDEN a single set of data 3 MOLDENMOLDENMOLDENMOLDENMOLDEN defaults used (Radius, polarizability, net charge, & bond interactions) single point 8 Common force fields ◦ AMBER ◦ GAFF ◦ CHARMM ◦ OPLS-UA ◦ OPLS-AA ◦ MM3 ◦ GROMACS ◦ UFF ◦ COSMOS-NMR 9 Potential Energy (PE) of the System PE = Ecovalent+Enoncovalent Covalent interactions Ecovalent = Ebond+Eangle+Edihedral+Eimproper dihedral+… Non-covalent interactions Enoncovalent = Eelectostatic+Evan der Waals + … 10 Covalent interactions ◦ Bond 2 E bond = ∑[kr (r − req ) ] ◦ Angles 2 E angle = ∑[kθ (θ −θeq ) ] € ◦ Dihedral angle V E = [ n (1− cos(φ − γ)] dihedral ∑ 2 € Ecovalent = Ebond + Eangle + Edihedral + Eimproper dihedral+ … € 11 Non-covalent interactions (electrostatic) qiq j E electrostatic = ∑ i< j εRij Ionic bonds: cation + anion: Na+ + Cl- -> NaCl single point point single Hydrogen bonds Hydrophobic interactions defaults used used defaults MOLDEN MOLDEN MOLDEN MOLDEN MOLDEN MOLDEN MOLDEN MOLDEN MOLDEN MOLDEN MOLDEN MOLDEN MOLDEN MOLDEN MOLDEN MOLDEN MOLDEN MOLDEN MOLDEN MOLDEN € 12 Non-covalent interactions (van der Waals) Aij Bij E vdW = ∑[ 12 − 6 ] i< j Rij Rij ◦ Attractive or repulsive forces Permanent dipole Induced dipole € permanent dipole and an induced dipole instantaneous induced dipole-dipole forces (London Dispersion Force) 13 ◦ Exact form of the equation for the potential energy is dependent on the particular program being used Energy of the System E = Ecovalent + Enoncovalent Ecovalent = Ebond+Eangle+Edihedral+Eimproper dihedral+… Enoncovalent = Eelectostatic+Evan der Waals+ … 14 Example: noscapine ◦ Geometry optimizations via energy minimization: find the most stable conformation (lowest energy) Global minima Local minima 15 Local Energy Minimization MOLDENMOLDENMOLDEN MOLDENMOLDEN MOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDEN MOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDEN MOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDEN MOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDEN defaults used Initial … local minimum defaults used first point last point 16 Simulated Annealing ◦ “Heat” or energy is provided followed by gradual or scheduled cooling so the system can explore all possible conformations RotationRotation of of dihedrala dihedral angle of SR-Noscapine angle with of axial noscapine CH3 -126841 -126842 -126843 -126844 -126845 -126846Energy -126847 Corrected Energy (kcal/mol) -126848 -126849 -200 -150 -100 -50 0 50 100 150 200 DihedralDihedral Angle C26-C 21angle-C17-N14 17 Global Energy Minimization MOLDENMOLDEN MOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDEN MOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDEN MOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDEN MOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDENMOLDEN Local vs. defaults used Global last point defaults used last point 18 Physical properties that can be calculated ◦ Local & global geometry (via energy minimization) ◦ Potential energy surfaces ◦ Binding affinities ◦ Protein folding process ◦ Protonation equilibria ◦ Active/binding site configuration ◦ … etc. 19 Common Computer Program Packages ◦ ACEMD: http://gpuradar.com/EN/Software/20/ACEMD ◦ AMBER: http://ambermd.org/ ◦ CHARMM: http://www.charmm.org/ ◦ BOSS: http://zarbi.chem.yale.edu/software.html#boss ◦ COSMOS: http://analyt.chem.msu.ru/preconcentration/pletnev/ cosmos_new/defaulte.htm ◦ GROMACS: http://www.gromacs.org/ ◦ NAMD: http://www.ks.uiuc.edu/Research/namd/ ◦ TINKER: http://dasher.wustl.edu/tinker/ 20 21 Molecular Mechanics (MM) Methods Force fields & potential energy calculations Example using noscapine Molecular Dynamics (MD) Methods Ensembles & trajectories Example using 18-crown-6 Quantum Mechanics (QM) Methods Schrödinger’s equation Semi-empirical (SE) Wave Functional Theory (WFT) Density Functional Theory (DFT) Hybrid QM/MM & MD Methods Comparison of hybrid methods 22 Studies the motions of atoms or particles Provides detailed time and space information of molecular behavior in phase space Important concept: Phase space p (momentum) q (position) 23 Allows insight into molecular motion on an atomic scale for: ◦ Small molecules ◦ Helium droplets ◦ Hydrogen clusters ◦ Nano science ◦ Proteins ◦ Bio-molecules ◦ Material science ◦ … etc. 24 Physical properties that can be calculated ◦ Structures Structure factor (radial distribution function) Solvation structure ◦ Energy Energy of a system as a function of time Binding affinities (free energy) ◦ Thermodynamics Follow reaction coordinate Thermodynamic quantities 25 Systems are ergodic ◦ system explores all microstates in time phase space Variables ◦ N: number of particles ◦ P: pressure ◦ T: temperature ◦ V: volume ◦ E: energy ◦ µ: chemical potential p (momentum) q (position) 26 Main types of Ensembles ◦ Micro-canonical ensemble [NVE] ◦ Canonical ensemble [NVT] ◦ Isothermal-Isobaric [NPT] 27 Micro-canonical Ensembles [NVE] ◦ Number of particles (N), volume (V), & total energy (E) kept constant during simulation ◦ Time evolution of the particle position ‘x’ and velocity ‘v’ Etotal = Ekinetic + Epotential 28 Canonical Ensembles [NVT] ◦ Number of particles (N), volume (V), & system temperature (T) kept constant during simulation ◦ Thermostat required to keep temperature constant Nosé-Hoover Berendsen Langevin dynamics 29 Isothermal-Isobaric Ensembles [NPT] Number of particles (N), pressure (P), & system temperature (T) during simulation Thermostat required to keep the temperature constant Barostat required to keep the pressure constant Ensemble required for biological membranes, lipid bilayers, & constant surface tension simulations Needed when pressure & temperature must be kept constant to simulate phenomena of interest 30 MD Hamiltonian at the MM level PE = Ecovalent + Enoncovalent ∧ 2 2 Vn H = ∑kr (r − req ) + ∑kθ (θ −θeq ) + ∑ [1+ cos(nφ − γ}]+ bonds angles dihedrals 2 bonds angles dihedral angles A B qiq j € [ ij − ij ] ∑ 12 6 + ∑ εR i< j Rij Rij i< j ij Lenard-Jones€ potential electrostatics € VDW forces € 31 Systems are ergodic ◦ system explores all microstates in time phase space Variables ◦ N: number of particles ◦ P: pressure ◦ T: temperature ◦ V: volume ◦ E: energy ◦ µ: chemical potential p (momentum) q (position) 32 Trajectories depend on equations of motions ◦ Uses MM potentials based on Newton’s equations of motion (EOM) Verlet algorithm for integrating EOM t = time Δt = time step a(t)Δt 2 + 2x(t) − x(t − Δt) x(t + Δt) = x = position Δt 2 a = acceleration € € 33 Constrained Dynamics ◦ Constraining some parts of the system being studied. To look at slower motions To freeze out some motions ◦ Constraining allows for an increase in the time step (cover more phase space) Ex. constrained hydrogen motion (SHAKE/RATTLE) 34 Give atoms initial position & select a Δt Calculate forces (F) and acceleration (a) Update atoms position Add time step (t=t+Δt) Repeat as long as required 35 Example: 18-crown-6 In non-polar solvents In polar solvents 18-crown-6 contains intra- 18-crown-6 contains inter- molecular hydrogen bonding molecular hydrogen bonding 36 Example: 18-crown-6 polar (H2O) solvent & non-polar (CHCl3) solvent conversion from inter- to intra-molecular hydrogen bonding 37 38 Physical properties that can be calculated ◦ Local & global geometry (via energy minimization) ◦ Potential energy surfaces ◦ Binding affinities ◦ Protein folding process ◦ Protonation equilibria ◦ Active/binding site configuration ◦ … etc. 39 Common Computer Program Packages ◦ AMBER: http://ambermd.org/ ◦
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