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 MOLDENMOLDENMOLDENMOLDENMOLDEN Ex: -CH3 is replaced withMOLDENMOLDENMOLDENMOLDENMOLDEN a single set of data

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/ ◦ CHARMM: http://www.charmm.org/ ◦ COSMOS: http://analyt.chem.msu.ru/preconcentration/pletnev/ cosmos_new/defaulte.htm ◦ GROMACS: http://www.gromacs.org/ ◦ p/f-Dynamo: http://www.pdynamo.org/mainpages/ ◦ ORCA: http://ewww.mpi-muelheim.mpg.de/bac/logins/ downloads_en.php ◦ TINKER: http://www.ks.uiuc.edu/Research/namd/ ◦ NAMD: http://www.ks.uiuc.edu/Research/namd/

40 41  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

42  Complexity of QM ◦ Electrons & nuclei are explicitly treated in the Schrödinger's Equation: ∧ H Ψ = EΨ

€

43 ∧  QM Hamiltonian H Ψ = EΨ ∧ 2 2 Z e2 H = − ∇ 2 − ∇ 2 − i ∑ 2m N ∑ 2m i ∑∑ 4πε | R − r | N N i e i j 0 i j € Kinetic (nuclei) Kinetic (electrons) e--nuc attraction

2 2 1 e 1 ZA ZBe € + ∑ + ∑ 4πε0 i< j | ri − rj | 4πε0 A

e--e- repulsion nuc-nuc repulsion

€

44  Types of Quantum mechanical methods

◦ Semi-empirical methods (SE)  Uses ab initio methods ◦ Parameters derived from experimental data, high level WFT, & DFT calculations ◦ Wave Functional Theory (WFT)  Approximately solves Schrödinger's equation, using a wave function to obtain the information of the system ◦ Density Functional Theory (DFT)  Approximately solves Schrödinger's equation, using an electronic density to obtain the information of the system

45  Properties that can be calculated

All OBSERVABLE properties in nature are theoretically accessible through Quantum Mechanical calculations

46  Common Computer Program Packages

◦ GAMESS-US: http://www.msg.chem.iastate.edu/gamess/ ◦ GAMESS-UK: http://www.cfs.dl.ac.uk/ ◦ GAUSSIAN: http://www.gaussian.com/ ◦ ORCA: http://ewww.mpi-muelheim.mpg.de/bac/logins/ downloads_en.php ◦ MOLCAS: http://www.teokem.lu.se/molcas/ ◦ NWCHEM: http://www.emsl.pnl.gov/capabilities/computing/ nwchem/ ◦ p/f-DYNAMO: http://www.pdynamo.org/mainpages/ ◦ MOPAC: openmopac.net

47 48  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

49 Molecular System MM region

Interface of MM & QM regions

QM region

50  Regions

◦ QM region  In most programs the QM region is treated with semi-empirical methods ◦ MM region  Many different force fields are available to treat the larger MM portion of the system ◦ MM & QM interface region  Care must be taken to pick appropriate place to separate the two regions (bonds are cut)

51  Why use QM/MM & MD

◦ Pros  Large systems can be investigated  Electronic information obtained for QM Region

◦ Cons  Computationally more expensive  Interface region between two levels must be carefully considered

52  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

53 54