EMPIRICAL FORCE FIELDS

What is a force field? A set of formulas (usually explicit) and parameters to express the conformational energy of a given class of molecules as a function of coordinates (Cartesian, internal, etc.) that define the geometry of a molecule or a molecular system. Features: Cheap Fast Easy to program Restricted to conformational analysis Non-transferable Results sometimes unreliable

All-atom empirical force fields: a very simplified representation of the potential energy surfaces Class I force fields

Multiplication of atom types in empirical force fields

Name Potential type References AMBER/OPLS all-atom, united-atom Weiner et al., 1984; 1986; Cornell et al., 1995; Jorgensen et al., CHARMmall-atom Brooks et al., 1983; MacKerrel et al., 1998; GROMOSall-atom van Gunsteren & Berendsen, 1987; Scott et al., ECEPP/3 all-atom; rigid valence geometry Nemethy et al., 1995; Ripoll et al., eppak/ DISCOVER (CVFF) all-atom Dauber-Osguthorpe, 1988; Maple et al., 1998 Force fields commonly used for protein simulations

d d0d0 d Es(d)Es(d) Bond distortion energy

Typical values of d 0 and k d Bondd 0 [A]k d [ kcal/(mol A 2 ) ] Csp 3 -Csp Csp 3 -Csp Csp 2 =Csp Csp 2 =O Csp 2 -Nsp C-N (amide)

Comparison of the actual bond-energy curve with that of the harmonic approximation

Anharmonic potential Morse potential (CVFF force field) Potentials that take into account the asymmetry of bond-energy curve d [A] E [kcal/mol] Harmonic potential Anharmonic potential Morse potential

 00  Eb()Eb() kk Energy of bond-angle distortion

Typical values of  0 and k  Angle  0 [degrees] k  [ kcal/(mol degree 2 ) ] Csp 3 -Csp 3 -Csp Csp 3 -Csp 3 -H H-Csp 3 -H Csp 3 -Csp 2 -Csp Csp 3 -Csp 2 =Csp Csp 3 -Csp 2 =O

Single bond between sp 3 carbons or between sp 3 carbon and nitrogen Example: C-C-C-C quadruplet dihedral angle [deg] Etor [kcal/mol] Double or partially double bonds Example: C-C(carboxyl)-C(amide)-C quadruplet Single bond between electronegative atoms (oxygens, sulfurs, etc.). Example: C-S-S-C quadruplet Basic types of torsional potentials

Potentials imposed on improper torsional angles A B X X 

Nonbonded Lennard-Jones (6-12) potential r [A] E nb [kcal/mol] -- r0r0  Lorenz-Berthelot combining rules

Sample values of  i and r 0 i Atom typer0r0  C(carbonyl) C(sp 3 ) N(sp 3 ) O(carbonyl) H(bonded with C) S

Other nonbonded potentials Buckingham potential potential used in some force fields (e.g., ECEPP) for proton…proton donor pairs

Coulombic (electrostatic) potential

Charge determination Mullikan population charges (ECEPP/3, other early force fields). Fitting to molecular electrostatic potentials + subsequent adjustment to reproduce potential- energy surfaces or experimental association energies, etc. Based on atomic electronegativities with corrections to topology and geometry (No and coworkers, J. Phys. Chem. B, 105, 3624–3634, 2001; Koca and coworkers, J. Chem. Inf. Model., 53, 2548–2558, 2013).

Charge determination: fitting to molecular electrostatic potential (MEP) maps

Ab initio calculationsFitted by using CHELP-SV Francl et al., J. Comput. Chem., 17, (1996)

Polarizable force fields

Energy contributionSource of parameters Bond and bond angle distortion Crystal and neutronographic data, IR spectroscopy TorsionalNMR and FTIR spectroscopy Nonbonded interactions Polarizabilities, crystal and neutronographic data Electrostatic energyMolecular electrostatic potentials All Energy surfaces of model systems calculated with molecular quantum mechanics Sources of parameters

Class II force fields (MM3, MMFF, UFF, CFF) Maple et al., J. Comput. Chem., 15, (1994)

Parameterization of class II force fields

Solvent in simulations  Explicit water TIP3P TIP4P TIP5P SPC  Implicit water Solvent accessible surface area (SASA) models Molecular surface area models Poisson-Boltzmann approach Generalized Born surface area (GBSA) model Polarizable continuum model (PCM)

O H H e e o Å O H H e 0.00 e e M 0.15 Å TIP3P modelTIP4P model  O = Å  O = kcal/mol  O = Å  O = kcal/mol

Solvent accessible surface area (SASA) models  i Free energy of solvation of atomu i per unit area, A i solvent accessible surface of atom i dostępna

Vila et al., Proteins: Structure, Function, and Genetics, 1991, 10,

Comparison of the lowest-energy conformations of [Met 5 ]enkefalin (H-Tyr-Gly-Gly-Phe-Met-OH) obtained with the ECEPP/3 force field in vacuo and with the SRFOPT model vacuumSRFOPT

vacuumSRFOPT Compariosn of the molecular sufraces of the lowest-energy conformation of [Met 5 ]enkefaliny obtained without and with the SRFOPT model

Molecular surface are model  Surface tension A molecular surface area

Generalized Born molecular surface (GBSA) model

Protein structure calculation/prediction and folding simulations Single energy minimization (wishful thinking at the early stage of force-field development). Global optimization of the PES (ignores conformational entropy). Molecular dynamics/Monte Carlo (take entropy into account but slow) and liable to non-convergence). Generalized ensemble sampling (MREMD).

Force field validation

Structure of gramicidiny S predicted by using the build-up procedure with energy minimzation with the ECEPP/3 force field (M. Dygert, N. Go, H.A. Scheraga, Macromolecules, 8, (1975). The structure turned out to be effectively identical with the NMR structure determined later.

Superposition of the native fold (cyan) and the conformation (red) with the lowest C  RMSD (2.85 Å) from the native fold Energy-RMSD diagram Global optimization of the energy surface of the N-terminal portion of the B-domain of staphylococcal protein A with all-atom ECEPP/3 force field + SRFOPT mean-field solvation model (Vila et al., PNAS, 2003, 100, 14812–14816)

First successful folding simulation of a globular protein by molecular dynamics Duan and Kollman, Science, 282, 5389, (1998)

Folding proteins at x-ray resolution using a specially designed ANTON machine (x-ray: blue, last frame of MD) simulation (red): villin headpiece (left), a 88 ns of simulations, WW domain (right), 58  s of simulations. Good symplectic algorithm; up to 20 fs time step. D.E. Shaw et al., Science, 2010, 330,