Molecular Mechanics force fields minimization

Force Fields good review: MacKerell (2004) JCompChem, 25:1584 FF typically contains terms for: –bonds and angles: harmonic/sigmoidal restraints –non-bonded: electrostatics: Coulomb term VDW: 12-6 Lennard-Jones example - CHARMM FF:

harmonic restraints around equilibrium distance –Hooke’s Law –frequency from spring constant are bonds really like springs (quadratic)? Morse potential Bond stretching and bending

Angle Restraints 3-atom angles: harmonic constraint torsion angles (4-atoms) –alkyl: sp3, tetrahedral, gauche –alkenyl: sp2, double-bonds, planar –“improper” dihedrals (restrain planar centers: Phe, peptide bond) can add cross terms for dependence of angle on adjacent bonds and angles...

Electrostatic terms Coulomb term dielectric: e vacuum =1 ; e water =80; e=2-4 in protein interior dipole moments vs. atom-centered partial charges? H-bonds: explicit or implicit (electrostatic)? for HCl---HCl

Partial Charges formal charges: 0, ±1 electronegativity, induction QM: solve wave equations, integrate orbital density (ESP) Mulliken charges –linear combination of molecular orbitals –tends to exaggerate charge separation Gasteiger charges (Gasteiger and Marsili, 1980) –iterative: redistribution of charges based on electronegativity I: ionization potentials, E: electron affinities, 0=neutral atoms, +=positive ions electronegativity of v’th orbital on atom i contribution to the atomic charge on the a-th step of iteration of charge; j are neighbors with higher electroneg.; k are less-electronegative neighbors

Hydrogen Bonds Directional Hydrogen Bonding in the MM3 Force Field (Lii and Allinger, 1994, 1998) –e HB is proportional to the difference of the first ionization potential between the hydrogen acceptor Y and the donor X, and also the bond moment of bond X-H –comparison with MP2-level ab initio calculations with 6-31G** basis set for predicting bond lengths etc. in organic molecules CHARMM – non-directional, electrostatic approximation

Common FF Parameterizations all-atom vs. united atom (only polar H’s) parameterize on small organic molecules –acetamide, cyclohexane... –predict vibrational spectra, melting temperatures, conformational/solvation energies... AMBER (Cornell 1995) OPLS (Jorgensen) Optimized Potential for Liquid Simulations MM3 (Allinger et al., 1989) MMFF94 (Merck) (Halgren, 1996) Charmm (Karplus) NAMD, Gromos, ECEPP, CFF...

Implicit-solvent: solvation parameters add terms (with derivatives) to energy function accessible surface area EFF1 (Lazaridis and Karplus, 1999) –for atom i, consider solvent-excluded volumes of atoms j around it, as function of contact distance –benefit for hydrophobic atoms, penalty for polar atoms Generalized Born –scale electrostatic interactions based on “effective radius” of atom, which depends on depth of burial in protein (integrate over shape of surface) –(more later)

QM/MM polarizability: –cation- , lone-pairs on sulfur –extra term in AMBER force field (see manual): handling metal cations –coordination geometry, charge-transfer –Edelman and Sobolev (motifs, induced fit) extended issues

Minimization 3N degrees of freedom, vector x= at minima, derivative equals 0:  E=dE/dx=0 steepest descent –calculate gradient  E with respect to each parameter –take small step in opposite direction how hard is it to calculate derivatives of force fields with respect to atomic coordinates? E=  bonds w(b-b 0 ) 2 =w  (((x i -x j ) 2 +(y i -y j ) 2 +(z i -z j ) 2 ) 1/2 +b 0 ) 2 dE/dx i =...

Conjugate Gradient Initialize at P0; g 0 = h 0 =  F(P 0 ); for i = 0 to n-1 P i+1 := minimum of F along the line h i through P i, i.e., choose i to minimize F(P i+1 )=F(P i + i h i ); g i+1 :=  F(P i+1 ); g i+1 := (g i+1 - g i )  g i+1 / g i  g i ; h i+1 := g i+1 +g i h i ; orthogonal directions line search convergence, n steps powell

Newton-Raphson iteration method for finding zero’s of f(x) –square roots: x 2 -5=0 -> x=sqrt(5) extend to finding zero’s of f’(x) second order method: Hessian

Broyden-Fletcher, Goldfarb, Shanno lbfgs minimizer (limited memory) in Phenix (python) second order, but avoids computing inverse of Hessian, which takes O((3N) 3 ) time approximation: Bg = H -1 g algorithm: –solve for s k –perform line search for optimal  k –calculate y k –update B BFGS

Simulated Annealing Li and Scherga (1987) – Monte Carlo method advantage: don’t have to compute derivatives make a “random” move, i.e. change some coords of atoms –accept change if energy decreases –accept probabilistically if energy increases –acceptance probability depends on temperature allows exploration of energy landscape –can get out of some local minima –higher temperatures allows more exploration –cooling forces search to proceed downhill atomic coordinates or torsion angles? –treat bond lengths as effectively fixed –rotation couples movements of sub-structures higher “radius of convergence” –Brunger, Adams, and Rice (1997)