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Molecular Mechanics

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:.

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Molecular Mechanics

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  1. Molecular Mechanics • force fields • minimization

  2. 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:

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

  4. 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...

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

  6. 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 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 electronegativity of v’th orbital on atom i I: ionization potentials, E: electron affinities, 0=neutral atoms, +=positive ions

  7. Hydrogen Bonds • Directional Hydrogen Bonding in the MM3 Force Field (Lii and Allinger, 1994, 1998) • eHB 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

  8. 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...

  9. 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)

  10. extended issues • QM/MM • polarizability: • cation-p, 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)

  11. Minimization 3N degrees of freedom, vector x=<xi> 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=Sbonds w(b-b0)2 =w S (((xi-xj)2+(yi-yj)2+(zi-zj)2)1/2+b0)2 dE/dxi=...

  12. Conjugate Gradient Initialize at P0; g0 = h0 = F(P0); for i = 0 to n-1 Pi+1 := minimum of F along the line hi through Pi, i.e., choose li to minimize F(Pi+1)=F(Pi+ lihi); g i+1 := F(Pi+1); gi+1 := (gi+1- gi)  gi+1 / gi  gi; hi+1:= gi+1+gi hi; • orthogonal directions • line search • convergence, n steps • powell

  13. Newton-Raphson iteration • method for finding zero’s of f(x) • square roots: x2-5=0 -> x=sqrt(5) • extend to finding zero’s of f’(x) • second order method: Hessian

  14. BFGS • 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-1g • algorithm: • solve for sk • perform line search for optimal ak • calculate yk • update B

  15. 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)

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