Molecular MechanicsForce FieldsBond stretching and bendingAngle RestraintsElectrostatic termsPartial ChargesSlide 7Hydrogen BondsCommon FF ParameterizationsImplicit-solvent: solvation parametersextended issuesMinimizationConjugate GradientNewton-Raphson iterationBFGSSimulated AnnealingMolecular Mechanics•force fields•minimizationForce 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 potentialBond stretching and bendingAngle 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: 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---HClPartial 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 electronegativityI: ionization potentials, E: electron affinities, 0=neutral atoms, +=positive ionselectronegativity of v’th orbital on atom icontribution to the atomic charge on the a-th step of iteration of charge; j are neighbors with higher electroneg.; k areless-electronegative neighborsHydrogen 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 approximationCommon 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 issuesMinimization•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=bonds w(b-b0)2 =w (((xi-xj)2+(yi-yj)2+(zi-zj)2)1/2+b0)2 dE/dxi=...Conjugate GradientInitialize at P0;g0 = h0 = F(P0);for i = 0 to n-1Pi+1 := minimum of F along the line hi through Pi,i.e., choose i to minimize F(Pi+1)=F(Pi+ i hi);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• powellNewton-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( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( )211...2k k k k kk k kk k kE x E x x x E x x x E xE x xE x x x E xx x E x E x* -� ��= + - + - +� � ��= + -� ��= -•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 k –calculate yk –update BBFGSSimulated 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
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