CHARMM force fields, parameterization strategies and future/ongoing force field developments Alexander D. MacKerell, Jr. University of Maryland, Baltimore School of Pharmacy MMTSB/CTBP 2006 Summer Workshop © Alexander D. MacKerell , 2006. Potential energy function (mathematical equations) Empirical force field equations and parameters relate chemical structure and conformation to energy © Alexander D. MacKerell , 2006. Common “additive” empirical force fields Class I CHARMM CHARMm (Accelrys) AMBER OPLS/AMBER/Schrödinger ECEPP (free energy force field) GROMOS Class II CFF95 (Accelrys) MM3 MMFF94 (CHARMM, Macromodel, MOE, elsewhere) UFF, DREIDING © Alexander D. MacKerell , 2006. State of the art additive force fields are typically all-atom models All atoms, including all hydrogens, explicitly represented in the model. Lone pairs included on hydrogen bond acceptors in some force fields. e.g., CHARMM22 and 27, AMBER94….03, OPLS/AA © Alexander D. MacKerell , 2006. Extended or united atom models (omit non-polar hydrogens) CHARMM PARAM19 (proteins) often used with implicit solvent models ACE, EEF, GB variants improper term to maintain chirality loss of cation - pi interactions OPLS AMBER GROMOS © Alexander D. MacKerell , 2006. Transition State Force Field Parameters Same approach as standard force field parameterization Require target data for transition state of interest: ab initio Metal Force Field Parameterization Only interaction parameters or include intramolecular terms Parameterization of QM atoms for QM/MM calculations © Alexander D. MacKerell , 2006. Polarizable “non-additive” force fields Include explicit term(s) in the potential energy function to treat induction/polarization of the charge distribution by the environment. Still under development. CHARMM Drude (MacKerell, Roux and coworkers) PIPF (Gao and coworkers) Cheq (Brooks and coworkers) AMBER Friesner/Berne et al. (Schrödinger Inc.) TINKER © Alexander D. MacKerell , 2006. Class I Additive Potential Energy Function Intramolecular (internal, bonded terms) 2 2 "Kb (b # bo ) + "K$ ($ #$o) + "K% (1+ cos(n% #&)) bonds angles torsions 2 2 + "K' (' #'o ) + "KUB (r1,3 # r1,3,o) impropers Urey#Bradley Intermolecular (external, nonbonded terms) ! 12 6 q q ,% R ( % R ( / i j + # .' min,ij * + 2' min,ij * 1 $ 4"Dr ij .' r * ' r * 1 nonbonded ij -& ij ) & ij ) 0 ! © Alexander D. MacKerell , 2006. Class II force fields (e.g. MM3, MMFF, UFF, CFF) 2 3 4 K b " b + K b " b + K b " b #[ b, 2( o) b, 3( o) b, 4( o) ] bonds 2 3 4 + K $ "$ + K $ "$ + K $ "$ #[ $ , 2( o) $ , 3( o) $ , 4( o) ] angles + #[K% ,1(1" cos%) + K% , 2(1" cos2%) + K% , 3(1" cos3%)] dihedrals 2 + #K & & impropers + # #K bb'(b " bo)(b'"bo ') + # #K$$ '($ "$ o)($ '"$ o ') bonds bonds' angles angles' + # #K b$ (b " bo)($ "$ o) bonds angles + # #(b " bo)[K% , b1 cos% + K% , b2 cos2% + K% , b3 cos3%] bonds dihedrals + # #(b'"bo ')[K% , b'1 cos% + K% , b'2 cos2% + K% , b'3 cos3%] bonds' dihedrals + # #($ "$ o)[K% ,$ 1 cos% + K% ,$ 2 cos2% + K% ,$ 3 cos3%] angles dihedrals + # # #($ "$ o)($ '"$ o ') cos% angles angles' dihedrals © Alexander D. MacKerell , 2006. ! Merck Molecular FF: Force field for drug-like molecules MMFF is a force field designed for pharmaceutical compounds as well as biological molecules. It may be considered one of the better general FFs, although its quality in treating proteins etc. is worse than CHARMM and other biological FFs. Therefore, MMFF is good for computing drug-receptor interactions but not for extensive minimizations etc. of proteins. The tutorial MMFF_Interaction gives an example of reading a drug molecule in Mol2 format, reading a protein structure and calculating the interaction energy. See mmff_inter_energy.inp © Alexander D. MacKerell , 2006. Intramolecular energy function and corresponding force field parameters 2 2 "Kb (b # bo ) + "K$ ($ #$o) + "K% (1+ cos(n% #&)) bonds angles torsions 2 2 + "K' (' #'o ) + "KUB (r1,3 # r1,3,o) + "VCMAP impropers Urey#Bradley %,( Equilibrium terms Force constants b : bonds o Kb: bonds θ : angles o Kθ: angles ! n: dihedral multiplicity Kφ: dihedral δ : dihedral phase o Kω: impropers ω : impropers o KUB: Urey-Bradley r1,3o: Urey-Bradley Aka. Internal or bonded terms © Alexander D. MacKerell , 2006. Diagram of intramolecular energy terms 2 Vbond = Kb (b " bo ) ! 1 2 Vdihedral = K" (1+ (cosn" #$)) 2 Vangle = K" (" #"o) ! 3 4 ! © Alexander D. MacKerell , 2006. 2 Vbond = Kb (b " bo ) Chemical type Kbond bo C-C 100 kcal/mole/Å2 1.5 Å C=C 200 kcal/mole/Å2 1.3 Å 2 ! C=-C 400 kcal/mole/Å 1.2 Å Bond Energy versus Bond length 400 300 Single Bond 200 Double Bond Triple Bond 100 Potential Energy, kcal/mol 0 0.5 1 1.5 2 2.5 Bond length, Å © Alexander D. MacKerell , 2006. Vdihedral = K" (1+ (cosn" #$)) Dihedral energy versus dihedral angle 20 ! 15 K=10, n=1 10 K=5, n=2 K=2.5, N=3 5 Potential Energy, kcal/mol 0 0 60 120 180 240 300 360 Dihedral Angle, degrees δ = 0˚ Note use of a Fourier series for a dihedral © Alexander D. MacKerell , 2006. 2 H Vimproper = K" (" #"o) H H! H 2 C VUrey"Bradley = KUB (r1,3 " r1,3o ) ! © Alexander D. MacKerell , 2006. 2D dihedral energy correction map to the CHARMM 22 φ,ψ backbone (CMAP) φ,ψ grid-based energy correction via bicubic interpolation i$1 j$1 4 4 & ) & ) " $ "L # $#L V CMAP = f (",#) = ,,cij ( + ( + i=1 j=1 ' %" * ' %# * Smooth first derivatives, continuous second derivatives Grid rectangle coefficients, c ! ij 1) Corner grid points H1 H8 O13 H14 " ! C4 N7 C12 C19 "f f H2 H15 " H3 C6 C9 N17 H16 2) First derivatives: , H10 O5 H20 C11 H18 "# "# 2 "f H21H22 3) Cross derivatives: "#"$ ! Use bicubic spline inte!rp olation to determine derivatives ! © Alexander D. MacKerell , 2006. Additive intermolecular energy function and corresponding parameters 12 6 q q ,% R ( % R ( / i j + # .' min,ij * + 2' min,ij * 1 $ 4"Dr ij .' r * ' r * 1 nonbonded ij -& ij ) & ij ) 0 qi: partial atomic charge D: dielectric constant ! ε: Lennard-Jones (LJ, vdW) well-depth Rmin: LJ radius (Rmin/2 in CHARMM) Combining rules (CHARMM, Amber) Rmin i,j = Rmin i + Rmin j εi,j = SQRT(εi * εj ) Aka. Nonbonded or external terms © Alexander D. MacKerell , 2006. Electrostatic energy 100 80 60 40 20 qi=1, qi=1 0 qi=1, qj=-1 0 2 4 6 8 10 -20 -40 -60 Interaction energy, kcal/mol -80 -100 Distance, Å © Alexander D. MacKerell , 2006. Treatment of hydrogen bonds??? Partial atomic charges -0.5 0.35 0.5 C O H N -0.45 © Alexander D. MacKerell , 2006. 12 6 *# R & # R & - " ,% min,ij ( ) 2% min,ij ( / ij ,% r ( % r ( / +$ ij ' $ ij ' . © Alexander D. MacKerell , 2006. ! Example of nonbond exclusions 1 2 3 4 nonbond (intermolecular) interactions between bonded atoms are treated with special rules 1,2 interactions: 0 1,3 interactions: 0 1,4 interactions: 1 or scaled > 1,4 interactions: 1 © Alexander D. MacKerell , 2006. Alternate intermolecular terms for the electrostatic (additive) or vdW interactions *$ '12 $ '10 - RHB,A# H RHB,A# H VHbond = "HB ,& ) #& ) / *cos(1A# H #D ) 0 r r Hbonds +,% A# H ( % A# H ( ./ 9 6 +$ R ' $ R ' . V = " -& min,ij ) *& min,ij ) 0 ! vdw # ij -& r ) & r ) 0 vdw ,% ij ( % ij ( / #aR 6 $ min,ij $ R ' ' & rij min,ij ) ! Vvdw = "ij e #& ) * & & r ) ) vdw % % ij ( ( © Alexander D. MacKerell , 2006. ! Limitation of additive force fields The use of Coulomb’s law with fixed atomic charges to treat the electrostatic interactions is a major simplification in current force fields. It is well known that the electron distribution of a molecule (and, thus, the atomic charges) changes as a function of the electrostatic field around the molecule. This is ignored in additive force fields. To compensate for this omission, the atomic charges are “enhanced” to mimic the polarization of molecules that occurs in a polar, condensed phase environment (e.g. aqueous solution, TIP3P water model dipole moment = 2.35 versus gas phase value of 1.85). This approximation has worked well in the current additive force fields; however, in many cases these models fail. To overcome this, next generation force fields are being developed that explicitly treat electronic polarization. © Alexander D. MacKerell , 2006. Methods to include electronic polarization in force fields Fluctuating charge (CHEQ) Induced dipoles (PIPF, Berne/Friesner, AMBER) Classical Drude Oscillator All methods require that the perturbation of the electronic distribution due to the surrounding electrostatic field be optimized in an iterative fashion. This is due to the change in the “charge distribution” of a system leading to a new electrostatic field which then requires additional re-adjustment of the charge distribution (SCF: self-consistent field calculation). Matrix diagonalization may also be used, but is frequently inaccessible due to the large number of atoms in biological systems. In the end the need to perform an SCF calculation leads to a large increase in computational demands. Special methods to minimize this limitation in MD simulations have been developed (see below). © Alexander D. MacKerell , 2006. Fluctuating Charge Model (CHEQ) Polarization is based on the movement of charge, q, between bonded atoms i and j in response to the surrounding electrostatic field. The extent of charge movement is based on the relative electronegativity, χ, and hardness, J, of the bonded atoms. The electrostatic energy is then obtained from the Coulombic interactions between the relaxed charges. 1 V(q ) = " q + J q2 ij ij ij 2 ij ij ' ' ' ' ' " ij = " i + " j Jij = Ji + J j + 2Jij Electrone!g ativity: attraction of an atom for electrons Hardness: work needed to transfer charge (resistance to charge movement) ! ! © Alexander D. MacKerell , 2006. Induced Dipole Model Each atom, i, carries a charge, qi, and a dipole moment, µi, such that electrostatic interactions between atoms i and j include: charge-charge interactions: 1/rij 2 charge-dipole interactions: 1/rij 3 dipole-dipole interactions: 1/rij Polarization included via relaxation of dipole moments in the electrostatic field, Ei, where αi is the polarizability of atom i % ( 0 induced 0 µi = " i (E i + E i ) = " i' E i + $Tijµ j * & i# j ) © Alexander D.