Empirical Force Fields:Overview and parameter optimization.

Alexander D. MacKerell, Jr.Department of Pharmaceutical Sciences

School of PharmacyUniversity of Maryland, Baltimore

Baltimore, MD 21201, USA

Adopted as Additional Instructions on F.F. for CHEM5021/8021 (J. Gao, Computational Chemistry)original material: http://www.pharmacy.umaryland.edu/faculty/amackere/param/force_field_dev.htm

Potential energy function(mathematical equations)

Empirical force field(equations and parameters that relate chemical structure and

conformation to energy)

Class ICHARMMCHARMm (Accelrys)AMBEROPLS/AMBER/SchrödingerECEPP (free energy force field)GROMOS

Class IICFF95 (Biosym/Accelrys)MM3MMFF94 (CHARMM, Macromodel, elsewhere)UFF, DREIDING

Common empirical force fields

Extended atom models (omit non-polar hydrogens)

CHARMM PARAM19 (proteins)

use with implicit solvent models

ACE, EEF, GB

improper term to maintain chirality

loss of cation - pi interactions

OPLS

AMBER

GROMOS

Transition State Force Field ParametersSame approach as standard force field parameterizationRequire target data for transition state of interest: ab initio

Metal Force Field ParameterizationOnly interaction parameters orInclude intramolecular terms

Parameterization of QM atoms for QM/MM calculations

Include explicit term in potential energy function to treat polarization of charge distribution by the environment.

Still under development.

PIPF (Gao and coworkers)AMBERFriesner/Berne et al. (Schrödinger Inc.)CHARMMTINKER

Polarizable (non-additive) force fields

Class I Potential Energy Function

Intramolecular (internal, bonded terms)

Kbbonds∑ b − bo( )2 + Kθ

angles∑ θ −θ o( )2 + Kφ

torsions∑ 1+ cos(nφ −δ )( )

+ Kϕimpropers∑ ϕ − ϕo( )2 + KUB

Urey −Bradley∑ r1, 3 − r1, 3, o( )2

Intermolecular (external, nonbonded terms)

qiq j

4πDrij

+ εijnonbonded

∑ Rmin,ij

rij

12

− 2Rmin,ij

rij

6

Class II force fields (e.g. MM3, MMFF, UFF, CFF)

Kb, 2 b − bo( )2 + Kb, 3 b − bo( )3 + Kb, 4 b − bo( )4[ ]bonds∑

+ Kθ , 2 θ −θ o( )2 + Kθ , 3 θ −θ o( )3 + Kθ , 4 θ −θ o( )4[ ]angles∑

+ Kφ ,1 1− cosφ( )+ Kφ , 2 1− cos2φ( )+ Kφ , 3 1− cos3φ( )[ ]dihedrals∑

+ Kχimpropers∑ χ 2

+ Kbb 'bonds'∑

bonds∑ b − bo( ) b '−bo '( )+ Kθθ '

angles'∑

angles∑ θ −θ o( ) θ '−θ o '( )

+ Kbθangles∑

bonds∑ b − bo( ) θ −θ o( )

+ b − bo( ) Kφ , b1 cosφ + Kφ , b2 cos2φ + Kφ , b3 cos3φ[ ]dihedrals∑

bonds∑

+ b '−bo '( ) Kφ , b '1 cosφ + Kφ , b' 2 cos2φ + Kφ , b' 3 cos3φ[ ]dihedrals∑

bonds'∑

+ θ −θ o( ) Kφ ,θ 1 cosφ + Kφ ,θ 2 cos2φ + Kφ ,θ 3 cos3φ[ ]dihedrals∑

angles∑

+ θ −θ o( ) θ '−θ o '( )cosφdihedrals∑

angles'∑

angles∑

Equilibrium termsbo: bondsθo: anglesn: dihedral multiplicityδo: dihedral phaseωo: impropersr1,3o: Urey-Bradley

Force constantsKb: bondsKθ: anglesKφ: dihedralKω: impropersKUB: Urey-Bradley

Intramolecular parameters

Kbbonds∑ b − bo( )2 + Kθ

angles∑ θ −θo( )2 + Kφ

torsions∑ 1+ cos(nφ −δ)( )

+ Kϕimpropers∑ ϕ −ϕo( )2 + KUB

Urey−Bradley∑ r1,3 − r1,3,o( )2

Vbond = Kb b − bo( )2

1 2

3 4

Vangle = Kθ θ −θo( )2

Vdihedral = Kφ (1+ (cosnφ −δ))

Intermolecular interactions between bonded atoms1,2 interactions: 01,3 interactions: 01,4 interactions: 1 or scaled> 1,4 interactions: 1

Bond Energy versus Bond length

0

100

200

300

400

0.5 1 1.5 2 2.5

Bond length, Å

Chemical type Kbond bo

C-C 100 kca l/mole/Å2 1.5 ÅC=C 200 kca l/mole/Å2 1.3 ÅC=-C 400 kca l/mole/Å2 1.2 Å

Vbond = Kb b − bo( )2

Dihedral energy versus dihedral angle

0

5

10

15

20

0 60 120 180 240 300 360

Dihedral Angle, degrees

Vdihedral = Kφ (1+ (cosnφ −δ))

δ = 0˚

H

C

HHH

Vimproper = Kϕ ϕ −ϕo( )2

VUrey−Bradley = KUB r1,3 − r1,3o( )2

qi: partial atomic chargeD: dielectric constantε: Lennard-Jones (LJ, vdW) well-depthRmin: LJ radius (Rmin/2 in CHARMM)Combining rules (CHARMM, Amber)

Rmin i,j = Rmin i + Rmin jεi,j = SQRT(εi * εj )

Intermolecular parameters

qiq j

4πDrij

+ εijnonbonded

∑ Rmin,ij

rij

12

− 2Rmin,ij

rij

6

Electrostatic Energy versus Distance

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 1 2 3 4 5 6 7 8

Distance, Å

Inte

racti

on

en

erg

y,

kca

l/m

ol

q1=1, q2=1

q1=-1, q2=1

Treatment of hydrogen bonds???

Partial atomic charges

C O H N0.5-0.5 0.35

-0.45

Lennard-Jones Energy

-0.5

-0.3

-0.1

0.1

0.3

0.5

0.7

0.9

1 2 3 4 5

Distance, Å

Series1

Rmin,ij

εi,j

εij

Rmin,ij

rij

12

− 2Rmin,ij

rij

6

Lennard-Jones Energy versus Distance

-0.5

-0.3

-0.1

0.1

0.3

0.5

0.7

0.9

1 2 3 4 5 6 7 8

Distance, Å

Inte

racti

on

En

erg

y,

kca

l/m

ol

e=0.2,Rmin=2.5

Rmin,i,j

eps,i,j

Vvdw = εij e−aRmin,ij

rij −Rmin,ij

rij

6

vdw∑

Vvdw = εijvdw∑ Rmin,ij

rij

9

−Rmin,ij

rij

6

Alternate intermolecular terms

VHbond = εHBRHB ,A− H

rA− H

12

−RHB,A− H

rA− H

10

Hbonds

∑ *cos θA− H −D( )

Extent of Parameter Optimization

Minimal optimizationby analogy (i.e. direct transfer of known parameters)dihedrals only??

Maximal optimizationtime-consumingrequires appropriate target data

Choice based on goal of the calculationsMinimal

database screeningNMR/X-ray structure determination

Maximalfree energy calculations (perturbations, potential of mean force)mechanistic studiessubtle environmental effects lead optimization

Solute-Solute

Solvent-SoluteSolvent-Solvent

"Interaction Triad"

IntermolecularIntramolecular

Self-Consistent Empirical Parameterization

Intermolecular Optimization

Partial Atomic Charges

VDW Parameters

Intramolecular Optimization

Bonds

Angles

Torsions

Impropers, Urey-Bradley

Parameter Optimization Complete

if intermolecular and intramoleculachanges < convergence criteria

Initial Geometry

if intermolecular change > conv.crit.

if intermolecular microscopic and macroscopic change < convergence criteria

if intramolecular change > conv.crit.

if intramolecular and intermolecular change > conv.crit.

NH

N

NHO

OH

NH

N

NHOOH

A B C

A) IndoleB) HydrazineC) PhenolLinking model compounds

When creating a covalent link between model compounds move the charge on deleted H into the carbon to maintain integer charge (i.e. methyl (qC=-0.27, qH=0.09) to methylene (qC=-0.18, qH=0.09)

1) Identify previously parameterized compounds2) Access topology information

i) Assign atom typesii) Connectivity (bonds)iii) Charges

CHARMM topology (parameter files)

top_all22_model.inp (par_all22_prot.inp)top_all22_prot.inp (par_all22_prot.inp)top_all22_sugar.inp (par_all22_sugar.inp)top_all27_lipid.rtf (par_all27_lipid.prm)top_all27_na.rtf (par_all27_na.prm)top_all27_na_lipid.rtf (par_all27_na_lipid.prm)top_all27_prot_lipid.rtf (par_all27_prot_lipid.prm)top_all27_prot_na.rtf (par_all27_prot_na.prm)toph19.inp (param19.inp)

NA and lipid force fields have new LJ parameters for the alkanes, representing increased optimization of the protein alkane parameters. Tests have shown that these are compatible (e.g. in protein-nucleic acid simulations). For new systems is suggested that the new LJ parameters be used. Note that only the LJ parameters were changed; the internal parameters are identical

From top_all22_model.inp

RESI PHEN 0.00 ! phenol, adm jr.GROUPATOM CG CA -0.115 !ATOM HG HP 0.115 ! HD1 HE1GROUP ! | |ATOM CD1 CA -0.115 ! CD1--CE1ATOM HD1 HP 0.115 ! // \\GROUP ! HG--CG CZ--OHATOM CD2 CA -0.115 ! \ / \ATOM HD2 HP 0.115 ! CD2==CE2 HHGROUP ! | |ATOM CE1 CA -0.115 ! HD2 HE2ATOM HE1 HP 0.115GROUPATOM CE2 CA -0.115ATOM HE2 HP 0.115GROUPATOM CZ CA 0.11ATOM OH OH1 -0.54ATOM HH H 0.43BOND CD2 CG CE1 CD1 CZ CE2 CG HG CD1 HD1 BOND CD2 HD2 CE1 HE1 CE2 HE2 CZ OH OH HHDOUBLE CD1 CG CE2 CD2 CZ CE1

HG will ultimately be deleted. Therefore, move HG (hydrogen) charge into CG, such that the CG charge becomes 0.00 in the final compound.

Use remaining charges/atom types without any changes.

Do the same with indole

Top_all22_model.inp contains all protein model compounds. Lipid, nucleic acid and carbohydate model compounds are in the full topology files.

Creation of topology for central model compound

Resi Mod1 ! Model compound 1Group !specifies integer charge group of atoms (not essential)ATOM C1 CT3 -0.27ATOM H11 HA3 0.09ATOM H12 HA3 0.09ATOM H13 HA3 0.09GROUPATOM C2 C 0.51ATOM O2 O -0.51GROUP ATOM N3 NH1 -0.47ATOM H3 H 0.31ATOM N4 NR1 0.16 !new atomATOM C5 CEL1 -0.15 ATOM H51 HEL1 0.15ATOM C6 CT3 -0.27ATOM H61 HA 0.09ATOM H62 HA 0.09ATOM H63 HA 0.09BOND C1 H11 C1 H12 C1 H13 C1 C2 C2 O2 C2 N3 N3 H3BOND N3 N4 C5 H51 C5 C6 C6 H61 C6 H62 C6 H63DOUBLE N4 C5 (DOUBLE only required for MMFF)

Start with alanine dipeptide.Note use of new aliphatic LJ parameters and, importantly, atom types. NR1 from histidine unprotonated ring nitrogen. Charge (very bad) initially set to yield unit charge for the group.

CEL1/HEL1 from propene (lipid model compound). See top_all27_prot_lipid.rtf

Note use of large group to allow flexibility in charge optimization.

N

NHO

Intermolecular Optimization Target Data

Local/Small MoleculeExperimental

Interaction enthalpies (MassSpec)Interaction geometries (microwave, crystal)Dipole moments

Quantum mechanicalMulliken Population AnalysisElectrostatic potential (ESP) based

CHELPG (g98: POP=(CHELPG,DIPOLE)Restricted ESP (AMBER)

Dimer Interaction Energies and Geometries (OPLS, CHARMM)

Global/condensed phase (all experimental)Pure solvents (heats of vaporization, density, heat capacity, isocompressibility)Aqueous solution (heats/free energies of solution, partial molar volumes)Crystals (heats of sublimation, lattice parameters, interaction geometries)

Additive Models: account for lack of explicit inclusion ofpolarizability via “overcharging” of atoms.

RESP: HF/6-31G overestimates dipole moments (AMBER)Interaction based optimization (CHARMM, OPLS)

local polarization includedscale target interaction energies (CHARMM)

1.16 for polar neutral compounds1.0 for charged compounds

Partial Atomic Charge Determination

For a particular force field do NOT change the QM level of theory. This is necessary to maintain consistency with the remainder of the force field.

Starting charges??peptide bondmethylimidazole (N-N=C)?Mulliken population analysis

Final charges (methyl, vary qC to maintain integer charge, always qH = 0.09)interactions with water (HF/6-31G*, monohydrates!)dipole moment

N

NO

H

CH3

H

CH3

Model compound 1-water interaction energies/geometries

Interaction Energies (kcal/mole) Interaction Distances (�)Ab initio Analogy Optimized Ab initio Analogy Optimized

1) O2...HOH -6.12 -6.56 -6.04 2.06 1.76 1.78

2) N3-H..OHH -7.27 -7.19 -7.19 2.12 1.91 1.89

3) N4...HOH -5.22 -1.16 -5.30 2.33 2.30 2.06

4) C5-H..OHH -3.86 -3.04 -3.69 2.46 2.51 2.44

Energetic statistical analysis Ave. Difference 1.13 0.06 RMS Difference 1.75 0.09 Dipole Moments (debeye)

5.69 4.89 6.00Ab initio interaction energies scaled by 1.16.

Comparison of analogy and optimized charges

N

NHO

Name Type Analogy OptimizedC1 CT3 -0.27 -0.27

H11 HA3 0.09 0.09H12 HA3 0.09 0.09H13 HA3 0.09 0.09C2 C 0.51 0.58O2 O -0.51 -0.50N3 NH1 -0.47 -0.32H3 H 0.31 0.33N4 NR1 0.16 -0.31C5 CEL1 -0.15 -0.25

H51 HEL1 0.15 0.29C6 CT3 -0.27 -0.09

H61 HA 0.09 0.09H62 HA 0.09 0.09H63 HA 0.09 0.09

Note charge on C6 methyl carbon.Non-integer charge is typically placed on the adjacent aliphatic carbon.

LJ (vdw) parameters

Direct transfer from available parameters is generally adequate

Test viaHeat of vaporizationDensity (Molecular Volume)Partial molar volumeCrystal simulations

For details of LJ parameter optimization see Chen, Yin and MacKerell, JCC, 23:199-213 (2002)

Geometries (equilibrium bond, angle, dihedral, UB and improper terms)microwave, electron diffraction, ab initiosmall molecule x-ray crystallography (CSD)crystal surveys of geometries

Vibrational spectra (force constants) infrared, raman, ab initio

Conformational energies (force constants)microwave, ab initio

Intramolecular optimization target data

Comparison of atom names (upper) and atom types (lower)

C2

NH

N4

N3O2

C5

OH

C

NH

NR1

NH1O

CEL1

OHH

HEL1

H3

H5

Bond and angle equilibrium term optimizationOnly for required parameters!

Experimental Crystal structure, electron diffraction, microwaveCrystal survey of specific moieties

QMFull compound if possible (HF or MP2/6-31G*)Fragments if necessary

NH

N

NHO

CH

OH

Bonds (list doesn’t include lipid-protein alkane nomenclature differences)NH1-NR1, NR1-CEL1

AnglesNR1-NH1-H, NR1-NH1-C, NH1-NR1-CEL1NR1-CEL1-CTL3, NR1-CEL1-HEL1

DihedralsCTL3-C-NH1-NR1, C-NH1-NR1-CEL1, O-C-NH1-NR1, NH1-NR1-CEL1-HEL1, NH1-NR1-CEL1-CTL3H-NH1-NR1-CEL1, NR1-CEL1-CTL3-HAL3

Bonds and angles for model compound 1

MP2/6-31G* CSD Analogy OptimizedBond lengths 1 2 1 2C-Na 1.385 1.382 1.37±0.03 1.35±0.01 1.342 1.344N-N 1.370 1.366 1.38±0.02 1.37±0.01 1.386 1.365N=C 1.289 1.290 1.29±0.02 1.28±0.01 1.339 1.289AnglesC-N-N 120.8 122.4 120.7±5.8 119.7±2.9 124.5 121.4N-N=C 116.0 116.6 114.5±5.3 115.8±1.6 119.6 115.6N=C-C 119.9 120.0 120.7±4.7 121.2±2.2 122.4 121.0The MP2/6-31G* results are for the 1) all-trans and 2) 0û, 180û, 180ûÊglobal minimum energy structures. TheCambridge structural database results represent mean±standard deviation for all structures with R-factor < 0.1 and1) the N7 and C10 sites undefined and 2) the N7 and C10 sites explicitly protonated. A) Not optimized as part ofthe present study.

NH1-NR1 from 400/1.38 to 550/1.36, NR1=CEL1 from 500/1.342 to 680/1.290: C-NH1-NR1 from 50.0/120.0 to 50.0/115.0, NH1- NR1-CEL1 from 50.0/120.0 to 50.0/115.0, NR1-CEL1-CT3 from 48.0/123.5 to 48.0/122.5. For planar systems keep the sum of the equilibrium angle parameters equal to 360.0

Bond, angle, dihedral, UB and improper force constants

Vibrational spectraFrequenciesAssignments

Conformational EnergeticsRelative energiesPotential energy surfaces

Vibrations are generally used to optimize the bond, angle, UB and improper FCs while conformational energies are used for the dihedral FCs. However, vibrations will also be used for a number of the dihedral FCs, especially those involving hydrogens and in rings.

# Freq Assign % Assign % Assign %1 62 tC2N 64 tN3N 462 133 tC1H3 50 tN3N 18 tC2N 173 148 tC1H3 46 tC6H3 254 154 dC2NN 44 dN3NC 28 dN4CC 165 205 tC6H3 59 tN4C 22 tN3N 216 333 tN4C 73 tC2N 227 361 dC1CN 45 dN4CC 21 dN3NC 168 446 rC=O 32 dN4CC 209 568 wNH 7710 586 dC1CN 21 dC2NN 20 rC=O 1811 618 wC=O 83 wNH 28 tC2N -2612 649 rC=O 27 dN4CC 1913 922 sC1-C 6214 940 wC5H 8015 1031 rCH3' 33 sC5-C 3116 1114 rCH3 6617 1139 rCH3' 76 wC=O 2018 1157 rCH3 61 wC5H 2119 1234 sC5-C 33 sN-N 3220 1269 sN-N 36 rCH3' 18

# Freq Assign % Assign %21 1446 rNH 3522 1447 rC5H 47 sC-N 1823 1527 dCH3 7724 1532 dCH3 8825 1599 dCH3a' 50 dCH3a 1726 1610 dCH3a 71 dCH3a' 2427 1612 dCH3a' 3028 1613 dCH3a 70 dCH3a' 2329 1622 dCH3a' 57 dCH3a 1930 1782 sN=C 7131 1901 sC=O 7832 3250 sCH3 76 sC5-H 2133 3258 sC5-H 78 sCH3 2134 3280 sCH3 9935 3330 sCH3a 75 sCH3a' 2536 3372 sCH3a' 10037 3377 sCH3a' 73 sCH3a 2438 3403 sCH3a 9939 3688 sN-H 100

Vibrational Spectra of Model Compound 1 from MP2/6-31G* QM calculations

Frequencies in cm-1. Assignments and % are the modes and there respective percentscontributing to each vibration.

Comparison of the scaled ab initio, by analogy and optimized vibrations for selected modes

g98 Analogy Optimized# Freq Assi % # Freq Assi % # Freq AssisN=N30 1782 sN=C 71 21 1228 sN=C 37 31 1802 sN=C

rC5H 36 sN-N30 1646 sN=C 28

sC5-C 24rC5H 18

sN-N19 1234 sC5-C 33 20 1113 sN-N 53 20 1200 rNH

sN-N 32 rNH 26 sN-N20 1269 sN-N 36 rC5H

rCH3' 18 23 1395 dCH3sN-N

31 1802 sN=CsN-N

dC2NN4 154 dC2NN44 5 207 dC2NN36 4 158 dC2NN

dN3NC28 tN4C 31 dN3NCdN4CC 16 dN4CC

10 586 dC1CN 21 12 607 dC1CN 26 11 574 dC1CNdC2NN20 dC2NN25 dC2NNrC=O 18 dN4CC

NH1-NR1 from 400/1.38 to 550/1.36NR1=CEL1 from 500/1.342 to 680/1.290:C-NH1-NR1 from 50.0/120.0 to 50.0/115.0,

NH

N

NHO

OH

N

NHO

Dihedral optimization based on QM potential energy surfaces (HF/6-31G* or MP2/6-31G*).

NH

N

NHOOHN

H

NH2O

HN

OH

Potential energy surfaces on compounds with multiple rotatable bonds

1) Full geometry optimization2) Constrain n-1 dihedrals to minimum energy values or trans

conformation3) Sample selected dihedral surface4) Repeat for all rotatable bonds dihedrals5) Repeat 2-5 using alternate minima if deemed appropriate

N

NHO

i)

ii)iii)

Note that the potential energy surface about a given torsion is the sum of the contributions from ALL terms in the potential energy function, not just the dihedral term

Creation of full compound

1) Obtain indole and phenol from top_all22_model.inp2) Rename phenol atom types to avoid conflicts with indole (add P)3) Delete model 1 terminal methyls and perform charge adjustments

i) Move HZ2 charge (0.115) into CZ2 (-0.115 -> 0.000) total charge on deleted C1 methyl (0.00) onto CZ2 (0.00 -> 0.00)

ii) Move HPG charge (0.115) into CPG (-0.115 -> 0.000) and move total charge on the C6 methyl (0.18) onto CPG (0.00 -> 0.18)

4) Add parameters by analogy (use CHARMM error messages)5) Generate IC table (IC GENErate)6) Generate cartesian coordinates based on IC table (check carefully!)

CZ2

C2

NH

N

NHO

C5CPG

OHN

NOC6H3

C1H3

H

H

Check novel connectivities1) dihedrals !2) bonds3) angles

NH

N

NHOOH

HN

OH

NH

NH2O

MP2/6-31G* versus HF/6-31G*

Addition of simple functional groups is generally straightforward once the full compound parameters have been optimized.

C

H

HH

NH

H

N

H

HH

C

O

O

O

H

C

O

NH2

OCH3F

Lead Optimization

1) Delete appropriate hydrogens (i.e. at site of covalent bond)2) Shift charge of deleted hydrogen into carbon being functionalized.3) Add functional group4) Offset charge on functionalized carbon to account for functional

group charge requirements1) Aliphatics: just neutralize added functional group, qH=0.092) Phenol OH: qC=0.11, qO=-0.54, qH=0.433) Aliphatic OH: qC=-0.04, qO=-0.66, qH=0.434) Amino: qC=0.16, qCH=0.05, qN=-0.30, qH=0.335) Carboxylate: qC=-0.37, qCO=-0.62, qO=-0.76

5) Internal parameters should be present. Add by analogy if needed.6) Optimize necessary parameters.

Perform above via the CHARMM PATCH (PRES) command

Summary

1) Junk in, junk out: Parameter optimization effort based on application requirements.

2) Follow standard protocol for the force field of interest (higher level QM is not necessarily better).

3) Careful parameter optimization of lead molecules4) Simple substitutions often require minimal or no

optimization.

Acknowledgements

Daxu YinIjen Chen

Nicolas FoloppeNilesh Banavali

NIHDoD HPC

NPACIPSC Terascale Computing

VT = Kb(b − b0)2

bonds∑ + Kθ (θ −θ0)2

angles∑ + Kφ 1+ cos(nφ −δ )[ ]

dihedrals∑

+ Kub(S − S0)2

1,3pairs∑ + Kw(w − w0)2

improper∑

+ εij

Rmin,ijrij

12

− 2Rmin,ij

rij

6

nonbonded

∑ +qiqj

4πDrij

AmberCHARMMGROMOS

OPLS

Class I potential energy function