Bio Molecular Simulation
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Transcript of Bio Molecular Simulation
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Simulation of Biomolecules
1. Force fields:Energy terms, Topology and parameter files
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Simulation of Biomolecules
1. Force fields:Energy terms, Topology and parameter files
2. Molecular Dynamics - Preceding Information:
Computation of nonbonded interactions, IntroductMechanics
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Simulation of Biomolecules
1. Force fields:Energy terms, Topology and parameter files
2. Molecular Dynamics - Preceding Information:
Computation of nonbonded interactions, IntroductMechanics
3. Molecular Dynamics I:The Idea, Integrating the equations of motion, MD
ensembles, Thermostats
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Simulation of Biomolecules
1. Force fields:Energy terms, Topology and parameter files
2. Molecular Dynamics - Preceding Information:
Computation of nonbonded interactions, IntroductMechanics
3. Molecular Dynamics I:The Idea, Integrating the equations of motion, MD
ensembles, Thermostats4. Molecular Dynamics II:
Langevin dynamics, Brownian dynamics
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Simulation of Biomolecules
1. Force fields:Energy terms, Topology and parameter files
2. Molecular Dynamics - Preceding Information:
Computation of nonbonded interactions, IntroductMechanics
3. Molecular Dynamics I:The Idea, Integrating the equations of motion, MD
ensembles, Thermostats4. Molecular Dynamics II:
Langevin dynamics, Brownian dynamics
5. Monte Carlo Simulations:
The Idea: Importance Sampling and the MetropolM M t C l i l ti i i
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Simulation of Biomolecules
1. Force fields:Energy terms, Topology and parameter files
2. Molecular Dynamics - Preceding Information:
Computation of nonbonded interactions, IntroductMechanics
3. Molecular Dynamics I:The Idea, Integrating the equations of motion, MD
ensembles, Thermostats4. Molecular Dynamics II:
Langevin dynamics, Brownian dynamics
5. Monte Carlo Simulations:
The Idea: Importance Sampling and the MetropolM M t C l i l ti i i
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Simulation of Biomolecules
Force fields:
Energy terms in all-atom force fields Non-covalent interactions
Parameterization of force fields Topology and parameter files Coarse-grained force fields
Knowledge-based force fields
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All-atom force fields
classical biomolecular simulations: approximation todynamics
classical force fields parameterized based on quant
simulations the potential ( force field) is a function of the cooparticles (N atoms)
energy expressed in terms of atom pairs, triples a
using the concept of chemical bonding:N 1 bond lengths, N 2 bond angles (involving torsion angles (between 3 bonds)
together with 6 degrees of freedom (DOF) for over
and rotation: 3N DOFs
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All-atom force fields
the potential can be expressed as
U =a U(1)(ra) +ab U
(2)(ra, rb)
+abc
U(3)(ra, rb, rc) +abcd
U(4)(ra, rb, r
Higher order terms arise from, e.g., polarisation winduce other multipoles. The latter interact with theon. Thus, the problem can only be solved iterative
An alternative expression to Eq. (1) is the division
and nonbonded contributions. Bonded contributions arsise from bond vibrations
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All-atom force fields
Nonbonded interactions are calculated between thdifferent molecules and for atoms of the same moseparated by more than thre bonds.
All nonbonded interactions are added up:
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Two-body potentials
Pairwise interactions are functions of the distance atoms a and b: rab = |ra rb|
Simple pairwise interactions:
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Two-body potentials: Lennard-Jones potentia
U(2)LJ = Um
rvdWij
r12
2rvdWij
r6
d
m
0.8 1 1.2
U
The attractive r6
term originates from quantum mto electron correlation, the so-called London or disinteractions.
The repulsive r12 term has a quantum origin in th
the electron clouds with each other (Pauli exclusiointernuclear repulsions.
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Two-body potentials: Lennard-Jones potentia
U(2)LJ = Um
rvdWij
r12
2rvdWij
r6
d
m
0.8 1 1.2
U
For r , U(2)
LJ 0: short range van der Waals ( At r = rvdWij the LJ potential has its minimum with
rvdWij = rvdWi + r
vdWj : sum of the vdW radii of atom
The potential depth is determined by the polarisabb f l t N i th t l t h
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Two-body potentials: Coulomb potential
U(2)coul = Kcoul qiqjr
distance (A
energy(kcal/mol)
2000
1000
0
1000
2000
0.1 0.2 0.0
For ionic interactions between fully or partially cha Repulsive if particles have the same charge, attrac
opposite charges.Th di l t i t t d ib th i
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Two-body potentials: Coulomb potential
U(2)coul = Kcoul
qiqjr
Unlike vdW interactions, the Coulomb interactions
with distance. The Coulomb potential constant is given by Kcoul
where the permittivity of a vacuum is 0 = 8.8542 C2 m1 J1.
With 1 m = 1010 , 1 C = 1/(1.6022 1019) esu (eelectrostatic charge unit for the elementary charge, e0,proton), and 1 J 6.0221 1023/4184 kcal mol1 f
Kcoul =1J m 332 kca
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Electrostatic interactions in molecules
Most interactions in molecules are electrostatic. An electric dipole consists of two charges q and
a distance l. The dipole is represented by a vecto
from the negative to the positive charge, q qmoment unit: 1 Debye = 1 D = 0.208 e0. In an external field the dominant molecular multipo All heteronuclear two-atomic molecules are polar d
difference of electronegativity of both atoms resultcharges (e.g., 1.08 D in HCl).
Depending on the symmetry, a multiatom molecul(e.g., 1.85 D in H2O).
Apolar molecules do not have a permanent electri
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Electrostatic interactions in molecules
Permanent and induced dipole moments are important in
TypicalType of Distance energy valuesinteraction dependence [kcal/mol] Emonopole-monopole 1/r 50 to 5 smonopole-dipole 1/r2 3.5 L
adipole-dipole 1/r3 0.5 b
London (vdW) 1/r6 0.1 a
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Hydrogen bonds
Hydrogen bonds derive from electrostatic interactithe positive partial charge of a H atom bound to anelectron-withdrawing donor (D) and the lone pair the acceptor atom (A): D H+ A whereand F.
The hydrogen bond energy depends on the geomA. The optimal DH A angle is 180.
The optimal H A AA angle (AA = anterior accon the D, A and AA elements and the hybridization
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Hydrogen bonds
Hydrogen bonds in proteins:
D AType distance [
amide-carbonyl >NH O=C< 2.90.1 hydroxyl-carbonyl OH O=C< 2.80.1 hydroxyl-hydroxyl OH OH 2.80.1
amide-hydroxyl >NH OH 2.90.1 amide-imidazole >NH N 3.10.2 ammonium-carboxyl NH+3
OOC 2.70.1 guanidinium-carboxyl
...NH+
2
OOC 2.70.1
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Two-body potentials: Bond length potentials
Bond length potentials model small-scale deviatioequilibrium bond length r0.
The harmonic potential using Hookes law is the s
molecular-mechanics formulation for bond vibratio According to Hookes law, the force F is proportiondisplacement, r r0, and the acceleration, d2r/dt
F(r) = kvib(r r0) = m d2
rdt2
, kvib =
From the angular frequency (number of radians which is connected to the wavelength via = 2c/mass m, the force constant kvib can be calculated
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Two-body potentials: Bond length potentials
From (2) the bond length potential follows:
U(2)HO =
kvib2
(r r0)2 (3)
r0
bondpotential
where the subscript HO is for harmonic oscillator. More realistic functions for bond vibration is, for ins
the Morse potential.
But since kvib usually large and and bond lengths (
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Three-body potentials: Bond angle potentials
Bond angle arrangement around each atom in a mgoverned by the hybridization stage of this atom, e sp hybridization linear geometry with 18
sp2
hybridization trigonal planar with 12 sp3 hybridization tetrahedral with 109.5
Exact orbits only exist if all bonded atoms are of the.g., as in methane. In propane, for instance, the C
angle is 112.5
, the HCH bond angles 107.5 109.5.
Electron lone pairs influence the geometry too. The bond angle is determined by
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Three-body potentials: Bond angle potentials
e1 =rb ra|rb ra|
, e2 =rc ra|rc ra|
The bond angle potential is often expressed as a h
function in terms of angle cosines:
U(3)HO() = kbend(cos cos 0)
2
Advantage of this trigonometric potential is its bouits ease of implementation and differentiation.
It avoids the calculation of inverse trigonometric fusingularity problems for linear bond angles.
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Four-body potentials: Bond torsion potentials
Origin of torsional potential: relief of steric congesstabilization and electronic repulsions (quantum meffects)
For example, in ethane the torsional strain is highemethyl groups are nearest (eclipsedor cis) state, athey are optimally separated (anti, transor stagge
The functional form is force fields is
U(4)() =n
Vn2
[1+cos (n 0)]
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Four-body potentials: Bond torsion potentials
Often 0 = 0, so that U(4)() =
nVn2 [1 cos n
The torsional angle is determined by the the posatoms involved, a, b, c, d and the three bonds i, j,
them:
(ra, rb, rc, rd) = arccos
(ei ej)(ej
sin ij sin
Combination of two- and threefold symmetry to repcis/transand trans/gaucheenergy differences:
U (4)()V2
[1 (2)] y[kcal/mol]
2 fold
3 fold
sum
O3
C3
C2
O2
3
4
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Four-body potentials: Bond torsion potentials
n and Vn are determined for low molecular weightexperimentally using NMR, IR, Raman and microwspectroscopy and/or quantum mechanical calclatiocompounds.
Examples for model compounds: hydrocarbons like ethane, propane or cyclohex
rotations about single CC bonds
ethylbenzene for rotations about CAC bonds(CA is an aromatic carbon here) alcohols like methanol and propanol for HC
CCOH, HCCO and CCCO seq
for serine and threonine) methylamine ethanal and N-methyl formamide
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Four-body potentials: Improper torsion
improper torsion is also known as out-of-plane b To enforce planarity or maintain chirality about cer
U(4)HO() =kimp
22 CA
where is the improper angle defined for the four for which the central atom a is bonded to b, c, d, asbetween the planes a b c and b c d.
Examples:
CCAN0 : for planarity around peptide bon C CA O 0 : for planarity of the carboxy grou
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Cross terms
Cross terms couple the as independent assumed the potential energy function arising from bonded
Schematic representation of various cross terms:
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Cross terms
For example, a stretch/bend cross term for a bondsequence abc allows bond lengths a b and b cincrease/decrease as abc decreases/increases.
A variation of the stretch/bend term is the Urey-Brawhich is commonly used in force fields (e.g., in CHUB potential is a simple harmonic function of the idistance between atoms i and k in the bonded seq
A torsion/torsion cross term is included in CHARMwhich accounts for the interdependence of the Raangles and .
Cross terms are corrections to the potential nergy
better agreement with the results from experimentchemical calculations
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Summary: Potential energy function
The total energy function for the CHARMM potential is:
U =
bondskb(b b0)
2 +
anglesk( 0)
2
+
dihedrals
n
Vn2
[1 + cos (n 0)] +
impropers
+
nonbondedpairs ij
UijrvdWij
r
12 2
r
vdW
ij
r
6++
UreyBradley
kUB(s s0)2 +
residues
UCMAP(, )
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Summary: Potential energy function
All force fields consider a molecule as a collectionheld together by some sort of elastic forces.
The atoms of a molecule may be thought of as joinmutually independent springs, restoring "natural" vlengths and angles.
All forces are defined in terms of potential energy internal coordinates of the molecules that constitu
force field. In these expressions, the sums extend over all bontorsions, and nonbonded interactions between all bound to each other or to a common atom.
More elaborate force fields may also include eithet (1 3 b d d i t ti ) i t
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Atom types in force fields
Transferability: The principle of transferability assumpotentials can be developed to incorporate all expfor model compounds and the applied successfullyof large biological molecules composed of the sam
subgroups. A reasonable assumption since bond lengths and
tend to adopt similar values in different molecular
To represent the different chemical environment (ecarbon in a phenyl ring) and hybridization, differenused for the same element.
In CHARMM22, for instance, there are around 57
carbons, 12 hydrogens, 11 nitrogens, 7 oxygens, 3heme iron one calcium cation one zinc cation
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Atom types in force fields
Examples of atom types defined in CHARMM:
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Atom types in force fields
Examples of atom typesas used in polypeptides inCHARMM:
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Force fields: Topology file
RESI ALA 0.00GROUPATOM N NH1 -0.47 ! |ATOM HN H 0.31 ! HN-NATOM CA CT1 0.07 ! | HB1
ATOM HA HB 0.09 ! | /
GROUP ! HA-CA--CB-HB2ATOM CB CT3 -0.27 ! | \
ATOM HB1 HA 0.09 ! | HB3ATOM HB2 HA 0.09 ! O=CATOM HB3 HA 0.09 ! |GROUP !ATOM C C 0.51ATOM O O -0.51BOND CB CA N HN N CABOND C CA C +N CA HA CB HB1 CB HB2 CB HB3DOUBLE O CIMPR N -C CA HN C CA +N OCMAP -C N CA C N CA C +N
DONOR HN NACCEPTOR O C
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Force fields: Parameter file
BONDS!!V(bond) = Kb(b - b0)**2!Kb: kcal/mole/A**2!b0: A!
!atom type Kb b0...CT3 CT1 222.500 1.5380...
ANGLES!
!V(angle) = Ktheta(Theta - Theta0)**2!Ktheta: kcal/mole/rad**2!Theta0: degrees!!V(Urey-Bradley) = Kub(S - S0)**2!Kub: kcal/mole/A**2 (Urey-Bradley)!S0: A!!atom types Ktheta Theta0 Kub S0
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Force fields: Parameter file
DIHEDRALS!!V(dihedral) = Kchi(1 + cos(n(chi) - delta))!Kchi: kcal/mole!n: multiplicity!delta: degrees
!!atom types Kchi n delta...C CT1 NH1 C 0.2000 1 180.00NH1 C CT1 NH1 0.6000 1 0.00...
0
180
ene
rgy[kcal/mol]
0.4
0.8
1.2
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Force fields: Parameter file
IMPROPER!!V(improper) = Kpsi(psi - psi0)**2!Kpsi: kcal/mole/rad**2!psi0: degrees!
!atom types Kpsi psi0...NH1 X X H 20.0000 0.00...
The X is a wildcard, i.e., N H 1 X X H includes the improper dihedral group, NH1 C CT1 H.
CMAP! 2D grid correction data. The following surfaces ! to the CHARMM22 phi, psi alanine, proline and gly! surfaces....
! alanine mapC NH1 CT1 C NH1 CT1 C NH1 24
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Force fields: Parameter file
NONBONDED!!V(Lennard-Jones) = Eps,i,j[(Rmin,i,j/ri,j)**12 - 2!epsilon: kcal/mole, Eps,i,j = sqrt(eps,i * eps,j)!Rmin/2: A, Rmin,i,j = Rmin/2,i + Rmin/2,j!
!atom epsilon Rmin/2...C -0.110000 2.000000CA -0.070000 1.992400...
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Parameterization
Parameterization process for potential energy funcdifficult task.
The combinations of parameters that can be usedUnrealistic choices for one group of parameters cacompensated for by adjustment of another in ordeset of structural and energetic data.
The energy terms should have clear physical signparameters calibrated by empirical fitting of crystabarriers of analogous small molecules, vibrational and quantum-chemical data.
Problems arise from
approximations made in the extension of datalarge systems
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Parameterization
In summary, much freedom and manipulation are parameterizing a given energy function.
Only if constructed and parameterized correctly wmodel allow reliable structural predictions.
Energy parameters are not transferable from one force fi
Need of improvement: Determination of partial charges.
Improvement of electrostatic potential, e.g., usmultipoles instead of atomic point charges onlyforce fields).
Solvent representation and interpretation of re
absence of solvent.
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Overview of all-atom force fields
Program Force fields Molecules Comments
AMBER ff94, ff96, ff98 proteins all-atom ff, ff96 cellation
ff99 proteins, RNA,
DNA
good nonbonded
bilization of helicff99SB proteins, RNA,DNA
fixed the overstaff99
ff03 proteins, RNA,DNA
reparameterizedprovision for po
ased towards hebsc0 proteins, RNA,
DNAreparameterizatand torsional forbackbone
GAFF any molecule general Amber ffor the ligands in
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Overview of all-atom force fields
CHARMM22 proteins all-atom quantum-rameterizused with
CHARMM22/CMAP proteins the CHA
CMAP baCHARMM27 proteins, lipids,RNA, DNA,sugars
based on
CHARMM force fields implemented in CHARMM, N
GROMOS GROMOS96,GROMOS43a2,GROMOS53a6
alkanes, pro-teins, sugars
united-atoff availabmodel
GROMOS force fields implemented in GROMOS anOPLS OPLS-UA proteins united-ato
models
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Coarse-grained force fields
Several nuclei (and electrons) are lumped into pseso-called beads.
For proteins one typically has 2 beads per amino abackbone and, depending on the amino acid, betwbeads for the sidechain.
Bonded and nonbonded interactions between beathose between atoms in all-atom models.
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Knowledge-based force fields
An alternative to physics-based force fields are knor statistical, energy functions, which derive from tknown protein structures.
The probabilities are calculated that residues appeconfigurations (e.g., rotamer conformations or bursurface environments), and that pairs of residues in a particular relative geometry.
These probabilities are converted into an effectiveenergy using the Boltzmann equation
G = kBT lnp(c1, c2)
p(c1)p(c2),
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Knowledge-based force fields
Advantage: any behaviour seen in known protein sbe modeled, even if a good physical understandinbehaviour does not exist.
Disadvantage: these energy functions are phenomcannot predict new behaviour absent from the trai
Examples: Go models, Associated Memory HamilWolynes), Rosetta (David Baker)
References:S. Wodak and M. Rooman, Generating and testing protein folds, Cur3, 249-259 (1993)M.J. Sippl, Knowledge-based potentials for proteins, Curr. Opin. Struc(1995)
R.L. Jernigan and I. Bahar, Structure-derived potentials and protein si
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Solvation
ExplicitExplicit vs.vs. Implicit SolventImplicit Solvent
li i l
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Every water molecule is modeled explicitly.
Commonwater models are:
TIP3P TIP5P
Explicit solvent interactions are replaced by anenergy term based on solvent's mean fieldbehavior.
Most implicit solvent models start from a continuumelectrostatic description for the solvent.
solvent-inaccessible lowdielectric cavity containingthe solute
high dielectric mediumfor the solvent
Explicit solvent : Implicit solvent :
ExplicitExplicit vs.vs. Implicit SolventImplicit Solvent
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Approximation !
explicit solvent continuum electrostatics
lost information :
Efficient ?
Enormous decrease in degrees of freedom.
But not every implicit-solvent implementation isefficient !
Implicit solvent 1 : EEF1Implicit solvent 1 : EEF1
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EEF1 = Effective Energy Function W(R) for proteins with coordinates R in solution.
Assumption 1 : sum over groups i
Assumption 2 :
Assumption 3 :
Implicit solvent 2 : Generalized Born modelsImplicit solvent 2 : Generalized Born models
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The reaction field potential can be computed by solving the Poisson-Boltzmann equation:
dielectric constantof the solvent
Debye-Hckel
screening factor
solute charge density
Generalized Born modelsGeneralized Born models
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Solving the PB equation for a spherical solute with radius R, charge q, and dielectric constantgives the Born formula(Born, Z. Phys. 1 (1920), 45) :
Inspired by the Born formula, the generalized Born (GB) theory was developped. The mostreliable GB formula is Still's equation (Still et al., JACS 112 (1990), 6127) :
Needs the Born radii as input, which are conventionally computed within Coulomb-fieldapproximation (CFA) :
Born radii
Born radiiBorn radii
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Calculation of Born radiiCalculation of Born radii
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pairwise summation :
GenBornDominy and Brooks, JPCB 103 (1999), 3765
ACE
Schaefer and Karplus, JPC 100 (1996), 1578Schaefer et al., JCC 22 (2001), 1857
GBr6
Tjong and Zhou, JPCB 111 (2007), 3055
numerical volume integration :
GBMVLee et al., JCP 116 (2002), 10606
GBSW
Im et al., JCC 24 (2003), 1691
Beyond Coulomb-field approximation ?
in GBSW :
in Gbr6 :
Coulomb-fieldapproximation :