1. Kirkwood-Buff Theory of Solutions and the
Development of Atomistic and Coarse-Grain Force
Fields
Nikos Bentenitis
Department of Chemistry and Biochemistry
Southwestern University
Georgetown, Texas, USA
bentenin@southwestern.edu
July 15, 2011

2. 1 Introduction
Challenges with common force fields for biomolecular
simulations
The Kirkwood-Buff theory
The application of the Kirkwood-Buff theory
2 Kirkwood-Buff derived all-atom force fields
The Kirkwood-Buff approach to developing force fields
Molecular dynamics engine
Details of molecular dynamics simulations
Kirkwood-Buff force fields developed to-date
Kirkwood-Buff force field for thiols, sulfides and disulfides
3 A coarse-grain force field for an ionic liquid in water
The structure of ionic liquids in water
Methodology for developing a coarse-grain force field for an
ionic liquid
The current state of development of the coarse-grain force
field

3. References
Molecular Theory of Solutions by Review article in Modeling Solvent
Arieh Ben-Naim Environments ed. Michael Feig

4. Force fields determine the quality of computer simulations
The structure of solutions explains solvation
Computer simulations can predict the structure of solutions
The quality of computer simulations depends on the quality of
the force fields which include several simplifications:
Transferable and additive intermolecular potentials
Effective charges (polarization is time-consuming)
Simplified water models:
AMBER TIP3P
CHARMM Modified TIP3P
GROMOS SPC
OPLS TIP3P, TIP4P

5. Effective charges of common force fields do not come from
experimental data of solutions at finite concentrations
van der Waals interactions
AMBER Density, ∆Hvap of pure liquids
CHARMM Ab initio interactions on rigid molecules
GROMOS Atomic polarizabilities
OPLS Thermodynamic properties and structure of pure liquids
Effective charges
AMBER Fit to gas-phase ab initio electrostatic potential surface
CHARMM Scaled gas-phase ab initio charges
GROMOS Pure liquids and ∆Hsolv
OPLS Thermodynamic properties and structure of pure liquids

6. Common force fields predict excessive aggregation of RNA
and NMA aqueous solutions
RNA in KCl solution simulated with Ammonium sulfate in water simulated
AMBER with GROMOS 45a3

7. The Kirkwood-Buff theory of solutions has attracted
considerable attention
Working definition of the Kirkwood-Buff theory
An exact theory that relates the structure of a solution to its
thermodynamic properties
Published in 1951 by
Kirkwood and Buff.
First applied to methanol
solutions in 1972 by
Ben-Naim.
Inverse theory was developed
in 1977 by Ben-Naim.

8. The Kirkwood-Buff integral is central to the theory
Definition of the KB integral
g (r )
R
G (R) = [g (r ) − 1]4πr 2 dr
0
G (r ) Note the r 2 !

9. The limiting value of the KB integral condenses
information on solution structure to the limit of large
distances
R
Gij (R) = lim [gij (r ) − 1] 4πr 2 dr
R→ ∞ 0
There are as many KB integrals as the species that are defined
Gij ’s are sensitive to solution structure
Gij ’s measure the affinity between species i and species j
The KB theory has several attributes
It can be applied to any stable mixture regardless of the number of
components
It applies to any molecule regardless of its size and complexity
It is easily calculated from computer simulations

10. The inverse KB theory connects thermodynamic properties
of solutions to the KB integrals
For a two component system the KB theory connects
three thermodynamic properties of the solution, and its
components
the isothermal compressibility of the solution, κT
the partial molar volumes of one component, either V 1 or V 2
the partial derivative of the chemical potential of one
component, either (∂µ1 /∂x1 )T ,P or (∂µ2 /∂x2 )T ,P
to three KB integrals
G11 , that measures the affinity among species 1
G22 , that measures the affinity among species 2
G12 = G21 , that measures the affinity among species 1 and
species 2

11. The inverse KB theory connects thermodynamic properties
of solutions to the KB integrals
For a two component system
1 ρ2 V 2 ρ
G11 = kB T κT − +
ρ1 ρ1 D
1 ρ1 V 1 ρ
G22 = kB T κT − +
ρ2 ρ2 D
V2
G12 = kB T κT − ρV 1
D
x1 ∂µ1
D=
kB T ∂x1 T ,P
ρ = ρ1 + ρ2

12. Ideal solutions may result from different radial distribution
functions
The chemical potential of an ideal solution in the mole-fraction
scale:
µi = µo (T , P) + kB T ln xi
i
The quantity:
x1 ∂µ1
D= = 1 ⇒ G11 + G22 − 2G12 = 0
kB T ∂x1 T ,P
Gij ’s do not need to be all zero
There are several ways by which the condition
G11 + G22 − 2G12 = 0 can be met

13. Ideal solutions may result from different radial distribution
functions
solvation shells at same distances but solvation shells of same magnitude but
of different magnitude at different distances

14. Kirkwood-Buff integrals depend on how well fitting
equations describe experimental activity coeffients.
Example: Ethanol in Water
Ben-Naim, A., J. Chem. Phys., 1977 Ben-Naim, A., Molecular Theory of Solutions, 2006

15. Excess coordination numbers are more convenient than KB
integrals for comparing theory with simulation
Working definition
Excess coordination numbers, Nij = ρi Gij , measure the excess (or
deficit) of species around a particle in a solution compared to that
in a random solution.
Excess coordination numbers are
less noisy at concentrations where the KB integrals are noisy
more intuitive to interpret

16. Paul Smith at Kansas State University was the first to
develop a force field based on the KB integrals

17. KB-derived force fields are based on a few principles
Principles
The force fields should be simple enough to allow large
long-time simulations of biomolecules
The number atom types should be kept to a minimum
Sources of data
Bond and angle parameters from the GROMOS force field
Lennard-Jones parameters of non-polar groups from the
GROMOS force field
Dihedral potentials from quantum mechanical calculations
Water model: SPC/E
Lennard-Jones parameters for polar groups are found by
reproducing the
density of the pure liquid for liquids solutes,
density of the pure crystal for solid solutes

18. Effective charge distributions are found by reproducing the
experimental excess coordination numbers

g11 sim sim
−→ G11 −→ N11

simulation −→ g22 sim sim
−→ G22 −→ N22


g12 sim sim
−→ G12 −→ N12
 exp exp
G11 −→ N11
∂µ1 
exp exp
experiment −→ κT , V 1 , −→ G22 −→ N22
∂x1 T ,P  exp exp
G12 −→ N12

sim ?exp
Nij = Nij

19. Gromacs is an effective tool for molecular dynamics
simulations
Gromacs
is efficiently parallelized for multi-processor, multi-core
computers
uses checkpoint files for accurate restarting of simulations
has a series of useful utility programs for the calculation of
self-diffusion coefficients
dielectric constants
radial distribution functions
is continuously developed (future versions will run on
computers with Graphical Processing Units)

20. Simulations are performed under standardized conditions
The NpT ensemble at 1 atm and experimental temperature is
used
Simulation boxes range between 75 – 1000 nm3
Equilibration of 1–2 ns and production runs of up to 10–40 ns
The Berendsen barostat, and the velocity-rescale thermostat
control pressure and temperature
Bonds are constrained using LINCS
Electrostatic interactions are calculated using the
particle-mesh-Ewald summation
Electrostatic and van der Waals interactions are calculated
with cut-off distances of 1.2 nm and 1.5 nm

21. Several Kirkwood-Buff derived force fields have been
developed to-date
Species Reference
Acetone Weerasinghe & Smith, 2003
Urea Weerasinghe & Smith, 2003
Na+ , Cl− , Weerasinghe & Smith, 2003
GuCl Weerasinghe & Smith, 2004
Amides Kang & Smith, 2005
tert-Butanol Lee & van der Vegt, 2005
Methanol Weerasinghe & Smith, 2006
Thiols, sulfides, disulfides Bentenitis, Cox & Smith, 2009
Li+ , K+ Hess & van der Vegt, 2009
Li+ , K+ , Rb+ , Cs+ Klasczyk & Knecht, 2010
Alkali metal halides Gee et. al, 2011
Aromatic amino-acids Ploetz & Smith (in press)

22. KB-derived force field agrees quantitatively with
experimental data for dimethylsulfide/methanol
(MSM/MOH) solutions
20
— MSM/MSM
— MOH/MOH
15 — MSM/MOH
ooo KBFF
10
••• Lubna et al. FF
£¥
¤
5
• KBFF for MOH incompatible
with Lubna et al.’s
• Quantitative disagreement at
0
high MSM mole-fractions
because of uncertainties in
-5 estimating experimental and
simulation excess coordination
numbers
-10
0.0 0.2 0.4
¡
¢
¡ 0.6 0.8
Excess coordination numbers as a function of
1.0
dimethylsulfide mole-fraction

23. KB-derived force field agrees quantitatively with
experimental data for methanethiol/methanol
(MSH/MOH) solutions
4
— MSH/MSH
— MOH/MOH
3 — MSH/MOH
ooo KBFF
2
¤
1
¦
¥ • Only one adjustable
parameter: charge on Sulfur
0
-1
-2
0.0 0.2 0.4
¡
¢
£ 0.6 0.8
Excess coordination numbers as a function of
1.0
methanethiol mole-fraction

24. KB derived force field agrees quantitatively with
experimental data for dimethyl disulfide/methanol
(DDS/MOH) solutions
8
— DDS/DDS
— MOH/MOH
6 k
— DDS/MOH
ooo KBFF
4
£
2
0
¥
¤ • Same single adjustable
parameter: charge on Sulfur,
same as for MSH
• Single parameter reproduces
-2
experimental KB integrals over
the entire concentration range
-4
-6
0.0 0.2 0.4
¡
¡
¢ 0.6 0.8
Excess coordination numbers as a function of
1.0
dimethyl disulfide mole-fraction

25. Ionic liquids show promise as “green” solvents
Ionic liquids
consist of organic cation and inorganic or organic anion
are liquid at room temperature with negligible vapor pressure
are promising “green” solvents
small amounts of solvents may change properties drastically
BF−
4
1-Butyl-3-methylimidazolium cation, bmim+

26. [bmim][BF4 ] and water show a high degree of aggregation
• Aggregation has been verified
by both vapor-pressure
measurements and by SANS
• The physical reason for this
aggregation is uncertain and
simulations may provide
insights
Problem
All-atom simulations require
large boxes
Solution
Coarse-graining should help
KB integrals as a function of [bmim][BF4 ]
mole-fraction

27. One mapping scheme for bmim+ has three beads of two
types

28. The approach by Villa, Peter & van der Vegt (2010,
JCTC) for benzene in water is the basis for the method
ꝏ AA-PMF r
ഠ ഠ ഠ
ഠ
◌ஂ ◌ஂ ◌ஂ ◌ஂ ഠ
CG-PMF (excl) r
ꝏ
ഠ ഠ
ഠ ഠ ഠ ഠ ഠ

29. The approach by Villa, Peter & van der Vegt (2010,
JCTC) for benzene in water is the basis for the method
CG AA CG
Vpmf = Vpmf - Vpmf ,excl

30. Potentials developed from a combination of iterative
Boltzmann inversion and potential of mean force
calculations

31. Potentials developed from a combination of iterative
Boltzmann inversion and potential of mean force
calculations
1 Select the Lopes et al. all-atom force field
2 Simulate pure water to get the water-water potential by
iterative Boltzmann inversion
3 Simulate one [bmim][BF4 ] ion-pair in water to get
1 the 3 bonded potentials of bmim+ by Boltzmann inversion
2 the 4 bead/water potentials by iterative Boltzmann inversion
4 For the bead-water potentials use ethane, [mmim]+ , and
BF− , calculate the potential of mean force between pairs of
4
all bead combinations in water
1 first, using an AA force field and
2 then, using the CG potentials from step 3.2, excluding the
same-bead potentials.
3 Subtract the potential from step 4.2 from that of step 4.1.

32. The potential for [mmim]+ , and BF− is typical
4
CG AA CG
Vpmf = Vpmf - Vpmf ,excl
10
5
U / kJ mol-1
0
−5
−10
0.2 0.4 0.6 0.8 1
r / nm

33. KB integrals from all-atom and coarse grain force fields do
not agree
100
All-atom Water/Water
0
Coarse-grain Water/Water
Gij (cm3/mol)
All-atom Ion/Ion
−200
Coarse-grain Ion/Ion
−300
0 0.025 0.05 0.075 0.1 0.125
xs

34. KB integrals from all-atom force field and experiment do
not agree
3000 Experimental Water/Water
2000
Gij (cm3/mol)
Experimental Ion/Ion
Simulation Water/Water
0
Simulation Ion/Ion
0 0.025 0.05 xs 0.075 0.1 0.125

35. Future work will focus on improvement of all-atom and
coarse-grain force fields
1 Improvement of the all-atom force field
Existing methdology using viscosity as a target property in
Florian M¨ller-Plathe’s group
u
Use of Kirkwood-Buff integrals. There has been a flood of
data recently on activity coefficients of ionic liquids in water
2 Improvement of the coarse-grain force field
Alternative mapping schemes
Alternative water-water potentials
Development of bead-water potentials using iterative
Boltzmann inversion with the KB integrals as the target
property

36. The work would not have been accomplished without the
help of
1 People
Paul Smith (Kansas State University)
Nico van der Vegt, Florian M¨ller-Plathe (Technical University
u
of Darmstadt)
Meagan Mullins, Alex Zamora and Nick Cox (Southwestern
University)
Emiliano Brini, Hossein Ali Karimi Varzaneh (Technical
University of Darmstadt)
2 Funding agencies
National Institutes of Health
Welch Foundation
Fleming Foundation

37. 100
All-atom Water/Water
0
Coarse-grain Water/Water
Gij (cm3/mol)
All-atom Ion/Ion
−200
Coarse-grain Ion/Ion
−300
0 0.025 0.05 0.075 0.1 0.125
xs