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Kirkwood-Buff Theory of Solutions and theDevelopment of Atomistic and Coarse-Grain Force
Fields
Nikos Bentenitis
Department of Chemistry and BiochemistrySouthwestern UniversityGeorgetown, Texas, USA
July 15, 2011
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1 IntroductionChallenges with common force fields for biomolecularsimulationsThe Kirkwood-Buff theoryThe application of the Kirkwood-Buff theory
2 Kirkwood-Buff derived all-atom force fieldsThe Kirkwood-Buff approach to developing force fieldsMolecular dynamics engineDetails of molecular dynamics simulationsKirkwood-Buff force fields developed to-dateKirkwood-Buff force field for thiols, sulfides and disulfides
3 A coarse-grain force field for an ionic liquid in waterThe structure of ionic liquids in waterMethodology for developing a coarse-grain force field for anionic liquidThe current state of development of the coarse-grain forcefield
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References
Molecular Theory of Solutions by
Arieh Ben-Naim
Review article in Modeling Solvent
Environments ed. Michael Feig
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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 ofthe force fields which include several simplifications:
Transferable and additive intermolecular potentialsEffective charges (polarization is time-consuming)Simplified water models:
AMBER TIP3PCHARMM Modified TIP3PGROMOS SPCOPLS TIP3P, TIP4P
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Effective charges of common force fields do not come fromexperimental data of solutions at finite concentrations
van der Waals interactions
AMBER Density, ∆Hvap of pure liquidsCHARMM Ab initio interactions on rigid moleculesGROMOS Atomic polarizabilitiesOPLS Thermodynamic properties and structure of pure liquids
Effective charges
AMBER Fit to gas-phase ab initio electrostatic potential surfaceCHARMM Scaled gas-phase ab initio chargesGROMOS Pure liquids and ∆Hsolv
OPLS Thermodynamic properties and structure of pure liquids
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Common force fields predict excessive aggregation of RNAand NMA aqueous solutions
RNA in KCl solution simulated with
AMBER
Ammonium sulfate in water simulated
with GROMOS 45a3
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The Kirkwood-Buff theory of solutions has attractedconsiderable attention
Working definition of the Kirkwood-Buff theory
An exact theory that relates the structure of a solution to itsthermodynamic properties
Published in 1951 byKirkwood and Buff.
First applied to methanolsolutions in 1972 byBen-Naim.
Inverse theory was developedin 1977 by Ben-Naim.
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The Kirkwood-Buff integral is central to the theory
g(r)
G (r)
Definition of the KB integral
G (R) =
∫ R
0[g(r)− 1]4πr2dr
Note the r2!
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The limiting value of the KB integral condensesinformation on solution structure to the limit of largedistances
Gij(R) = limR→ ∞
∫ R
0[gij(r)− 1] 4πr2dr
There are as many KB integrals as the species that are definedGij ’s are sensitive to solution structureGij ’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 ofcomponentsIt applies to any molecule regardless of its size and complexityIt is easily calculated from computer simulations
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The inverse KB theory connects thermodynamic propertiesof solutions to the KB integrals
For a two component system the KB theory connects
three thermodynamic properties of the solution, and itscomponents
the isothermal compressibility of the solution, κTthe partial molar volumes of one component, either V 1 or V 2
the partial derivative of the chemical potential of onecomponent, either (∂µ1/∂x1)T ,P or (∂µ2/∂x2)T ,P
to three KB integrals
G11, that measures the affinity among species 1G22, that measures the affinity among species 2G12 = G21, that measures the affinity among species 1 andspecies 2
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The inverse KB theory connects thermodynamic propertiesof solutions to the KB integrals
For a two component system
G11 = kBTκT −1
ρ1+ρ2V 2ρ
ρ1D
G22 = kBTκT −1
ρ2+ρ1V 1ρ
ρ2D
G12 = kBTκT − ρV 1V 2
D
D =x1
kBT
(∂µ1∂x1
)T ,P
ρ = ρ1 + ρ2
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Ideal solutions may result from different radial distributionfunctions
The chemical potential of an ideal solution in the mole-fractionscale:
µi = µoi (T ,P) + kBT ln xi
The quantity:
D =x1
kBT
(∂µ1∂x1
)T ,P
= 1⇒ G11 + G22 − 2G12 = 0
Gij ’s do not need to be all zero
There are several ways by which the conditionG11 + G22 − 2G12 = 0 can be met
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Ideal solutions may result from different radial distributionfunctions
solvation shells at same distances butof different magnitude
solvation shells of same magnitude butat different distances
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Kirkwood-Buff integrals depend on how well fittingequations describe experimental activity coeffients.Example: Ethanol in Water
Ben-Naim, A., J. Chem. Phys., 1977 Ben-Naim, A., Molecular Theory of Solutions, 2006
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Excess coordination numbers are more convenient than KBintegrals for comparing theory with simulation
Working definition
Excess coordination numbers, Nij = ρiGij , measure the excess (ordeficit) of species around a particle in a solution compared to thatin a random solution.
Excess coordination numbers are
less noisy at concentrations where the KB integrals are noisy
more intuitive to interpret
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Paul Smith at Kansas State University was the first todevelop a force field based on the KB integrals
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KB-derived force fields are based on a few principles
Principles
The force fields should be simple enough to allow largelong-time simulations of biomoleculesThe number atom types should be kept to a minimum
Sources of data
Bond and angle parameters from the GROMOS force fieldLennard-Jones parameters of non-polar groups from theGROMOS force fieldDihedral potentials from quantum mechanical calculations
Water model: SPC/E
Lennard-Jones parameters for polar groups are found byreproducing the
density of the pure liquid for liquids solutes,density of the pure crystal for solid solutes
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Effective charge distributions are found by reproducing theexperimental excess coordination numbers
simulation −→
g11 −→ G sim
11 −→ Nsim11
g22 −→ G sim22 −→ Nsim
22
g12 −→ G sim12 −→ Nsim
12
experiment −→ κT ,V 1,
(∂µ1∂x1
)T ,P
−→
G exp11 −→ Nexp
11
G exp22 −→ Nexp
22
G exp12 −→ Nexp
12
Nsimij
?=Nexp
ij
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Gromacs is an effective tool for molecular dynamicssimulations
Gromacs
is efficiently parallelized for multi-processor, multi-corecomputers
uses checkpoint files for accurate restarting of simulations
has a series of useful utility programs for the calculation of
self-diffusion coefficientsdielectric constantsradial distribution functions
is continuously developed (future versions will run oncomputers with Graphical Processing Units)
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Simulations are performed under standardized conditions
The NpT ensemble at 1 atm and experimental temperature isused
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 thermostatcontrol pressure and temperature
Bonds are constrained using LINCS
Electrostatic interactions are calculated using theparticle-mesh-Ewald summation
Electrostatic and van der Waals interactions are calculatedwith cut-off distances of 1.2 nm and 1.5 nm
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Several Kirkwood-Buff derived force fields have beendeveloped to-date
Species ReferenceAcetone Weerasinghe & Smith, 2003Urea Weerasinghe & Smith, 2003Na+, Cl−, Weerasinghe & Smith, 2003GuCl Weerasinghe & Smith, 2004Amides Kang & Smith, 2005tert-Butanol Lee & van der Vegt, 2005Methanol Weerasinghe & Smith, 2006Thiols, sulfides, disulfides Bentenitis, Cox & Smith, 2009Li+, K+ Hess & van der Vegt, 2009Li+, K+, Rb+, Cs+ Klasczyk & Knecht, 2010Alkali metal halides Gee et. al, 2011Aromatic amino-acids Ploetz & Smith (in press)
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KB-derived force field agrees quantitatively withexperimental data for dimethylsulfide/methanol(MSM/MOH) solutions
0.0 0.2 0.4 0.6 0.8 1.0����-10
-5
0
5
10
15
20
���
Excess coordination numbers as a function ofdimethylsulfide mole-fraction
— MSM/MSM— MOH/MOH— MSM/MOHo o o KBFF• • • Lubna et al. FF
• KBFF for MOH incompatiblewith Lubna et al.’s• Quantitative disagreement athigh MSM mole-fractionsbecause of uncertainties inestimating experimental andsimulation excess coordinationnumbers
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KB-derived force field agrees quantitatively withexperimental data for methanethiol/methanol(MSH/MOH) solutions
0.0 0.2 0.4 0.6 0.8 1.0����-2
-1
0
1
2
3
4
���
Excess coordination numbers as a function ofmethanethiol mole-fraction
— MSH/MSH— MOH/MOH— MSH/MOHo o o KBFF
• Only one adjustableparameter: charge on Sulfur
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KB derived force field agrees quantitatively withexperimental data for dimethyl disulfide/methanol(DDS/MOH) solutions
0.0 0.2 0.4 0.6 0.8 1.0����-6
-4
-2
0
2
4
6
8
���
Excess coordination numbers as a function ofdimethyl disulfide mole-fraction
k
— DDS/DDS— MOH/MOH— DDS/MOHo o o KBFF
• Same single adjustableparameter: charge on Sulfur,same as for MSH• Single parameter reproducesexperimental KB integrals overthe entire concentration range
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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
1-Butyl-3-methylimidazolium cation, bmim+
BF−4
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[bmim][BF4] and water show a high degree of aggregation
KB integrals as a function of [bmim][BF4]mole-fraction
• Aggregation has been verifiedby both vapor-pressuremeasurements and by SANS• The physical reason for thisaggregation is uncertain andsimulations may provideinsights
ProblemAll-atom simulations requirelarge boxes
SolutionCoarse-graining should help
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One mapping scheme for bmim+ has three beads of twotypes
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The approach by Villa, Peter & van der Vegt (2010,JCTC) for benzene in water is the basis for the method
ഠꝏ
ꝏ
rCG-PMF (excl)
ഠ
ഠ
ഠ ഠ
ഠ
ഠ ഠ ഠ ഠ
ഠ
ഠ◌ ◌◌◌
AA-PMF r
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The approach by Villa, Peter & van der Vegt (2010,JCTC) for benzene in water is the basis for the method
V CGpmf = V AA
pmf - V CGpmf ,excl
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Potentials developed from a combination of iterativeBoltzmann inversion and potential of mean forcecalculations
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Potentials developed from a combination of iterativeBoltzmann inversion and potential of mean forcecalculations
1 Select the Lopes et al. all-atom force field
2 Simulate pure water to get the water-water potential byiterative Boltzmann inversion
3 Simulate one [bmim][BF4] ion-pair in water to get1 the 3 bonded potentials of bmim+ by Boltzmann inversion2 the 4 bead/water potentials by iterative Boltzmann inversion
4 For the bead-water potentials use ethane, [mmim]+, andBF−4 , calculate the potential of mean force between pairs ofall bead combinations in water
1 first, using an AA force field and2 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.
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The potential for [mmim]+, and BF−4 is typical
V CGpmf = V AA
pmf - V CGpmf ,excl
U /
kJ
mol
-1
−10
−5
0
5
10
r / nm0.2 0.4 0.6 0.8 1
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KB integrals from all-atom and coarse grain force fields donot agree
Coarse-grain Water/Water
Coarse-grain Ion/Ion
All-atom Water/Water
All-atom Ion/Ion
Gij
(cm
3 /mol
)
−300
−200
0
100
xs
0 0.025 0.05 0.075 0.1 0.125
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KB integrals from all-atom force field and experiment donot agree
Simulation Water/Water
Simulation Ion/Ion
Experimental Water/Water
Experimental Ion/Ion
Gij
(cm
3 /mol
)
0
2000
3000
xs0 0.025 0.05 0.075 0.1 0.125
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Future work will focus on improvement of all-atom andcoarse-grain force fields
1 Improvement of the all-atom force fieldExisting methdology using viscosity as a target property inFlorian Muller-Plathe’s groupUse of Kirkwood-Buff integrals. There has been a flood ofdata recently on activity coefficients of ionic liquids in water
2 Improvement of the coarse-grain force fieldAlternative mapping schemesAlternative water-water potentialsDevelopment of bead-water potentials using iterativeBoltzmann inversion with the KB integrals as the targetproperty
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The work would not have been accomplished without thehelp of
1 People
Paul Smith (Kansas State University)Nico van der Vegt, Florian Muller-Plathe (Technical Universityof Darmstadt)Meagan Mullins, Alex Zamora and Nick Cox (SouthwesternUniversity)Emiliano Brini, Hossein Ali Karimi Varzaneh (TechnicalUniversity of Darmstadt)
2 Funding agencies
National Institutes of HealthWelch FoundationFleming Foundation
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Coarse-grain Water/Water
Coarse-grain Ion/Ion
All-atom Water/Water
All-atom Ion/Ion
Gij
(cm
3 /mol
)
−300
−200
0
100
xs
0 0.025 0.05 0.075 0.1 0.125