Kirkwood-Buff Theory of Solutions and theDevelopment of Atomistic and Coarse-Grain Force

Fields

Nikos Bentenitis

Department of Chemistry and BiochemistrySouthwestern UniversityGeorgetown, Texas, USA

bentenin@southwestern.edu

July 15, 2011

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

References

Molecular Theory of Solutions by

Arieh Ben-Naim

Review article in Modeling Solvent

Environments ed. Michael Feig

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

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

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

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.

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!

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

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

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

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

Ideal solutions may result from different radial distributionfunctions

solvation shells at same distances butof different magnitude

solvation shells of same magnitude butat different distances

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

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

Paul Smith at Kansas State University was the first todevelop a force field based on the KB integrals

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

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

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)

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

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)

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

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

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

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

[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

One mapping scheme for bmim+ has three beads of twotypes

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

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

Potentials developed from a combination of iterativeBoltzmann inversion and potential of mean forcecalculations

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.

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

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

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

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

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

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