Kirkwood Buff Theory

The Kirkwood-Buff theory ( KB theory ) is a theory of the solution of molecules . It combines macroscopic material properties with molecular properties .

principle

The Kirkwood-Buff theory uses statistical mechanics methods to calculate thermodynamic quantities from pair distribution functions between all molecules in a multi-component solution. KB theory can be used to validate molecular modeling and to explain the mechanism of various physical processes. Furthermore, the KB theory is applied to various biological issues.

A reversal of the KB theory was developed, the reverse KB theory, in which molecular properties can be determined from thermodynamic measurements.

Pair distribution function

→ Main article : Radial distribution function

The pair distribution function describes a localized order in a mixture. The pair distribution function of the components and in or is defined as follows: ${\displaystyle i}$ ${\displaystyle j}$ ${\displaystyle {\boldsymbol {r}}_{i}}$ ${\displaystyle {\boldsymbol {r}}_{i}}$

${\displaystyle g_{ij}({\boldsymbol {R}})={\frac {\rho _{ij}({\boldsymbol {R}})}{\rho _{ij}^{\text{bulk}}}}}$

with the local density of the components compared to the component , the density of the component and the radius vector between molecules . It follows: ${\displaystyle \rho _{ij}({\boldsymbol {R}})}$ ${\displaystyle j}$ ${\displaystyle i}$ ${\displaystyle \rho _{ij}^{\text{bulk}}}$ ${\displaystyle j}$ ${\displaystyle {\boldsymbol {R}}=|{\boldsymbol {r}}_{i}-{\boldsymbol {r}}_{j}|}$

${\displaystyle g_{ij}({\boldsymbol {R}})=g_{ji}({\boldsymbol {R}})}$

Assuming spherical symmetry, the function simplifies to:

${\displaystyle g_{ij}(r)={\frac {\rho _{ij}(r)}{\rho _{ij}^{\text{bulk}}}}}$

with the distance between two molecules. ${\displaystyle r=|{\boldsymbol {R}}|}$

The free energy is occasionally determined to further describe the interrelationships between the molecules. The average force potential between two components is related to the pair distribution function as follows:

${\displaystyle PMF_{ij}(r)=-kT\ln(g_{ij})}$

with PMF ( potential of mean force ) as a measure of the effective interrelationship between two components in a solution.

Kirkwood-Buff integrals

The Kirkwood-Buff integral (KBI) between the components and is defined as the space integral over the pair distribution function: ${\displaystyle i}$ ${\displaystyle j}$

${\displaystyle G_{ij}=\int \limits _{V}[g_{ij}({\boldsymbol {R}})-1]\,d{\boldsymbol {R}}}$

In the case of spherical symmetry, the equation simplifies to:

${\displaystyle G_{ij}=4\pi \int _{r=0}^{\infty }[g_{ij}(r)-1]r^{2}\,dr}$

Thermodynamic Relations

Two-component system

Various relationships ( , and ) can be derived for a system with two components. ${\displaystyle G_{11}}$ ${\displaystyle G_{22}}$ ${\displaystyle G_{12}}$

The partial molar volume of component 1 is:

${\displaystyle {\bar {\nu }}_{1}={\frac {1+c_{2}(G_{22}-G_{12})}{c_{1}+c_{2}+c_{1}c_{2}(G_{11}+G_{22}-2G_{12})}}}$

with the molar concentration and ${\displaystyle c}$ ${\displaystyle c_{1}{\bar {\nu }}_{1}+c_{2}{\bar {\nu }}_{2}=1}$

The compressibility : ${\displaystyle \kappa }$

${\displaystyle \kappa kT={\frac {1+c_{1}G_{11}+c_{2}G_{22}+c_{1}c_{2}(G_{11}G_{22}-G_{12})^{2}}{c_{1}+c_{2}+c_{1}c_{2}(G_{11}+G_{22}-2G_{12})}}}$

with as the Boltzmann constant and the temperature . ${\displaystyle k}$ ${\displaystyle T}$

The osmotic pressure in relation to the concentration of component 2: ${\displaystyle \Pi }$

${\displaystyle \left({\frac {\partial \Pi }{\partial c_{2}}}\right)_{T,\mu _{1}}={\frac {kT}{1+c_{2}G_{22}}}}$

with as the chemical potential of component 1. ${\displaystyle \mu _{1}}$

The chemical potentials in relation to concentrations, at constant temperature ( ) and constant pressure ( ) are: ${\displaystyle T}$ ${\displaystyle P}$

${\displaystyle {\frac {1}{kT}}\left({\frac {\partial \mu _{1}}{\partial c_{1}}}\right)_{T,P}={\frac {1}{c_{1}}}+{\frac {G_{12}-G_{11}}{1+c_{1}(G_{11}-G_{12})}}}$

${\displaystyle {\frac {1}{kT}}\left({\frac {\partial \mu _{2}}{\partial c_{2}}}\right)_{T,P}={\frac {1}{c_{2}}}+{\frac {G_{12}-G_{22}}{1+c_{2}(G_{22}-G_{12})}}}$

or alternatively, in relation to the amount of substance :

${\displaystyle {\frac {1}{kT}}\left({\frac {\partial \mu _{2}}{\partial \chi _{2}}}\right)_{T,P}={\frac {1}{\chi _{2}}}+{\frac {c_{1}(2G_{12}-G_{11}-G_{22})}{1+c_{1}\chi _{2}(G_{11}+G_{22}-2G_{12})}}}$

Coefficient of preferred interactions

The relative inclination of the solution or interaction of a substance in a solvent is the binding coefficient (engl. Preferential interaction coefficient ) described. In an aqueous solution of a solvent and two dissolved components ( Solut and Cosolut ) the effective preferential interaction coefficient of the water is related to the preferential hydration coefficient , which assumes a positive value in the case of solubility . According to the Kirkwood-Buff theory, the preferential hydration coefficient at low concentrations of the cosolute is: ${\displaystyle \Gamma }$ ${\displaystyle \Gamma _{W}}$

${\displaystyle \Gamma _{W}=M_{W}\left(G_{WS}-G_{CS}\right)}$

with the molarity of water and and as water, solute and cosolute. ${\displaystyle M_{W}}$ ${\displaystyle W,S}$ ${\displaystyle C}$

In the most general case, the preferred hydration is a function of the KBI of the solute with both the solvent and the cosolute. Under some assumptions and in various application examples, the equation simplifies to:

${\displaystyle \Gamma _{W}=-M_{W}G_{CS}}$

Then the only relevant function is . ${\displaystyle G_{CS}}$

history

The theory was developed in 1951 by John G. Kirkwood and Frank P. Buff . The reverse KB theory was described by Arieh Ben-Naim in 1977 .

literature

• A. Ben-Naim: Molecular Theory of Water and Aqueous Solutions, Part I: Understanding Water . World Scientific, 2009, ISBN 978-981-283-760-8 , p. 629.
• E. Ruckenstein, IL. Shulgin: Thermodynamics of Solutions: From Gases to Pharmaceutics to Proteins . Springer, 2009, ISBN 978-1-4419-0439-3 , p. 346.
• W. Linert: Highlights in Solute – Solvent Interactions . Springer, 2002, ISBN 978-3-7091-6151-7 , p. 222.

Individual evidence

1. ↑ ^{a b} J.G. Kirkwood, FP Buff: The Statistical Mechanical Theory of Solutions. I . In: J. Chem. Phys. . 19, 1951, pp. 774-777. bibcode : 1951JChPh..19..774K . doi : 10.1063 / 1.1748352 .
2. ↑ KE Newman: Kirkwood-Buff solution theory: derivation and applications . In: Chem. Soc. Rev. . 23, 1994, pp. 31-40. doi : 10.1039 / CS9942300031 .
3. ↑ D. Harries, J. Rösgen: A practical guide on how osmolytes modulate macromolecular properties. . In: Biophysical Tools for Biologists: Vol 1 in Vitro Techniques . Elsevier Academic Press Inc, 2008, pp. 679-735, doi : 10.1016 / S0091-679X (07) 84022-2 .
4. ↑ S. Weerasinghe, GM Baeee, M. Kang, N. Bentenitis, Smith, PE: Developing Force Fields from the Microscopic Structure of Solutions: The Kirkwood-Buff Approach. . In: Modeling Solvent Environments: Applications to Simulations of Biomolecules . Wiley-VCH, 2010, pp. 55-76, doi : 10.1002 / 9783527629251.ch3 .
5. ↑ Veronica Pierce, Myungshim Kang, Mahalaxmi Aburi, Samantha Weerasinghe, Paul E. Smith: Recent Applications of Kirkwood - Buff Theory to Biological Systems . In: Cell Biochem Biophys. . 50, 2008, pp. 1-22. doi : 10.1007 / s12013-007-9005-0 .
6. ↑ ^{a b} A Ben-Naim: Inversion of the Kirkwood-Buff theory of solutions: Application to the water-ethanol system . In: J. Chem. Phys. . 67, 1977, pp. 4884-4890. bibcode : 1977JChPh..67.4884B . doi : 10.1063 / 1.434669 .
7. ^ PE Smith: On the Kirkwood-Buff inversion procedure . In: J. Chem. Phys. . 129, 2008, p. 124509. bibcode : 2008JChPh.129l4509S . doi : 10.1063 / 1.2982171 .
8. ↑ VA Parsegian: Protein-water interactions. . In: Int. Rev. Cytol. . 215, 2002, pp. 1-31. doi : 10.1016 / S0074-7696 (02) 15003-0 .
9. ↑ L. Sapir, D. Harries: Is the depletion force entropic? Molecular crowding beyond steric interactions. . In: Curr. Opin. Coll. Int. Sci. . 20, 2015, pp. 3–10. doi : 10.1016 / j.cocis.2014.12.003 .
10. ↑ S. Shimizu, N. Matubayasi: Preferential solvation: Dividing Surface vs Excess Numbers. . In: J. Phys. Chem. B. . 118, 2014, pp. 3922-3930. doi : 10.1021 / jp410567c .