Partial molar volume of proteins

The partial molar volume, which is a very important quantity to probe the response of the free energy (or stability) of protein to pressure, including the so-called pressure denaturation, is not a canonical thermodynamic quantity for the (V, T) ensemble, since volume is an independent thermodynamic variable of the ensemble. The partial molar volume of protein at infinite dilution can be calculated from the Kirkwood-Buff equation [20] generalized to the site-site representation of liquid and solutions [21,22],... 

The example is the partial molar volume of protein, which can be calculated using (10.19) from h(r), or equivalently from c(r) obtained from the 3D-RISM equation. The partial molar volume of several proteins in water which appear frequently in the literature of protein research is plotted against the molecular weight in Fig. 10.1. [23] By comparing the results with the experimental ones plotted in the same figure, one can readily see that the theory is capable of reproducing the experimental results in quantitative level. At a glance, the results seem to be reproduced by just simple consideration... 

Fig. 10.1. Partial molar volume of proteins plotted against the molecular weight. The theoretical results (black circles) show quantitative agreement with the experimental ones (crosses)...

The present paper is devoted to the derivation of a relation between the preferential solvation of a protein in a binary aqueous solution and its solubility. The preferential binding parameter, which is a measure of the preferential solvation (or preferential hydration) is expressed in terms of the derivative of the protein activity coefficient with respect to the water mole fraction, the partial molar volume of protein at infinite dilution and some characteristics of the protein-free mixed solvent. This expression is used as the starting point in the derivation of a relationship between the preferential binding parameter and the solubility of a protein in a binary aqueous solution. 

In this expression Vatom and Vcavities are the volumes of the atoms and the cavities respectively and AVhydrat.on is the volume change of the solution resulting from the interactions of the protein molecule with the solvent. More defined models for the partial molar volumes of proteins are discussed by Chalikian [27,28]. Care should be taken if quantities derived Ifom the volume (such as compressibility and thermal expansion) are interpreted on the molecular level. The experimental results may depend on the sensitivity range of the method used. Global measurements such as ultrasonics detect the whole molar volume, while some local probes may feel only the change of the protein interior volume. 

T. Imai, A. Kovalenko and F. Hirata. Partial molar volume of proteins studied by the three-dimensional reference interaction site model theory. J. Phys. Ghem. B 109, 2005, 6658-6665. 

To evaluate molar masses from equilibrium runs it is necessary to know (very accurately) the value of the partial specific volume of the species. The partial spedfic volumes of proteins and nucleic acids are not dependent on conformation so they can be evaluated from the amino acid or nucleotide composition (Laue et al. 1992). A satisfactory working value for proteins is v = 0.735 10-3 m3 kg-1. Partial specific volumes can also be measured by comparison of equilibrium runs carried out in H20, D20 and H280, using the different densities of the three isotopic forms of water. 

One can also write the following expression for the partial molar volume of a protein at infinite dilution in a mixed solvent (V ) in terms of the Kirkwood-Buff theory of solution ... 

It shonld be noted that only the derivative Ji and the partial molar volume of the protein depend on the protein characteristics all the other quantities in Eq. (9) can be determined from the characteristics of the protein-free mixed solvent. Equation (9) shows that the preferential binding parameter F23 can be decomposed into the sum of two terms. One of them, depends on the protein nature, reflected in 72i and V, and the other one depends only on the properties of the protein-free mixed solvent. 

In Eq. (8), y2 is the protein solubility in mole finction, J 1 = limxj o ( ), Xi is the mole fraction of component i, 7, is the activity coefficient of component i in a mole Iraction scale, Vi is the partial molar volume of component i. 

Equations 3 and 5 allow one to calculate the Kirkwood-Buff integrals G12 and G23 using experimental data regarding the preferential binding parameters r2 and the partial molar volume of a protein at infinite dilution in a mixed solvent 49-50,52 jjjg Kirkwood—Buff integrals Gn and Go can be evaluated on the basis of the properties of protein-free mixed solvent water + cosolvent. It should be mentioned that recently the Kirkwood—Buff theory was used to analyze the effects of various cosolvents on the properties of aqueous protein solutions. " ... 

An analysis of the cosolvent concentration dependence of the osmotic second virial coefficient (OSVC) in water—protein—cosolvent mixtures is developed. The Kirkwood—Buff fluctuation theory for ternary mixtures is used as the main theoretical tool. On its basis, the OSVC is expressed in terms of the thermodynamic properties of infinitely dilute (with respect to the protein) water—protein—cosolvent mixtures. These properties can be divided into two groups (1) those of infinitely dilute protein solutions (such as the partial molar volume of a protein at infinite dilution and the derivatives of the protein activity coefficient with respect to the protein and water molar fractions) and (2) those of the protein-free water—cosolvent mixture (such as its concentrations, the isothermal compressibility, the partial molar volumes, and the derivative of the water activity coefficient with respect to the water molar fraction). Expressions are derived for the OSVC of ideal mixtures and for a mixture in which only the binary mixed solvent is ideal. The latter expression contains three contributions (1) one due to the protein—solvent interactions which is connected to the preferential binding parameter, (2) another one due to protein/protein interactions (B p ), and (3) a third one representing an ideal mixture contribution The cosolvent composition dependencies of these three contributions... 

Using eq 8, one can calculate B2 from the properties of protein-free mixed solvents, such as the concentrations c, (i = 1,3), the partial molar volumes of the components y (i = 1, 3), the isothermal compressibility kj, and 7n as well as those of infinitely dilute proteiu mixtures such as the partial molar volume of a protein at infinite dilution Vf and the derivatives 721 nd 722. 

The difference between the ternary ideal mixture approximation (eq 9) and eq llA consists in the molar volume of the protein the partial molar volume of the protein at infinite dilution (Vf) in eq 11A and the hypothetical molar volume of a pure protein (V5) in eq 9. 

Various Systems. 3.3.1. Water (l)-Malate Dehydrogenase (Hm MalDH) (2)—NaCl (3). For water (1)—Hm MalDH (2)—NaCl (3), experimental data for both F and OSVC are available. 3,44 -pbe partial molar volumes of the components of the protein-free mixed solvent (Vi and V3) were calculated from the densities of the water—NaCl... 

Malate dehydrogenase. " 2 was taken equal to the partial molar volume of the protein in a 1 M NaCl solution. 

The observed partial molar volumes of macromolecules (see Table 4.1 for examples) will depend on a number of factors. For globular proteins, for example, it will depend on how well the native protein is folded and how many interior void spaces there might be within the structure, as well as the hydration effects arising from interactions with solvent at the surface. 

The partial molar volume is a thermodynamic quantity that plays an essential role in the analysis of pressure effects on chemical reactions, reaction rate as well as chemical equilibrium in solution. In the field of biophysics, the pressure-induced denaturation of protein molecules has continuously been investigated since an egg white gel was observed under the pressure of 7000 atmospheres [60]. The partial molar volume is a key quantity in analyzing such pressure effects on protein conformations When the pressure in increased, a change of the protein conformation is promoted in the direction that the partial molar volume reduces. A considerable amount of experimental work has been devoted to measuring the partial molar volume of a variety of solutes in many different solvents. However, analysis and interpretation of the experimental data are in many cases based on drastically simplified models of solution or on speculations without physical ground, even for the simplest solutes such as alkali-halide ions in aqueous solution. Matters become more serious when protein molecules featuring complicated conformations are considered. 

Compared to the effort devoted to experimental work, theoretical studies of the partial molar volume have been very limited [61, 62]. The computer simulations for the partial molar volume were started a few years ago by several researchers, but attempts are still limited. As usual, our goal is to develop a statistical-mechanical theory for calculating the partial molar volume of peptides and proteins. The Kirkwood-Buff (K-B) theory [63] provides a general framework for evaluating thermodynamic quantities of a liquid mixture, including the partial molar volume, in term of the density pair correlation functions, or equivalently, the direct correlation functions. The RISM theory is the most reliable tool for calculating these correlation functions when the solute molecule comprises many atoms and has a complicated conformation. 

In this section, we combine the K-B theory with the RISM theory and derive an equation that allows us to calculate the partial molar volume of a polyatomic solute immersed in molecular solvent. The equation is then applied to calculations for the series of 20 amino acids occurring naturally in living systems. These amino acids are chosen because they are of great biological interests as ingredients of proteins and they have... 

By combining the K-B theory with the RISM theory, we have derived the equation for calculating the partial molar volume of a polyatomic solute in solvent. We have calculated the VM-values of the 20 amino acids, constituents of natural proteins. The calculated values are always smaller than the corresponding experimental values. Moreover, the discrepancy becomes larger as the number of the atoms in the amino-... 

FST provides a clear theoretical framework for the solvation of biomolecules in pure water. In the absence of osmolytes, the change in partial molar volume of a protein, provided by Equation 11.4, reduces to (Ben-Naim 1992 Shimizu 2004 Shulgin and Ruckenstein 2005a Pierce et al. 2008)... 

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