An important characteristic of a solution of a protein (component 2) in a mixture water (1) + cosolvent (3) is the preferential binding parameter (1–6)

(1)

(2)

The preferential binding parameter was determined experimentally (5–7) and provides information regarding the interactions between a protein and the components of the mixed solvent. As a rule (1–5), the protein is preferentially hydrated, for cosolvents such as glycerol, sucrose, etc., which can stabilize at high concentrations the protein structure and preserve its enzymatic activity (3–5), and the protein is preferentially solvated by cosolvents (such as urea), which can cause protein denaturation.

In literature (8) a number of different definitions of the preferential binding parameter (coefficient) have been employed. They can be connected by thermodynamic relations for ternary mixtures (8). In this Comment the preferential binding parameter will be mostly defined by Eqs. 1 and 2.

Because the preferential binding parameter is a meaningful physical quantity, attempts have been made to relate it to a general theory of solutions, such as the Kirkwood-Buff theory of solutions (9). Several authors reported results in this direction (10–17). The authors of this Comment derived the following equation for (16):

(3)

(4)

Equation 3 differs from the expression of employed in Shimizu (10,11):

(5)

In a recent article in this journal (17), the Kirkwood-Buff theory of solutions was used to express the preferential binding coefficient defined as

(6)

(7)

As noted in Schurr et al. (17) the preferential binding coefficient defined by Eq. 6 differs from the preferential binding parameter defined by Eq. 2.

However, Eqs. 3 and 5 are different equations even though they are based on the same definition of the preferential binding parameter and have the same theoretical basis: the Kirkwood-Buff theory of solutions. To make a selection between Eqs. 3 and 5 a simple limiting case, the ideal ternary mixture, will be examined using the traditional thermodynamics, and the results will be compared to those provided by Eqs. 3 and 5.

Let us consider an ideal ternary mixture. According to the definition of an ideal mixture (18), the activities of the components (a_i) are equal to their mol fractions () and their partial molar volumes are equal to those of the pure components ().

Because

(8)

(9)

For isothermal-isobaric conditions

(10)

(11)

When is a constant, Eqs. 10 and 11 lead to

(12)

(13)

By inserting Eqs. 12 and 13 into Eq. 9 at infinite dilution of component 2, one obtains the following expression for of an ideal ternary mixture:

(14)

On the other hand, expressions for for an ideal ternary solution can be also derived by combining Eq. 3 or Eq. 5 with the following Kirkwood-Buff integrals for ideal ternary mixtures (16):

(15)

(16)

(17)

Equation 3 leads to

(18)

(19)

One can see that the result obtained on the basis of Eq. 3 (Eq. 18) coincides with Eq. 14 derived from general thermodynamic considerations, whereas that based on Eq. 5 does not. The numerical difference between the two expressions is very large because the molar volume of a protein is, usually, much larger than the molar volume of the cosolvent.

Whereas the above discussion involves the quantity which is usually determined experimentally (2–7), is related to through the equation (1,16)

(20)

(21)

(22)

By combining Eqs. 3, 20, 21, and 22, one obtains after some algebra the following simple expression:

(23)

Whereas and depend on the protein characteristics, and depend only on the characteristics of the protein-free mixed solvent.

For usual cosolvents (organic solvents, salts, etc.), one can use the following approximation of Eq. 23 in the dilute cosolvent range:

(24)

Indeed, and are much smaller than the Kirkwood-Buff integrals for the pairs involving the protein ( and ). Table 1 provides their values for the system water (1) + lysozyme (2) + urea (3) (pH 7.0, 20°C).

TABLE 1 Numerical values of the Kirkwood-Buff integrals for the water (1) + lysozyme (2) + urea (3) (pH 7.0, 20°C) system

However, when and are large, and this occurs when the cosolvent is, for example, a polymer ( (cm³/mol) for the system water/polyethylene glycol 2000 at a weight fraction of polyethylene glycol of 0.02 (21)), the complete Eq. 23 should be used. This conclusion is valid for all large cosolvent molecules (polymers, biomolecules, etc.).

Let us consider the biochemical equilibrium between infinitely dilute native (N) and denaturated (D) states of a protein in a mixed solvent. The changes of the preferential binding parameters and in this process are given by

(25)

(26)

(27)

Equations 25 and 26 follow from Eqs. 3 and 23 by taking into account that and are characteristics of the protein-free mixed solvent at infinite protein dilution.

The equilibrium constant K of biochemical equilibrium between infinitely dilute native (N) and denaturated (D) states of a protein in a mixed solvent can be expressed in terms of (22)

(28)

Using for and expressions from Shulgin and Ruckenstein (16), Eq. 28 can be also rewritten in the form

(29)

We are indebted to Prof. J. Michael Schurr (Dept. of Chemistry, University of Washington, Seattle, WA) for helpful comments regarding this manuscript and for drawing our attention to the fact that the coefficients and are different.