Choice of Concentration Scale

Consider initially an experiment in which a solution of protein (A) is brought into dialysis equilibrium with solvent (s). Since aqueous solutions are being considered, it will be assumed that both solvent and solution are incompressible, a simplification that allows the partial molar volumes of all constituents to be regarded as constants. In thermodynamic studies, the chemical potential (fji,) of a solute component is considered to be the sum of a standard state value (|jl°) and a part that, under ideal conditions, depends logarithmically upon solute concentration. Although the solute concentration may be measured on the mole-fraction, molal, molar, or weight-based scale, the choice of concentration scale serves only to dictate the value of the standard state chemical potential [28,29], For very dilute solutions, all concentration scales are related to each other in approximately linear fashion, whereupon the variation of chemical potential can be taken into account through the logarithmic term, regardless of the concentration variable used [28,29]. In considerations of thermodynamic nonideality, however, the interest is on effects that arise when this approximation fails. Consequently, it becomes necessary to account for all the additional terms that appear in the thermodynamic equations, either as the result of molecular interactions or as a result of the nonlinear relationship between concentration scales. Polynomial expansions in concentration allow this to be done in a self-consistent manner.

The addition of protein to solvent at constant temperature gives rise to one of two situations, depending on the nature of the experiment. In equilibrium dialysis and classical osmometry, for example, the chemical potential of solvent in the protein-containing phase (a) and the solvent phase ((3) remains equal to that of solvent at atmospheric pressure (P); that is, (p,?)r,p+n = (p-f)r,/>. On the other hand, the laboratory constraints of fixed temperature and pressure that apply to the determination of partial specific volumes from density measurements on undialyzed solutions lead to a situation in which p" differs from the value that applied before the addition of protein.

In partition studies such as equilibrium dialysis, the expression relating the thermodynamic activity zi of a solute component to its chemical potential may be written [30] as

where the thermodynamic activity of solute is a molar quantity and is therefore most appropriately expressed as the product of its molar (or comparable weight-based) concentration C, and the corresponding activity coefficient 7,. By adopting this concentration scale, it is possible to express the activity coefficient of any solute component in terms of virial coefficients, B,y and so on, which appear in the usual expression for the osmotic pressure, namely,

K1 i j where the index i encompasses all solute species, and j > i. (Note that the range of the index j in the summation differs from that used by us on previous occasions. The current notation is consistent with that used by Hill and other authors.) The result obtained is

which has the advantage that the coefficients By, etc., find simple statistical-mechanical interpretation in terms of the physical interactions between pairs, triplets, etc., of molecules [30,31]. In this formulation, 7, is a molar activity coef ficient of solute component /', but it is obtained in terms of a standard state with a lower pressure (P) than that (P + II) of the solution.

Alternatively, in circumstances where the chemical potential of solute is being defined under conditions of fixed temperature and pressure, the expressions analogous to Eq. (8) must now be written as

MSNS

where the thermodynamic activity of solute, a„ is a molal quantity and is therefore described most simply as the product of an activity coefficient y, and the molal concentration m,. In the definition of molality [Eq. (11c)], NA and Ns denote the respective numbers of solute and solvent molecules, while Ms is the molar mass of solvent expressed in kg/mol to retain customary convention that the units of w, are mol/kg solvent. The counterparts of Eqs. (9) and (10) are

j*i where y, is a molal activity coefficient that is obtained by defining chemical potential in terms of a standard state at the same pressure but with a different chemical potential of solvent. The coefficients C,y are not normally expressible in simple statistical-mechanical terms, but for incompressible solutions the two sets of osmotic virial coefficients conform to the identities

where ps is the solvent density. These relationships allow the two thermodynamic activities to be expressed as polynomial expansions of either concentration scale. Specifically,

= In mi + In p.s + (2Cu + M,v,ps)m,- + £(C,y + M,v,ps)m, + •••

= In Q - In p, + (2B,v - MiV/)C, + £ (Bv - M;v,)C; + -

Although zi may be expressed in terms of either molar or molal concentration and an appropriate activity coefficient, its definition as a molar activity remains unchanged; the activity coefficients therefore differ [Eqs. (15a,b)]. Furthermore, Eqs. (8) and (10) provide a valid description of the dependence of solute chemical potential upon protein concentration only when solutions are compared at the same chemical potential of solvent. Likewise, Eqs. (11) and (13) require the chemical potentials of the protein in different solutions to be compared at the same pressure.

The distinction between the alternative definitions of the thermodynamic activities zi and a, is a subtlety that must be taken into account in the interpretation of partial specific volume measurements. We therefore make a direct attack on analysis of the effect of small solute on the thermodynamic activity of the protein in such measurements.

C. Direct Thermodynamic Interpretation

As mentioned in Sec. II.A, the apparent partial specific volume <f>A is measured under conditions of constant chemical potentials of solvent and small solute. It is now clear that under these circumstances it is most convenient to express concentrations on the molar scale. We shall therefore derive all expressions in such terms [22],

For incompressible solutions, the density may be expressed as the sum of the weight concentrations of all species, including solvent -, that is, p = 2C,M„ The density p of a protein solution (a) and the density po of the diffusate ((3) with which it is in dialysis equilibrium are therefore given by p = MaQ + MmCS, + ps(l - MmVmC'ù - MavaCa) po = MmC& + ps(l - MmVmCm)

where p.v refers to the density of unsupplemented solvent. These expressions are readily combined to yield

Because of the identity of small solute activities (zM) in the two phases, the concentration difference may be written as

Activity coefficients are now replaced by expressing them as power series in concentration, whereupon

where we have retained terms up to first order in solute concentration. With these substitutions, the ratio of activity coefficients becomes

Because Cm — Cm —> 0 in the limit of zero protein concentration, Eqs. (17), (18), and (20) may be combined to obtain

Ma where the value of Cm may now be taken as that of the diffusate.

Provided the concentration of small solute is confined to the region wherein the expansion of activity coefficients may be truncated after the linear term, the linear dependence of 1 — cJjApo upon Cm has a slope of (1 — vmPs)Mm(Bam/Ma). The thermodynamic nonideality of the system may thus be expressed experimentally in terms of the second virial coefficient (B^m), which, for an inert nonelectrolyte as small solute, may be identified with the protein-small solute covolume. Specifically,

where N is Avogadro's number, and ra and rm are the respective exclusion radii of A and M, both assumed spherical.

Having outlined this direct procedure [22] for interpreting partial specific volume measurements in terms of thermodynamic nonideality, we are now in a position to make comparisons with the classical approach in terms of protein solvation.

D. Equivalence of Treatments

In order to establish equivalence of the preferential solvation and statistical-mechanical treatments of thermodynamic nonideality, the relationship between and the second virial coefficient will be obtained by manipulating Eq. (2a) to a form more in keeping with Eq. (21). The first step is to introduce the solvent density (p.y) into Eq. (2a) by expressing the solution density as a power series in solute concentration, the molal equivalent of Eq. (16) being p = pil + I Mimjjll + £ Mivumpsj (23a)

where the subscript i spans all species, including solvent. On taking into account that is obtained in the limit of zero solute concentration (mA —> 0), substitution of Eq. (23b) into Eq. (2a) gives

1 - <|>/ipo = (1 - vaps) + (1 - vmPj) [ ~ vApsMMmM) + •■•] (24)

The next step is to express the partial derivatives appearing in Eq. (7) as polynomial expansions in solute concentrations, but it is only necessary to retain the leading terms in each to obtain the description of correct to first order in concentration. With that proviso, the three partial derivatives in Eq. (7) become j./ahiU =Cam+... (25)

Q->° \ OniA ) T, P. mu limfe^) = 2Cmm + ■ • • (26)

c„-> 0 \ dmm j t, p, ma lim =RTps[ 1 + -] (27)

The last derivative follows from Eq. (9), while the other two emanate from Eq. (15c) on noting that In yM = In a\t ~ In mM. Substitution of Eqs. (25)-(27) into Eq. (7) then gives

& = ~ TT [(Cam + MMVMPs)mM + •• •] [1 - ■ ■ ■] (28b)

Ma whereupon Eq. (24) becomes 1 - 4>Apo = (1 - vxp*)

- (1 - vMp.v) 77— [(Cam + MmvmPs + MA vaPs)mM + ■■•] <~29'> Ma

Conversion from molal to molar concentration on the basis that m\i(CA = 0) = — [1 + MMVMPsCM + ■■•]

and substitution of [BAM ~ (Mmvm + MavA)/2]ps for Cam [Eq. (14c)] then yields Eq. (21), the expression derived in Sec. C.

Finally, the formal relationship between and Bam is obtained either directly from Eqs. (14c) and (28b) or by equating the leading terms in Eqs. (21) and (24). Either approach shows that whereupon the solvation parameter, -Ç^/mMAÎM, becomes (Bam - Mava)Ps/Ma, correct to zero order in concentration.

From a thermodynamic viewpoint, Bam and £m/wm provide different means of expressing deviations from thermodynamic ideality that arise from interactions between molecules of the protein (A) and small solute (M). Analysis of results in terms of Bam allows readier access to the composition dependence of the thermodynamic activity coefficient of the protein; and when the only significant interactions between molecules of protein and small solute are of the excluded volume type, Bam represents the covolume, (Uam) [Eq. (22)]. In this case, is seen to be a measure of the difference between the volume in the region of the protein that is inaccessible to the small solute (Uam) and the corresponding volume that is inaccessible to solvent (Mava). As Eisenberg [24,25,32] has noted, interpretation of £m as a measure of preferential solvation is equivocal. Like the excluded volume interpretation of Bam, it entails the consideration of a thermodynamic parameter in terms of postulated molecular interactions. Although such resort to model-dependent interpretations of thermodynamic data represents a departure from classical protocol, the approach is justified in the event that the model provides a helpful explanation of the thermodynamic behavior of the system under consideration.

In order to gain further insight into the prediction of covolumes for protein-small nonelectrolyte systems, we now employ the statistical-mechanical approach for reanalysis of published densimetric studies of thermodynamic non-ideality in protein solutions containing high concentrations of a range of small nonelectrolytes.

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