INTRODUCTION

TOP ABSTRACT INTRODUCTION THEORY AND METHODS RESULTS AND DISCUSSION CONCLUSION REFERENCES

Altering the solvent environment by adding large quantities of cosolvent can cause significant changes in the structure and stability of biological macromolecules. For example, several-molar concentrations of urea, guanidine HCl, or alcohol cause protein denaturation, whereas sucrose stabilizes protein native states; alcohols have long been used to promote DNA condensation. (Technically, molecules such as urea, guanidine HCl, and sucrose are cosolutes, being solid in their pure form. However, at typical concentrations, they make up a significant fraction of the solution8 M urea is 43 wt% urea and 2 M sucrose is 55 wt% sucrose. These cosolute molecules bathe and solvate the macromolecular solute just as much as water does; they also solvate water molecules as water solvates them. In that sense, these cosolutes behave like solvent molecules. To emphasize this point, and to put cosolvents and cosolutes on an equal footing, we refer to all species which solvate as "cosolvent" molecules. Only in the Theory and Methods section discussion on obtaining molecular hard-sphere radii, where we need to emphasize the solid nature of the pure substances, do we use the term "cosolute.") Also, when a macromolecule changes structure, parts of it experience a change in solvent environment, e.g., when proteins denature and the protein interior moves from a primarily hydrophobic milieu to an aqueous one. Despite the importance of understanding solvation effects and much research effort along these lines, how (co)solvents interact with proteins and DNA is still not well understood.

To probe solvent-macromolecule interactions, one would like to measure the solvation free energy, G_solv, the free energy of interaction between solute and solvent. Since G_solv is difficult to obtain experimentally, one measures instead the free energy of transfer, G_tr, of the macromolecule (or its constituent parts) from one solvent environment (denoted A) to another (B): G_tr = G_solv(B)  G_solv(A). How does one interpret G_tr? Let us first return to G_solv and dissect it into more meaningful partsa part due to "soft" (e.g., dispersion, hydrogen-bonding, dipole, and electrostatic) interactions, denoted here as G_solvⁱ, and a part due to "excluded-volume" interactions (G_solv^ev). G_solv = G_solv^ev + G_solvⁱ. (G_solv^ev and G_solvⁱ are defined more exactly in Theory, below) G_solv^ev describes the work of making room for the solute, i.e., of creating a hard cavity. Not only is G_solvⁱ dependent on the solvent environment, but so is G_solv^ev. The price of creating a fixed-size cavity depends on the amount of unoccupied or free volume. Creating a cavity in a dense environment is generally more difficult than in one that has a lot of free volume. Now the transfer free energy, G_tr, can also be split into soft and excluded-volume parts: G_tr = G_tr^ev + G_trⁱ, where G_tr^ev = G_solv^ev(B)  G_solv^ev(A), and likewise for G_trⁱ. G_tr^ev, which we call the "free energy of cavity transfer" from environment A to B, is the difference in free energy between creating a cavity in B versus in A. A positive (negative) value indicates that it is harder (easier) to create a cavity in environment B. G_trⁱ = G_solvⁱ(B)  G_solvⁱ(A) embodies the difference in soft interactions between the two environments.

Scaled-particle theory (SPT) is commonly used to calculate G_tr^ev. Since SPT was designed to capture the packing interactions of a hard-sphere solute in a fluid of hard spheres (there are no soft interactions), it would seem to be an ideal theory for calculating G_solv^ev and G_tr^ev. The only input parameters necessary are the solute radius and the concentrations and radii of the solvent species.

However, there are indications that using SPT for making (semi)quantitative calculations of G_tr^ev may be problematic:

1. It has been shown that solvation energies (G_solv^ev values), as calculated by SPT (Morel-Desrosiers and Morel, 1981; Wilhelm and Battino, 1972; Lucas, 1976; Pierotti, 1976; Crovetto et al., 1982; Postma et al., 1982; Ben-Naim et al., 1989; Madan and Lee, 1994; Prévost et al., 1996) and by more realistic models (Postma et al., 1982; Pohorille and Pratt, 1990; Madan and Lee, 1994; Prévost et al., 1996; Floris et al., 1997), are strongly dependent on the solute radius (Morel-Desrosiers and Morel, 1981; Lucas, 1976; Pierotti, 1976; Crovetto et al., 1982; Postma et al., 1982; Ben-Naim et al., 1989; Pohorille and Pratt, 1990; Madan and Lee, 1994; Prévost et al., 1996; Floris et al., 1997) and especially on the solvent radius (Morel-Desrosiers and Morel, 1981; Wilhelm and Battino, 1972; Lucas, 1976; Pohorille and Pratt, 1990; Madan and Lee, 1994). A change of 2% in solvent radius results in a change of ~15% in G_solv^ev (Wilhelm and Battino, 1972). Preliminary results (Lucas, 1976) and evidence from heat capacities of transfer (Desrosiers and Desnoyers, 1976) and partition coefficients (Watarai et al., 1982) suggest that the transfer free energy, G_tr^ev, is also sensitive to solvent size. Unfortunately, determining the radii of real molecules, which are not spherical, is somewhat ambiguous. Different experimental and theoretical methods yield different values (see, e.g., Gogonea et al., 1998). These two factsthe sensitivity of G_solv^ev to solvent size and the ambiguity in obtaining these sizessuggest that calculating actual numbers for G_tr^ev values using SPT might be problematic.

2. One of the contributions to G_solv^ev is the mechanical pressure-volume (pV) work of displacing solvent or the atmosphere around it. However, which pressure value to use, the hard-sphere pressure (p_hs) needed to maintain the system of hard spheres at the experimental fluid density or atmospheric pressure (p_atm) is not yet clear (Shimizu et al., 1999). The choice presumably depends on how G_solv is dissected and on which interactions are being apportioned to the excluded-volume part of the free energy (G_solv^ev). (Note that even if p_hs is used, soft interactions are still included implicitly in G_solv^ev. Soft interactions determine the experimental solvent densities, which are then used as input parameters in SPT calculations.) Unfortunately, which pressure value is used does make a significant difference in G_solv^ev (Pierotti, 1976), and possibly in G_tr^ev, since at fluid densities p_hs is typically orders of magnitude greater than p_atm (Pierotti, 1976).

3. The last potential difficulty regards obtaining the water molarity in an aqueous mixed solvent (n_w^mix). For a specific solvent one can get n_w^mix from the experimental solution density () plus the cosolvent molarity (n_c^mix). However, for making calculations on generic cosolvents, n_w^mix must be obtained theoretically. Some researchers (Berg, 1990, Guttman et al., 1995, Saunders et al., 2000) use the approximation of applying the Gibbs-Duhem relation at constant temperature and pressure to the SPT portion of the equation of state; this is thermodynamically equivalent to holding p_hs fixed (Guttman et al., 1995) to the value of pure water, p_hs^wat. (The Gibbs-Duhem relation, of course, applies to the entire equation of state, but not necessarily to a subset of it.) How good is this approximation for the purpose of calculating G_tr^ev?

In this work, we determined the uncertainties in G_tr^ev due to ambiguities in SPT input parameters. Are these uncertainties small enough such that G_tr can be usefully separated into excluded-volume and soft-interaction terms? We performed calculations and comparisons for the transfer of amino acid solutes from water to aqueous solutions of ethanol, ethylene glycol, sucrose, and urea to compare with the experimental results of Nozaki and Tanford (1971, 1965) and Bolen and colleagues (Liu and Bolen, 1995; Wang and Bolen, 1997). We have addressed the above three particular concerns as follows:

1. To determine the degree of uncertainty in G_tr^ev caused by uncertainties in molecular radii, we varied the input solvent radii within the range of representative solvent radii from the literature and looked at the spread in the G_tr^ev values.

2. To see how choice of pressure affects G_tr^ev, we calculated G_tr^ev using both p_atm and p_hs.

3. To check the approximation of fixing the hard-sphere pressure at p_hs^wat to determine n_w^mix, we compared the predicted solvent densities with the experimental values as well as the G_tr^ev values calculated with the predicted versus the experimentally determined n_w^mix values.

In addition, we discuss why the work of formation of a hard cavity is so dependent on solvent size.

	THEORY AND METHODS

TOP ABSTRACT INTRODUCTION THEORY AND METHODS RESULTS AND DISCUSSION CONCLUSION REFERENCES

Theory

First, let us more carefully define G_solv^ev. The solute-insertion process can be separated into three steps. In step 1 all soft interactions are turned off; only hard interactions remain. However, the solvent density is kept fixed at the fluid density. In step 2, a hard cavity in which to place the solute is created within the solvent of hard particles. In step 3, the soft interactions, both solvent-solvent and solute-solvent, are turned back on. The free energy associated with step 2 is G_solv^ev; that associated with both steps 1 and 3 is G_solvⁱ. G_solv = G_solv^ev + G_solvⁱ. Since step 2 involves no explicit soft interactions, any hard-particle theory of fluids can be used to calculate G_solv^ev.

(Note, there is another common separation of G_solv into a cavity and a soft-interaction term: G_solv = G_solv^cavity + G_solv^interaction. G_solv^cavity is the work of creating a hard cavity in a solvent whose solvent-solvent interactions are on; G_solv^interaction is the conditional free energy of turning on the solute-solvent soft interactions, once the cavity has been created. One of the advantages of this dissection of G_solv^ev is that one can easily write analytic formulas for G_solv^cavity and G_solv^interaction in terms of ensemble averages. For more details, see section 3.5 of Ben-Naim (1987). However, because the solvent-solvent soft interactions are always on, there are solvent reorganization and redistribution terms in G_solv^cavity which are not present in G_solv^ev and which are, unfortunately, hard to ascertain. Hence, there is an enthalpic component to G_solv^cavity, whereas G_solv^ev is purely entropic. To examine only solvent-size effects, it would seem more useful to determine G_solv^ev.)

SPT has commonly been employed to calculate G_solv^ev. The fundamental idea behind the theory was described by Reiss (1966) this way: "[T]he most important problem in the theory of liquids is concerned with the packing of hard cores... . In this model, the soft intermolecular potential (or the non-hard-core part of the potential) acts primarily to establish the overall density of the fluid, while the internal structure is determined by the packing of the hard cores. Thus it might be said that the soft potential determines the volume of a container which in turn is filled with a hard sphere fluid... . [S]caled particle theory ... is geometric in nature and deals in a rigorous manner with the problem of the packing in a sufficiently dense fluid of molecular hard cores."

The derivation of SPT involves finding the probability, P(R), of inserting a hard spherical cavity of radius R with its center at an arbitrary (fixed) location in a fluid whose m species have hard cores of radii R_i. P(R) is simply related to the work of inserting the same cavity, G_solv^ev(R) (Tolman, 1938):

(1)

where k is Boltzmann's constant and T is the absolute temperature. There are several exact conditions on P(R) and G_solv^ev(R) for very small cavities and macroscopic cavities (Reiss, 1966). Combined with conditions on the smoothness of derivatives one obtains the following result for G_solv^ev(R) (Lebowitz et al., 1965):

(2)

(3)

(4)

(5)

where

(6)

n_i is the number density of species i, and p is the pressure. Note that ₃ has a physical meaning. ₃ = _i=1^m n_i R_i³ = (fractional volume occupancy)  (packing fraction) and 1  ₃ = (fractional free volume). We see in G_solv^ev(R) the familiar pV term, the work of creating a macroscopic cavity of volume V (Eq. 5), as well as a surface-tension term  R² (Eq. 4) with a curvature correction  R¹ (Eq. 3).

The appropriate pressure to use in Eq. 5 is not yet clear (Shimizu et al., 1999 and references therein). Both p_atm and p_hs have been used, yielding very different values for G_solv^ev(R) (Pierotti, 1976). The functional form of p_hs is given by Lebowitz et al. (1965):

(7)

In general, p_hs is very high at fluid densities (Pierotti, 1976). For example, pure water's hard-sphere pressure, p_hs^wat, is 8000 atm at 25°C (obtained by using Eqs. 6 with one species and 7 with n_w = n_w^wat = 55.342 M and R_w = 1.38 Å). To cover both possibilities in this work, we calculate G_solv^ev(R) with p set to both p_hs and p_atm.

Lastly, we point out that G_solv^ev(R) is the work of inserting a hard cavity at a fixed site in the solvent. Translational and liberational entropies are not included. In comparing to experimental transfer data, the corresponding value is the difference in standard state chemical potential of the solute, µ° = µ°(B)  µ°(A), on the number-density (molarity) scale (Ben-Naim, 1978). (The translational entropy present in µ° cancels in transfer processes.) We have converted experimental data reported on the mole-fraction scale to molarity scale.

Molecular hard-sphere radii are not well defined

To calculate the work of cavity formation, SPT requires only the water and cosolvent number densities and their hard-sphere radii as inputs. However, these radii are not well defined. For both water and cosolvent, there are fairly wide ranges of reasonable values.

The hard-sphere radius of water as measured by experiment is typically around 1.35 Å, but different experiments give values ranging from 1.25 to 1.46 Å (Pierotti, 1965). In theoretical studies of water, the following radii have been used: 1.35 Å (a hard-sphere fluid with water's fractional free volume and number density has this radius; Pohorille and Pratt, 1990), 1.38 Å (Lee, 1985; obtained from solubility experiments of Pierotti (1965, 1976); see also next paragraph), 1.40 Å (the most probable water oxygen-oxygen distance; Prévost et al., 1996), 1.44-1.5 Å (obtained by fitting SPT with the water radius as an adjustable parameter, to free-energy data obtained via simulations of simple point charge (SPC) (Postma et al., 1982) and transferable intermolecular potential 4 point (TIP4P) (Floris et al., 1997) water models), and 1.58 Å (the Lennard-Jones  parameter divided by 2; Prévost et al., 1996). Solvent probe radii of 1.4 Å (Lee and Richards, 1971; Shrake and Rupley, 1973) and 1.5 Å (Connolly, 1983) have been used to determine the solvent-accessible surface areas of macromolecules. There is not one unique hard-sphere radius for water.

Since many (co)solvents are less studied than water, their hard-sphere radii can be even more ambiguous. Some researchers (Wilhelm and Battino, 1972; Morel-Desrosiers and Morel, 1981) have argued that the most self-consistent hard-sphere radius for use in SPT is measured via a technique pioneered by Pierotti (1965). Solubilities of a series of nonpolar, spherical solutes (e.g., noble gases) are measured. When the data are extrapolated to zero polarizability, only the hard-sphere interaction remains. Matching to SPT yields the solvent's hard-sphere radius. These experiments are non-trivial, and hard-sphere radii have been obtained by other methods: fitting pressure-density data to a hard-sphere plus Lennard-Jones equation of state (Ben-Amotz and Herschbach, 1990; Ben-Amotz and Willis, 1993); fitting surface-tension (Mayer, 1963), isothermal compressibility (Mayer, 1963), and heats of vaporization data (Pierotti, 1976) to SPT; from cell theories of liquids (Salsburg and Kirkwood, 1953; Kobatake and Alder, 1962); and from gas-phase virial coefficients and viscosities (Hirschfelder et al., 1964 and references therein). The radii obtained by all of these experimental methods implicitly include solvent-solvent interactions and are therefore effective radii. Unfortunately, none of these methods can be used to obtain the hard-sphere radii of many biologically interesting cosolutes such as urea or sucrose since the methods assume that the molecule of interest is a liquid (or a gas) in its pure form. For cosolutes, the only available techniques measure the length dimensions of a molecule in isolation. These techniques include calculating molecular van der Waals volumes (Bondi, 1964; Edward, 1970; Gogonea et al., 1998), as well as actually measuring lengths on a space-filling model (Goldstein and Solomon, 1960; Schultz and Solomon, 1961). Radii from these methods do not include any solvent-solvent interactions. However, the relationship between molecular lengths and the hard-sphere radius of an equivalent sphere has not been fully determined (Gogonea et al., 1998).

Table 1 lists the hard-sphere radii of cosolvents (cosolutes) by various methods. For common organic solvents, for which experimental values are available, the radii vary by several tenths of an angstrom. This is not surprising since these molecules are not spherical and the solvent-solvent interactions implicitly included in the experimental values are experiment-dependent. Values obtained from molecular-length calculations tend to be larger than experimental values. Radius data for cosolutes are limited. We presume that if there were some way of obtaining them from experiment, there would be a similar variation in radii as for the (co)solvents.

SPT parameters used in this work

Because the cosolvents studied here are of comparable size to water, we treat water explicitly in our SPT calculations. We chose the following radii for water and cosolvents. The water radii (R_w), 1.35, 1.38, and 1.40 Å, represent those used in water studies (Pohorille and Pratt, 1990; Lee, 1985; Prévost et al., 1996) as well as the value measured from solubility experiments (Pierotti, 1965, 1976). The cosolvent radii, R_c, we chose are ethanol, 2.00, 2.15, and 2.30 Å; ethylene glycol, 2.20, 2.30, and 2.40 Å; sucrose, 3.85, 4.00, and 4.15 Å; and urea, 2.15, 2.25, and 2.35 Å. The radii values of ethanol and ethylene glycol span the observed range from experimental data and molecular-length calculations; for sucrose and urea, we have chosen a range of radii representing a possible spread of values around those obtained from molecular-length calculations.

The radii of the solutesthe amino acids, triglycine (3gly; all in their zwitterionic form), and diketopiperazine (DKP)were obtained using van der Waals volume increments (Edward, 1970) and are listed in Table 2. We do not vary solute radii because it has been noted that G_solv^ev values are not as sensitive to solute radii as to solvent radii (Morel-Desrosiers and Morel, 1981) and preliminary evidence (data not shown) indicate that this is also true of G_tr^ev values.

Except in the section where n_w^mix was approximated by holding p_hs constant, n_w^mix values were obtained from the following experimentally measured cosolvent molarities and solution densities: 60 vol% (10.3 M) ethanol and 30 vol% (5.36 M) ethylene glycol: 0.9096 and 1.0405 g/ml, respectively, at 20°C (obtained by interpolating data from Wolf et al. (1985); 1 M sucrose: 1.127100 g/ml at 25°C (Liu and Bolen, 1995); 2 M urea: 1.028 g/ml at 25°C (D. W. Bolen and M. Auton, University of Texas Medical Branch, personal communication). The number densities of pure water (n_w^wat), 55.407 and 55.342 M at 20 and 25°C, respectively, were obtained from the corresponding mass densities (Weast, 1987), 0.9982063 and 0.9970480 g/ml. Note that, here, both water and cosolvent number densities are fixed by experiment; they are not adjustable parameters.

In studies of generic cosolvents, n_w^mix cannot be measured experimentally, and the approximation of holding p_hs constant has been used to obtain n_w^mix (Berg, 1990; Guttman et al., 1995; Saunders et al., 2000). Below, we test this approximation against calculations done with n_w^mix values obtained from experiment (per the previous paragraph). n_w^mix obtained from holding p_hs fixed was calculated by numerically solving the equation p_hs^mix(R_w, R_c, n_w^mix, n_c^mix) = p_hs^wat(R_w, n_w^wat) for n_w^mix and taking the real root. p_hs^wat and p_hs^mix were calculated using Eqs. 6 and 7 with one species (i = water) and two species (i = {water, cosolvent}), respectively. n_w^wat is 55.407 M (20°C) and 55.342 M (25°C). Note that n_w^mix is a function of R_w, R_c, n_w^wat, and n_c^mix. Hence, for transfer to a given solvent mixture (fixed n_c^mix), n_w^mix will vary as solvent radii are varied.

	RESULTS AND DISCUSSION

TOP ABSTRACT INTRODUCTION THEORY AND METHODS RESULTS AND DISCUSSION CONCLUSION REFERENCES

To determine what role excluded volume plays in transfer processes, we have used SPT to model the excluded-volume portion, G_tr^ev, of the total transfer free energy, G_tr (= G_tr^ev + G_trⁱ), of solutes from water to mixed solvents. We then compared the G_tr^ev values and uncertainties with experimentally measured G_tr values to see whether G_trⁱ can be usefully determined. In particular, we determined how uncertainties in solvent radii and in solvent density, necessary for the input parameters, translate into uncertainties in G_tr^ev. Also, we checked how G_tr^ev values calculated using an approximate method of obtaining n_w^mix compare to those calculated using experimentally determined n_w^mix values. The systems we studied are amino-acid solutes transferring from water to 60 vol% ethanol, 30 vol% ethylene glycol, 1 M sucrose, and 2 M urea.

G_tr^ev is very sensitive to solvent radii

Cosolvent: ethylene glycol

View larger version (26K): [in this window] [in a new window]   FIGURE 1   Gtr and Gtrev values for the transfer of amino-acid solutes from water to 30% ethylene glycol. For any solute, the overall spread/uncertainty in Gtrev (several kcal/mol) due to variation in solvent radii is as large or larger than the Gtrev value itself, and much larger than Gtr. For each solute (along the x-axis) there are plotted one experimental apparent Gtr () (Nozaki and Tanford, 1965) (converted to the molarity scale), and nine (for 3 × 3 water/cosolvent radius pairs) SPT-calculated Gtrev values (three open symbols, , , and , for Rc = 2.2, 2.3, and 2.4 Å, respectively, with rectangular bars through them). The rectangular bars indicate how Gtrev varies as water's radius is changed but the cosolvent radius is kept fixed. The open symbol is located at the Gtrev value with Rw = 1.38 Å; the ends of the rectangular bars are at the Gtrev values with Rw = 1.35 and 1.40 Å. Whether Gtrev with Rw = 1.35 Å is at the upper or lower end of the rectangular bar (the value with Rw = 1.40 Å being at the opposite end) is independent of the solute and is only a function of cosolvent size. The water radii are marked on the figure only for the first solute (in this case 3gly) for clarity. Therefore, for any particular solute and choice of Rc, if Gtrev with Rw = 1.35 Å and the same Rc is at the top (bottom) of the bar for 3gly, then it is also at the top (bottom) for that solute. For example, for the transfer of tyrosine, if Rc = 2.2 Å (), Gtrev with Rw = 1.35 Å is at the top of the rectangular bar, as it is for 3gly with Rc = 2.2 Å, and is equal to 0.039 kcal/mol; with Rw = 1.38 Å and 1.40 Å, the values are, from the locations of  and the bottom of the corresponding bar, 0.26 and 0.51 kcal/mol, respectively. It turns out that for the three cosolvent radii considered here, increasing water's radius always increases favorability of transfer (Gtrev with Rw = 1.40 Å is always at the bottom of the bar). This is not the case for all cosolvent radii. See for example, the data for transfer to aqueous sucrose and urea (Figs. 3 and 4). (The experimental Gtr were measured at 25°C, whereas our calculations were at 20°C, the temperature at which solution density data were available (Wolf et al., 1985). However, the qualitative conclusions we make should not be affected by a 5° temperature change.)

Cosolvents ethanol, sucrose, and urea: results are qualitatively the same as for ethylene glycol

View larger version (23K): [in this window] [in a new window]   FIGURE 2   Gtr and Gtrev values for the transfer of amino acid solutes from water to 60% ethanol. Experimental apparent Gtr values () (Nozaki and Tanford, 1971) were again converted to the molarity scale. SPT-calculated Gtrev (p = patm) were obtained with Rc = 2.00 (), 2.15 (), and 2.30 Å (). Gtr values were measured at 25°C, whereas Gtrev were calculated at 20°C, the temperature at which density data were available (Wolf et al., 1985). See caption of Fig. 1 for more details on interpreting the figure.

View larger version (25K): [in this window] [in a new window]   FIGURE 3   Gtr and Gtrev values for the transfer of amino-acid solutes from water to 1 M sucrose. Experimental apparent Gtr values () are from Liu and Bolen (1995). SPT-calculated Gtrev (p = patm) with Rc = 3.85 (), 4.00 (), and 4.15 Å () are indicated by open symbols. Both Gtr and Gtrev were obtained at 25°C. See caption of Fig. 1 for more details on interpreting the figure.

View larger version (26K): [in this window] [in a new window]   FIGURE 4   Gtr and Gtrev values for the transfer of amino acid solutes from water to 2 M urea. Experimental apparent Gtr values () are from Wang and Bolen (1997). SPT-calculated Gtrev (p = patm) were obtained with Rc = 2.15 (), 2.25 (), and 2.35 Å (). Both Gtr and Gtrev were obtained at 25°C. See caption of Fig. 1 for more details on interpreting the figure.

Why is G_tr^ev so sensitive to solvent radii?

(8)

(9)

(10)

(11)

View larger version (148K): [in this window] [in a new window]   FIGURE 5   The volume available for cavity insertion is significantly smaller than the free volume. This is a random snapshot of a two-dimensional binary fluid of hard disks, indicated by black circles, with a total fractional occupancy (packing fraction) of 0.40. The smaller disks have a radius of 1, the larger a radius of 2. The gray regions are areas that are unoccupied, yet unavailable for the insertion of a cavity with a radius of 1.5. The volume available for insertion (white regions) is significantly smaller than the "free" but unavailable (gray) regions. If one attempts to insert a cavity the same size as the larger disks (radius 2), there is no region in this configuration available for insertion.

Why is the sign of G_tr^ev not predictable?

G_tr^ev is also very sensitive to solvent density

The experimental solvent density, , is used in conjunction with the experimental cosolvent molarity to determine the molarity of water in the mixed solvent, n_w^mix. Here, we show that G_tr^ev is also extremely sensitive to , as previously noted by (Berg, 1990). Fig. 6 shows G_tr^ev values for the transfer of amino acids to 30 vol% ethylene glycol, calculated with  set to the experimental density and to the experimental density plus and minus 1%. A change of density of 1% yields a change in G_tr^ev of 0.2 to 0.4 kcal/mol, depending on the solute. For transfer to 60 vol% ethanol, 1 M sucrose, and 2 M urea, the change in G_tr^ev due to a 1% change in  is 0.2 to 0.3, 0.2 to 0.5, and 0.2 to 0.4 kcal/mol, respectively. These variations are comparable to the experimental G_tr magnitudes themselves.

View larger version (18K): [in this window] [in a new window]   FIGURE 6   Gtrev is also sensitive to . Plotted are data for the transfer of amino acids to 30 vol% ethylene glycol, with Rw = 1.38 Å and Rc = 2.3Å. Gtrev was calculated with  set to the experimental density (, same as in Fig. 1) and to the experimental density plus 1% (+) and minus 1% (×). The solid circles () are the experimental apparent Gtr values (Nozaki and Tanford, 1965) converted to the molarity scale. A change of density of ±1% yields a change in Gtrev of 0.2 (gly) to 0.4 (trp) kcal/mol.

G_tr^ev is so sensitive to  for essentially the same reason it is also sensitive to the solvent radii: changing  alters the amount of space occupied by solvent, which in turn strongly alters the work of inserting a cavity. To avoid substantial errors in G_tr^ev due to uncertainties in , it is necessary to measure  to high precision, e.g., with a precision densitometer.

p_hs versus p_atm: the behavior of G_tr^ev is qualitatively the same

It is still not clear which pressure, p_hs or p_atm, to use in the pV term in the cavity formation work (Eq. 5) (Shimizu et al., 1999 and references therein). On the one hand, Neff and McQuarrie (1973) advocate the use of p_hs as a consistent separation of the interaction into hard and soft parts. In contrast, Pierotti (1976) has argued, SPT "is used primarily as a means of determining the reversible work required to introduce a hard-sphere molecule into a real fluid whose molecules behave as hard cores but whose volume and pressure ... are determined by the real intermolecular potentials... ." Pierotti (1976) thus uses p_atm and then the pV term becomes negligible. The choice presumably depends on how G_solv is dissected and on which interactions are being apportioned to the excluded-volume portion of the free energy (G_solv^ev). We point out that if SPT with p = p_hs is used, soft interactions are still included in G_solv^ev. The soft interactions determine the experimental fluid density which is then used as an input parameter.

To gauge how choice of pressure affects G_tr^ev, we recalculated the G_tr^ev data in Figs. 1-4, this time using p = p_hs in place of p_atm. Fig. 7 shows the results for the transfer of amino acid solutes from water to 30% ethylene glycol. Comparing to the calculations with p = p_atm (Fig. 1), we make the same observations as before (section 3.1): G_tr^ev is very sensitive to solvent radii; the sign of G_tr^ev can be either positive or negative; uncertainties in G_tr^ev due to uncertainties in solvent radii are larger than experimental G_tr values. The main difference between G_tr^ev values calculated with p = p_hs versus p_atm is that both the uncertainties and the magnitudes of G_tr^ev are a factor of two or three larger with p = p_hs. Similar conclusions can be drawn for the transfer of amino acids to 60% ethanol, 1 M sucrose, and 2 M urea (data not shown).

View larger version (24K): [in this window] [in a new window]   FIGURE 7   Gtr and Gtrev values for the transfer of amino-acid solutes from water to 30% ethylene glycol. Same as Fig. 1 except that p was set to phs.

Approximating n_w^mix by holding p_hs constant: G_tr^ev values and uncertainties are an order of magnitude smaller

For calculations of G_tr^ev from water to a generic aqueous solvent, n_w^mix cannot be obtained from experiment but must be calculated via some other method. Applying the Gibbs-Duhem relation to the SPT portion of the equation of state, equivalent to holding p_hs fixed at pure water's value (p_hs^wat) (Guttman et al., 1995), is one method of obtaining n_w^mix for any aqueous mixed solvent (Berg, 1990; Guttman et al., 1995).

Formally, this is an approximation. The Gibbs-Duhem relation certainly applies to the entire equation of state, but not necessarily to a part of it. Keeping p_hs constant leads to the unrealistic conclusion that the (very positive) hard-sphere pressure and the (very negative) pressure due to the soft interactions must combine to make up the (nearly zero) atmospheric pressure. If p_hs is fixed, then, since atmospheric pressure is constant, the pressure due to soft interactions must also be constant, irrespective of the cosolvent. However, different cosolvents have different soft interactions, so this cannot be true.

In practice, does this approximation predict reasonable n_w^mix values? Our first test was to calculate solution densities using the approximate n_w^mix values and compare them to experimental densities. Table 3 lists the calculated  values of 30% ethylene glycol (20°C), and the percentage differences from the experimental value. Tables 4, 5, and 6 show the same for 60% ethanol (20°C), 1 M sucrose (25°C), and 2 M urea (25°C), respectively. (Note that n_w^mix and hence the calculated  values depend on the radii of both water and cosolvent.) The constant-p_hs approximation predicts the solution density fairly well, to within a few percentage points of the real value, for aqueous ethylene glycol, sucrose, and urea. The approximation is less good for aqueous ethanol, where the deviations from the correct value can be more than 10%. The poorer quality of the approximation for 60% ethanol is probably due to its higher weight concentration of cosolvent (480 mg/ml, as opposed to 330, 340, and 120 mg/ml for 30% ethylene glycol, 1 M sucrose, and 2 M urea, respectively).

View this table: [in this window] [in a new window]   TABLE 3   The calculated densities, , of 30% ethylene glycol (20°C) using nwmix values obtained by holding phs fixed at phswat, and the percent differences from the experimental

Our second test was to compare our previous G_tr^ev values calculated with the experimentally obtained n_w^mix values, which we denote in this section as G_tr^ev(ex), with those obtained using the approximate n_w^mix values, G_tr^ev(cp). Figs. 8-11 show the G_tr^ev(cp) values for transfer of amino acid solutes from water to aqueous ethylene glycol, ethanol, sucrose, and urea, respectively. Comparing them to the corresponding G_tr^ev(ex) values (Figs. 1-4), we observe that the G_tr^ev(ex) and G_tr^ev(cp) are qualitatively different. 1) G_tr^ev(ex) values and uncertainties are an order of magnitude larger than G_tr^ev(cp) values and uncertainties. The latter are typically one-tenth of a kcal/molcomparable to the experimental G_tr values. With the constant-p_hs approximation, G_trⁱ can be usefully determined. 2) G_tr^ev(cp) values are still sensitive to solvent radii. The uncertainties in G_tr^ev(cp) are still significant relative to the G_tr^ev(cp) magnitudes. 3) For the four mixed solvents studied, G_tr^ev(cp) > 0. In other words, transfer to the mixed solvent is always unfavorable. This was also the case in previous studies with different solutes and solvents (Berg, 1990