INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS CONCLUSIONS REFERENCES

Mixed solvent systems are important to the understanding of the structure and stabilities of macromolecules. For example, the cosolvents urea and guanidine hydrochloride have long been used to study protein denaturation, sucrose is added to stabilize proteins, and alcohols are condensing agents for DNA. (Technically, molecules such as urea, guanidine hydrochloride, and sucrose are cosolutes. However, at typical concentrations, they make up a significant fraction of the solution8M urea is 43 wt % urea and 2 M sucrose is 55 wt % sucrose. These cosolute molecules bathe and solvate the macromolecular solute as water does. To emphasize the cosolvents' role in solvation and to treat them on an equal footing with water, we refer to both cosolvents and cosolutes as "cosolvent" molecules.)

To understand cosolvent-macromolecule interactions, it is necessary to determine where the cosolvent molecules are; do they tend to lie within the bulk solvent or do they prefer to associate with the macromolecule? In particular, what is the distribution (or the composition) of solvent at the macromolecular surface? Currently, there is little information on this. The primary experimental means of obtaining atomic-resolution data on solvent structure are x-ray diffraction and nuclear magnetic resonance (NMR) experiments (Wiithrich et al., 1996). Both types of experiments are fairly complex and require specialized equipment. Also, they tend not to see all solvent molecules equally. Crystallography detects the more ordered waters (Levitt and Park, 1993) whereas NMR more easily sees those with longer residence times and those that are closer to the surface (Otting et al., 1991; Otting, 1997). (In several NMR studies of proteins in mixed solvents, only a few cosolvent molecules could be unambiguously identified (Liepinsh and Otting, 1997; Ponstingl and Otting, 1997; Liepinsh et al., 1999).) Other techniques, such as hydrodynamic, small-angle x-ray and neutron scattering, calorimetric, dielectric, and vapor-pressure absorption isotherm experiments, provide only low-resolution data on surface-associated solvent (see, e.g., Kuntz and Kauzmann, 1974; Pessen and Kumosinsky, 1985; Rupley and Careri, 1991; Svergun et al., 1998). Atomic-resolution molecular mechanics simulations pose as an alternative to experiment; in practice, however, equilibrating the mixed solvent in the presence of a macromolecule is time-consuming and, to date, only two works (Tirado-Rives et al., 1997; Sprous et al., 1998) have published the distributions of neutral cosolvents at a macromolecular surface.

Measurement of the preferential interaction coefficient, ₃,

(1)

is a potentially powerful tool for examining the solvent at a macromolecular surface. (We use the following notation: component 1 is water, 2 is the macromolecular solute, which we sometimes refer to simply as the "solute," and 3 is the cosolvent. We use the term "solvent" to mean the mixed solvent, not water; m_i is the molality of species i.) Experiments to measure ₃ are straightforward (although tedious) and can be done by a variety of methods (see, e.g., Casassa and Eisenberg, 1964; Kuntz and Kauzmann, 1974; Eisenberg, 1976; Schellman, 1990; Timasheff, 1993, 1998; Zhang et al., 1996; and references therein). Many ₃ data have already been gathered on proteins and DNA in the presence of many cosolvents as discussed in several review articles (Kuntz and Kauzmann, 1974; Timasheff, 1993, 1998).

To understand what ₃ tells us about the water and cosolvent distributions (or compositions) at a solute surface, let us first describe in molecular terms what ₃ measures by using the "local-bulk domain" (AKA "two-domain") model (see Record et al., 1998 and references therein), depicted in Fig. 1. The "bulk domain" consists of the region where the solvent displays the same properties as the mixed solvent in the absence of macromolecular solutes. That region is sufficiently far away from any solute so that the solvent doesn't "know" anything about the presence of solutes. In the "local domain" the solvent properties have in some way been affected by the solute. When all species are uncharged and when ₃ is measured at dilute concentration of solute, ₃ can be interpreted as (Record and Anderson, 1995; Inoue and Timasheff, 1972; Reisler et al., 1977)

(2)

where B_i is the number of i molecules (per solute) in the local domain (i = water or cosolvent; m is the cosolvent molality of the mixed solvent in the absence of solute; m₁ is a constant, equal to 55.5 mol/kg for water). Record and Anderson (1995) put this equation on a more rigorous footing by showing that it is exact for the common situation of measurement of ₃ by dialysis equilibrium even when m₂ is not negligibly small. (They also proposed functional forms for how B₃ and B₁ depend on m, in terms of coefficients of partition and stoichiometry of solvent-cosolvent exchange. However, representation in terms of these coefficients is not necessarily unique.) It is not necessary to make any assumptions regarding the nature of the interaction between either solvent species and the solute (Na and Timasheff, 1981); the interactions can be attractive or repulsive or the solute may merely change the structure of the mixed solvent such that the local density of either or both solvent species is altered. One can also write for the preferential hydration, (see Schellman, 1990 and references therein),

(3)

₁ is merely a different way of presenting the information in ₃. In this work we consider only uncharged species. The equations for ₃ and ₁ are modified when charges are present (see, e.g., Record et al., 1998).

View larger version (20K): [in this window] [in a new window]   FIGURE 1   The two-domain model. The "bulk domain" is the region where the solvent has characteristics of bulk solvent, as if there were no solute present. The solvent in the "local domain" is in some way altered by the presence of the solute. There is no well-defined boundary between the local and the bulk domains (Kuntz and Kauzmann, 1974).

(Note that in the literature there is another valid interpretation of Eqs. 2 and 3, in which the B_i values are heuristically thought of as "effective total numbers of [cosolvent] and water molecules in contact with sites on the [solute] surface" (Timasheff, 1998). The exact interpretation used here and by others (Arakawa and Timasheff, 1982b; Record and Anderson, 1995; Hammou et al., 1998) is different in that the B_i values count the number of real solvent molecules in the entire local domain, not solvent molecules effectively contacting the solute surface.)

The sign of ₃ tells us about the local solvent composition as compared to that of the bulk solvent. Dividing Eq. 2 by B₁ (which is positive by definition), we see that the sign of ₃ is determined by the quantity B₃/B₁  m/m₁. B₃/B₁ and m/m₁ are the cosolvent:water number ratios in the local domain and of the bulk binary solvent, respectively. When ₃ < 0 (₁ > 0), the solvent composition (but not necessarily the solvent density) of the local domain is depleted in cosolvent and/or enriched in water relative to that of the bulk binary solvent. This is what is meant by the term "preferential hydration." Likewise, when ₃ > 0 (₁ < 0) the solvent composition of the local domain is depleted in water and/or enriched in cosolvent relative to bulk, and is described by the term "preferential binding" (although there may be no actual binding going on) or "preferential accumulation" of cosolvent. "Preferential solvation" by water/cosolvent is another commonly used term.

Unfortunately, it is not yet clear how to interpret the magnitude of ₃. Because solvent interactions are relatively weak, there is unlikely to be a sharp physical boundary between the local and bulk domains (Kuntz and Kauzmann, 1974). Hence, one cannot say definitively which solvent molecules are counted in B_i and contributing to ₃. Also, for the purpose of understanding solute-solvent interactions, information on the solvent population right next to the solute is desired. However, experimental evidence indicates that the local domain can extend over quite a large territory, even when the interactions are short-ranged. For example, changes in water structure due to solid hydrophobic surfaces can be sensed tens of nanometers away (see, e.g., Blokzijl and Engberts, 1993; Vogler, 1998). Even the water distribution around proteins (Komeiji et al., 1993; Lounnas et al., 1994; Makarov et al., 1998), peptides (Gerstein and Lynden-Bell, 1993), and small solutes (e.g., argon (Blokzijl and Engberts, 1993) and N-methylacetamide (Beglov and Roux, 1996)) as well as the distributions of urea and trifluoroethanol around a methane solute (Smith, 1999) display two or more peaks, corresponding to two or more layers of structured solvent. Thermodynamic experiments suggest that the protein perturbs solvent beyond the first water shell (see Lounnas et al., 1994 and references therein). In summary, not only is it not clear which solvent molecules are counted in B_i, but also the counted solvent molecules may not be contacting, nor binding, nor necessarily even close to the solute. Because of these two issues, it is not yet clear how to turn ₃ data into quantitative information on the solvent population near the solute.

The broad goal of our research is to get quantitative information on the solvent distribution (or composition) at the solute surface from ₃. In particular, we would like to obtain the number of cosolvent (water) molecules within the first cosolvent (water) shell at the solute surface. We call this quantity B(B), "fs" standing for first shell. To obtain B and B from ₃ and B_i data two hurdles need to be overcome.

The first hurdle is that the relationship between B and B_i needs to be determined. The latter measures the number of i's in the entire local domain, which clearly encompasses i's first shell but likely extends further (i.e., B_i  B). However, in practice, because the solvent molecules that are most affected by the solute lie close to it, one expects that the major contribution to ₃ should be due to the first-shell solvent molecules. If ₃ is mostly a function of the B values, then B_i should in some way also be dominated by B. However, this relationship between B_i and B has not yet been worked out. Another way to phrase the issue is that the exact definition of B_i is not very useful for understanding solvent behavior near a solute, as B_i counts the solvent over a poorly demarked and potentially large territory. Can we find an approximate definition of B_i based on the nearby solvent distribution that, when plugged into Eq. 2, gives a good first-order approximation to ₃? If so, then we can get a sense of how ₃ samples the properties of the nearby solvent distribution.

Second, a validated method of obtaining B₁ and B₃ from ₃ is needed. With one experimental quantity (₃) and two desired unknowns (B₁ and B₃), some assumptions on the B_i values or on the solvent distributions must be made. For example, 1) it has been suggested that B₁ be obtained from (or compared to) independent experiments on the macromolecular solute in the absence of cosolvent (Inoue and Timasheff, 1972; Lee and Timasheff, 1974, 1981; Lee and Lee, 1981; Na and Timasheff, 1981; Arakawa and Timasheff, 1982a, b, 1985; Timasheff, 1998; Courtenay et al., 2000a), e.g., by the NMR method of Kuntz (1971) or by the vapor pressure measurements of Bull and Breese (1968), or that B₁ be obtained using a different cosolvent (e.g., a "completely excluded" one (Courtenay et al., 2000a)). The assumption here is that B₁ is unchanged by the addition of (or a change in the) cosolvent. 2) Various researchers have made comparisons of B₁ values to the amount of water in a monolayer (Zhang et al., 1996; Hammou et al., 1998; Courtenay et al., 2000a) suggesting that B₁ ~ B and that the local domain extends only through the first shell of the water, the mixed solvent further out behaving (mostly) like bulk. 3) To explain the effects of denaturants on proteins, it has been assumed that the cosolvent partition coefficient and water-cosolvent exchange stoichiometry are independent of cosolvent concentration (Courtenay et al., 2000b). 4) A steric-exclusion model (Arakawa and Timasheff, 1985; Bhat and Timasheff, 1992) attributes preferential hydration to the cosolvent molecules' larger-than-water size. The cosolvent molecules cannot closely approach the solute, and the region right up against the solute is assumed to be filled only with pure bulk water. 5) In the analysis of osmotic stress experiments, which often aim to measure the number of water molecules "bound" or "released" in a reaction like the binding of a ligand to a protein (Parsegian et al., 1995), it is assumed that B₃ = 0 (see Parsegian et al., 1995; Timasheff, 1998, references therein, and below). Parsegian et al. (1995) discuss under what conditions this approximation is valid, but not all works have followed their careful guidelines. (To translate the language of osmotic stress to that of preferential interactions, see Table 1 of Parsegian et al., 2000; also, what is called N_ew, the "excess number of waters," is equivalent to ₁; similarly, N_es, the "excess number of [cosolvent molecules]," is ₃. By equating N_ew with a difference in number of waters associated with the product versus with the reactant, the implicit assumption is that B₃ = 0.) 6) By assuming that B₁ and B₃ values are constant, independent of solvent composition, the B_i values can be readily obtained from the slope and intercept of linear ₃ versus m (or ₁ versus 1/m) plots (Reisler et al., 1977; Lee and Lee, 1979, 1981; Na and Timasheff, 1981; Lee and Timasheff, 1981; Eisenberg, 1994).

Clearly, various assumptions have been used to understand ₃ (or ₁) data or to extract B₁ and B₃ values from these data; to date, none of these assumptions has been validated against surface solvent distributions.

To overcome these two hurdles, what is needed is a way to measure, on the same solute-mixed solvent system, both ₃ and the water and cosolvent distributions around the solute. From the latter, B_i and B values can be calculated. As discussed above, obtaining solvent distribution data on mixed solvents from either experiment or from atomic-resolution simulations is difficult. We therefore turn to modeling simple systems. In this work, we have performed Monte Carlo simulations on two-dimensional (2D) systems consisting of a 2D "box" with two hard "walls"; the box contains a binary mixture of small (component 1) and large (component 3) circular hard disks (see the sample configuration in Fig. 2 A). The hard wall mimics a macromolecular surface (component 2). The small and large hard disks are simplified representations of the water and cosolvent molecules, respectively. The number and size ratios of the small-to-large disks are varied to study different solvent compositions and cosolvents of different sizes. Because of the absence of directed soft interactions, we can investigate excluded-volume contributions to ₃. We chose to model a 2D system because the lower dimensionality lets us simulate a system with many fewer molecules (a ratio of ~N^1/3 fewer than in three dimensions) with a considerable savings in computational time. This is critical because it is difficult to gather sufficient statistics to precisely determine ₃ (which is a measure of fluctuations). The time savings allows us to examine many different water:cosolvent compositionsincluding those fairly dilute in cosolventand several different cosolvent sizes. Equilibration was not a serious issue (as it is for more realistic simulations) except in cases of very dilute cosolvent and/or large cosolvent molecules. Because our conclusions are qualitative, originating from a sound physical basis, extending our results to three dimensions and to complex-shaped molecules is straightforward.

View larger version (61K): [in this window] [in a new window]   FIGURE 2   Sample configurations of the ms (A) and the bulk (B) boxes. The small light gray circles represent the water molecules, the large dark gray circles the cosolvent molecules. The dotted box boundaries represent the periodic boundary conditions, and the solid boundaries the hard walls. Here, R3/R1 = 1.6 and r = 0.5825.

For each of the many solvent compositions and cosolvents we studied, we analyzed the surface solvent distributions and made comparisons to ₃. We found the following: 1) although the local domain in principle can extend quite far from the solute surface, in practice, its boundary can be demarked by a cosolvent (not water) monolayer, if solvent structure is short-ranged. (The cosolvent monolayer is thicker than the water monolayer because cosolvent molecules are larger than water molecules.) 2) The solvent structure associated with the second and further layers of solvent can mostly be ignored (if solvent structure is short-ranged). 3) Combining 1) and 2) yields a recipe for B_i (see also Fig. 3 B):

(4)

(5)

where is the average water number density in the bulk mixed solvent, and (V  V) is the difference between the volume of space corresponding to the cosolvent first shell (V) and that corresponding to the water first shell (V). (V  V) is the amount of water that would be in the region of space between the end of the water first shell and the end of the cosolvent first shell if it were filled with bulk mixed solvent. 4) Including only first-shell waters in B₁ is not a good approximation (B₁  B). These results are, of course, specific to the simple 2D solvent/solute system we studied. However, because the main assumption behind these conclusions is that the degree of solute-induced solvent structure associated with the first cosolvent shell is much larger than that associated with the second and further shells (and likewise for water), these results may serve as a first-order approximation for real solvents whenever the solvent-structure induced by the solute is short-ranged and neither solvent species is strongly repelled by the solute. Lastly, we also investigated a steric-exclusion model of preferential hydration (Arakawa and Timasheff, 1985; Bhat and Timasheff, 1992). The model's approximations on the solvent distribution are nonphysical and the predictions of ₁ values are not accurate.

View larger version (21K): [in this window] [in a new window]   FIGURE 3   (A) Sample (x) data, normalized by the bulk density in the mixed solvent. This distribution is typical in that there is an unambiguous, large first peak corresponding to the first layer of solvent, and often several smaller peaks corresponding to the second and further layers. In this example, R3/R1 = 2.3 and r = 0.1853. (B) Approximated (x), same as A with the bumps and wiggles associated with the second and further layers of solvent flattened to bulk densities. For this approximate surface solvent distribution, the local domain's outer boundary is at x. Therefore, Bi counts the number of solvent molecules out to x. B1 and B3 are represented by the vertically and horizontally hatched regions, respectively.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS CONCLUSIONS REFERENCES

Model of the solvent and solute

In our simplified model, the water and cosolvent are small and large hard circular 2D disks of radii R₁ and R₃, respectively. The macromolecular solute's surface is modeled as a flat hard 2D "wall" of effectively infinite length. The only interactions are excluded-volume in nature. The solvent molecule disks cannot overlap with each other or with the solute hard wall. Fig 2 A displays a sample configuration.

We chose to model this simple system primarily for the savings in computational time, as mentioned in the Introduction. Representing the solvent and solute as low-resolution hard circular disks and as a hard flat surface, respectively, also speeds computation and allows us to examine excluded-volume effects on ₃ without the complication of shape effects. Future work would involve simulations of the hard sphere model in three dimensions to make direct comparisons with experiment, and might also add soft interactions (see, e.g., Silverstein et al., 1998) and examining differently shaped solutes and solvent.

In this work we performed simulations with several different cosolvent radii: R₃/R₁ = 1.3 (which is approximately the cosolvent:water hard-sphere-radius ratio for methanol); 1.6 (ethanol, ethylene glycol, trifluoroethanol, urea); 2.0, 2.3 (glucose); 3.0 (sucrose); and 4.0 (which is the ratio of the radius of gyration of PEG 200 to the water radius). To obtain the values of R₃/R₁, the hard-sphere radii are averages from values tabulated in Tang and Bloomfield, 2000, except that of trifluoroethanol is an average from the values calculated using the methods described in Edward, 1970 and Ben-Amotz and Willis, 1993, and the radius of gyration of PEG 200 is from Bhat and Timasheff, 1992.

Monte Carlo simulations

We mimic a dialysis equilibrium by simulating two different boxes, one (denoted the "ms" box) containing the hard-wall macromolecular solute surface plus binary solvent; and the other box ("bulk"), containing only the binary solvent. Fig. 2, A and B show sample configurations of the ms and bulk boxes, respectively. Each box, separately, is simulated under the grand canonical ensemble with the parameters (µ₁, µ₃, V). This means that the water and the cosolvent in the ms box are in chemical equilibrium with the water and the cosolvent of an infinite bath of mixed solvent with parameters (µ₁, µ₃); the bulk box's solvent is also in exchange equilibrium with an infinite bath of mixed solvent with the same parameters (µ₁, µ₃). Because the solvent of both the ms and bulk boxes are in chemical equilibrium with the same infinite mixed-solvent bath, the ms and bulk boxes are, in effect, in chemical equilibrium with each other. This is how we mimicked a dialysis equilibrium.

Because both boxes (separately) were simulated under the grand canonical ensemble, the numbers of water and cosolvent molecules (N₁ and N₃, respectively) in each box fluctuate, but their equilibrium averages N₁ and N₃ are well defined. Because of the constraint of not allowing overlap of solvent with the hard walls, there is a very small net reduction of particles from the ms box relative to the bulk box (N^ms_i < N^bulk_i), which is what ₃ detects.

In this paper we examine many different binary solvents, each specified by the variables (R₃/R₁, r, ), where r is the average cosolvent:water number ratio in the bulk binary solvent (r  N/N = m/m₁), and  is the average packing fraction (fractional volume occupancy) of the bulk solution (  (NR₃² + NR₁²)/V). Except where indicated, all studies were performed with  = 0.39-0.40. We chose this value because, at this density, the amount of surface solvent structure is as much as or more than that seen by experiment and high-resolution simulations (see, e.g., Burling et al., 1996; Tirado-Rives et al., 1997; Sprous et al., 1998; Pettitt et al., 1998). In any case, changing the overall packing fraction does not greatly affect the ₃ values. For the solvent system of (R₃/R₁ = 2, r = 0.12) reducing the packing fraction to 0.33 reduces the magnitude of ₃ by 14%; increasing  to 0.5 increases |₃| by 17%. For the solvent sytem of (R₃/R₁ = 2, r = 0.25), the changes in |₃| are 11% and +7%, respectively.

Below, we describe the details of the boxes and the simulations. Table 1 lists the simulation parameters for each of the mixed solvents. The ms box has two opposing hard walls at left and right, and periodic boundary conditions at top and bottom. The opposing hard walls are sufficiently far apart that they are separated by a large region of solvent that is bulk-like (i.e., whose density is uniform at the same value as in the bulk box). The opposing hard walls are thus not interacting with each other and the limit of dilute solute is maintained. In the bulk box we have implemented periodic boundary conditions in all directions. Starting configurations were created by one by one, placing each solvent molecule at a random location; if there was an overlap, a new random location was selected repeatedly until a nonoverlapping location for the molecule was found. The starting values of N_i were the same for both the ms and bulk boxes.

The Monte Carlo moves for each solvent species, i, consisted of 1) displace steps, in which a random molecule of species i is displaced to a new location; the new location is random within a square with sides of length 2R₁ centered at the molecule's original location; 2) create moves, in which a molecule of species i is created at a random location in the box; and 3) destroy moves, in which a random molecule of species i is removed from the box. The sequence of events during the Monte Carlo sequence is 1) a species (water or cosolvent) is chosen randomly; 2) a type of move (displace, create, or destroy) is chosen randomly; 3) the move is accepted or rejected based on a Metropolis criterion (see, e.g., Allen and Tildesley, 1987). In practice, because there are only hard interactions, the displace move is accepted as long as there is no overlap. The acceptance probability for the create move is min[1, z_iV/(N + 1)], where z_i is the activity of species i and N is the number of i's in the box before the move is attempted. z_i  exp(µ_i/(kT))/ (_i is the thermal de Broglie wavelength _i = (h²/2m_ikT)^1/2). The acceptance probability for the destroy move is min[1, N/z_iV]. 4) The move is performed. In the case of displace and create moves, if there is an overlap, the move is rejected. Checking for overlaps was the most time-consuming calculation, and to do it less frequently, this step was performed after step 3. 5) The cycle of steps 1-4 is repeated until the desired number of Monte Carlo trial moves has been executed. (Note that this sequence of events is different from in a more typical scenario when there are soft interactions. In the latter case, the acceptance criterion depends on the total energy of the system, which depends on the configuration of molecules; the move must then be attempted before the acceptance criterion is calculated and hence steps 3 and 4 must be reversed.) Because our solvent consists of a mixture of hard disks, we took advantage of scaled-particle theory to set the solvent activities, z_i, such that the equilibrium values of were very close to the values we desired. As µ_i = kTln() + W_i (W_i is the work of inserting an i at a fixed site; see, e.g., (Ben-Naim, 1974), z_i =  exp(W_i/(kT). We obtained W_i/(kT) from scaled-particle theory in two dimensions (Lebowitz et al., 1965). To increase the computational speed, we used the cell method of neighbor lists (see Allen and Tildesley, 1987) for the waters. For each binary solution (R₃/R₁, r, ), we performed at least six simulations with different random seeds to obtain statistics and error bars. Error bars were calculated as the standard error.

Calculation of ₃ and B^fs

During each simulation run, for each box (ms or bulk), we gathered N₁ and N₃ data. For the ms box, we also calculated ₁(x) and ₃ (x), the average densities of water and cosolvent molecule centers, respectively, at a distance x from the hard wall. Fig. 3 A, shows an example of such a distribution.

We calculated ₃ based on its definition. If we divide Eq. 1 by the molecular weight of water, we see that ₃ can be recast in terms of the cosolvent(solute):water number ratios, r_i (N_i/N₁): ₃ = r₃/r₂. Because the ms box is essentially the bulk box with two solutes (two hard walls, far apart) added to it, we can directly calculate ₃ by approximating the derivative by a difference:

(6)

where  indicates the difference between the ms and bulk boxes (e.g., r  r^ms  r^bulk). Let us now determine r₃ and r₂:

(7)

and

(8)

Because the ms box is dilute in solute, N₃ N and N₁ N. Inserting Eqs. 7 and 8 into Eq. 6, and performing a Taylor's expansion keeping only first-order terms in N₃/N and N₁/N, we obtain

(9)

(The factor of 2 arises because there are two solutes.)

Note that this method of obtaining ₃ from a grand canonical simulation is different than that of Record and co-workers (Mills et al., 1986; Olmsted et al., 1989, 1991, 1995) who simulated without explicit water. Because our simulations contain explicit water, they are more useful for examining effects where the interactions with water are comparable in magnitude with interactions with cosolvent.

B is defined as the number of i molecules within the first peak of _i(x):

(10)

where x is the location of the first minimum in the _i(x) distribution (see, e.g., Fig. 3 A) and demarks the "end" of the first shell of species i; x is the thickness of a monolayer of i. Note that, as the cosolvent is larger than water, a cosolvent monolayer is thicker than a water monolayer (x > x).

In this paper, all lengths are reported in units of the water radius (i.e., a water radius is "1"). In the limit of dilute solute, the amount of solvent accumulated at a surface depends linearly on the surface area if the surface is uniform. Therefore, ₃, ₁, B_i, and B are all proportional to the length of the hard wall. In this paper, ₃, ₁, B_i, and B values are all reported per unit length (=R₁) of hard wall.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS CONCLUSIONS REFERENCES

How B_i depends on the solvent near the solute

In practice, the local domain is demarked by the outer edge of a cosolvent monolayer; B_i depends mostly on B.

View larger version (13K): [in this window] [in a new window]   FIGURE 4   The bumps and wiggles in the solvent distribution associated with the second and further solvent layers can be ignored. If the peaks and valleys associated with these further solvent shells have been flattened to bulk density (see, e.g., Fig. 3 B), the resulting approximate preferential interaction coefficient () compares excellently to the exact 3. The key indicates the values of R3/R1. Except where shown, the y-error bars are smaller than the points.

Assuming that B₁ counts only the water in a monolayer leads to the prediction of the wrong sign for ₃

View larger version (13K): [in this window] [in a new window]   FIGURE 5   Assuming that B1 is the amount of water in a water monolayer predicts the wrong sign for 3. If it is assumed that Bi is the amount of i in an i monolayer, then B  rB is the corresponding preferential interaction coefficient. Comparing these values with the exact 3 values, we see that for all the mixed solvents we investigated, B  rB values are positive, whereas the 3 values are negative. (The key indicates the values of R3/R1; y-error bars for R3/R1 = 1.6 and 2.3 were omitted because they are smaller than the point.)

How thick is a cosolvent monolayer?

View larger version (9K): [in this window] [in a new window]   FIGURE 6   How thick is a cosolvent monolayer? This figure accompanies the heuristic argument in the text that describes where the peaks and valleys in the surface cosolvent distribution should be. It shows the molecular configurations corresponding to the first two peaks of the cosolvent distribution. The dark gray circles correspond to the cosolvent molecules for which 3(x) is observed; the light gray circle corresponds to the water that serves as a spacer between the wall and the observed cosolvent molecule.

View larger version (12K): [in this window] [in a new window]   FIGURE 7   The thickness of a cosolvent monolayer increases linearly as a function of cosolvent size. The outer delimiter of the cosolvent monolayer, x, is shown as a function of cosolvent size (in units of R1). For a given cosolvent, x depends only weakly on the solvent composition, so data for all solvent compositions are plotted together (x and x shift slightly closer to the solute surface as the cosolvent concentration increases; data not shown). The heavy solid line is our underestimate x > R3 + R1. x ~ R3 + 2R1 (light solid line) provides a rough rule of thumb for locating the end of the first cosolvent layer. The best-fit straight line, x = aR3 + bR1 where a = 0.99 ± 0.02 and b = 1.83 ± 0.05, is also shown (dashed line).

When determining B₁, count waters from the same region of space as when determining B₃