Preferential Solvation - American Chemical Society

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Preferential Solvation: Dividing Surface vs Excess Numbers Seishi Shimizu*,† and Nobuyuki Matubayasi*,‡,§ †

York Structural Biology Laboratory, Department of Chemistry, University of York, Heslington, York YO10 5YW, United Kingdom Division of Chemical Engineering, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan § Elements Strategy Initiative for Catalysts and Batteries, Kyoto University, Katsura, Kyoto 615-8520, Japan ‡

ABSTRACT: How do osmolytes affect the conformation and configuration of supramolecular assembly, such as ion channel opening and actin polymerization? The key to the answer lies in the excess solvation numbers of water and osmolyte molecules; these numbers are determinable solely from experimental data, as guaranteed by the phase rule, as we show through the exact solution theory of Kirkwood and Buff (KB). The osmotic stress technique (OST), in contrast, purposes to yield alternative hydration numbers through the use of the dividing surface borrowed from the adsorption theory. However, we show (i) OST is equivalent, when it becomes exact, to the crowding effect in which the osmolyte exclusion dominates over hydration; (ii) crowding is not the universal driving force of the osmolyte effect (e.g., actin polymerization); (iii) the dividing surface for solvation is useful only for crowding, unlike in the adsorption theory which necessitates its use due to the phase rule. KB thus clarifies the true meaning and limitations of the older perspectives on preferential solvation (such as solvent binding models, crowding, and OST), and enables excess number determination without any further assumptions. perspective (stoichiometric binding model of solvation)1,2,18−23 was superseded by the second (adsorption and dividing surface approach to solvation),3,4 and was refined further by the third (KB theory).5−17 The third perspective, KB theory, has since then been applied successfully to clarify the molecular mechanism of protein denaturation, stabilization, and binding in the presence of various cosolvents.5,9−17 Despite all the successes of the new KB perspective, however, the following fundamental questions remain unclear and unaddressed: (i) Can the KB theory be applicable to the systems much larger than macromolecular solutes, such as supramolecules, micelles, membranes, and surfaces, to determine the excess numbers? Where do we draw a line between solutes and surfaces? (ii) What really is meant by the number of water molecules released upon binding and allosteric effect estimated by OST, which are derived from the dividing surface concept imported from the adsorption theory?1−5 These questions arise from a similarity in form between the preferential solvation and the Gibbs adsorption theories.3,4 Indeed, Gibbs adsorption has been addressed using molecular distribution functions or KB theory.23,24 1. Solvation vs Adsorption in the Historical Perspective. What is fundamentally important in the OST perspective is a parallel between solvation and adsorption: the cosolventinduced modulation of the solvation free energy (or surface tension) is understood by the competition between the excess

I. INTRODUCTION Chemical processes are profoundly influenced by the solvent water.1−5 However, rarely is water pure and unmixed: the additional component, the cosolvent, can drastically alter the rates and equilibria.1−5 Hence, cosolvents are exploited routinely for solubilization and extraction in chemical and pharmaceutical industry, as well as for protein denaturation, stabilization, and crystallization in biosciences.1−5 This paper aims to establish a new perspective for the understanding of the cosolvent effect, and therefore a clear guideline on how to interpret thermodynamic measurements. The goal is to obtain molecular-level information on solute− water and solute−cosolvent interactions solely from experimental data. This has been made possible by the recent development in the application of a rigorous statistical thermodynamic theory, the Kirkwood−Buff (KB) theory (1951)6 and its inversion procedure by Ben-Naim (pioneered in the 1970s),7,8 to the interpretation of thermodynamic measurements.5,9−17 Why, then, is a new perspective necessary now? This is because the existing perspectives, which have provided fruitful insights in their long history, have nevertheless introduced significant controversies and confusions in the molecular-based interpretation of thermodynamic data.1−4 A prime showcase of all the different perspectives is the debate regarding how to estimate hydration changes which take place upon biomolecular processes, such as binding and allosteric effect.1−5 This can be done, according to the osmotic stress technique (OST), through the use of “inert” cosolvents (frequently referred to as osmolytes in protein biophysics) to modulate the equilibria.3,4 The theoretical foundation of OST has long been debated.1−5 It is in this debate that the first historical © 2014 American Chemical Society

Received: October 25, 2013 Revised: March 17, 2014 Published: April 1, 2014 3922

dx.doi.org/10.1021/jp410567c | J. Phys. Chem. B 2014, 118, 3922−3930

The Journal of Physical Chemistry B

Article

numbers (or surface excesses) of water and cosolvent.1−17 However, here is the problem: the measurable quantity (the modulation of solvation free energy or surface tension) should be explained by the two unknowns (excess numbers or surface excesses of water and cosolvent).1,2,23,25 These two unknowns cannot be determined independently only from a single measurement.1,2 How can we resolve this indeterminate problem? The answer for the adsorption theory is the Gibbs dividing surface.3,4,23,25,26 By appropriately positioning the dividing surface, the surface excess of water can be made zero.3,4,23,25,26 The surface-tension change due to the introduction of the cosolvents can therefore be attributed entirely to the surface excess of the cosolvent.3,4,27−31 The indeterminate problem can thus be circumvented elegantly by introducing the dividing surface. In OST, the modulation of the solvation free energy is entirely attributed to the excess number of water.3,4,27−31 Hence, the number of water molecules released upon binding or allosteric effect can be directly obtained.3,4,27−31 This was made possible, albeit implicitly in the beginning, by introducing the dividing surface into the preferential solvation theory.3,4 This major progress from the older binding perspective, however, has stirred significant controversy,1−5 as will be reviewed in the following. 2. Estimating the Change of Hydration. How many water molecules are released when a ligand binds a protein or when an allosteric transition takes place? This number, according to OST, can be experimentally accessible by measuring the “volume of hydration”, namely, the dependence of the reaction Gibbs energy change upon the “osmotic pressure” (whose meaning will be clarified below).3,4,27−31 To this end, OST exploits “inert” cosolvents called osmolytes, such as sugars, polyols, or polyethylene glycols.3,4 Since these osmolytes are strongly excluded from biomolecular surfaces (such as grooves, cavities, binding sites, and channels), the dividing surface can be placed outside the hydration shell, into which the osmolytes do not enter.3,4 Hence, the hydration shell is subjected to the “osmotic pressure”.3,4 It is for this purpose that the dividing surface is used as the theoretical foundation for the estimation of hydration changes. However, the use of the dividing surface around the biomolecular solutes has been subjected to considerable controversy.1−5 The most important experimental reason behind the debate is the accumulated evidence that the hydrostatic pressure and the “osmotic pressure” modulate the same reaction very differently (even to the opposite direction in many cases), leading to a widely discrepant estimation in hydration change.32−35 Which pressure should we really use to estimate the hydration change? Thus, the newest perspective, KB theory, has been proposed to clarify this puzzle.5,9−17 This rigorous statistical thermodynamic approach recognized the need for two independent thermodynamic quantities for the determination of the two unknowns: excess numbers of water and osmolyte.5,9−13 The hydrostatic pressure and the “osmotic pressure” have thus provided complementary information toward the determination of the two excess numbers.5,9−13 3. Hydration and Crowding. A question, however, remains unresolved: what really is the physical meaning of the number of water molecules determined by OST?1−5 This question becomes even more puzzling upon realization that the experimental result analyzed by OST can also be rationalized

independently from a molecular crowding perspective: the exclusion of osmolytes from the solute surface is a major cause of the “osmotic pressure” dependence of the chemical reaction (see section III for a full discussion).36,37 In cases that OST’s “volume of hydration” is exactly the same as the excluded volume (namely, the volume from which the osmolytes are excluded) in the molecular crowding perspective, it is even more imperative to clarify what OST’s “volume of hydration” really means. In order to answer this question, we need to clarify, based upon KB theory, the physical meaning of the dividing surface, which is the theoretical foundation of OST.

II. THEORY 1. A Solute or a Surface? A species u has been introduced at infinite dilution into a two-component solvent mixture. This u may be a micelle, membrane, or bubble. Is u, then, a solute or a surface? Indeed, what really is the difference between a surface and a solute? Gibbs’s phase rule gives a clear answer to these questions; for the two-component mixture in the presence of u, the degree of freedom (f) is related to the number of phases p by f = 4 − p.26,38 (Note that the f degrees of freedom can be changed without changing the number of macroscopically observable distinct phases that are present in the system.) It is this rule that gives rise to an experimental criterion to distinguish a solute from a surface. Solute. If three thermodynamic parameters (all of temperature, pressure, and composition) are independently changeable (f = 3), p = 1 according to the phase rule. The solvent mixture remains in a single phase. Surface. If two thermodynamic parameters are changeable independently ( f = 2, one of temperature, pressure, and composition is dependent), p = 2 according to the phase rule. The solvent mixture separates into two phases. 2. Preferential Solvation: KB Theory. Here we clarify that information regarding solute−water and solute−cosolvent interactions can be determined from thermodynamic data when the system is of single phase. The theory presented here is a generalization of the tradition which goes back to Wyman (who discovered the thermodynamic linkage relationships),18,19 to Casassa and Eisenberg (who have developed a statistical thermodynamic theory of multicomponent solutions),20 to Schellman and Tanford (who have pioneered solvent binding and exchange models),21,22 and to Timasheff (who has developed the concept of preferential solvation).1,2 The main focus of these theories was the cosolvent (osmotic)-induced equilibrium shift.1−4,18−22 It is in the KB-based generalization that the complementary nature of the “osmotic” (cosolvent) and hydrostatic effects has been clarified, as will be explained below. Consider a three-component solution consisting of solute (i = u), water (i = 1), and cosolvent (i = 2) molecules. Now, let us divide the solution into two parts: the first part (called the “solute’s vicinity”) contains a solute molecule, and the other part (called the “bulk”) is far away from the solute. To explore the thermodynamic consequence of this soluteinduced concentration change of the solvent species, let us first write down the Gibbs−Duhem equations for each part, vicinity (represented by *) and bulk:5,9,38,39 Nu*dμu + N1*dμ1 + N2*dμ2 − V *(dP + dΠ) + S*dT = 0 (1) 3923



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N1dμ1 + N2dμ2 − V dP + S dT = 0

change of μ1 under constant temperature and pressure is brought about by changing the solvent composition (i.e., increasing the cosolvent concentration). This is the quantity that OST measures or, to be precise, the difference of νu1 between two configurational states. (Note that −(∂μu/ ∂μ1)T,P,nu→0 = −(∂μ*u /∂μ1)T,P,nu→0 follows from −RT((∂ ln nu)/∂μ1)T,P,nu→0 = 0, since nu is kept constant during differentiation.) Partial Molar Volume.

(2)

where Ni and μi are respectively the number and the chemical potential of the species i, S is the entropy, P is the pressure, Π is the osmotic pressure due to the presence of a solute, and T is the temperature. Now we combine eqs 1 and 2; by introducing the concentration (number per volume, ni = Ni/V) and the entropy density (s = S/V), the dP term cancels out elegantly: nudμu − RT dnu + (n1* − n1)dμ1 + (n2* − n2)dμ2 + (s* − s)dT = 0

⎛ ∂μ ⎞ Vu = ⎜ u ⎟ ⎝ ∂P ⎠T , N

(3)

Here, in addition, we have used the van’t Hoff equation for the osmotic pressure (Π = RTnu), where R is the gas constant, which holds true at the infinite dilution of the solute. Thus, the Gibbs−Duhem eqs 1 and 2 have now been rewritten explicitly in terms of the concentration change (n*i − ni) in the solute’s vicinity.5,39 Now we introduce the concept of the excess solvation number of the species i around the solute defined as

Nui =

ni* − ni nu

i

⎛ ∂μ* ⎞ ⎛ ∂ln(Nu/V ) ⎞ = ⎜⎜ u ⎟⎟ + RT ⎜ ⎟ ⎝ ⎠T , N ∂P ⎝ ∂P ⎠T , N i i

= −V1Nu1 − V2Nu2 + RTκT

where Vi is the partial molar volume of the species i and κT is the isothermal compressibility of the solution, which is usually negligibly small. This quantity can be measured by densitometry or from high pressure experiments.5,34 These two experiments, νu1 and Vu, are independent; this is guaranteed, as a consequence of the phase rule, by the fact that μ1 and P are independent thermodynamic parameters which can be varied at constant temperature. Thus, Nu1 and Nu2, which contain crucial information on solute−water and solute−cosolvent interactions, can be determined from two independent experiments.5,9−15 3. Gibbs Adsorption Isotherm. In stark contrast to the case of solutes, the surface excesses of water and cosolvent cannot be determined independently, as we demonstrate in the following. We start from the Gibbs−Duhem equations, one with the surface, two without the surface in the gas and the liquid phases, as25

(4)

which have the following microscopic interpretation through the solute−solvent distribution function, gui(r), as a function of solute−solvent distance r: Nui = niGui ,

Gui = 4π

∫0

∞

[gui(r ) − 1]r 2 dr

(5)

Here Gui is referred to as the Kirkwood−Buff (KB) integral.5,39 We can now express the solvation free energy μu* (defined in eq 6) in terms of μ1, μ2, and T. This is done by combining eqs 3 and 4 into the following form: −dμu* ≡ −d(μu − RT ln nu) = Nu1dμ1 + Nu2dμ2 +

s* − s dT nu

(6)

which is the fundamental KB relationship. The same relationship can also be represented in terms of μ1, P, and T, which can be obtained, via eq 2, by eliminating μ2: ⎛N ⎞ ⎛ n s* − −dμu* = ⎜Nu1 − 1 Nu2⎟dμ1 − ⎜ u2 S − n2 ⎠ nu ⎝ ⎝ N2 Nu2 dP + n2

s⎞ ⎟d T ⎠ (7)

(10)

N1l dμ1 + N2ldμ2 − V l dP + S l dT = 0

(11)

N1g dμ1 + N2g dμ2 − V g dP + S g dT = 0

(12)

−A dγ = N1sdμ1 + N2sdμ2 + (S* − S l − S g)dT

⎛ ∂μ* ⎞ ⎛ ∂μ ⎞ n = −⎜⎜ u ⎟⎟ = Nu1 − 1 Nu2 νu1 = −⎜⎜ u ⎟⎟ n2 ⎝ ∂μ1 ⎠T , P , n → 0 ⎝ ∂μ1 ⎠T , P , n → 0 def

= n1(Gu1 − Gu2)

A dγ + N1*dμ1 + N2*dμ2 − V *dP + S*dT = 0

where A is the surface area and γ is the surface tension; the superscripts l, g, and *, respectively, represent the liquid, gas, and in the presence of the interface. Now we exploit the fact that the system’s volume does not change by the presence of the surface, V* − Vl − Vg = 0, for the purpose of eliminating the dP term. This is in contrast to the case of preferential solvation, where there were only two Gibbs−Duhem equations, in which a mere division by V could eliminate the dP term. Combining eqs 10−12, we obtain

On the basis of the above two different representations (eqs 6 and 7), we can now show that both μ1 and P dependences of μu* (which have often been referred to as the effects of “osmotic pressure” and the hydrostatic pressure, respectively) correspond to the following well-known experiments: Preferential Hydration Parameter.

u

(9)

(13)

where Nsi is the surface excess, defined as25

u

Nis = Ni* − Nig − Nil

(8)

This quantity is experimentally accessible by the measurement of the solvation free energy μu* as a function of μ1. Here the

(14)

Now, in a strict parallel to eq 7, we convert the variables using eqs 11 and 12, which yields 3924



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⎡ ⎛ nl − ng ⎞ ⎤ 1 ⎟N s⎥dμ1 −A dγ = ⎢N1s − ⎜⎜ 1l g⎟ 2 ⎢⎣ ⎝ n2 − n2 ⎠ ⎥⎦

The debate on whether the excess numbers are indeterminates1−5 has thus been resolved clearly through the phase rule and the Gibbs−Duhem equations.

⎡ ⎤ ⎛ Sl Sg ⎞ ⎢ ⎜ V l − V g ⎟ s⎥ l g + ⎢ (S * − S − S ) − ⎜ l N2 dT ⎜ n2 − n2g ⎟⎟ ⎥⎥ ⎢⎣ ⎝ ⎠ ⎦

III. THE “OSMOTIC PRESSURE” VS THE HYDROSTATIC PRESSURE In the preceding section, there was no restriction on the solute size, and the KB theory can now be applied to systems much larger than those studied before, such as ion channels and actin polymerization.27−31,40−44 What OST measures is the osmolyte-induced equilibrium shift. To analyze this in the framework of the KB theory, we now introduce the difference in the solvation free energy μ*u between the two configurational states, which is denoted as Δμ*u . Since OST deals with the infinite dilution limit of the osmolytes, we obtain, from eqs 8 and 9, the following useful relationships exact at this limit:5,9

(15)

nl1 and ng1 are the densities of species 1 in the bulk liquid and gas phases, respectively, and nl2 and ng2 are those for species 2. Equation 15 is analogous to eq 7, which may inspire an expectation that KB theory and the Gibbs adsorption isotherm may be analogous. Thus, can the two surface excesses (Ns1 and Ns2) be determined directly from experiments, through the μ1 and/or P dependence of the surface tension? Here we show that it is impossible to determine both of the surface excesses of water and cosolvents from experiments. This is because they are not independent physical quantities. Indeed, an inspection of eq 15 reveals that there are only two independent variables for the adsorption isotherm, instead of three for the preferential solvation. In fact, the surface excesses are related to the density profile ρsi (z) of species i along the surface normal by23,24 N1s −

⎛ ∂Δμ* ⎞0 u ⎟ 0 0 0 0 ⎜⎜ ⎟ ≡ V1 Δνu1 = ΔGu1 − ΔGu2 ∂Π ⎝ ⎠T , P ⎛ ∂Δμ* ⎞0 u ⎟ ⎜⎜ ≡ ΔV u0 = −ΔGu01 ⎟ P ∂ ⎝ ⎠T , Ni

n1l − n1g

N2s n2l − n2g ⎡ ⎤ n l − n1g s (ρ (z) − n2α)⎥ = dz⎢(ρ1s (z) − n1α) − 1l g 2 n2 − n2 ⎣ ⎦

since is equal to 1/n1 at the dilution limit. We emphasize here again that the so-called “osmotic pressure” dependence in 0 the OST literature actually refers to V10Δνu1 , which, as Timasheff pointed out, has nothing to do with the standard definition of the osmotic pressure through dialysis equilibrium.1,2 It is for this reason that the “osmotic pressure” dependence in eq 19 is different from the rigorous KB expressions for dialysis equilibrium.45 Thus, a comparison between the “osmotic pressure” and the hydrostatic pressure is actually to compare the two quantities in eq 19.5 The same formalism can also be applied to the transition states.3,4,27−30 1. Strongly Excluded Osmolytes around the Ion Channels.27−29,40−42 The effects of the osmotic and hydrostatic pressures are compared in terms of ΔG0u1 and V01Δν0u1 for the opening (activation) of the Na+ channel (Table 1), the opening (activation) of the K+ channel (Table 2), and the band 1 → 2 transition of alamethicin (Table 3), with polyethylene glycols (PEGs), sucrose, glucose, and sorbitol as osmolytes. The hydration change (ΔG0u1) is an order of magnitude smaller than V01Δν0u1 for PEGs, sucrose, glucose, and sorbitol (Tables 1−3). This means |V01Δν0u1| = |ΔG0u2 − ΔG0u1| ≫ |ΔG0u1|, hence

∫

(16)

nαi

where z is the coordinate along the normal and is a weighted sum of the liquid density nli and gas density ngi with arbitrary α and is defined as niα = αnil + (1 − α)nig

(17)

Equation 16 is indeed the Gibbs relative excess and can be determined experimentally. Further, it is a convergent integral for any α and the integral value does not depend on α. Equation 16 reduces to eq 6 of ref 24 when the gas density ngi is set to zero. In contrast, the KB-type integral expressed as

∫ dz(ρis (z) − niα)

(19)

V01

(18)

is not well-defined for any value of α, because the density profile ρsi (z) does not have a unique, asymptotic value when separated far from the surface. When α is taken to be 1, the integral converges when it is done only over the liquid side. When α = 0, on the other hand, the convergence is assured when the integral is conducted only over the gas side. The integral of eq 18 is divergent for any α, however, when the domain of integration extends over both the liquid and gas sides. In other words, the KB-type integral does not exist in the form of eq 18, but it does only in a combined form of eq 16. Although a surface is often and rightly considered as the limit of large solute in a single-phase system, the absence of unique asymptotics in the density profile differs sharply from the case of the solute. To process a combined form of KB-type integral (eq 16), some kind of dividing surface needs to be introduced just as has been done by Gibbs, while it is in principle not necessary when the KB-type integral can be independently determined, for example, through eqs 8 and 9.

Table 1. KB Parameters for the Osmolyte Effect upon the Opening (Activation) of the Na+ Channela (in Å3) cosolvent formamide ethylene glycol glycerol erythritol sorbitol glucose sucrose PEG 600

V01Δν0u1 c

91.8 198c 332b 482b 580b 630b 802b 973b

ΔG0u1 = −ΔV0u

ΔG0u2

−58 −58c −58c −58c −58c −58c −58c −58c

−150 −256 −390 −540 −638 −688 −860 −1031

c

a

Na+ channel from the squid giant axon. bKukita (1997).27 cConti et al. (1982).40 Measurements at 10 °C. 3925



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case; molecular crowding is therefore not an excellent approximation. Hence, in contrast to the case of strongly excluded osmolytes, there is no simple one-to-one correspondence between the pressures and the KB parameters; the osmotic pressure here is simply a measure of V01Δν0u1 = ΔG0u1 − ΔG0u2, namely, the difference in the KB parameters. Therefore, the molecular crowding,36,37 i.e., eq 20, becomes a less accurate approximation for the closure of ion channels induced by weakly excluded osmolytes. However, osmolyteinduced equilibrium shift is still in the same direction, because −ΔG0u2 is still larger in magnitude than ΔG0u1. 3. Osmolyte Binding to Actin.31,43,44 The actin polymerization is enhanced by the presence of osmolytes (i.e., negative V01Δν0u1), as has been expected at first sight by the molecular crowding perspective (Table 4).3,4,31,43,44 This enhancement,

Table 2. KB Parameters for the Osmolyte Effect upon the Opening (Activation) of the K+ Channela (in Å3) cosolvent ethylene glycol erythritol glucose sorbitol sucrose

V01Δν0u1

ΔG0u1 = −ΔV0u

ΔG0u2

−61 −61c −61c −61c −61c

−526 −1593 −2099 −2099 −2579

b

c

465 1532b 2038b 2038b 2518b

a + K channel from the squid giant axon. bKukita (2011).29 cConti et al. (1982).41 Measurements at 10 °C.

Table 3. KB Parameters for the Osmolyte Effect upon the Band 1 → 2 Transition of Alamethicin (in Å3) cosolvent PEG 2000 PEG 3400

V01Δν0u1

ΔG0u1 = −ΔV0u

ΔG0u2

−81 −81b,c

−2281 −2381

a,c

2200 2300a,c

b,c

Table 4. KB Parameters for the Osmolyte Effect on the Association of F-Actins in the Presence of Ca2+ (in Å3)

a

Vodyanoy et al. (1993).30 bBruner and Hall (1983).42 cPE lipid under 100 mV transmembrane voltage. Measurements at 10 °C.

|ΔG0u2| ≫ |ΔG0u1|. For these osmolytes, eq 19 is now reduced to the following form: ⎛ ∂ΔGu ⎞0 ⎜ ⎟ ≡ V10Δνu01 ≈ −ΔGu02 ⎝ ∂Π ⎠T , P ⎛ ∂ΔGu ⎞0 ⎜ ⎟ ≡ ΔV u0 = −ΔGu01 ⎝ ∂P ⎠T , Ni

cosolvent

V01Δν0u1

ΔG0u1 = −ΔV0u

ΔG0u2

glycerol TMAO sorbitol glucopyranoside sucrose glucose

−120 −221a −304a −320a −368a −392a

−490 −490b −490b −490b −490b −490b

−370 −269 −186 −170 −122 −98

a

b

a Fuller and Rand (1999).31 bIkkai and Ooi (1966)44 and Matthews et al. (2005).43

(20)

We have thus clarified the difference between the osmotic and hydrostatic pressures:1−5,32−35 • The “osmotic pressure” probes ΔG0u2, the change of channel−osmolyte interaction. • The hydrostatic pressure probes ΔG0u1, the change of channel hydration. • The “osmotic pressure” modulates the equilibrium much stronger than the hydrostatic pressure. We also note that, in contrast to the previous examples,9,32−35 the two pressures modulate the channel opening to the same direction. Now, what are the characteristics of such osmolytes that satisfy eq 20? Since the ion channels expand when they open, the accompanying large negative ΔG0u2 therefore indicates the increase in the contribution from the negative gu2 − 1 to the KB integral, due to the existence of a large region in which gu2 = 0. This means that the osmolytes are excluded from the channel, more so from the open state than from the closed state. The large negative ΔG0u2 thus constructed increases Δμu* of ion channel opening, thereby inducing closure of the ion channels. The above conclusion, i.e., the dominance of the osmolyte exclusion in the osmolyte-induced prevention of ion channel opening, is in full agreement with the molecular crowding perspective;36,37 the “osmotic pressure” for the strongly excluded osmolytes probes crowding, i.e., the osmolyteexcluded volume around the biomolecule. 2. Weakly Excluded Osmolytes around the Ion Channels.27−29,40−42 For osmolytes such as formamide, ethylene glycol, and glycerol, ΔG0u1 is of comparative order to ΔG0u2, even though the latter is still negative. The osmolytes are still excluded from channels but not strongly (Tables 1−3). Hence, the approximation |ΔG0u2| ≫ |ΔG0u1| is not valid in this

according to the molecular crowding, would be due to the decrease in the osmotic-excluded volume, namely, the negative −ΔG0u2 (= positive ΔG0u2). ΔG0u2, however, is negative (Table 4). This indicates the osmolyte−actin binding and its reduction when actin polymerizes, in stark contrast to the expectation from the molecular crowding. Actually, since G0u1 and G0u2 are expected to reduce in magnitude upon polymerization and is more so for the larger one, ΔG0u1 < ΔG0u2 < 0 indicates that G0u1 > ΔG0u2 > 0. In other words, osmolytes are considered to bind actin. However, their binding is not as strong as water: ΔG0u1 is more negative than ΔG0u2 (Table 4). That osmolytes bind actin weaker than water is the reason why the osmolyte enhances actin polymerization. Osmolytes do not have to be excluded to do the job; they only have to bind weaker than water. We thus observe here the preferential hydration1,2 rather than crowding. This preferential hydration scenario1,2 is outside the molecular crowding perspective, and therefore is outside the fundamental assumption underlying OST.

IV. THE USE OF THE DIVIDING SURFACE IN PREFERENTIAL SOLVATION Now we turn to the question of considerable experimental significance: what does the number of water, which has been inferred from OST, really mean?1−5 To answer this question, let us go back to the theoretical basis of OST, the dividing surface, imported from the adsorption theory.3,4 The idea of the dividing surface was first introduced by Gibbs.26 Its main purpose was to circumvent the indeterminate problem inherent in the Gibbs adsorption theory for mixed solvents: only one measurable quantity for the two excess numbers for the two-component solvents.23,25 The dividing surface has elegantly circumvented this indeterminate prob3926



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lem.23,25 Ironically, this very elegance has introduced a new, implicit unknown: the precise position of the surface. Here we are going to address these two historical questions simultaneously. We show that, although the position of the dividing surface cannot in principle be determined for surfaces, it can, by the help of KB theory, be clearly introduced for the solutes. 1. Basic Formalism. Now we divide the solution into two parts, the solvation shell Σ and the bulk B. The dividing surface is the boundary between the two parts. The step distribution function, as defined below, is introduced to define the dividing surface: gΣ(r ) = 0

for

r∈Σ

gΣ(r ) = 1

for

r∈B

VS =

νu01 = NuS1 = n10GuS01 (27)

The physical interpretation of the dividing surface, again, is clear: the volume enclosed by the osmotic dividing surface is introduced in parallel to the volume enclosed by the Gibbs dividing surface; the partner of the Kirkwood−Buff integral is changed from water to the osmolyte. In this sense, VS is a volumetric indicator of solute−osmolyte interaction. For strongly excluded osmolytes, let us now explore the physical meaning of the KB paremeters with G and S dividing surfaces. Using eqs 25 and 27 under |ΔG0u2| ≫ |ΔG0u1|, we obtain

(21)

Using the step distribution function, the excess numbers in Σ can now be defined as NuiΣ

= ni

∫ dr[gui(r) − gΣ(r)] =

niGuiΣ

∫ dr[gui(r) − 1] = ni ∫ dr[gui(r ) − gΣ(r )] + ni ∫ dr[gΣ(r ) − 1]

Nui = ni

(28)

GuS01 = νu01/n10 = Gu01 − Gu02 ≈ −Gu02 = V S

(29)

(23)

Here, VΣ is the volume inside the dividing surface. We also emphasize that the aim of the dividing surface (eq 21) is not to model or approximate the surface but to eliminate one of the two excess numbers by its appropriate positioning, as we will show below.23,25 We emphasize here that νui and Vu are observables and are independent of the choice of Σ. 2. Two Different Dividing Surfaces. There are two possibilities in the positioning of the dividing surface, which has been introduced to the preferential solvation theory. The Gibbs dividing surface (Σ = G) is chosen such that the excess number of water becomes zero:23,25

NuG1 = GuG1 = 0

Figure 1. (a) The meaning of the hydration number according to OST: GSu1 is the integration of the difference (green) between gu1(r) and the osmotic dividing surface gS(r) (red, eqs 21 and 26); gS(r) has been drawn in relation to gu2(r) (pink dotted). (b) At the strongly excluded limit of the osmolytes, GSu1 approaches VS, the volume confined by the dividing surface (green). (c) The meaning of the excess osmolyte number calculated using the Gibbs dividing surface. The difference between gu2(r) and the Gibbs dividing surface gG(r) (orange, eqs 21 and 24) is shown in green for +ve and gray for −ve contribution. gu1(r) is drawn for comparison as a blue dotted line. (d) At the strongly excluded limit of the osmolytes, GGu2 approaches −VS, i.e., the volume confined by the dividing surface (gray).

(24)

The consequence of this choice can be clarified for the first time by the KB theory. Using eqs 8, 9, 23, and 24, we obtain VG =

0 0 0 0 0 S GuG0 2 = − νu1/ n1 = Gu2 − Gu1 ≈ Gu2 = − V

These results vividly clarify what OST really measures. First, the number of water molecules inferred by OST is 0 S n01GS0 u1 = n1V . This actually is the number of bulk water molecules contained within the osmolyte-excluded volume (Figure 1). There is no information about the hydration water structure which is contained in gu1(r) outside the excluded volume. OST, therefore, measures the crowding effect.

(22)

These new excess numbers NΣui can be related to Nui in the following manner:

= NuiΣ − niV Σ

Vu − RTκT + V1NuS1 ⎯⎯⎯⎯⎯→ − Gu01 + (Gu01 − Gu02) = − Gu02 n2 → 0 n1V1 + n2V2

Vu − RTκT + V2NuG2 ⎯⎯⎯⎯⎯→ V u0 − RTκT = −Gu01 n2 → 0 n1V1 + n2V2

νu01 = −n10GuG20 (25)

This result has a clear physical interpretation: the volume enclosed by the Gibbs dividing surface is the nonideal part of the solute’s partial molar volume. The osmotic dividing surface (Σ = S) is chosen to satisfy NuS2 = GuS2 = 0

Second, even if we adopt the Gibbs over osmotic dividing surface, we obtain the same thing: the osmolyte-excluded volume, VS. This is because the preferential hydration 0 0 parameter is simply ν0u1 = −n01GG0 u2 ≈ −n1Gu2. When the excluded volume of the osmolyte is large and its contribution to the KB integral G0u2 overwhelms those from the other domains, G0u2 is the negative of the osmolyte-excluded volume VS (Figure 1). The reason why we obtain the same information for

(26)

NSu2,

such that information on hydration, can be inferred only from the measurement of ν0u1.3,4 The consequence of this choice is apparent by using eqs 8, 9, 23, and 26: 3927



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The major advantage of this perspective is a direct connection between the solute−solvent radial distribution functions and thermodynamic quantities. Consequently, the power and applicability of this approach are not limited to thermodynamic measurements alone. X-ray and neutron scattering experiments,46 for which gui(r) is accessible, have now gained a powerful link to thermodynamics, which has been impossible without the KB framework.

strongly excluded osmolytes regardless of the choice of the dividing surface is illustrated clearly in Figure 1. Third, hydration structure contained in gu1(r) can only be accessible via Gu1, which requires the simultaneous equations (eqs 8 and 9) to be solved. For weakly excluded osmolytes, bound osmolytes and denaturants, in contrast, neither eq 25 nor eq 27 leads to simplification like eqs 28 and 29, which have a simple physical interpretation. Thus, the simultaneous KB eqs 8 and 9 should be solved for a clear understanding of the solute−water and solute−osmolyte interactions. This is especially important when the osmolytes actually do bind the biomolecule, which cannot be known a priori other than solving the simultaneous KB equations.

VI. CONCLUDING REMARKS What is the fundamental difference between the preferential solvation and Gibbs adsorption theories which are similar in mathematical form? Can we apply the preferential solvation theory, which has been reformulated rigorously using KB theory,5−17 to larger systems than has been studied so far, such as to supramolecules, micelles, and membranes?16,17,24 The phase rule has given a clear answer to these questions: KB theory’s condition and the limit of applicability is the existence of three independent thermodynamic parameters for the system. This clear criterion has made it possible to extend the applicability of the KB theory to supramolecules, such as ion channels and actin polymerization. KB theory thus enabled the determination of the excess numbers of water and cosolvents around the supramolecular solute based solely upon the experimental data. However, these numbers cannot be determined independently for surfaces, because there are fewer independent thermodynamic parameters. The phase rule thus clarified the condition upon which KB theory can determine the two excess numbers independently (cf. question i in the Introduction). The apparent similarity between the preferential solvation and Gibbs adsorption theories did in the past inspire major progress in the theoretical prespective; introduction of the dividing surface into preferential solvation has made it possible to determine the number of water molecules released upon binding and allosteric transition through the use of the strongly excluded osmolytes, as has been promulgated by the OST.3,4,27−31 Such numbers of water, however, have been subjected to controversy;1−5,32−35 their meaning has not been completely clarified even after the advent of the new, KB-based view.5−17 Here, through a rigorous KB-based analysis of the dividing surface in the case of strongly excluded osmolyte, we have clarified what such numbers really meanthe number of bulk water contained within the osmolyte-excluded dividing surface, within which hydration structure as reflected by gu1(r) is completely overshadowed. In addition, yet another difficulty in the application of the dividing surface is the fact that it is impossible to know a priori whether the osmolyte in question is really excluded from the biomolecular surface. This doubt can be dispelled only through a direct determination of the KB integrals. Thus, we continue to advocate a direct determination of the KB integrals, Gu1 and Gu2, which contain explicitly, albeit in integrated form, both gu1(r) and gu2(r). In this way, the new perspective is superior to the old ones; it enables, in addition, an alternative approach to thermodynamic measurements through X-ray and neutron scattering.46

V. THE EVOLUTION OF PERSPECTIVE AS VIEWED FROM KB THEORY 1. Solvent Binding Perspective. Let us now clarify the foundation and validity of the oldest perspective, the solvent binding (and exchange) models1,2,18−22 from a KB perspective. The solvent binding model is based on the following approximation: Nui = ni

∫0

∞

d r [⃗ gui(r ) − 1] ≈ ni

∫R

R uimax min ui

d rg⃗ ui(r )

(30)

max where Rmin ui and Rui determine the range of the solvation shell. This approximation, that (excess number) = (coordination number), essentially assumes the following: (i) the excluded volume contribution, namely, the contribution to the KB integral from 0 ≤ r ≤ Rmin ui , is negligible; (ii) the long-ranged contribution to the KB integral, namely, from r ≥ Rmax ui , is negligible; (iii) the solvent binding is strong enough to ensure max gui(r) ≫ 1 in the solvation shell Rmin ui ≤ r ≤ Rui . The above conditions clarify the limits of applicability of the solvent binding models; they are applicable only when the solvent is attracted strongly to the hydration shell, strong enough to overshadow other contributions. This is why the solvent binding model has been shown to be powerless in the face of strongly excluded osmolytes. 2. Dividing Surface Perspective. The dividing surface perspective serves as the theoretical foundation of OST.3,4,27−31 For strongly excluded osmolytes (|ΔG0u2| ≫ |ΔG0u1|), OST becomes exact and equivalent to the crowding theory. On the contrary, the osmotic dividing surface is not useful for weakly excluded osmolytes and denaturants; for weakly excluded 0 0 osmolytes, neither ΔGu1 nor ΔGu2 is negligible, which complicates the interpretation of OST. However, the characteristics of the strongly excluded osmolytes, |ΔG0u2| ≫ |ΔG0u1|, pose a serious limitation on the estimation of hydration changes. Information regarding biomolecular hydration, as reflected in gu1(r) and hence in G0u1, is overshadowed in the face of the dominant ΔG0u2, the very dominance of the osmolyte exclusion. Thus, the resultant number of water molecules is simply the number of bulk water molecules contained within the dividing surface, from which the osmolytes are excluded. 3. KB Perspective. Unlike the solvent binding perspective (limited to the solvents which are strongly attracted to the solute) and the OST/dividing surface perspective (limited to strongly excluded osmolytes), KB theory can be applied to any solute, solvent, and cosolvent molecules, as long as P and μ1 are independently changeable at a given temperature.

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Corresponding Authors

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*Phone: +81 774 38 3071. Fax: +81 774 38 3074. E-mail: [email protected].

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Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We are grateful to Professor Masakazu Sekijima for his hospitality during S.S.’s stay in his laboratory at the Tokyo Institute of Technology and to Dr Fumio Kukita for sending us the raw data for Table 2. This work is supported by the Grantsin-Aid for Scientific Research (Nos. 21300111 and 23651202) from the Japan Society for the Promotion of Science, by the Grant-in-Aid for Scientific Research on Innovative Areas (No. 20118002) and the Elements Strategy Initiative for Catalysts & Batteries from the Ministry of Education, Culture, Sports, Science, and Technology, and by the Nanoscience Program, the Computational Materials Science Initiative, the Theoretical and Computational Chemistry Initiative, the HPCI Strategic Program, and the Strategic Programs for Innovative Research of the Next-Generation Supercomputing Project.

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