Volumetric Properties of Solvation in Binary Solvents - The Journal of

J. Phys. Chem. B 2009, 113, 2443–2450

2443

Volumetric Properties of Solvation in Binary Solvents Soyoung Lee and Tigran V. Chalikian* Department of Pharmaceutical Sciences, Leslie Dan Faculty of Pharmacy, UniVersity of Toronto, 144 College Street, Toronto, Ontario M5S 3M2, Canada ReceiVed: October 8, 2008; ReVised Manuscript ReceiVed: December 15, 2008

The volumetric properties of solutes, including partial molar volume, compressibility, and expansibility, are determined by and, therefore, sensitive to the entire spectrum of solute-solvent interactions. However, applications of volumetric measurements to the study of solvation of solutes in binary solvents consisting of the principal solvent and a cosolvent are almost nonexistent. This deficiency partially reflects the lack of an adequate theoretical framework to rationalize the measured volumetric observables in terms of solute-principal solvent and solute-cosolvent interactions. To address this deficiency, we present in this work a simple statistical thermodynamic model describing solute thermodynamics in binary solvents. Based on the model, we derive relationships that link the partial molar volumetric properties of solutes with the extent and intensity of interactions of a solute with the principal solvent and the cosolvent. If the approximation of homogeneous solvation is introduced, the derived formalism can be simplified to a form that can be readily employed for practical treatment of experimental volumetric data on small solutes and/or individual atomic groups. As an example, we use this simplified formalism to rationalize the volumetric properties of the zwitterionic amino acid glycine in concentrated urea solutions. In general, the theoretical development presented in this work opens the way for systematic applications of volumetric measurements to quantitative characterization of solute-solvent interactions in complex solvents. Introduction Volumetric observables, such as the partial molar volume, compressibility, and expansibility of a solute, are linked to the properties of solute-solvent interactions.1 In recognition of this fact, volumetric measurements have been widely applied to studying the hydration of various organic and inorganic solutes in water.2-7 On the other hand, the use of volumetric measurements for investigating the solvation properties of solutes in binary solvents consisting of water and water-soluble cosolvents has been extremely rare and unsystematic.8-11 Such watersoluble cosolvents are frequently referred to as osmolytes, since many of them are involved in the response of both prokaryotic and eukaryotic cells to the stress of high osmolality environment. The mode of action of osmolytes is based on modulating the equilibria of macromolecular processes due to their differential affinity for individual macromolecular domains.12-16 The lack of volumetric measurements in binary solvents is unfortunate, since, as shown below, volumetric data provide unique, quantitative insights into the nature of solute-solvent and solute-cosolvent interactions. The theoretical link between the partial molar volume, V°, of a solute and solute-solvent and solute-cosolvent interactions have been outlined via the Kirkwood-Buff integrals

Gij )

∫0∞ [gij(r) - 1](4πr2) dr

where gij(r) is the radial distribution function.17-20 The Kirkwood-Buff integral, Gij, represents the average excess (or deficiency) of particles i around the central particle j normalized * To whom correspondence should be addressed. Telephone: (416) 9463715. Fax: (416) 978-8511. E-mail: [email protected].

per number density of i. For the limit of a dilute solution, the partial molar volume of a solute in the mixture of the principal solvent and cosolvent is given by V2 ) βT,0RT - C1V°1G12 C3V°3G32, where the subscripts 1, 2, and 3 refer to the principal solvent, solute, and cosolvent, respectively; βT,0 is the coefficient of isothermal compressibility of the solvent; C1 and C3 are the molarities of the principal solvent and cosolvent, respectively; and V°1 and V°3 are the partial molar volumes of the principal solvent and cosolvent, respectively.19,20 We have recently begun a program in which we systematically employ ultrasonic velocimetric and densimetric measurements to study solvation of proteins and low molecular weight protein analogues in concentrated solutions of denaturing and stabilizing osmolytes. As the first stage of this research, we develop a theoretical framework for rationalization of data on the partial molar volume, compressibility, and expansibility of a solute in binary solvents. To this end, we extend our previously developed statistical thermodynamic formalism to solvation in solvent-cosolvent mixtures.21 We derive equations that connect the affinities, stoichiometry, and volumetric properties of solute-solvent and solute-cosolvent interactions with the partial molar volume, compressibility, and expansibility of a solute. Our derived formalism enables one to quantify and separate the relaxation contributions to partial molar expansibility and compressibility that originate from a shift in the number of interacting solvent and cosolvent molecules under the influence of temperature and pressure. In general, our results open the way for systematic applications of volumetric measurements to characterize solute-solvent interactions in complex solvents. Materials and Methods Urea and glycine were purchased from (Sigma-Aldrich Canada, Oakville, Ontario, Canada) and used without further

10.1021/jp8089159 CCC: $40.75  2009 American Chemical Society Published on Web 02/05/2009

2444 J. Phys. Chem. B, Vol. 113, No. 8, 2009

Lee and Chalikian

TABLE 1: Volumetric Properties of Water as a Function of Urea Concentration [urea], M 3

-1

[U], cm mol V°, cm3 mol-1 K°S, 10-4 cm3 mol-1 bar-1

0

2

4

6

8

0 18.07 8.09

-0.59 ( 0.04 18.06 ( 0.01 7.95 ( 0.03

-0.95 ( 0.03 18.03 ( 0.01 7.71 ( 0.03

-1.41 ( 0.05 17.99 ( 0.01 7.54 ( 0.03

-1.74 ( 0.04 17.94 ( 0.02 7.48 ( 0.03

TABLE 2: Volumetric Properties of Urea as a Function of Concentration [urea], M 3

-1


0

2

4

6

8

17.4 ( 0.1 44.1 ( 0.1 -3.0 ( 0.2

14.5 ( 0.5 44.64 ( 0.03 0.8 ( 0.3

11.3 ( 0.2 44.93 ( 0.04 3.9 ( 0.1

8.9 ( 0.3 45.14 ( 0.08 5.9 ( 0.2

6.4 ( 0.4 45.44 ( 0.05 7.8 ( 0.3

TABLE 3: Volumetric Properties of Glycine as a Function of Urea Concentration [urea], M 3

-1


0

2

4

6

8

35.3 ( 0.1 43.0 ( 0.2 -26.8 ( 0.2

32.7 ( 0.1 44.1 ( 0.2 -20.2 ( 0.2

29.8 ( 0.7 45.1 ( 0.3 -14.8 ( 0.6

27.6 ( 0.2 46.0 ( 0.2 -10.9 ( 0.2

25.6 ( 0.2 46.6 ( 0.2 -8.1 ( 0.2

purification. Before use, urea and glycine were dried for 72 h under vacuum in the presence of P2O5. Concentrated urea solutions (with concentrations of 2, 4, 6, and 8 M) were prepared by weighing 10-50 g of urea and adding preestimated amounts of water to achieve the desired molalities, m. The molar concentration, C, of urea in solution was computed from the molal value, m, using C ) [1/(mFW) + φV/1000]-1, where FW is the density of water and φV is the apparent molar volume of urea. These solutions were used as solvents in which 10-20 mg of glycine, water, or urea was dissolved. The concentration of glycine or changes in concentrations of water and urea were determined by weighing 10-20 mg of a solute with a precision of (0.03 mg and subsequent dissolution of the material in a known amount of water or concentrated urea solution. Solution sound velocities, U, were measured at 25 °C at a frequency of 7.2 MHz using the resonator method and a previously described differential technique.22-25 The analysis of the frequency characteristics of the ultrasonic cells required for sound velocity measurements was performed by a HewlettPackard Model E5100A network/spectrum analyzer (Mississauga, ON, Canada). For the type of ultrasonic resonators used in this work, the accuracy of the sound velocity measurements is about ((1 × 10-4)%.23,26,27 The key acoustic parameter of a solute determined from our ultrasonic velocimetric measurements is the relative molar sound velocity increment, [U], which is equal to (U - U0)/(U0C), where U and U0 are the sound velocities in a solution and the solvent, respectively. All densities were measured at 25 °C with a precision of ((1.5 × 10-4)% using a vibrating tube densimeter (DMA-5000, Anton Paar, Graz, Austria). The partial molar volumes, V°, of the solutes were calculated from V° ) M/F - (F - F0)/(FF0m), where M is the molecular weight of solute, m is the molal concentration (for glycine) or an increase in molal concentration (for water and urea) of solute, and F and F0 are the densities of the solution and the solvent, respectively. The values of [U] were used in conjunction with the V° values derived from densimetric measurements to calculate the partial molar adiabatic compressibility, K°S, using the relationship K°S ) βS,0(2V° - 2[U] - M/F0), where βS,0 ) F0-1U0-2 is the coefficient of adiabatic compressibility of the solvent. The values of F0, U0, and βS,0 were directly determined for each urea solution from our densimetric and acoustic measure-

ments. Each densimetric or ultrasonic velocimetric experiment was repeated three to five times, with the average values of [U], V°, and K°S being reported. Experimental Results Tables 1, 2, and 3 present, respectively, the volumetric properties of water, urea, and glycine in 0, 2, 4, 6, and 8 M urea. Errors were estimated as the standard deviations for each measurement set. Data presented in Tables 1-3 were determined in the concentration range of 1-3 mg/mL. For most solutes, the reported concentration dependences of the apparent molar volumes and compressibilities within a narrow concentration range on the order of ∼5 mg/mL are rather weak.28,29 Therefore, within the limits of our experimental error, our measured quantities correspond to the partial molar volume, V°, and the adiabatic compressibility, K°S, of the studied compounds at each experimental urea concentration. For glycine, the determined values of V° and K°S correspond to extrapolation to infinite dilution in each experimental urea solution; V° ) limNf0 (∂V/ ∂N)P,T and K°S ) limNf0 (∂KS/∂N)P,T. On the other hand, the partial molar volumetric properties of water and urea were determined at very substantial initial concentrations. Therefore, their measured V° and K°S values correspond to partial molar volumes, V° ) (∂V/∂N)P,T, and adiabatic compressibilities, K°S ) (∂KS/∂N)P,T, at finite N (the number of moles of water or urea in solution). Thermodynamics of Solvation in Binary Mixtures We consider a system that includes a solute dissolved in a binary mixture of the principal solvent (water) and the cosolvent (osmolyte). The chemical potential of a solute in such a system is described by the equation19,20,30

µl ) µ°g + ∆G* + RT ln [Sl] ) µ°l + RT ln [Sl] (1) where µ°g is the standard chemical potential of solute in the ideal gas phase, µ°l ) µ°g + ∆G* is the standard chemical potential of a solute in the liquid phase, [Sl] is the molar concentration of a solute in the liquid phase that exists in equilibrium with the solute in the gas phase, and ∆G* is the Gibbs free energy of solvation.

Thermodynamics of Solvation in Binary Solvents

J. Phys. Chem. B, Vol. 113, No. 8, 2009 2445

Solvation is defined here as a process in which a solute is transferred from a fixed position in the ideal gas phase to a fixed position in the liquid (solution) phase.19,20,30 The Gibbs free energy of solvation, ∆G*, is the reversible work required for such a transfer and equals the average binding energy (coupling work) of a solute with the solvent components at some fixed configuration:

∆G* ) -RT ln([Sl]/[Sg])

(2)

S0 + iW + jO T SWiOj

(R1)

where S0 signifies a solute species that, while being in solution, interacts with neither the principal solvent (water) nor the cosolvent. Each reaction is characterized by a binding free energy, ∆Gij, that is contributed by the formation of direct solute-solvent and solute-cosolvent interactions and disruption of preexisting solvent-solvent and solvent-cosolvent (and, possibly, cosolvent-cosolvent) interactions. Therefore, ∆Gij may exhibit a range of values extending from negative (favorable interactions) to positive (unfavorable interactions). The net concentration of a solute in solution can be expressed via the binding polynomial: n/r

n-jr

∑ ∑ j)0

a1ia3j exp(-∆Gij /RT)

(3)

i)0

where a1 and a3 are the activities of the principal solvent and the cosolvent, respectively; n is the maximum number of solvent-binding sites of a solute; r is the number of solvent molecules displaced by a cosolvent molecule upon its association with the solute. Combining eqs 2 and 3, one obtains the following expression for solvation Gibbs free energy:

∆G* ) -RT ln([Sl]/[Sg]) ) n/r

-RT ln[([S0]/[Sg])

n-jr

∑ ∑ j)0

a1ia3j exp(-∆Gij /RT)] )

i)0

n/r

-RT ln([S0]/[Sg]) - RT ln[

n-jr

∑ ∑ j)0

a1ia3j ×

i)0

exp(-∆Gij /RT)] ) GC + GI (4)

n/r

n-jr

j)0

i)0

∑ ∑

∆V* ) (∂∆G*/∂P)T ) VC +

where [Sg] is the molar concentration of a solute in the ideal gas phase that exists in equilibrium with the solute in the liquid phase.19,20,30 To evaluate the concentration of a solute in the liquid phase, we use an approach analogous to that we have employed in our previous work.21 We consider the interactions between a solute and the solvent components as a set of chemical reactions, in which a solute associates with i water (W) and j osmolyte (O) molecules:

[Sl] ) [S0]

is the free energy of direct interactions of solute with the components of the solvent. The term GC is unfavorable and corresponds to the free energy of formation of a cavity in the solvent that is sufficiently large to accommodate the solute. Solvation-induced changes in volume, ∆V*, and enthalpy, ∆H*, can be determined by differentiating eq 4 with respect to pressure and temperature:

Rij∆Vij ) VC + 〈∆V〉(5)

∆H* ) -RT2(∂(∆G*/RT)/∂T)P ) HC + n/r

n-jr

j)0

i)0

∑ ∑

Rij∆Hij ) HC + 〈∆H〉 (6)

where VC ) (∂GC/∂P)T is the volume of the cavity enclosing the solute;

/

n/r

Rij ) a1ia3j exp(-∆Gij /RT) [

n-jr

∑ ∑ j)0

a1ia3j ×

i)0

exp(-∆Gij /RT)]

is the fractional composition of the solute species SWiOj; HC ) -RT2[∂(GC/RT)/∂P]T is the enthalpy of cavity formation. ∆Vij ) (∂∆G ij/∂P)T, ∆Hij ) -RT2[∂(∆G ij/RT)/∂P]T, 〈∆V〉 ) ∑jn/r) 0 n - jr ∑ni )- 0jr Rij∆Vij, and 〈∆H〉 ) ∑n/r j ) 0 ∑i ) 0 Rij∆Hij are, respectively, the ensemble average changes in volume and enthalpy accompanying direct solute-solvent interactions. The expression for the partial molar volume, V°, of a solute can be derived by differentiating eq 1 with respect to pressure and taking into account eq 5:

V° ) (∂µl /∂P)T ) (∂∆G*/∂P)T + RT(∂ ln [Sl]/∂P)T + βT,0RT ) VC +

n/r

n-jr

j)0

i)0

∑ ∑

Rij∆Vij + βT,0RT ) VC + 〈∆V〉 + βT,0RT (7)

where βT,0 is the coefficient of isothermal compressibility of the solvent. The partial molar isothermal compressibility, K°T, of a solute is the negative pressure slope of its partial molar volume, V°:

K°T ) -(∂V°/∂P)T ) -(∂VC /∂P)T n/r

n-jr

∑ ∑ j)0

i)0

Rij(∂∆Vij /∂P)T -

n/r

n-jr

j)0

i)0

∑ ∑

∆Vij(∂Rij /∂P)T RT(∂βT,0 /∂P)T (8)

where GC ) -RT ln([S0]/[Sg]) is the free energy of solvation of the noninteracting solute species and

[

GI ) -RT ln

n/r

n-jr

∑ ∑ j)0

i)0

]

a1ia3j exp(-∆Gij /RT)

Taking into account that KC ) -(∂VC/∂P)T is the intrinsic compressibility of the cavity enclosing a solute, Kij ) -(∂∆Vij/ ∂P)T is the compressibility effect of formation of the solvated complex with i water and j cosolvent molecules, and (∂Rij/∂P)T ) (Rij/RT)(〈∆V〉 - ∆Vij), one obtains the following relationship:


Lee and Chalikian

K°T ) KC + 〈∆KT〉 + (〈∆V2〉 - 〈∆V〉2)/RT RT(∂βT,0 /∂P)T (9) n-r n-jr where 〈∆V2〉 ) ∑j)0 ∑i)0 Rij∆Vij2. The partial molar expansibility, E°, of a solute is the temperature slope of its partial molar volume, V°:

where ∆KT,1 ) -(∂∆V1/∂P)T and ∆KT,3 ) -(∂∆V3/∂P)T are the elementary changes in compressibility accompanying a soluteprincipal solvent association event and a solute-cosolvent association event, respectively. Differentiation of eq 12 with respect to temperature yields the relationship for expansibility, E°:

E° ) EC + n1∆E1 + n3∆E3 + n/r

E° ) (∂V°/∂T)P ) (∂VC /∂T)P + n/r

n-jr

∑ ∑ j)0

n/r

Rij(∂∆Vij /∂T)P +

i)0

n-jr

∑ ∑ j)0

(1/RT2)[∆V1∆H1(

(∆V1∆H3 + ∆V3∆H1)(

i)0

R[(βT,0 + T(∂βT,0 /∂T)P] (10)

E° ) EC + 〈∆E〉 + (〈∆H∆V〉 - 〈∆H〉〈∆V〉)/RT2 + R[βT,0 + T(∂βT,0 /∂T)P] (11) While the “interaction” terms 〈∆V〉, 〈∆KT〉, and 〈∆E〉 in eqs 7, 9, and 11 all linearly correlate with the extent and intensity of solute-solvent and solute-cosolvent interactions, the fluctuation terms (〈∆V2〉 - 〈∆V〉2)/RT and (〈∆H∆V〉 - 〈∆H〉〈∆V〉)/ RT2 in eqs 9 and 11 are principally nonlinear and reflect changes in the extent of solute-solvent and solute-cosolvent interactions under the influence of pressure and temperature. In the next section, we clarify this point by considering a simplified scenario of homogeneous solvation. Homogeneous Solvation. The analysis presented above can be simplified when a solute with identical and independent (noninteracting) solvent-binding sites is considered. For such noncooperative binding, ∆Vij ) i∆V1 + j∆V3, while eq 7 reduces to the form

∑ ∑

i2Rij - n12) +

j)0 i)0 n/r n-jr

∆Vij(∂Rij /∂T)P +

Since EC ) (∂VC/∂T)P is the intrinsic expansibility of the cavity enclosing the solute, Eij ) (∂∆Vij/∂T)P is the expansibility effect of the formation of the solvated complex with i water and j cosolvent molecules, and (∂Rij/∂T)P ) (Rij/RT2)(∆Hij 〈∆H〉), one arrives at the expression

n-jr

∑ ∑ j)0

∆V3∆H3(

n/r

n-jr

j)0

i)0

∑ ∑

ijRij - n1n3) +

i)0

j2Rij - n32)] + R[βT,0 + T(∂βT,0 /∂T)P]

(14) where ∆E1 ) (∂∆V1/∂T)P and ∆H1 are, respectively, the elementary changes in expansibility and enthalpy accompanying a solute-principal solvent association event, while ∆E3 ) (∂∆V3/∂T)P and ∆H3 are, respectively, the elementary changes in expansibility and enthalpy accompanying a solute-cosolvent association event. Note that the partial molar volume, V°, of a solute at a given solvent composition linearly correlates with the composition of its solvation shell (represented by the numbers n1 and n3). However, the partial molar isothermal compressibility, K°T, and expansibility, E°, in addition to the linear terms (n1∆KT,1 + n3∆KT,3) and (n1∆E 1 + n3∆E3), contain the nonlinear fluctuation terms n/r

(1/RT)[∆V12(

n-jr

∑ ∑ j)0

i2Rij - n12) +

i)0

2∆V1∆V3(

n/r

n-jr

j)0

i)0

∑ ∑

ijRij - n1n3) + n/r

∆V32(

∑ ∑ j)0

V° ) VC + n1∆V1 + n3∆V3 + βT,0RT

n-jr

j2Rij - n32)]

i)0

(12) and

where ∆V1 and ∆V3 are the elementary changes in volume accompanying a solute-principal solvent association event and a solute-cosolvent association event, respectively; the effective numbers of principal solvent, n1, and cosolvent, n3, molecules in the solvation shell of a solute (solvation numbers) are given n/r n-jr n/r n-jr ∑i)0 iRij and n3 ) ∑j)0 ∑i)0 jRij, respectively. by n1 ) ∑j)0 Differentiation of eq 12 with respect to pressure yields the relationship for isothermal compressibility, K°T:

(1/RT2)[∆V1∆H1(

n/r

n-jr

j)0

i)0

∑ ∑

i2Rij - n12) +

(∆V1∆H3 + ∆V3∆H1)(

n/r

n-jr

j)0

i)0

∑ ∑

ijRij - n1n3) + n/r

∆V3∆H3(

n-jr

∑ ∑ j)0

j2Rij - n32)]

i)0

K°T ) KC + n1∆KT,1 + n3∆KT,3 + n/r

(1/RT)[∆V12(

n-jr

∑ ∑

i2Rij - n12) +

j)0 i)0 n/r n-jr

2∆V1∆V3(

∑ ∑

j)0 i)0 n/r n-jr j2Rij ∆V32( j)0 i)0

∑ ∑

ijRij - n1n3) + n32)] - RT(∂βT,0 /∂P)T (13)

that do not correlate with the composition of the solvation shell. These terms reflect the relaxation of the composition of the solvation shell of a solute following a change in pressure or temperature. Cavity Volume. Any significant change in the cavity volume, VC, accompanying transfer of a solute from pure water to a concentrated cosolvent solution would affect the values of V°, K°S, and E°. The cavity volume, VC, in a mixed solvent can be computed based on scaled particle theory (SPT):31


VC ) (βT,0 /R0T)HC + NAπdS3 /6


(15)

where R0 is the coefficient thermal expansibility of the solvent; NA is Avogadro’s number; dS is the diameter of a solute molecule. The enthalpy of cavity formation, HC, is given by the relationship9

HC ) [R0RT2 /(1 - ξ3)][ξ3 + 3ξ2dS /(1 - ξ3) + 3ξ1dS2 /(1 - ξ3) + 9ξ22dS2 /(1 - ξ3)2] (16) where ξk ) Cidi ; k has the values of 1, 2, and 3; m is the number of solvent components; Ci and di are, respectively, the molar concentration and the molecular diameter of the ith solvent component. SPT-based calculations, which treat the solute and solvent molecules as hard spheres, are critically sensitive to the specific choice of the diameters di in eqs 15 and 16.32 As discussed by Tang and Bloomfield,32 several hard-sphere diameters of the urea molecule have been reported in the literature. In addition, ambiguity also exists with the size of a water molecule. As pointed out by Graziano, the size of a water molecule depends on its interactions with the neighbors; the effective diameter of a hydrogen-bonded molecule is 2.8 Å, while the diameter of a water molecule engaged only in van der Waals interactions is 3.2 Å.33 Significantly, there are both hydrogen bonding and purely van der Waals interactions in liquid water. We used eqs 15 and 16 to compute a change in the cavity volume of a spherical particle with a diameter ranging from 4 to 10 Å accompanying its transfer from water to 2, 4, 6, and 8 M urea. In our analysis, we employed the hard-sphere diameter of water of 2.74 Å34,35 and the diameter of urea of 4.41 Å. The latter has been evaluated from SPT-based calculations of changes in volume and compressibility accompanying the waterto-urea transfer of alkali-metal halides.9,34 The coefficients of isothermal compressibility, βT,0, of urea solutions required in eq 15 are not known. Instead, we used the coefficients of adiabatic compressibility, βS,0, which was determined from our densimetric and ultrasonic velocimetric measurements. The values of βS,0 are equal to 44.8 × 10-6, 40.2 × 10-6, 36.8 × 10-6, 33.9 × 10-6, and 32.2 × 10-6 bar-1 at 0, 2, 4, 6, and 8 M urea, respectively. Given the large specific heat capacity and small coefficient of thermal expansion of aqueous solutions, the difference between βS and βT should not exceed a few percent. Hence, the substitution of βT with βS should not cause significant inaccuracies. Table 4 presents our SPT-based-calculations results. Inspection of data listed in Table 4 reveals that the cavity volumes of spherical molecules with diameters ranging between 4 and 10 Å change insignificantly (on the order of ∼1 cm3 mol-1 or less) with an increase in urea concentration. Therefore, in the analysis below, we assume that the volume of the cavity enclosing a solute remains constant as the concentration of urea in solution increases from 0 to 8 M. Hence, the urea-induced changes in the partial molar volume and, by extension, compressibility and expansibility of solutes, in particular, glycine (see below), predominantly reflect alterations in solute-solvent and solute-cosolvent interactions. The cavity volume, VC, for glycine can be calculated by approximating it by a sphere with a volume equal to its van der Waals volume. With the van der Waals volume of glycine of 40.5 cm3 mol-1,36 the diameter of the approximating sphere is 5.04 Å. Our SPT-based calculations yield the cavity volumes, m (πNA/6)∑i)1

k

TABLE 4: Scaled Particle Theory-Based Calculations of the Cavity Volume, VC (cm3 mol-1), for a Spherical Solute with a Diameter dS in Pure Water and in Urea Solutions dS (Å) water 2 M urea 4 M urea 6 M urea 8 M urea

4

6

8

10

41.9 41.2 41.2 41.3 42.1

113.0 111.6 111.5 111.8 113.5

237.9 235.6 235.4 236.0 238.8

431.8 428.2 427.9 428.9 433.2

VC, of glycine of 73.1, 72.1, 72.0, 72.2, and 73.4 cm3 mol-1 at 0, 2, 4, 6, and 8 M urea, respectively. Comparison of these values with the partial molar volumes, V°, of glycine presented in Table 3 in conjunction with eq 7 reveals interaction volumes, 〈∆V〉, which range from -29 to -27 cm3 mol-1 depending on the urea concentration. These estimates are in numerical agreement with previous data37,38 and suggest strong electrostriction in the vicinity of zwitterionic glycine. Practical Treatment of Experimental Data In experimental studies, one is more interested in changes in thermodynamic properties accompanying substitution of water by an osmolyte in the solvation shell of a solute rather than the binding of water or cosolvent to the dry (unsolvated) solute species. To this end, reaction R1 can be modified and, following the seminal works of Schellman,14,39-41 presented as an exchange reaction in which osmolyte replaces water of hydration:

SWnh + jO T SWnh-rjOj + rjW

(R2)

where SWnh denotes an ensemble of purely hydrated solute species with no osmolytes in their solvation shells. The collective species SWnh is effectively solvated by nh water molecules. By modifying eq 3, the concentration of SWnh is given by n [SWnh] ) [S0]∑i)0 [a1i exp(-∆Gi0/RT)], where ∆Gi0 is the free energy of the formation of a hydrated complex containing i waters and no osmolyte in its solvation shell. The effective number of water molecules solvating the purely hydrated species n n iRi, where Ri ) a1i exp(-∆Gi0/RT)/∑i)0 [a1i is nh ) ∑i)0 exp(-∆Gi0/RT)]. The exchange model implies that the effective numbers of bound water, n1, and osmolyte, n3, molecules are related via n1 + rn3 ) nh. While this approximation leaves out a number of solute configurations present in reaction R1 (in particular, solute species partially solvated by cosolvent molecules and not containing any water molecules in their solvation shell), the statistical weight of such configurations is not large. Therefore, the exchange model represents a useful tool and adequately describes the statistical distribution of solvated solute species. With these notions, eq 3 can be rewritten in the following form: nh/r

[Sl] ) [SWnh]

∑ (a3/a1r)j exp(-∆∆Gj/RT)

(17)

j)0

where ∆∆G j is the average exchange free energy of the binding of j osmolyte molecules to the purely hydrated species SWnh with concomitant expulsion of rj waters. When the assumption of identical noninteracting binding sites is introduced, the net exchange free energy is given by ∆∆G j ) j∆∆G with the intrinsic binding constant k ) -RT ln(∆∆G).


Lee and Chalikian

∆E° ) E° - E°H2O ) -γ1nh∆E°1 - γ1∆V°1(∂nh /∂T)P +

Equation 17 simplifies to the form

[Sl] ) [SWnh][1 +

(a3 /a1r)k]nh/r

(18)

By summing the contributions of all solute species, its partial molar volume is given by the relationship nh/r

V° ) VH2O + ∆V

∑ jRj

(19)

j)0

where VH2O is the partial molar volume of the species SWnh at a given osmolyte concentration; ∆V ) (∂∆∆G /∂P)T is the elementary change in volume accompanying replacement of r water molecules in the solvation shell of a solute by a molecule of osmolyte; and Rj ) [(nh/r)!/(j!(nh/r - j)!)](a3/a1r)jkj/[1 + (a3/ a1r)k]nh/r is the fractional composition of the solute species associated with j osmolyte molecules. The partial molar volume of the purely hydrated species in water is described by the relationship V°H2O ) Vhc - nhV°1 + βT,0RT, where Vhc is the partial molar volume of the hydrated complex (solute plus nh waters). In a concentrated osmolyte solution, the partial molar volume of the purely hydrated species can be presented as VH2O ) V°H2O - nh∆V°1, where ∆V°1 is the excess partial molar volume of water (the partial molar volume of water in an osmolyte solution is given by V1 ) V°1 + ∆V°1). An implicit assumption of this presentation is that water of hydration is unaffected by a decrease in water activity in the bulk.41 Strong solute-solvent interactions may, indeed, minimize the effect of lowered water activity on the partial molar volume of waters of hydration and, hence, the purely hydrated species SWnh in general. Alternatively, the effect may be significant in the case of weak solute-solvent interactions. To account for it, one can introduce an adjustable parameter γ1 (ranging between 0 and 1) to VW ) V°W - γ1nh∆V°1. In fact, - γ1nh∆V°1 is the volume of the transfer of the purely hydrated solute species from water to an osmolyte solution. A change in volume associated with the transfer of solute from water to a concentrated osmolyte solution is given by the expression nh/r

∆V° ) V° - V°H2O ) -γ1nh∆V°1 + ∆V

∑ jRj ) j)0

-γ1nh∆V°1 + ∆V(nh /r)(a3 /a1r)k/[1 + (a3 /a1r)k] (20) An equation for the transfer change in compressibility can be obtained by differentiating eq 20 with respect to pressure:

∆K°T ) K°T - K°T, H2O ) -γ1nh∆K°T,1 + γ1∆V°1(∂nh /∂P)T + ∆KT(nh /r)(a3 /a1r)k/[1 + (a3 /a1r)k] + ∆V2(nh /r)(a3 /a1r)k/RT[1 + (a3 /a1r)k]2 (21) where (∂nh/∂P)T ) -na1k1∆V1/RT(1 + a1k1)2; k1 is the intrinsic binding constant for the water-solute association event; ∆KT is the elementary change in compressibility accompanying replacement of r water molecules by an osmolyte molecule in the solvation shell of the solute. An equation for the transfer change in expansibility can be obtained by differentiating eq 20 with respect to temperature:

∆E(nh /r)(a3 /a1r)k/[1 + (a3 /a1r)k] + ∆V∆H(nh /r)(a3 /a1r)k/RT2[1 + (a3 /a1r)k]2 (22) where (∂nh/∂T)P ) na1k1∆H1/RT2(1 + a1k1)2 and ∆E is the elementary change in expansibility accompanying replacement of r water molecules by an osmolyte molecule in the solvation shell of the solute. Comments on ∆V, ∆KT, and ∆E. The volume, ∆V, compressibility, ∆KT, and expansibility, ∆E, of an elementary exchange reaction can be presented by ∆V ) V(SWnh-rO) VH2O + rV°1 - V°3, ∆KT ) KT(SWnh-rO) - KT,H2O + rK°T,1 K°T,3, and ∆E ) E(SWnh-rO) - EH2O + rE°1 - E°3, where SWnh-rO signifies the complex of a solute with (nh - r) water molecules and a single cosolvent molecule. The subscript H2O refers to the purely hydrated, osmolyte-free solute species SWnh. V°1, K°T,1, and E°1 are, respectively, the partial molar volume, isothermal compressibility, and expansibility of bulk water; V°3, K°T,3, and E°3 are, respectively, the partial molar volume, isothermal compressibility, and expansibility of bulk cosolvent. Since the volumetric properties of the solvent components in the bulk depend on the cosolvent concentration, the values of ∆V, ∆KT, and ∆E also should be cosolvent-dependent. This dependence reflects the differential cosolvent-induced changes in the properties of the solvent components in the bulk and in the solvation shell of a solute. In line with these arguments and following the discussion presented in the previous section, an adjustable parameter γ3 in addition to γ1 needs to be introduced to the expressions for ∆V, ∆KT, and ∆E to account for the effect of the bulk solvent on the properties of the solvation shell:

∆V ) ∆V0 + γ1r∆V°1 - γ3∆V°3

(23)

∆KT ) ∆KT,0 + γ1r∆K°T,1 - γ3∆K°T,3

(24)

∆E ) ∆E0 + γ1r∆E°1 - γ3∆E°3

(25)

where ∆V0, ∆KT,0, and ∆E0 refer to the elementary exchange reaction in an ideal solution; ∆V°3, ∆K°T,3, and ∆E°3 are the excess volume, isothermal compressibility, and expansibility of the cosolvent; γ1 and γ3 are the correction factors reflecting the influence of the bulk solvent on the properties of solvating water and cosolvent, respectively. The values of γ1 and γ3 may change from 0 (the properties of the solvent components in the solvation shell change in parallel with those in the bulk) to 1 (the properties of the solvation shell are independent of the properties of the bulk phase). Analysis of Glycine Data. In this section, we apply the formalism derived above to analyzing the partial molar volume and adiabatic compressibility of glycine in urea solutions. Glycine, a highly charged molecule, is a good candidate for being treated as a solute with a homogeneous solvation shell, while urea is one of the most extensively studied and yet poorly understood biological cosolvents.42 Due to the zwitterionic nature of glycine and the closely located geometry of the positively and negatively charged termini, its solvation shell is dominated by strong solute-solvent interactions. Assuming the homogeneity of the solvation shell, we can apply eqs 20-22 to analysis of the binding of urea to glycine. The precondition of this analysis is the knowledge of the hydration number of glycine, nh, and the number of water molecules, r, expelled to


Figure 1. Urea concentration dependence of the change in the partial molar volume of glycine. The experimental points are fitted using eqs 20 and 23 (solid line).

the bulk per bound urea molecule. The hydration number, nh, of glycine has been estimated to be ∼15 from the temperature dependences of its partial molar volume and adiabatic compressibility.37,38 This value roughly corresponds to the number of water molecules within the first coordination sphere of glycine. Further, as a good approximation, the value of r can be estimated to be ∼2 as the ratio of the cross-sectional areas of urea to water molecules. By using the spherical approximation of water and urea molecules, their cross-sectional area ratio can be calculated as r ) [VW(urea)/VW(water)]2/3, where the van der Waals volumes, VW, of water (12.0 cm3 mol-1) and urea (32.8 cm3 mol-1) molecules can be evaluated with the additive approach and the group contributions presented by Bondi.36 In our treatment, we use equations derived for isothermal compressibility to analyze our adiabatic compressibility results. The relationship between K°T and K°S is as follows: K°T ) K°S + (TR2/F0cP,0)(2E°/R0 - C°P/F0cP,0), where F0 is the density of the solvent, cP,0 is the specific heat capacity at constant pressure of the solvent, E° is the partial molar expansibility of a solute, and C°P is the partial molar heat capacity of a solute.43 Due to a small R0 and a large cP,0 of water-based solvents, the difference between K°T and K°S in aqueous solutions is not large. Therefore, the above-developed equations for isothermal compressibility can be applied to adiabatic compressibility. Figures 1 and 2 graphically present the urea dependences of the partial molar volume and adiabatic compressibility of glycine listed in Table 3. The dependences were fitted with eqs 20, 21, 23, and 24 to evaluate the urea-glycine binding constant, k, and the exchange volume, ∆V0, and the adiabatic compressibility, ∆KS,0. The second term of eq 21 was neglected due to a small value of ∆V°1. Given the small size of water molecules and strong solute-solvent interactions in the vicinity of glycine, its waters of hydration can be considered to be predominantly influenced by the solute and relatively insensitive to the properties of the solvent in the bulk. It is a good approximation, therefore, to take γ1 in eqs 23 and 24 equal to 1. On the other hand, urea is bulkier than water and can potentially form up to eight hydrogen bonds with its neighbors. Consequently, despite its being engaged in charge-dipole interactions with glycine, urea can still develop numerous interactions with solvent in the bulk. Therefore, the thermodynamic properties of solvating urea molecules should be, generally, influenced to a significant degree by the bulk solvent (γ3 is smaller than 1). In our treatment, we


Figure 2. Urea concentration dependence of the change in the partial molar adiabatic compressibility of glycine. The experimental points are fitted using eqs 21 and 24 (solid line).

TABLE 5: Binding Constant, k, and Changes in Volume, ∆V0, and Adiabatic Compressibility, ∆KS,0, Accompanying Replacement of Two Water Molecules by a Urea Molecule in the Solvation Shell of Glycine k, M ∆V0, cm3 mol-1 ∆KS,0, 10-4 cm3 mol-1 bar-1

0.117 ( 0.009a 0.64 ( 0.02 2.89 ( 0.08

a Determined from volume data. ssibility data.

b

0.112 ( 0.010b

Determined from compre-

assumed γ3 to be zero. It should be noted, however, that the use of γ3 of 0.1, 0.25, and 0.5 did not increase (and, in fact, reduced) the correlation coefficient of the fit of experimental data (see below), thereby lending credence to the γ3 ) 0 assumption. The activity of water, as the principal solvent, was taken equal to its mole ratio a1 ) [W]/([W] + [urea]), where the actual concentration of water at each experimental urea concentration was determined from [W] ) (F - [urea]Murea)/MW; F is the density of the urea solution (data not shown) and Murea and MW are the molecular weights of urea and water, respectively. As the principal solvent, water exhibits an activity of 1 in its pure state. Our evaluated urea dependence of a1 (not shown) was approximated by a1 ) 1 - 0.0173[urea] - 7.1 × 10-4[urea]2. The activity of urea, a3, was assumed to be equal to its molar concentration. The urea dependences of the excess volume and compressibility of water presented in Table 1 were approximated by second-order polynomials: ∆V°1 (cm3 mol-1) ) -0.0036[urea] - 0.0016[urea]2 and ∆K°S,1 (10-4 cm3 mol-1 bar-1) ) -0.11 [urea] + 0.0004 [urea]2. These analytical dependences were incorporated into eqs 20-22 to fit the plots in Figures 1 and 2. Table 5 presents our evaluated values of k, ∆V0, and ∆KS,0. Inspection of data presented in Table 5 reveals that replacement of two water molecules by a urea molecule in the solvation shell of glycine is accompanied by increases in both volume (0.64 ( 0.02 cm3 mol-1) and compressibility [(2.89 ( 0.08) × 10-4 cm3 mol-1 bar-1]. These observations are consistent with a decrease in electrostriction when water is replaced by urea as a solvent in the vicinity of a charged solute. The binding constant, k, represents a quantitative measure of the affinity of a cosolvent for a solute or an individual atomic group of a solute. The value of k for urea-glycine association determined from the volume (0.117 ( 0.009 M) is in excellent

2450 J. Phys. Chem. B, Vol. 113, No. 8, 2009 agreement with that derived from the compressibility data (0.112 ( 0.010 M). The average of 0.115 ( 0.010 M is on the order of the binding constants determined for the association of urea with proteins and protein groups.44 Cosolvent-dependent volumetric investigations on larger sets of low molecular weight protein analogues are necessary to produce the wealth of thermodynamic information on the interactions of osmolytes with different functional groups (in particular, amino acid side chains and the peptide group). Such investigations and the ensuing results will complement and represent a valuable addition to the existing data on the free energy of transfer of solutes from water to cosolvent solution.16,45-48 Taken together, the emerging volumetric insights combined with the existing data will lead to a better understanding of the mechanisms of osmolyte-induced stabilization/ destabilization of proteins. We have already initiated such studies involving N-acetyl amino acid amides and a variety of osmolytes. Concluding Remarks Based on a simple yet rigorous statistical thermodynamic model, we derived a set of equations that link volumetric observables, including the partial molar volume, compressibility, and expansibility of a solute, with the number and intensity of solute-solvent interactions in binary solvents. The model and the ensuing volumetric relationships can be simplified if the approximation of homogeneous solvation is introduced. With such an approximation, our derived formalism can be employed for practical treatment of experimental volumetric data on small solutes and/or individual atomic groups. However, such an analysis requires knowledge of a number of important parameters relevant to the solvation complex that cannot be measured directly and, therefore, must be evaluated based on nonthermodynamic assumptions. These parameters include the hydration number, nh, of the purely hydrated species, the number of molecules of the principal solvent replaced by a cosolvent molecule, r, and the correction factors γ1 and γ3 that account for the dependence of the properties of solvent in the solvation shell of a solute on the properties of solvent in the bulk. As an example, we use the formalism developed in this work to describe the urea dependences of the partial molar volume and adiabatic compressibility of the zwitterionic amino acid glycine. Our combined volume and compressibility results yield an average intrinsic urea-glycine association constant of 0.115 ( 0.010 M. Assuming that a bound urea molecule replaces two water molecules from the solvation shell of glycine, each urea-glycine association event is accompanied by increases in both volume (0.64 cm3 mol-1) and compressibility (2.89 × 10-4 cm3 mol-1 bar-1). These observations are consistent with a decrease in electrostriction when water is replaced by urea as a solvent in the vicinity of a charged solute. In general, the theoretical development presented in this work opens the way for systematic application of volumetric measurements to characterization of solute-solvent interactions in complex solvents. Acknowledgment. The authors acknowledge stimulating discussions with Drs. Robert B. Macgregor, Jr., and Jen Vo¨lker. We also would like to thank Dr. Giuseppe Graziano for his advice regarding the SPT calculations. This work was supported by a grant from the Natural Sciences and Engineering Research Council of Canada to T.V.C. S.L. gratefully acknowledges her graduate support from the CIHR Protein Folding Training Program.

Lee and Chalikian References and Notes (1) Chalikian, T. V.; Breslauer, K. J. Curr. Opin. Struct. Biol. 1998, 8, 657–664. (2) Zamyatnin, A. A. Annu. ReV. Biophys. Bioeng. 1984, 13, 145– 165. (3) Sarvazyan, A. P. Annu. ReV. Biophys. Biophys. Chem. 1991, 20, 321–342. (4) Chalikian, T. V.; Sarvazyan, A. P.; Breslauer, K. J. Biophys. Chem. 1994, 51, 89–107. (5) Chalikian, T. V.; Breslauer, K. J. Biopolymers 1998, 48, 264–280. (6) Chalikian, T. V. Annu. ReV. Biophys. Biomol. Struct. 2003, 32, 207– 235. (7) Lin, L. N.; Brandts, J. F.; Brandts, J. M.; Plotnikov, V. Anal. Biochem. 2002, 302, 144–160. (8) Joliceur, C.; Riedl, B.; Desrochers, D.; Lemelin, L. L.; Zamojska, R.; Enea, O. J. Solution Chem. 1986, 15, 109–128. (9) Desrosiers, N.; Desnoyers, J. E. Can. J. Chem. 1976, 54, 3800– 3808. (10) Perron, G.; Desrosiers, N.; Desnoyers, J. E. Can. J. Chem. 1976, 54, 2163–2183. (11) Mathieson, J. G.; Conway, B. E. J. Chem. Soc., Faraday Trans. 1 1974, 70, 752–768. (12) Timasheff, S. N. AdV. Protein Chem. 1998, 51, 355–432. (13) Timasheff, S. N. Biochemistry 2002, 41, 13473–13482. (14) Schellman, J. A. Biophys. J. 2003, 85, 108–125. (15) Courtenay, E. S.; Capp, M. W.; Saecker, R. M.; Record, M. T., Jr. Proteins 2000, 4, 72–85. (16) Bolen, D. W. Methods 2004, 34, 312–322. (17) Hall, D. G. Trans. Faraday Soc. 1971, 67, 2516–2524. (18) Ben-Naim, A. J. Chem. Phys. 1975, 63, 2064–2073. (19) Ben-Naim, A. Statistical Thermodynamics for Chemists and Biochemists; Plenum Press: New York, London, 2002. (20) Ben-Naim, A. Molecular Theory of Solutions; Oxford University Press: Oxford, 2006. (21) Chalikian, T. V. J. Phys. Chem. B 2008, 112, 911–917. (22) Eggers, F.; Funck, T. ReV. Sci. Instrum. 1973, 44, 969–977. (23) Sarvazyan, A. P. Ultrasonics 1982, 20, 151–154. (24) Eggers, F. Acustica 1992, 76, 231–240. (25) Eggers, F.; Kaatze, U. Meas. Sci. Technol. 1996, 7, 1–19. (26) Sarvazyan, A. P.; Selkov, E. E.; Chalikyan, T. V. SoV. Phys. Acoust. 1988, 34, 631–634. (27) Sarvazyan, A. P.; Chalikian, T. V. Ultrasonics 1991, 29, 119–124. (28) Høiland, H. In Thermodynamic data for biochemistry and biotechnology; Hinz, H.-J., Ed.; Springer-Verlag: Berlin, Heidelberg, New York, Tokyo, 1986; pp 17-44. (29) Høiland, H. In Thermodynamic data for biochemistry and biotechnology; Hinz, H.-J., Ed.; Springer-Verlag: Berlin, Heidelberg, New York, Tokyo, 1986; pp 129-147. (30) Ben-Naim, A. J. Phys. Chem. 1978, 82, 792–803. (31) Pierotti, R. A. Chem. ReV. 1976, 76, 717–726. (32) Tang, K. E. S.; Bloomfield, V. A. Biophys. J. 2000, 79, 2222– 2234. (33) Graziano, G. Chem. Phys. Lett. 2006, 432, 84–87. (34) Desrosiers, N.; Lucas, M. J. Phys. Chem. 1974, 78, 2367–2369. (35) Likhodi, O.; Chalikian, T. V. J. Am. Chem. Soc. 1999, 121, 1156– 1163. (36) Bondi, A. J. Phys. Chem. 1964, 68, 441–451. (37) Kharakoz, D. P. Biophys. Chem. 1989, 34, 115–125. (38) Chalikian, T. V.; Sarvazyan, A. P.; Breslauer, K. J. J. Phys. Chem. 1993, 97, 13017–13026. (39) Schellman, J. A. Biophys. Chem. 1990, 37, 121–140. (40) Schellman, J. A. Biopolymers 1990, 29, 215–224. (41) Schellman, J. A. Biopolymers 1994, 34, 1015–1026. (42) Stumpe, M. C.; Grubmuller, H. J. Am. Chem. Soc. 2007, 129, 16126–16131. (43) Blandamer, M. J.; Davis, M. I.; Douheret, G.; Reis, J. C. R. Chem. Soc. ReV. 2001, 30, 8–15. (44) Makhatadze, G. I. J. Phys. Chem. B 1999, 103, 4781–4785. (45) Qu, Y.; Bolen, C. L.; Bolen, D. W. Proc. Natl. Acad. Sci. U.S.A. 1998, 95, 9268–9273. (46) Auton, M.; Bolen, D. W. Biochemistry 2004, 43, 1329–1342. (47) Auton, M.; Bolen, D. W. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 15065–15068. (48) Auton, M.; Holthauzen, L. M.; Bolen, D. W. Proc. Natl. Acad. Sci. U.S.A. 2007, 104, 15317–15322.

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