5910
J. Phys. Chem. B 1999, 103, 5910-5916
Parallel Change with Temperature of Water Structure and Protein Behavior Irving M. Klotz* Northwestern UniVersity, 2145 Sheridan Road, EVanston, Illinois 60208-3113 ReceiVed: April 21, 1999; In Final Form: May 19, 1999
Dome-shaped temperature dependencies have been observed widely for the unfolding of proteins and for ligand binding by proteins. These curves have been rationalized in terms of hydrophobic interactions. Similar dependencies have been found for transfer equilibria of many nonpolar molecules to water, and these too have been ascribed to hydrophobic effects. Less well-recognized is the variation with temperature of the ionization constants of formic acid and of related small carboxylic acids, which are also dome-shaped; it is difficult to attribute such behavior in these molecules to hydrophobic interactions. What all of these equilibria do have in common is that they are taking place in water. In protein studies, little attention has been paid to the variation with temperature of the structure of pure liquid water. Vibrational spectroscopic studies of water upon warming above 4 °C and upon cooling the supercooled liquid have disclosed and tracked similar changes in the molecular structure with the rise or with the fall of temperature. Furthermore, the densities of liquid water from -31 to +60 °C fit strikingly on a dome-shaped curve, which is also a manifestation of structural changes. If one adopts a phenomenological formulation of unfolding transformations that couples the conformational rearrangement in a protein with the equilibria between closely-packed and more open forms of water, one can derive an equation for protein unfolding that incorporates the dome-shaped temperature behavior of the solvent water. The equation fits the experimental observations over the full range of temperatures from cold denaturation to warm denaturation and is concordant with a dome-shaped temperature dependence. This outcome appears without any consideration of the effect of temperature on the interactions of specific residues of the protein with solvent molecules.
In accounting for the behavior of proteins in solution, investigators focus largely on the chemical nature and molecular structure of the macromolecule. Thus, in interpreting thermal unfolding of proteins, they consider interactions of peptide residues and of side chains with other functional groups in the protein and with solvent molecules. Little attention has been paid to the dependence of the structure of water itself on temperature. Taking cognizance of such effects on pure liquid water, one is led to a perspective on thermal unfolding of proteins that reveals novel features that probably play a role in the behavior of a variety of molecules, as well as of proteins, dissolved in aqueous solution. Dome-Shaped Temperature Effects on Proteins In recent times, it has become abundantly evident that loss of activity or denaturation may occur on cooling a protein as well as on heating it.1-7 (See also citations in listed references.) Cold denaturation/heat denaturation experiments are frequently summarized in terms of the variation of the standard free energy of unfolding, ∆G°, with temperature. A recent representative study following unfolding of barstar, the inhibitor of barnase,7 is illustrated in Figure 1. Clearly there is a maximum in ∆G° near 25 °C. It is not always realized that the temperature at which the ∆G° of unfolding reaches its maximum (positive) value is not the temperature of the maximum stability of the protein.8 Hence, the temperature of minimum equilibrium unfolding should be established from a stability curve showing the equilibrium * Telephone: (847) 491-3546. Fax: (847) 491-7713.
Figure 1. Free-energy changes for unfolding of wild-type barstar at pH 7.4 (adapted from ref 7).
constant, or ln K, as a function of temperature. Figure 2 (adapted from ref 6) presents such data for the reversible denaturation of T4 lysozyme. For this protein, the maximum stability occurs at a temperature near 10 °C. Cold denaturation sets in at lower temperatures, heat denaturation at higher temperatures. A dome-shaped (or inverted-dome, bowl-shaped) dependence on temperature is also manifested in protein interactions with other molecular species. A detailed study of the interactions of lac repressor with the symmetric Osym operator12 is summarized in Figure 3. Maximum stability of the complex, ln K, is manifested near 20 °C. Increasing dissociation appears at lower temperatures or at higher ones. It should be noted, however, that the minimum for -∆G° (Figure 3) occurs near or above
10.1021/jp9913057 CCC: $18.00 © 1999 American Chemical Society Published on Web 06/24/1999
Water Structure and Protein Unfolding
Figure 2. Unfolding of phage T4 lysozyme by cooling or by heating. The equilibrium constant, as ln K, for unfolding is presented over the range of temperature -10 to 50 °C (adapted from ref 6).
J. Phys. Chem. B, Vol. 103, No. 28, 1999 5911
Figure 4. Solubility of (liquid) benzene in water, expressed in terms of mole fraction X2 of dissolved solute. Figure adapted from ref 12.
of ∆H with temperature, with the values of ∆H° actually changing their sign.8 From the temperature dependence of ∆H°, one can also calculate ∆Cp° for the formation of the protein complexes. The experimentally observed values of ∆Cp° have generally been interpreted primarily in terms of hydrophobic interactions of the protein.12,13 (See also citations in these references.) Dome-Shaped Temperature-Dependent Equilibria of Small Molecules
Figure 3. Variation with temperature of the stability of the complex between lac repressor and the Osym operator. The lower panel presents ln K as a function of temperature; the upper panel follows ∆G°. The temperature of maximum stability of the complex, the peak of ln K, is near 20 °C. If there is a minimum in ∆G°, it is g40 °C (adapted from ref 12).
40 °C, substantially above the temperature of maximum stability of this protein-DNA complex. Other complexes of proteins also show dome-shaped, temperature-dependent stabilities.13 For example, studies of the interactions of glyceraldehyde-3-phosphate dehydrogenase with NAD13,14 point to a turnaround in ln K near 0 °C. The corresponding values of the unitary ∆G° vary little with temperature. A dome-shaped (or inverted-dome, bowl-shaped) dependence of ln K on temperature must be a manifestation of a variation
Because of their interest as simple models for residues within proteins, the transfer equilibria of many nonpolar small molecules from a gas, liquid, or solid phase to water have been measured or unearthed from earlier literature. These solubilities are, in essence, equilibrium constants. When their dependencies on temperature are also known, one can plot ln K versus T. Particularly detailed data are available for the transfer of benzene, from the pure liquid state, to water,12,15,16 and these are presented in Figure 4. Again we see an extremum, near 16 °C, with solubilities increasing at lower or higher temperatures. Clearly the ∆H° for the process is zero near 16 °C and changes sign as the temperature moves from colder to warmer temperatures. The ∆Cp° value derived from the temperature dependence is similar to that found for the formation of the lac repressorDNA complex and has been rationalized as a manifestation of hydrophobic interactions.12 Dome-shaped temperature dependencies are also observed in equilibria17 of entities to which it is difficult to ascribe hydrophobic interactions. A class of such equilibria is the ionization of small carboxylic acids: formic, acetic, propionic, butyric. Exceptionally precise and thorough studies of the ionization of these acids were carried out a half-century ago.20-23 The dependence of ln K of ionization of acetic acid on temperature is presented in Figure 5. A similar dome-shaped dependence is characteristic of each of the other carboxylic acids in this series. For acetic acid, the maximum occurs slightly below 25 °C.24 It is difficult to view the equilibrium between CH3CO2H, CH3CO2-, and H+ as a manifestation of hydrophobic effects. The CH3 substituent might make a minor contribution, but clearly that must be trivial; the replacement of CH3 by an H to give formic acid, HCO2H, produces minuscule effects on the shape of the graph of ln K versus temperature, the maximum occurring at 25 °C. In a complementary direction, the addition of a CH2 substituent to acetic acid to give propionic acid,
5912 J. Phys. Chem. B, Vol. 103, No. 28, 1999
Klotz
Figure 6. Variation of density of liquid water with temperature (data from ref 25 and 51).
Structure of Liquid Water As Revealed by Vibrational Spectroscopy
Figure 5. Temperature dependence of the ionization of acetic acid in water. The upper panel illustrates the dome-shaped dependence of the equilibrium constant on temperature. The lower panel demonstrates that ∆G° for the same equilibrium increases almost linearly with temperature and does not manifest a minimum (or maximum) in the large range of temperatures studied.
CH3CH2CO2H, shifts the maximum of the ln K versus temperature graph to 20 °C, a minor change. What all the equilibria summarized in Figures 1-5 do have in common is that they are taking place in water. Dome-Shaped Variation of Density of Liquid Water with Temperature It is widely known that liquid water has a maximum in density at 4 °C. Obviously that means that a graph of its density versus temperature is dome-shaped. Since liquid water is normally transformed into ice at 0 °C, just slightly below 4 °C, one does not usually visualize the density-temperature curve as domeshaped. However, it has been possible to measure the densities for supercooled liquid water down to temperatures below -30 °C.25 Figure 6 summarizes some classical and recent data. The similarity in shape of Figure 6 with the curves of Figures 1-5 is striking. It is of interest to note that supercooled liquid water has a larger molar volume V1, 18.3 cm3 mol-1 at -30 °C, than does the maximum density water, 18.02 cm3 mol-1 at 4 °C. In fact, the V1 of liquid water at -30° is about the same as that for water at 60 °C, 18.33 cm3 mol-1. The variation in density of liquid water with temperature must reflect changes in the arrangements of water molecules in that solvent. These could have marked effects on the behavior of solutes, of any molecular size, dissolved in water. It is pertinent, therefore, to consider what is known about changes in the structure of liquid water at different temperatures.
That changes in properties of liquid water are manifestations of shifts in equilibria between different molecular species was suggested at least as early as a century ago, by Roentgen (1892).26,27 With the development of vibrational spectroscopic probes, it became possible to gain insights into the nature of these species and their reversible interconversions. Raman spectra of liquid water were studied by Segre´ 70 years ago.28 A weak low-frequency band at 170 cm-1 and the strong OH stretching vibration near 3250 cm-1 were observed to decrease in intensity with a rise in temperature. Such behavior was ascribed to a breakdown of the intermolecular bonding in the liquid. That view has been progressively amplified since that time in extensive investigations of the infrared as well as the Raman spectra of liquid water in a wide range of circumstances. Changes upon Warming Water. With improvements in the accuracy of measurement of intensities, it has been possible to follow the changes with temperature precisely. Variations upon heating liquid water have been examined in the infrared as well as by Raman scattering.29-43 (See also citations in these references.) An illustrative example is shown in Figure 7, which displays changes in intensity of overtone infrared vibrations as the temperature is varied from 7 to 60 °C. The trends in absorbances fit expectations for the temperature dependence of a system involving equilibria between hydrogen-bonded and distorted or non-hydrogen-bonded species.44 As the temperature rises, there is a large increase in intensity of the “free” O-H band at 1.416 µm (7061 cm-1) accompanied by a diminished intensity of absorption at 1.525 µm (6557 cm-1) of the bonded species. Moreover, the existence of an isosbestic point (at 1.468 µm, 6812 cm-1) lends support to the assumption that a chemical equilibrium between different groups or species of water molecules is present in the liquid. Detailed investigations of the Raman scattering from liquid water over the range of temperatures38 from 3 to 72 °C showed similar progressive shifts of spectra in the O-H stretching region of 3200-3600 cm-1. Again, an isosbestic point, at 3425 cm-1, was clearly present and was ascribed to an equilibrium between hydrogen-bonded and bent or stretched (“non-hydrogen bonded”) nearest-neighbor O-O pairs. Interestingly, from the temperature
Water Structure and Protein Unfolding
J. Phys. Chem. B, Vol. 103, No. 28, 1999 5913 changes in molecular structure, becoming much more open in the low-temperature liquid state. Analytical Treatment of Protein Unfolding in Cool and Warm Water Representation of Structural Equilibria of Liquid Water. The physical properties of water clearly establish that the liquid attains its most compact structure at 4 °C. Whatever the polyhedral conformation at that temperature, rearrangements occur in the liquid at lower temperatures and at higher temperatures. At the molecular level, all the structures are in flux and in equilibrium with their cohorts. To simplify an analytical treatment of the equilibria, we shall assume that just three ensembles of structures are involved: WD, the most compact or dense form, that dominant at 4 °C; WH, that dominant at warm temperatures; WC, that dominant at cold temperatures. The equilibria between them can then be represented by the following equations:
WC h WD h WH Figure 7. Absorbance versus wavelength in micrometers (in overtone region of the infrared spectrum40) of a 6.0 M solution of H-OD in D2O as a function of temperature.
dependence of the Raman curves, one can extract a ∆H° of 2.6 kcal/mol for the opening of an O-H‚‚‚O, in close agreement with the ∆H° of 2.4 kcal/mol obtained from the overtone infrared spectra.40 Similar structural conclusions have been reached from very recent femtosecond infrared pump-probe spectroscopy studies43 of liquid water at ambient temperature. With this technique, one can follow the dynamics of orientational relaxation of water molecules in the liquid. Two different relaxation time constants were discovered, with τR values of 0.7 and 13 ps, respectively. It is clear that two distinct ensembles of water molecules are present, one with a fast orientational relaxation and the other with a slower one. The slower τR is observed upon excitation with the lower frequency end of the 3200-3600-cm-1 infrared radiation, associated with strongly hydrogen-bonded water molecules. Infrared radiation of the higher frequency end of the absorption band, associated with open and distorted hydrogen bonds, manifests a fast relaxation process. Thus, these observations also indicate that liquid water at ambient temperatures can be viewed as a mixture of two distinct ensembles of water molecules. Changes upon Supercooling Liquid Water. It is possible to produce supercooled liquid water down to the stability limit temperature, -46 °C.25 At this temperature, the molar volume (V1) of the liquid is 19.2 cm3 mol-1, so the volume of voids in the structure is much larger than that in maximum density water (V1 ) 18.02 cm3 mol-1) or of the liquid at any temperature up to 100 °C (V1 ) 18.80 cm3 mol-1). The Raman spectra of the supercooled liquid37,45 also show progressive shifts with drops in temperature, reflecting changes in the structure of the liquid. These have been interpreted in terms of hexagonal, pentagonal, and tetragonal ring structures for the water molecule clusters.37,46 Polyhedral water clathrates have molar volumes of 19.019.3 cm3 mol-1, close to that (19.2 cm3 mol-1) for the most highly supercooled liquid. Regardless of which polyhedral clathrate-like structures best describe the thermodynamic and spectroscopic characteristics of the low-temperature liquid, it is obvious that water cooled below 4 °C undergoes marked
KCD h
KDH h
(WD) (WC) (WH) (WD)
(1)
(2)
(3)
Since the range of denaturation by warm temperatures is clearly separated from that produced by cold (note Figures 1 and 2), let us examine the right-hand side of eq 1 as an independent equilibrium. Then we can write
(WH) ) (WD)KDH
(4)
(WH) + (WD) ) (WT)
(5)
where (WT) is the total water concentration, to a first approximation a constant over the temperature range being examined. Simple algebraic steps lead to
KDH (WH) ) (WT) 1 + KDH
(6)
Nature of an Equilibrium Constant for Solutes in Solution. In solution, an equilibrium constant, K, for a transformation involving solutes is defined thermodynamically by the familiar relationship
K ) e-∆Gh 2°/RT
(7)
where ∆G h 2° is the change in partial molar free energies (or the change in chemical potentials, ∆µ2) of the solute species. For a particular solute species i, G h 2,i is not the molar free energy of the solute; it is not a manifestation of the molecular character of the solute per se. Rather, it is the partial molar free energy,
G h 2,i )
∂Gsolution ∂n2,i
(8)
that is, the change in free energy of the entire solution as dn2 moles of solute i is added.
5914 J. Phys. Chem. B, Vol. 103, No. 28, 1999
Klotz
This distinction can be accentuated in more tangible form by considering the partial molar volume, V h 2, of a solute in solution, which is defined by
V h2 )
∂Vsolution ∂n2
(9) B h)
Generally, V h 2 is not equal to the molar volume of the pure solute and is not a measure of the molecular volume of a mole of solute molecules in solution. Rather, it reflects changes in the entire solution. For example, hydrogen exhibits a V h 2 of 26 cm3 -1 3 -1 mol in water, 50 cm mol in ether, and 38 cm3 mol-1 in acetone, in contrast to 25 000 cm3 mol-1 in the gas phase (at 1 atm and 25 °C). Even more surprising are V h 2 values for some electrolyte solutes dissolved in water. For NaCl, V h 2 is 16.4 cm3 -1 3 -1 mol compared to 27 cm mol for the pure crystal. Particularly striking is the behavior of Na2CO3; V h 2 in water is -6.7 cm3 mol-1 (compared to 42 cm3 mol-1 for the pure solute). Clearly the volume occupied by the atoms of Na2CO3 in water is not negative. What the negative value of V h 2 signifies is that if Na2CO3 is added to water, the volume of the liquid solution shrinks. Clearly there must be a strong interaction of the solvent with the solute. From a molecular perspective, one presumes that this solute actuates a pronounced shrinkage in the volume of the solvent. Focusing on the unfolding of a protein solute in solution, one must recognize that in view of eq 7, if G h 2 represents the change in free energy of the entire solution upon addition of dn2 moles of solute, so does the equilibrium constant K for unfolding. Thus, a K expressed only in terms of concentrations of solutes overlooks the presence of solvent. Such an oversight can be especially deceiving when one examines the temperature dependence of an equilibrium in aqueous solution, for the pure water itself undergoes striking changes in structure. Analytic Coupling of Solvent with Unfolding Equilibria. In a phenomenological formulation of interactions between solvent water and protein solute in solution, one can introduce the perspective delineated above by assuming that the multiple binding of WH (or WC) water molecules leads stepwise to progressive unfolding of the protein P. As Figure 2 illustrates, unfolding in the warm region from 15 to 40° is separated from that in the cold region below 5 °C. So for the warm regime, we write
P + H2O ) P(H2O)
K1 )
P(H2O) + H2O ) P(H2O)2
K2 )
· · ·
· · ·
· · ·
P(H2O)n-1 + H2O ) P(H2O)n
· · · Kn )
From the steep rise of unfolding with increasing temperature, it is evident that the binding of water by the protein is highly cooperative. Under such circumstances, eq 11 can be transformed into48
[P(H2O)] (P)(H2O) [P(H2O)2] [P(H2O)](H2O)
[P(H2O)n] [P(H2O)n-1](H2O) (10)
The total moles of bound (H2O) per mole of total protein, B, can be expressed as47 B) K1(H2O) + 2K1K2(H2O)2 + ‚‚‚ + j(K1K2‚‚‚Kj)(H2O)j + ‚‚‚ 1 + K1(H2O) + K1K2(H2O)2 + ‚‚‚ + (K1K2‚‚‚Kj)(H2O)j + ‚‚‚
(11)
A12enλBh (H2O)2 B ) n 1 + A 2enλBh (H O)2 1
(12)
2
where A12 is the binding constant for the uptake of the first pair of water ligands and λ is a factor that reflects the enhancement of affinities as B (or B h ) increases. If (WH) is the form of (H2O) that is the denaturant at warm temperatures, we can insert it into eq 12. With some straightforward rearrangements,48 we then obtain
[1 -Bh Bh ] ) ln(A
ln
2 1 )
+ 2 ln(WH) + λnB h
(13)
By replacing WH by eq 6, we reach
[ ]
ln
[
]
KDH B h ) ln(A12) + 2 ln WT + λnB h (14) 1-B h 1 + KDH
Thus, it is evident that the experimentally-measured unfolding equilibrium constant, represented by [B h /(1 - B h )], depends on the water structure equilibrium constant, KDH, as well as on A12, which is a measure of the binding of WH molecules by protein followed by the unfolding of protein to P(H2O)j.48 With increasing temperature, KDH increases (see Figure 7 for example), as will B h . (We assume WT and λ are approximately nonvarying with temperature.) The sharp increase in slope of ln[B h /(1 - B h )] with increasing temperature reflects the contribution of the change in water structure49
d ln KDH ∆H°DH ) dT RT2
(15)
and of the binding-unfolding step
d ln A1 ∆H°A1 ) dT RT2
(16)
as well as of dB h /dT, which is implicitly coupled to KDH and A12. Turning to the cold denaturation region, we can write a series of stepwise equilibria analogous to those in eq 10 except that cold (H2O) should be designated by (WC). That is the form of water that is formed from the dense one, WD, as the temperature is lowered below 4 °C, and WC is the one that is assumed to facilitate the cold unfolding. An equation for B h can then be obtained that is analogous to eq 13 but with (WC) replacing (WH):
ln
[ ]
B h ) (ln A1,C2) + 2 ln(WC) + λnB h 1-B h
(17)
To proceed to the analogue of eq 14, we replace WC in eq 17 by rearranging eq 2 to give
WC )
WD KCD
WC + WD ) WT
(18) (19)
Water Structure and Protein Unfolding
J. Phys. Chem. B, Vol. 103, No. 28, 1999 5915
1 WC ) WT 1 + KCD ln
[ ]
[
(20)
]
B hC 1 ) ln A1,C2 + 2 ln WT + λnB h 1-B hC 1 + KCD
(21)
The refolding in the cold region, as the temperature rises above the coldest reached, occurs in parallel with the transformation of the water structure, WC f WD. The extent of refolding with temperature rise, measured by ln[B h C/(1 - B h C)], thus depends on the water structure equilibrium constant (KCD) as well as on A1,C2, the intrinsic strength of binding by protein of the WC structural form of cold water molecules. With rising temperature in the cold domain, WC decreases, since it is converted to WD. In analogy to eqs 15 and 16, we can write
∆H°DC d ln KCD ∆H°CD )) 2 dT RT RT2
(22)
d ln A1,C ∆H°A1,C ) dT RT2
(23)
Concordance of Experiments with Analytical Formulation. Equation 14 and eq 21 each expresses the dependence of ln[B h ]/1 - [B h ] implicitly on temperature and explicitly on B h, which is also a function of temperature. Of the three terms on the right-hand side of eq 14, the third, λnB h , varies most strongly with temperature. That B h is highly sensitive to changes in temperature is evident from Figure 2, for B h covers a range from nearly 0 to nearly 1. In contrast, the terms with equilibrium constants A12 and KDH or their logarithms change much more gradually with temperature. Thus, eqs 14 and 21 suggest that a graph of ln[B h /(1 - B h )] versus B h might be approximately linear over a substantial range of B h . Figure 8 reveals that this is indeed so for the unfolding of T4 lysozyme, in both the cold and the warm regimes. Particularly noteworthy is the linearity of the graph over the range from 0.2 to 0.8 for B h . Also striking is the fit of data points from the cold range on the same curve as that for the warm range. Evidently the coefficient λn is the same and a constant for both temperature regimes. This implies further that λ, the enhancement factor for the successive affinity constants, is the same throughout the temperature range covered, for it is likely that n, the total moles of bound H2O on the unfolded protein molecule, is the same. Within the temperature range 6-18 °C for T4 lysozyme (see Figure 2), there are simultaneous overlapping contributions to unfolding from the WC and the WH forms of water. Consequently, it is not surprising that the points in the region of B h below 0.2 in Figure 8 should deviate from the line presenting independent, nonoverlapping effects of the WC and WH forms, respectively. In the region of B h above 0.8 in Figure 8, encompassing temperatures below 7 °C or above 35 °C, there is also a progressive deviation from linearity. A likely major contribution to this divergence comes from a systematic error in each of the calculated values of B h due to the difficulty of defining a baseline for the fully unfolded form of the protein.50 The baseline of the fully native form is even more uncertain for T4 lysozyme, for as is evident in Figure 2, as one descends along the warm limb toward colder temperatures and the denaturing WH decreases in concentration, the colder denaturing species WC begins to be generated and to manifest its effect on the protein conformation.
Figure 8. Unfolding of T4 lysozyme presented in terms of the natural logarithm of the operative binding ratio [B h /(1 - B h )] as a function of the fraction unfolded, B h . O, data in the region of heat denaturation; b, data in the region of cold denaturation; 0, data in the overlapping region.
Transformations in the structure of liquid water above and below the temperature of maximum density offer a simple explanation for the inversion in sign of ∆H° over the course of the denaturation from very cold to very warm temperatures (see Figure 2). As the temperature increases along the left limb of the curve, WC is progressively transformed to WD, and along the right limb, WD is converted to WH. The ∆H° for the WC ) WD equilibrium would be expected to be opposite in sign to that for the WD ) WH equilibrium. For the latter, estimates from changes in infrared spectra40 and in Raman spectra38 give a ∆H° near 2.5 kcal mol-1 for the conversion by heat of the more compact form of water to the more disordered structure, WD h WH. For WC h WD, one might reasonably expect a ∆H° of -2.5 kcal mol-1. The opposite slopes of the warm and cold limbs of the curve in Figure 2 can thus be understood in terms of the antipodal transformations in structure of the water solvent. Conclusions Vibrational spectroscopies of liquid water and its macroscopic properties such as molar volumes reveal molecular rearrangements in this solvent over a wide range of temperatures. These structural changes must play a leading role in the temperature dependence of equilibria of solutes in aqueous solution. In protein unfolding, the dome-shaped variation with temperature of the equilibrium constant for reversible denaturation is strikingly similar to the dome-shaped curve for the molar volume of pure liquid water from -30 °C upward. If one adopts a thermodynamic, phenomenological formulation of unfolding transformations that couples the conformational rearrangement in a protein with the equilibria between dense and more open forms of water, one can derive an equation for unfolding that manifests the dome-shaped temperature behavior of solvent water. It seems evident, therefore, that one must recognize the impact on a solute equilibrium of changes in the character of the solvent per se before assigning temperature effects to specific molecular interactions of solute with solvent. Acknowledgment. I have benefited from many profitable discussions with Dr. R. C. Dougherty and with Dr. G. E. Walrafen. References and Notes (1) Christensen, L. K. C. R. TraV. Lab. Carlsberg, Ser. Chim. 1952, 28, 37.
5916 J. Phys. Chem. B, Vol. 103, No. 28, 1999 (2) Maier, V. P.; Tappel, A. L.; Volman, D. H. J. Am. Chem. Soc. 1955, 77, 1278. (3) Schellman, J. A. C. R. TraV. Lab. Carlsberg, Ser. Chim. 1958, 30, 395. (4) Brandts, J. F. J. Am. Chem. Soc. 1964, 86, 4291. (5) Shortle, D.; Meeker, A. K.; Freire, E. Biochemistry 1988, 27, 4761. (6) Chen, B.; Schellman, J. A. Biochemistry 1989, 28, 685. (7) Scho¨ppe, A.; Hinz, H.-J.; Agashe, V. R.; Ramachandran, S.; Udgaonkar, J. B. Protein Sci. 1997, 6, 2196. (8) For any equilibrium that varies with temperature, the temperature of the extremum (maximum or minimum) in the standard Gibbs free-energy change does not correspond to the temperature at which the equilibrium constant attains its minimum or maximum value. Specific examples of such divergences are described in this paper. The generality of the statement arises from the differing thermodynamic relationships9,10 d ln K/dT ) ∆H°/ RT 2 and (∂∆G°/∂T)p ) -∆S°. On the other hand, the variation in the Planck function (∆Y ) -∆G/T) with temperature parallels the variation of the equilibrium constant. (For a more detailed discussion, see refs 9-11.) (9) Klotz, I. M.; Rosenberg, R. M. Chemical Thermodynamics; J. Wiley and Sons: New York, 1994, Chapter 7. (10) Rosenberg, R. M.; Klotz, I. M. J. Chem. Ed., in press. (11) Schellman, J. A. Biophys. J. 1997, 73, 2960. (12) Ha, J.-H.; Spolar, R. S.; Record, M. T., Jr. J. Mol. Biol. 1989, 209, 801. (13) Sturtevant, J. M. Proc. Natl. Acad. Sci. U.S.A. 1977, 74, 2236. (14) Niekamp, C. W.; Sturtevant, J. M.; Velick, S. F. Biochemistry 1977, 16, 436. (15) Arnold, D. S.; Plank, C. A.; Erickson, E. E.; Pike, F. P. Ind. Eng. Chem., Chem. Eng. Data Ser. 1958, 3, 252. (16) Franks, F.; Gent, M.; Johnson, H. H. J. Chem. Soc. 1963, 2716. (17) It is of interest to recall also that equilibria-dependent rates of hydrogen-isotope exchange in proteins show inverted-dome dependencies (on pH), which have also been interpreted in terms of unfolding of (the helices in) the biomacromolecule. Studies with polyamides that cannot form helical structures18 and with polypeptides retaining helical structures19 established, however, that dome-shaped dependencies (on pH) still dominated the isotope-exchange reaction. In this case, at low pH values, H2O was transformed to H3O+ and at higher pH values, H2O was converted to OH-, which catalyzed the isotope exchange at low and higher pH values, respectively. (18) Scarpa, J. S.; Mueller, D. D.; Klotz, I. M. J. Am. Chem. Soc. 1967, 89, 6024. (19) Leichtling, B. H.; Klotz, I. M. Biochemistry 1966, 5, 4026. (20) Harned, H. S.; Ehlers, R. W. J. Am. Chem. Soc. 1933, 55, 652; 2379. (21) Harned, H. S.; Embree, N. D. J. Am. Chem. Soc. 1934, 56, 1042. (22) Harned, H. S.; Sutherland, R. O. J. Am. Chem. Soc. 1934, 56, 2039. (23) Harned, H. S.; Owen, B. B. The Physical Chemistry of Electrolyte Solutions, 3rd ed.; Reinhold Publishing Co.: New York, 1958.
Klotz (24) For the ionization of acetic acid in water, Harned and Owen23 obtained the following analytic expressions for their data: log K ) -A/T + D - CT; ∆G° ) 2.303RA - 2.303RDT + 2.303RCT 2. From the first equation, one can show that Tmax(K) ) (A/C)1/2 and from the second that Tmax(∆G°) ) D/2C. Clearly the temperature of maximum in K is different from that for the maximum in ∆G°. (25) Speedy, R. J. J. Phys. Chem. 1987, 91, 3354. (26) Roentgen, W. H. Ann. Phys. 1892, 45, 91. (27) In the last sentence of his paper, Roentgen “recalls” that thoughts similar to his (“a¨nliche Gedanken”) were also expressed by Maxwell and by Pfaundler. (28) Segre´, E. Atti Accad. Lincei 1931, 13, 929. (29) Suhrmann, R.; Breyer, F. Z. Phys. Chem. 1933, B20, 17. (30) Waldron, R. J. Chem. Phys. 1957, 26, 809. (31) Busing, W. R.; Hornig, D. F. J. Phys. Chem. 1961, 65, 284. (32) Hornig, D. F. J. Chem. Phys. 1964, 40, 3119. (33) Walrafen, G. E. J. Chem. Phys. 1962, 36, 1035. (34) Walrafen, G. E. J. Chem. Phys. 1964, 40, 3249. (35) Walrafen, G. E. J. Chem. Phys. 1966, 44, 1546. (36) Walrafen, G. E. In Encyclopedia of Earth Science; Academic Press: New York, 1992; Vol. 4, pp 463-470. (37) Walrafen, G. E. In Supercooled Liquids; Fourkas, J. J., Kivelson, D., Mohanty, U., Nelson, K. A., Eds.; American Chemical Society: Washington, DC, 1997; pp 287-308. (38) Walrafen, G. E.; Fisher, M. R.; Hokmabadi, M. S.; Yang, W.-H. J. Chem. Phys. 1986, 85, 6970. (39) Swenson, C. A. Spectrochim. Acta 1965, 21, 987. (40) Worley, J. D.; Klotz, I. M. J. Chem. Phys. 1966, 45, 2868. (41) Franks, F. Water, A ComprehensiVe Treatise, Volume 1, The Physics and Physical Chemistry of Water; Plenum Press: New York, 1972. (42) D’Arrigo, G.; Maisano, G.; Mallamace, F.; Migliardo, P.; Wanderlingh, F. J. Chem. Phys. 1981, 75, 4264. (43) Woutersen, S.; Emmerichs, U.; Bakker, H. J. Science 1997, 278, 658. (44) Pimentel, G. C.; McClellan, A. L. The Hydrogen Bond; W. H. Freeman and Co.: San Francisco, CA, 1960. (45) Hare, D. E.; Sorensen, C. M. J. Chem. Phys. 1992, 96, 13. (46) Dougherty, R. C.; Howard, L. N. J. Chem. Phys. 1998, 109, 7379. (47) Klotz, I. M. Ligand-Receptor Energetics: A Guide for the Perplexed; J. Wiley and Sons: New York, 1997. (48) Klotz, I. M. Proc. Natl. Acad. Sci. U.S.A. 1996, 93, 14411. (49) From eq 6, it follows that dWH/dT ) WT/(1 + KDH)2 dKDH/dT, 1/WH dWH/dT ) [1 + KDH/WTKDH][WT/(1 + KDH)2] dKDH/dT ) 1/(1 + KDH) d ln KDH/dT, and d ln WH/dT ) 1/(1 + KDH) ∆H°DH/dT. (50) Allen, D. L.; Pielak, G. T. Protein Sci. 1998, 7, 1262. (51) Lange, N. A. Handbook of Chemistry, 6th ed.; Handbook Publishers: Sandusky, OH 1946; pp 1358-1359.
