J. Phys. Chem. 1996, 100, 7283-7292
7283
Planck-Benzinger Thermal Work Function: Determination of the Thermodynamic Stability of Chymotrypsinogen A and Ribonuclease A in Glycerol† Paul W. Chun* Department of Biochemistry and Molecular Biology, College of Medicine, UniVersity of Florida, GainesVille, Florida 32610-0245 ReceiVed: January 12, 1996X
A method is described for evaluating the temperature-invariant enthalpy, ∆H°(T0), for chymotrypsinogen A and ribonuclease A in an aqueous glycerol solution in the standard state, to determine the effect of glycerol on the thermodynamic stability of these two proteins. Chymotrypsinogen A has a temperature-invariant enthalpy of 210 kcal mol-1 at low pH in the absence of glycerol. In 10% aqueous glycerol solution, ∆H°(T0) is 239 kcal mol-1; in 40% glycerol, this value is 249 kcal mol-1. The temperature-invariant enthalpy of R-chymotrypsin dimerization is only 33 kcal mol-1, while that required for the conformational transition of ribonuclease A at low pH is approximately 60 kcal mol-1.9-13 Solvents such as glycerol tend to lower the melting temperature, 〈Tm〉, in chymotrypsinogen A at low pH, possibly due to preferential interaction as a result of solvent ordering in the associated state, rather than a conformational thermal transition from the native to denatured state. Using the Planck-Benzinger thermal work function, it is possible to determine the different types of thermodynamic compensation taking place in these systems, that is, cases in which there is (i) thermodynamic compensation dominated by the temperature-invariant enthalpy, ∆H°(T0); (ii) balanced thermodynamic compensation between ∆H°(T0) and the heat integrals; (iii) thermodynamic compensation among ∆H°(T0), the heat integrals, and T∆S°(T), typical of biological systems; and (iv) thermodynamic compensation dominated by T∆S°(T), the bound energy change, and the heat integrals.
I. Introduction Benzinger1
In 1971, T. H. proposed a thermal work function to take into account both Boltzmann statistical energy effects and the energies of quantum-mechanical bonds. While the latter are usually not altered significantly in micromolecular reactions, it was Benzinger’s conjecture that the large-scale and long-range changes of conformation which accompany protein folding or assembly might generate significant energy differences due to the cumulative alteration of their numerous covalent bond structures. It is probably true that the Giauque function2-5 (see Appendix I) could have been used and would have been appropriate: it simply was not. Rather, Benzinger pursued a separate path, although one which can be related to the work of Giauque and other chemists. Benzinger arrived at the expression
∆W°(T) ) ∆H°(T0) - ∆G°(T)1
(see Appendix II)
where ∆W°(T) represents the thermal energy difference responsible for breaking or forming noncovalent bonds in macromolecular interactions, ∆H°(T0) is the temperature-invariant enthalpy term arising from noncovalent interaction energies (including van der Waals, electrostatic, hydrogen bonding, and hydrophobic effects), and ∆G°(T) is the Gibbs free energy change.6,7 Traditionally, the Gibbs-Helmholtz expression has been applied to the formation or breaking of covalent bonds in a chemical reaction, and the heat of reaction has only one term, * Address correspondence to P. W. Chun, Box 100245, Health Science Center, Department of Biochemistry and Molecular Biology, University of Florida, Gainesville, FL 32610-0245. † This work was supported by NSF Grant BMD 83-12101-(02) and in part by a faculty development award, Division of Sponsored Research, University of Florida. X Abstract published in AdVance ACS Abstracts, April 1, 1996.
0022-3654/96/20100-7283$12.00/0
the heat integral. In the case of macromolecular interactions on the scale involved in most biological systems, however, the difference between the heat capacities of product and reactant may be substantial enough to totally obscure any difference between the temperature-invariant enthalpy and the heat integrals.1,9-13 Thus, the Gibbs-Helmholtz expression cannot be applied to accurately describe the noncovalent chemical bond forces operating in a biological system. The Planck-Benzinger thermal work function, based on ∆G°(T) and ∆W°(T), however, yields the thermodynamically stable function ∆H°(T0), obtainable at several different temperatures. The temperature-invariant enthalpy is a measure of the chemical force that gives molecules the cohesiveness to form biological structures and represents the basic energy level of that interaction. The major advantage of this type of evaluation is the separation of temperature-invariant interactions from those which are temperature-dependent. Most biochemical interactions can be more easily understood in this context than via the ordinary Gibbs-Helmholtz approach: for example, solvent interactions, the effects of solvent additives such as glycerol, sucrose, or Hofmeister anions on protein interactions, sitespecific interactions of macromolecules, and distinguishing between denaturation and protein self-association. Glycerol has been used for many years to stabilize the activity of enzymes and the native structure of proteins.14-24 Previous studies have shown that a number of proteins are preferentially hydrated in aqueous glycerol solution, suggesting that the preferential exclusion of glycerol from the domain of protein molecules is due to enhanced solvent ordering.25-29 This communication describes a method of evaluating the temperature-invariant enthalpy, ∆H°(T0), for chymotrypsinogen A and ribonuclease A in an aqueous glycerol solution in the standard state, to determine the effect of glycerol on the thermodynamic stability of these two proteins. The significance of the thermodynamic compensation operating among ∆H°(T0), © 1996 American Chemical Society
7284 J. Phys. Chem., Vol. 100, No. 17, 1996
Chun TABLE 1: Evaluation of ∆H°(T0) and the Expansion Coefficients of the Planck-Benzinger Thermal Work Function in Chymotrypsinogen A in 0.04 M Glycine Buffer at pH 2.0 as a Function of Glycerol Concentration, Based on Data Reported by Gekko and Timasheff26 f
Figure 1. Thermodynamic plot of the standard Gibbs free energy change of chymotrypsinogen A as a function of temperature in different glycerol concentrations at pH 2.0, based on data reported by Gekko and Timasheff26 in the temperature range 312-320 K. The experimental data were evaluated with the general linear model procedure of statistical analysis of the IMSL subroutine. The solid lines represent fitted data. F ) 0.001; thus the goodness of fit of the experimental data was 99.5% or better in each case.
% glycerol
R [∆H°(T0)] (kcal mol-1)
β (kcal mol-1 K-2)
γ (kcal mol-1 K-3)
0a 10b 20c 30d 40e
210.42 238.85 245.79 245.816 249.420
-5.249 × 10-3 -6.09 × 10-3 -6.303 × 10-3 -6.345 × 10-3 -6.468 × 10-3
9.941 × 10-6 1.170 × 10-6 1.218 × 10-5 1.231 × 10-5 1.259 × 10-5
a R-square ) 0.9995; SD ) 0.01352; PR > F ) 0.0001. b R-square ) 0.9996; SD ) 0.01147; PR > F ) 0.0001. c R-square ) 0.9992; SD ) 0.02659; PR > F ) 0.0002. d R-square ) 0.9994; SD ) 0.02435; PR > F ) 0.0001. e R-square ) 0.9992; SD ) 0.02722; PR > F ) 0.0002. f Compiled using the general linear model procedure of statistical analysis of an IMSL subroutine adapted for use in an IBM or Macintosh PC (Computer-aided Analysis of Biochemical Processes: CAABP, 1991). Each data point between 0 and 340 K was evaluated with extrapolation of F-statistics.31,32 F ) 0.0001; thus the goodness of fit of experimental data was 99.8% or better in each case.
the heat integrals and T∆S°(T), the bound energy change, in these systems is discussed. II. Methods and Procedures To analyze the thermodynamic processes operating in a particular biological system, and the thermal transition taking place, it is necessary to extrapolate the thermodynamic parameters over a much broader temperature range, and the enthalpy, entropy, and heat capacity terms are evaluated as partial derivatives of the Gibbs free energy function defined by Helmholtz and Kelvin’s expression,7,30 assuming that the heat capacity integral upon which these expressions are based is a continuous function.
∂∆G°(T)/∂T ) -∆S°(T), {∂∆G°(T)/T}/∂T ) -∆H°(T)/T 2 ∂∆H°(T)/∂T ) ∆C°p(T), ∂∆S°(T)/∂T ) ∆C°p(T)/T The interplay among ∆H°(T0), the heat integrals, and T∆S°(T) during the thermodynamic compensation process in a biological system can be visualized by this method of analysis. (I) Why the Third-Order Polynomial Functions for the Gibbs Free Energy, ∆G°(T) ) r + βT 2 + γT 3, in the Standard State Fit Equally Well to Any Noncovalent Macromolecular Interaction. Planck’s value for the average energy per mode of oscillation8,30 is j ) (E/Z) ) (πkT)4/5(hν)3, and at low temperatures, Einstein’s formula reduces the specific heat at constant volume to CV ) (hν/kT)2ehν/kT. Adapting Planck’s expression for , the total energy of the system is E ) (3N0/5)(πkT)4/(hνm)3. When Planck’s equation is multiplied by 3N0, the low-temperature specific heat of simple solid becomes CV ) 12(π4/5)(N0k)(kT/hνm)3. With a proper substitution of θ ) hνm/k and R ) N0k, the low-temperature specific heat becomes CV ) (12π4R/5)(T/θ)3. Clearly, the energy and specific heat are universal functions of (kT/hνm) or (T/θ). The cuberoot relation has proven reliable when used as an extrapolation method for measuring the difference between the heat content, or entropy, of a simple solid at a certain low temperature and the corresponding value at absolute zero. (II) Computational Procedure. In this treatment, the Gibbs free energy data in the standard state, shown in Figures 1, 4, and 6, were fitted with a three-term linear polynomial (eq 1) in
Figure 2. Thermodynamic plot of the Planck-Benzinger thermal work function: The conformational transition of chymotrypsinogen A as a function of glycerol concentration at pH 2.0, based on data reported by Gekko and Timasheff26 in the temperature range 312-320 K. Each data point between 0 and 450 K was evaluated with extrapolation of F-statistics.
the 280-343 K temperature range in which experiments had been conducted (chymotrypsinogen A,26 R-chymotrypsin,28,29 ribonuclease A26). The rationale for selecting the third-order Gibbs free energy polynomial function, given by eq 1, may be found in the fundamentals of quantum theory.30 To analyze the thermodynamic processes operating in a particular biological system, and the thermal transition taking place, it is necessary to extrapolate the thermodynamic parameters over a much broader temperature range. This does not necessarily imply that these processes continue unaltered outside the data range. Other thermodynamic parameters were determined by manipulation of the Gibbs polynomial function,
∆G°(T) ) R + βT 2 + γT 3
(1)
where ∆H°(T0) ) R, using the general linear model procedure of the International Mathematical Subroutine Library’s (IMSL) mathematical and statistical subroutines.31,32 Techniques of error-minimizing statistical analysis31,32 were also applied to determine the polynomial function which would give the best possible fit in the maximum number of cases, while still exhibiting zero slope at 0 K, consistent with the third law of thermodynamics.7,8,30 Once evaluated as shown in Tables 1,
Thermodynamic Stability of Proteins
J. Phys. Chem., Vol. 100, No. 17, 1996 7285
TABLE 2: Comparison of ∆H°(T0), ∆H°(Ts), and ∆H°(Tm) for Chymotrypsinogen A Conformational Transition as a Function of Glycerol Concentrationa % glycerol
∆H°(T0) at 0 K (kcal mol-1)
∆H°(T0) at 〈Th〉 (kcal mol-1)
∆H°(T0) at 〈Ts〉 (kcal mol-1)
∆H°(T0) at 〈Tm〉 (kcal mol-1)
〈Th〉 (K)
〈Ts〉 (K)
〈Tm〉 (K)
0 10 20 30 40
210.42 ( 3.53 238.85 ( 2.18 245.75 ( 1.89 245.82 ( 1.95 249.40 ( 3.54
210.86 ( 1.59 238.26 ( 1.32 246.48 ( 2.03 244.73 ( 1.41 248.07 ( 1.59
210.23 ( 0.43 238.68 ( 3.63 250.11 ( 3.59 249.60 ( 3.14 252.60 ( 1.37
210.45 ( 1.56 238.46 ( 0.93 244.32 ( 2.31 246.57 ( 1.46 249.16 ( 1.24
315 315 315 315 315
350 345 345 345 340
380 375 370 365 365
a Compiled using the general linear model procedure analysis of the IMSL subroutine.31,32 The goodness of fit of experimental data was 99.5% or better in each case.
TABLE 3: Evaluation of ∆H°(T0) and the Expansion Coefficients of the Planck-Benzinger Thermal Work Function for r-Chymotrypsin Dimerizationa R [∆H°(T0)] (kcal mol-1)
β (kcal mol-1 K-2)
γ (kcal mol-1 K-3)
32.3530
-1.257 × 10-3
2.7672 × 10-6
a Compiled using the general linear model procedure of statistical analysis of the IMSL subroutine adapted for use in the PC (CAABP).32 The goodness of fit of the experimental data was 99.9% or better in each case. R-square ) 0.000; SD ) 0.02656; PR > F ) 0.001.
3, and 5, the coefficients R, β, and γ were used in other thermodynamic polynomials. The goodness of fit of the experimental data was 99.5% or better in each case.31,32 Thermodynamic parameters ∆H°(T), ∆C°p(T), T∆S°(T), and ∆W°(T) were then defined as follows:
∆H°(T) ) R - βT 2 - 2γT 3
(2)
∆C°p(T) ) -2βT - 6γT 2
(3)
T∆S°(T) ) -2βT 2 - 3γT 3
(4)
∆W°(T) ) -βT 2 - γT 3
(5)
assuming that the heat capacity integral upon which these expressions are based is a continuous function. A built-in constraint in the extrapolation procedure is that the values for ∆G°(T) and ∆H°(T) determined from the polynomial functions intersect at 0 K with zero slope on a thermodynamic plot, thus obeying Planck’s definition of the Nernst heat theorem, consistent with the third law of thermodynamics.8 By definition, the value of ∆H°(T0) will always be positive. Other polynomial functions failed to meet all three restrictions of ∆G°(T) and ∆H°(T) intersecting at 0 K with zero slope and ∆H°(T0) being positive and thus were discarded. Values of these thermodynamic parameters were generated by fitting the coefficients R, β, and γ, as shown in Figures 2, 3A-E, 5, 7, and 8A-E, using the International Mathematical Subroutine Library (IMSL) program, in which each equation was iteratively executed in Do loops in steps of 1 K, and the values were plotted and overlaid for each set of experimental conditions. This IMSL program was incorporated into software developed for the computer-aided analysis of biochemical processes, and each data point was evaluated with extrapolation of F-statistics.32 The linear polynomial function for ∆G°(T) provided a reasonably good fit for the three systems. The ∆G°(T) data collected over the limited temperature range of 280-343 K were extrapolated over a wider range to evaluate the various thermodynamic parameters, as shown in Figures 2, 3A-E, 5, 7, and 8A-E. III. Results (I) Evaluation of ∆H°(T0) at 〈Ts〉 and 〈Tm〉, 〈Th〉, and 0 K and Analysis of the Planck-Benzinger Thermal Work
Function, ∆W°(T). Plots of the Gibbs free energy, ∆G°(T), and the Planck-Benzinger thermal work function, ∆W°(T), as a function of temperature for chymotrypsinogen A and ribonuclease A at different percent glycerol values are shown in Figure 2. The temperature-invariant enthalpy, ∆H°(T0), may be evaluated at four points on these curves: 〈Th〉, 〈Ts〉, 〈Tm〉, and 0 K. A plot of the Gibbs polynomial function, ∆G°(T) ) R + βT 2 + γT 3, as a function of temperature as shown in Figures 3A-E and 5, exhibits an initial zero value of ∆G°(T) at 〈Th〉, the harmonious temperature at which ∆H°(T) and T∆S°(T) intersect and ∆C°p(T) approaches zero, and a minimum value for ∆G°(T) at 〈Ts〉, the stable temperature at which T∆S°(T) ) 0. Here 〈Tm〉 is the melting temperature at which the heat integrals are equal to T∆S°(Tm) ≡ ∆H°(Tm) (Figures 3A-E and 5). The nature of the thermodynamic compensation which takes place between 〈Th〉 and 〈Tm〉 may be characterized by evaluating the temperature-invariant enthalpy at 〈Th〉 or at 0 K, the values of ∆W°(T) and ∆G°(T) exhibit a positive maximum and negative minimum, respectively, at 〈Ts〉, therefore, the temperatureinvariant enthalpy, ∆H°(T0) ) ∆W°(Ts)max + ∆G°(Ts)min at 〈Ts〉, and the temperature-invariant enthalpy at the melting temperature is, by Benzinger’s definition, ∆H°(T) ) ∆H°(T0) + ∫TT0∆C°p(T) dT, where ∆H°(T) and T∆S°(T) are of the same magnitude, ∆W°(T) ) ∆H°(T0), and ∆G°(T) approaches zero. The results of this study make it apparent that the nature of the biochemical thermodynamic compensation which takes place between 〈Th〉 and 〈Tm〉 may be characterized by evaluating ∆H°(T0) and the heat integrals. Between 〈Th〉 and 〈Ts〉, ∆H°(T0) is compensated by T∆S°(T), the bound energy of the system, while between 〈Ts〉 and 〈Tm〉, it is the heat integrals which are compensated by T∆S°(T). (II) Conformational Transition of Chymotrypsinogen A as a Function of Glycerol Concentration. The self-association of chymotrypsinogen A above pH 8.0 and at low ionic strength in the vicinity of the isoelectric point has been studied by several researchers,33,34 with both monomer-dimer-trimer and isodesmic models proposed as modes of association.34 Osborne and Steiner35 observed that the self-association of chymotrypsinogen A is reduced by decreasing the pH below 8.0 or by an increase in electrolyte level. All these studies suggest, however, that the modes of self-association operate in this system independent of temperature, as determined from the weight-average molecular weight distribution as a function of concentration.36 In examining X-ray crystallographic data for the dimeric forms of chymotrypsinogen A and R-chymotrypsin, Wang et al.37 found that R-chymotrypsin exhibits a 2-fold axis of symmetry. No such symmetry was observed in the case of the chymotrypsinogen A dimer. In this study, values for ∆H°(T0) for chymotrypsinogen A at pH 2.8 in the absence of glycerol were found to be 210.42 ( 3.53 and 210.23 ( 0.43 kcal mol-1 at 0 K and at 〈Ts〉. As shown in Figure 3A-E and Table 2, in aqueous glycerol solutions of varying concentrations from 10 to 40%, ∆H°(T0) values for
7286 J. Phys. Chem., Vol. 100, No. 17, 1996
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Figure 3. Thermodynamic plot of the conformational transition of chymotrypsinogen A in 0% glycerol at pH 2.0. Each data point between 0 and 450 K was evaluated with extrapolation of F-statistics.31,32 Thermodynamic Compensation for Chymotrypsinogen A Conformational Transitiona
A B C D E a
% glycerol
〈Th〉 (K)
〈Tm〉 (K)
0 10 20 30 40
320 315 315 315 315
380 375 370 365 360
T∆S°(Th) ∆H°(Th) (kcal mol-1) 101.97 108.45 110.08 104.89 102.10
100.63 109.05 108.61 105.69 103.77
T∆S°(Tm) ∆H°(Tm) (kcal mol-1) -145.97 -170.68 -125.16 -104.89 -114.71
-146.98 -168.89 -124.77 -105.69 -113.96
∫TT0∆C°p(T) dT (kcal mol-1) -145.74 -170.68 -124.16 -105.69 -113.96
Compiled using GLM procedure of statistical analysis of the IMSL subroutine.
chymotrypsinogen A increase with increasing percent of glycerol, while 〈Th〉 remains relatively constant at 315 K, 〈Ts〉 drops from 350 to 340 K, and 〈Tm〉 decreases from 380 to 365 K. In a 10% glycerol solution, ∆H°(T0) at 0 K increases to 238.85 ( 2.18, ∆H°(T0) at 〈Th〉 is 238.26 ( 1.32, ∆H°(T0) at 〈Ts〉 is 238.68 ( 3.63, and ∆H°(T0) at 〈Tm〉 is 238.46 ( 0.93 kcal mol-1. It is apparent that the four values of ∆H°(T0) deviate by less than 0.68% at each glycerol concentration. At a glycerol concentra-
tion of 40%, ∆H°(T0) at 0 K is 249.40 ( 3.54; at 〈Th〉, 248.07 ( 1.59; at 〈Ts〉, 252.60 ( 1.37; and at 〈Tm〉, 249.16 ( 1.24, only a slight increase over values in 20 or 30% glycerol, as seen in Table 2. It is readily apparent that chymotrypsinogen A is highly stable in glycerol solution. As shown in Figure 3A, in the absence of glycerol, values for ∆H°(T) and T∆S°(T) converge at a point above 〈Ts〉, designated 〈Th〉, or the harmonious temperature, and remain
Thermodynamic Stability of Proteins
Figure 4. Thermodynamic plot of the standard Gibbs free energy change of R-chymotrypsin dimerization in 0.178 M NaCl (or in 0.1 M NaCl), 0.01 M acetate buffer, pH 4.10, based on data reported by Aune and Timasheff22 and by Aune et al.29 in the temperature range 277312 K. The experimental data for the standard Gibbs free energy change as a function of temperature were evaluated with the general linear model procedure of statistical analysis of the IMSL subroutine.32 The solid line is the fitted data. F ) 0.001; thus the goodness of fit of the experimental data was 99.5% or better in each case.
equivalent until they reach 〈Tm〉. The values for T∆S°(Th) and ∆H°(Th) in the absence of glycerol were determined to be 101.97 and 100.63 kcal mol-1, respectively. As shown in Figure 3BE, the values for T∆S°(Th) and ∆H°(Th) in the presence of glycerol were determined to be as follows (in kcal mol-1): no glycerol, 101.97 and 100.63 at 320 K; 10% glycerol, 108.45 and 109.05 at 315 K. At 〈Tm〉, T∆S°〈Tm〉 and ∆H°(Tm) values were determined to be -145.97 and -146.98 at 320 K in no glycerol and -170.68 and -168.89 at 315 K in 10% glycerol, respectively. It is apparent that T∆S°(Th) = ∆H°(Th) and T∆S°〈Tm〉 = ∆H°(Tm) for chymotrypsinogen A in no glycerol or 10% glycerol in the temperature range 315-375 K, indicative of balanced thermodynamic compensation among ∆H°(T0) and the heat integral, and T∆S°(T). In chymotrypsinogen A at higher glycerol concentrations (Figure 3C-E), the thermodynamic quantity ∆H°(T0) and heat integral ∫TT0∆C°p(T) dT are not identical. In these cases, the magnitude of the heat integral is much smaller than that of ∆H°(T0) and becomes increasingly smaller with increased glycerol concentration. These seem to be systems in which the thermodynamic compensation is dominated by the temperature-invariant enthalpy. The melting temperature of chymotrypsinogen A in the absence of glycerol was found to be 380 K, decreasing to 365 K in 40% glycerol. A similar decrease in 〈Tm〉, from 330 to 305 K, was observed with decreasing pH from 3.0 to 1.0.13,44 This lowering of the melting temperature would seem to be indicative of denaturation, but the relatively high value of the temperature-invariant enthalpy for chymotrypsinogen A suggests that the shift in 〈Tm〉 is the result of a conformational fluctuation of the associated state due to preferential interaction, rather than lying in the two-state transition theory of a denatured or unfolded state assumed by earlier workers.44-52 It has also been established that the temperature-invariant enthalpy of R-chymotrypsin dimerization is 33.6 kcal mol-1, equivalent to 0.13 kcal/residue. It is obvious from the values for the temperature-invariant enthalpy that these inactive and active forms of the same enzyme must differ markedly in structure and stability. (III) r-Chymotrypsin Dimerization in the Absence of Glycerol. Above pH 6.0, R-chymotrypsin undergoes selfassociation of a higher order than monomer-dimer, as has been
J. Phys. Chem., Vol. 100, No. 17, 1996 7287 reported by several researchers.38-42 In this study, values for four quantities, ∆H°(T0) at 〈T0〉, 〈Th〉, 〈Ts〉, and 〈Tm〉, for R-chymotrypsin dimerization at pH 4.10 as shown in Table 4 and Figure 5, were found to be 32.35 ( 0.13, 32.35 ( 0.17, 32.39 ( 0.11, and 32.36 ( 1.92 kcal mol-1, respectively. Values for 〈Ts〉 and 〈Tm〉 were determined to be 305 and 345 K, as shown in Table 4. As seen in Figure 5, ∆H°(T) and T∆S°(T) intersect at 〈Th〉, where ∆G°(T) and ∆C°p(T) approach zero. Thermodynamic compensation typical of many biological systems is observed up to 〈Tm〉, and the range of 〈Th〉 varies from 225 to 345 K, as shown in Figure 5. ∆H°(T0) at 〈Tm〉 was determined to be 32.86 kcal mol-1. It has been previously reported, in earlier examinations of four other self-associating protein systems,9-13 that ∆H°(T0) at 0 K is 26.83 ( 1.24, 50.76 ( 1.89, 29.79 ( 3.98, and 71.24 ( 1.24 kcal mol-1 for bovine liver L-glutamate dehydrogenase, S-CAM-apo A-II, glucagon, and tubulin (Mg2+), respectively. The range of 〈Th〉 in each of these systems is the same as that for R-chymotrypsin dimerization10 and ribonuclease A at low pH.12 (IV) Conformational Transition of Ribonuclease A as a Function of Glycerol Concentration. In the case of ribonuclease A in glycerol solution, values for the Gibbs free energy change as a function of temperature, ∆G°(T), becoming linear over the experimental temperature range, as shown in Figure 6. Hence no maximum or minimum values for ∆W°(Ts) and ∆G°(Ts) are observed, and the thermodynamic compensation takes place between 〈T0〉 and 〈Tm〉, as shown in Figure 8A-E. At 〈Tm〉, ∆H°(T) and T∆S°(T) are equal and positive, so the system as a whole loses entropy. Evaluation of ∆H°(T0) and the expansion coefficients of the Planck-Benzinger thermal work function in ribonuclease A as a function of temperature in different glycerol concentrations in 0.04 M glycine buffer-glycerol mixture at pH 2.8, based on data reported by Gekko and Timasheff26 (Figure 6), are shown in Table 5. Using the expansion coefficients of R, β, and γ (Table 5), obtained using the Gibbs free energy function, eq 1, the following thermodynamic functions may be generated: no glycerol
40% glycerol
∆G°(T) ) 45.34 (5.0 × 10-4) + (1.26 × 10-7)T 3 ∆H°(T) ) 45.34 + (5.01 × 10-4)T 2 (2.52 × 10-7)T 3 T∆S°(T) ) (10.2 × 10-4)T 2 (3.78 × 10-7)T 3 ∆C°p(T) ) (10.2 × 10-4)T (7.56 × 10-7)T 2 ∆W°(T) ) (5.01 × 10-4)T 2 (1.26 × 10-7)T 3
∆G°(T) ) 33.48 (9.04 × 10-5)T 2 + (7.36 × 10-7)T 3 ∆H°(T) ) 33.48 (9.04 × 10-5)T 2 (14.7 × 10-7)T 3 T∆S°(T) ) (18.04 × 10-5)T 2 (22.05 × 10-7)T 3 ∆C°p(T) ) (18.04 × 10-5)T (44.1 × 10-7)T 2 ∆W°(T) ) (9.04 × 10-5)T 2 (7.35 × 10-7)T 3
With increasing glycerol concentration, ∆H°(T0) values decrease from 45 to 33 kcal mol-1. In 40% glycerol, the magnitude of the temperature-invariant enthalpy is the same as that for R-chymotrypsin dimer, as seen in Table 6 and Figures 7 and 8A-E. It would appear that with increasing temperature, ribonuclease A association tends to shift toward the dimeric form, which is consistent with findings reported by Crestfield et al.39 As glycerol concentration increases to 40%, 〈Tm〉 remains unchanged. The values for the heat integrals were determined to be (in kcal mol-1) as follows: no glycerol, 39.47; 10% glycerol, 39.99; 20% glycerol, 50.24; 30% glycerol, 53.63; 40% glycerol, 52.36, respectively, as seen in Figure 8A-E. At low concentrations of glycerol, below 10%, thermodynamic compensation in ribonuclease A at pH 2.8 is balanced;
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TABLE 4: Comparison of ∆H°(T0) at 0 K, ∆H°(T0) at 〈Tc〉, ∆H°(T0) at 〈Ts〉, and ∆H°(T0) at 〈Tm〉 for r-Chymotrypsin Dimerizationa ∆H°(T0) at 〈T0〉 (kcal mol-1)
∆H°(T0) at 〈Th〉 (kcal mol-1)
∆H°(T0) at 〈Ts〉 (kcal mol-1)
∆H°(T0) at 〈Tm〉 (kcal mol-1)
〈Th〉 (K)
〈Ts〉 (K)
〈Tm〉 (K)
32.35 ( 0.13
32.35 ( 0.17
32.39 ( 0.11
32.36 ( 1.92
225
305
345
a
subroutine.31,32
Compiled using the general linear model procedure of statistical analysis of the IMSL The goodness of fit of the experimental data was 99.9% or better in each case. Observed values for 〈Tm〉 are consistent with experimental values.
Figure 5. Thermodynamic plot of R-chymotrypsin dimerization in 0.178 M NaCl (or in 0.1 M NaCl), 0.01 M acetate buffer, pH 4.10, based on data shown in Figure 4 in the temperature range 0-380 K. Each data point between 0 and 380 K was evaluated with extrapolation of F-statistics.31,32 Thermodynamic Compensation for R-Chymotrypsin Dimerizationa 〈Th〉 (K)
〈Tm〉 (K)
T∆S°(Tm)
225
345
-68.76
∆H°(Tm)
∆W°(Tm)max + ∆G°(Ts)min (kcal mol-1)
T ∫T0 ∆C°p(T) dT (kcal mol-1)
-69.31
32.86
-68.76
(kcal mol-1)
a
Compiled using the general linear model procedure analysis of the IMSL subroutine. The goodness of fit of experimental data was 99.5% or better in each case.31,32
Figure 6. Thermodynamic plot of the Gibbs free energy change in ribonuclease A as a function of temperature at different glycerol concentrations in 0.04 M glycine buffer-glycerol mixture at pH 2.8, based on data reported by Gekko and Timasheff26 in the temperature range 293-333 K. The solid lines represent fitted data. F ) 0.001; thus the goodness of fit of experimental data was 99.5% or better in each case.
at higher glycerol concentrations, thermodynamic compensation is operating among ∆H°(T0), the heat integrals, and T∆S°(T). IV. Discussion In previous studies by a number of researchers,44-52 the conformational thermal transition of ribonuclease A, R-chy-
Figure 7. Thermodynamic plot of the Planck-Benzinger thermal work function: The conformational thermal transition of ribonuclease A as a function of glycerol concentration in 0.04 M glycine buffer-glycerol at pH 2.8. Each data point between 0 and 500 K was evaluated with extrapolation of F-statistics.31,32
motrypsin, and chymotrypsinogen A at low pH has been analyzed in terms of a two-state transition, from the native to the denatured state. The equilibrium constant has been calculated from spectroscopic data. Values for the temperatureinvariant enthalpy obtained from this study indicate that these molecules are highly stable at low pH in glycerol solution, rather than suggest that they are thermally denatured under these conditions.
Thermodynamic Stability of Proteins
J. Phys. Chem., Vol. 100, No. 17, 1996 7289
Figure 8. Thermodynamic plot of the conformational thermal transition of ribonuclease A as a function of glycerol concentration in 0.04 M glycine buffer-glycine mixture of pH 2.8. Each data point between 0 and 500 K was evaluated with extrapolation of F-statistics.31,32 Thermodynamic Compensation for Ribonuclease A Conformational Transitiona
A B C D E
% glycerol
∆H°(T0) at 〈T0〉 (kcal mol-1)
∆H°(T0) at 〈Tm〉 (kcal mol-1)
〈Tm〉 (K)
T∆S°(Tm) (kcal mol-1)
∆H°(Tm) (kcal mol-1)
∫TT0∆C°p(T) dT (kcal mol-1)
0 10 20 30 40
45.34 45.58 37.31 33.48 33.48
44.39 44.20 36.94 34.74 31.94
320 310 315 315 315
84.91 85.57 87.55 87.11 86.84
85.87 85.87 87.92 87.84 87.98
39.47 39.99 50.24 53.63 52.36
a Thermodynamic plot of the conformational thermal transition of ribonuclease A in 0.5 M sucrose in 4 × 10-2 M glycine, pH 2.8. Each data point between 0 and 360 K was evaluated with extrapolation of F-statistics.
Using the Planck-Benzinger thermal work function, it is possible to determine the different types of thermodynamic compensation taking place in these systems, that is, cases in which (i) thermodynamic compensation is dominated by the temperature-invariant enthalpy, ∆H°(T0); (ii) there is balanced thermodynamic compensation between ∆H°(T0) and the heat
integrals; (iii) there is thermodynamic compensation among ∆H°(T0), the heat integrals, and T∆S°(T), typical of many biological systems; and (iv) thermodynamic compensation is dominated by T∆S°(T), the bound energy change, and the heat integrals. This last case is typical of the protein unfolding process, but was not observed in these studies.
7290 J. Phys. Chem., Vol. 100, No. 17, 1996
Chun
TABLE 5: Evaluation of ∆H°(T0) and the Expansion Coefficients of the Planck-Benzinger Thermal Work Function of Ribonuclease A as a Function of Glycerol Concentration in 0.04 M Glycine Buffer-Glycerol Mixture at pH 2.826 f % glycerol
R [∆H°(T0)] (kcal mol-1)
β (kcal mol-1 K-2)
γ (kcal mol-1 K-3)
0a 10b 20c 30d 40e
45.34 45.58 37.31 33.48 33.48
-5.01 × 10-4 -4.89 × 10-4 -2.34 × 10-4 -1.12 × 10-4 -9.04 × 10-5
1.26 × 10-7 9.46 × 10-8 -4.38 × 10-7 -6.92 × 10-7 -7.35 × 10-7
a R-square ) 0.9995; SD ) 0.01342; PR > F ) 0.0001. b R-square ) 0.9990; SD ) 0.02436; PR > F ) 0.0002. c R-square ) 0.9990; SD ) 0.02418; PR > F ) 0.0002. d R-square ) 0.9992; SD ) 0.01329; PR > F ) 0.0001. e R-square ) 0.9997; SD ) 0.01192; PR > F ) 0.0001. f Compiled using the general linear model procedure of a statistical analysis of the IMSL subroutine adapted for use in an IBM or Macintosh PC. Each data point between 0 and 500 K was evaluated with extrapolation of F-statistics.31,32
TABLE 6: Comparison of ∆H°(T0) and ∆H°(Tm) for Ribonuclease A Conformational Transition as a Function of Glycerol Concentrationa % glycerol
∆H°(T0) at 〈T0〉
∆H°(T0) at 〈Tm〉
〈Tm〉
0 10 20 30 40
45.34 ( 1.12 45.58 ( 1.68 37.31 ( 7.37 33.48 ( 1.53 33.48 ( 1.01
44.39 ( 1.02 44.20 ( 1.46 36.94 ( 1.78 34.74 ( 1.07 31.94 ( 1.16
320 315 315 315 315
a Compiled using the general linear model procedure of the IMSL subroutine.31,32 F ) 0.001; thus the goodness of fit of experimental data was 99.8% or better in each case.
TABLE 7: Preferential Hydration Parameters of Proteins in a Glycerol-Water Mixture at 20 °Ca Gekko & Timasheff25,26 % (∂g3/∂g2)T,µ1,µ3 ) (∂g1/∂g2)T,µ1,µ3 ) Bull & Breese49 Kuntz50 glycerol ξ3 (g/g) ξ1 (g/g) A1 A1 Chymotrypsinogen A 0 10 20 30 40
-0.040 ( 0.012 -0.081 ( 0.008 -0.123 ( 0.010 -0.161 ( 0.032
0.285 0.258 0.229 0.195
0.290 0.258 0.229 0.195
0.273 0.273 0.273 0.273
R-Chymotrypsin 0 10 20 30 40
-0.025 ( 0.010 -0.057 ( 0.011 -0.101 ( 0.012 -0.146 ( 0.029
0.176 0.182 0.189 0.176
0.308 0.308 0.308 0.308
d ln Ka ) (∂ ln Ka/∂ ln A3) d ln A3
Ribonuclease A 0 10 20 30 40
-0.020 ( 0.010 -0.045 ( 0.011 -0.080 ( 0.020 -0.128 ( 0.024
0.140 0.144 0.149 0.159
surface area of the protein is smaller.26 It would seem that compaction of the chymotrypsinogen A molecule occurs in the aqueous glycerol environment as a result of the exclusion volume effect. The system will tend to reduce preferential exclusion by decreasing the surface area of solvent-protein contact through enhancement of the protein self-association. X-ray crystallographic data show that chymotrypsinogen A exists as an asymmetric dimer.37 In R-chymotrypsin dimerization in the absence of glycerol, the compensatory temperature, 〈Th〉, ranges from 220 to 325 K. At 〈Th〉, ∆W°(T), ∆H°(T), and T∆S°(T) intersect, where ∆G°(T) and ∆C°p(T) approach zero, and ∆H°(T0) is of the same magnitude as ∆H°(T0) at 〈Ts〉. This temperature range is typical of this form of thermodynamic compensation, studies suggest. The thermodynamic compensation process operating in the temperature range between 〈Th〉 and 〈Tm〉 is typical of the majority of the self-associating systems previously examined.9-11 This compensation is a playoff between the thermodynamic aspects of the self-associating protein molecules: for instance, in the case of R-chymotrypsin monomer-dimer equilibrium and the interaction thermodynamics between solvent and these protein molecules. Throughout the region encompassing 〈Th〉, 〈Ts〉, and 〈Tm〉, the relative importance of solvent-solute interaction changes continuously as a function of temperature. At 〈Th〉, the interaction forces between solvent and solute are more important than at 〈Tm〉. At 〈Th〉, ∆H°(T) and T∆S°(T) are of equal magnitude and positive, resulting in a loss of entropy of the system. At 〈Tm〉, these quantities are equal and negative, so the system as a whole gains entropy. The extent of the contribution of ∆H°(T) and T∆S°(T) changes between monomer and dimer. Gekko and Timasheff26 found the amount of glycerol repelled by R-chymotrypsin in aqueous glycerol solution to be about the same as for chymotrypsinogen A and observed no change in the degree of preferential hydration with increasing glycerol concentration, suggesting that the solvent ordering of the immediate domain of the protein is relatively unchanged. This means that increasing the concentration of glycerol raises the chemical potential of the protein. In fact, it has been shown28,29 that introduction of a solvent component 3 with negative ξ3 can increase the association constant Ka, where (∂g3/∂g2)T,µ1,µ3 ) ξ3 ) g1A3 - g3A1. Here A3 is total solvation, i.e., the actual amount of glycerol in the immediate domain of the protein (grams of glycerol/gram of protein). A1 is total hydration (grams of water/ gram of protein calculated on the basis of amino acid composition). g3 is grams of glycerol per gram of water.54-57
0.355 0.355 0.355 0.355
0.322 0.322 0.322 0.322
a (∂g /∂g ) 3 2 T,µ1,µ3 ) ξ3 is the preferential interaction parameter of the glycerol component with protein (grams of glycerol bound to grams of protein).55-57 (∂g1/∂g2)T,µ1,µ3 ) ξ1 is the preferential hydration parameter of the H2O component with protein (grams of H2O preferentially bound to grams of protein).55-57 (∂g3/∂g2)T,µ1,µ3 ) ξ3 ) g1A1 - g3A1, where A3 is total solvation, i.e., the actual amount of glycerol in the immediate domain of the protein (grams of glycerol/ grams of protein). A1 is the total hydration (grams of water/grams of protein).55-57
The preferential hydration parameters of chymotrypsinogen A in aqueous glycerol environment reported by Gekko and Timasheff26 and others,53,54 seen in Table 7, indicate that glycerol is increasing repelled as its concentration is increased, while preferential hydration decreases, suggesting that the
(6)
where A3 is the activity of the added component and (d ln Ka/d ln A3) ) ∆ν3 is the preferential binding of component 3 (glycerol in this case). As is apparent in the case of chymotrypsinogen A, the preferential exclusion of solvent from the domain of protein molecules results in enhanced solvent ordering. Such solventordering phenomena have an influence on the temperatureinvariant enthalpy which cannot be ignored, and the values of this quantity will change with different solvent additives. V. Conclusion The Planck-Benzinger thermal work function is universally applicable in the analysis of the nature of the temperatureinvariant enthalpy in any biological system. It has been established that the two components of the heat of reaction are the temperature-invariant enthalpy, ∆H°(T0), and the heat
Thermodynamic Stability of Proteins integral. In the literature to date, however, values for the heat of reaction have only considered the heat integrals, while the contribution of the temperature-invariant quantity ∆H°(T0) has not been fully assessed. This gives only a partial picture of the processes taking place within that system. Solvents or solvent additives such as glycerol or Hofmeister anions tend to lower the melting temperature, 〈Tm〉, in chymotrypsinogen A and ribonuclease A9,13 at low pH, as a result of a conformational fluctuation in the associated state, rather than a transition from native to denatured state. Differences in the temperature-invariant enthalpy can be attributed to specific changes in solvent ordering in the immediate domain of the protein, resulting in increased compaction or increased association of the molecules. It is apparent, therefore, that the two-state transition theory52 does not suffice to explain the processes taken place in these systems and thus must be subject to reconsideration. It is recommended to researchers that the broader PlanckBenzinger thermal work function be used instead of the GibbsHelmholtz expression in the thermodynamic analysis of biological systems. Using the Planck-Benzinger thermal work function, it is possible to determine levels of thermodynamic compensation among the temperature-invariant enthalpy, ∆H°(T0), the heat integrals, and bound energy change, T∆S°(T), operating in these systems and thus visualize the nature of the biochemical processes taking place.
J. Phys. Chem., Vol. 100, No. 17, 1996 7291 terms and therefore is a close approximation of the chemical bond energy between reactants and products. But in the case of macromolecular interactions in a biological system,9-13 differences between the heat capacities of products and reactants may be substantial enough to totally obscure any differences between the chemical bond term and the heat integral. Thus, the heat of reaction cannot be used to accurately represent the chemical bond forces associated with these systems. It was Benzinger1 who recognized that the enthalpy term ∆H°(T) is a composite term representing both the temperatureinvariant enthalpy and the heat of a given reaction in the standard state, such that
∆H°(T) ) ∆H°(T0) + ∫T ∆C°p(T) dT T
(7)
0
∆H°(T0), the chemical bond energy at 0 K, is a virtual quantity invariant with temperature and with integrals of the difference in heat capacities between reactants and product together contributing to the Gibbs free energy function.6 Therefore, the heat integral must be incorporated into the original GibbsHelmholtz expression7,8 to relate the chemical bond energy to the other thermodynamic variables:
∆G°(T) ) ∆H°(T) - ∫T ∆C°p(T) dT - [T∆S°(T) T 0
∫TT∆C°p(T) dT]
(8)
0
Acknowledgment. I wish to thank Dr. Robert J. Hanrahan, Department of Chemistry, and Dr. Robert Cohen, Department of Biochemistry and Molecular Biology, University of Florida, for providing the information on the Gibbs free energy function as defined by Giauque.
A proper substitution of eq 7 and
∆S°(T) ) ∫T [∆C°p(T)/T] dT T 0
into eq 8 yields Appendix I The Giauque Function and the Planck-Benzinger Thermal Work Function. It should be further noted that for the Gibbs free energy for an ideal diatomic molecule in the standard state, G°effective ) ∆G°total - E°0 ) -NkT ln f/e, where G°effective is the thermodynamic potential. What counts are the effective minimum values at chemical equilibrium obtainable from the partition function, f.5,30 The Gibbs free energy function was defined as G°T - E°0/T and subsequently defined by Giauque2-5 as G°T - H°0/T. Letting G°T - H°0/T ) -Ψ°/T, then (G°T H°0) ) -Ψ°. For an ideal gas or pure solid, this function would be analogous to the Planck-Benzinger thermal work function, ∆W°(T). The Giauque function, (G°T - H°0) ) -Ψ°, has been sterile in terms of its application to biological systems. By contrast, the equivalent (although independently derived) PlanckBenzinger work function has been quite fruitful. Values for -Ψ° for pure solids are tabulated in the National Bureau of Standards’ Circular 50059 and JANAF thermodynamic tables.60 Appendix II The Planck-Benzinger Thermal Work Function. In an examination of the heat of reaction, it could be said that it is a composite of two expressions which have fundamentally different origins. One of the terms deals with the difference in heat capacities between products and reactants, while the other has a direct relationship to chemical bonding. In dealing with the chemical kinetics of small molecules, differences in heat capacities often play a minor role. For example, Arrhenius plots of ln krate versus (1/T) are linear over a moderate temperature range, but such graphs would be nonlinear (indeed, sometimes are) if ∆Cp is substantial. Consequently, the heat of reaction consists primarily of the contribution from the chemical bonding
∆G°(T) ) ∆H°(T0) - {T∫T (∆C°p(T)/T) dT T 0
∫TT∆C°p(T) dT}
(9)
0
Since the only manipulation of the Gibbs-Helmholtz expression was to add and subtract the same quantity, eq 9 is a general thermodynamic relationship not subject to any assumptions and a direct consequence of the first, second, and third laws of thermodynamics. Max Planck derived the characteristic function φ ) S - H/T,8 which may also be expressed as
φ ) ∫(Cp/T) dT - 1/T∫Cp dT
(10)
having the dimension of entropy, not of heat. This function cannot serve as a term in an expression of chemical equilibrium. Those criteria are met by formulation of the Planck-Benzinger thermal work function.
∆W°(T) ) {T∫T [(∆C°p(T)/T)] dT - ∫T ∆C°p(T) dT} (11) T
T
0
0
∆W°(T), which has been designated as the Planck-Benzinger thermal work function,9,10,13 represents the thermal energy difference responsible for breaking or forming the noncovalent bonds in macromolecular interactions and may be defined as
∆W°(T) ) ∆H°(T0) - ∆G°(T) These invariant functions, of course, relate only to reactions actually experimentally measured. This new state function clearly posits that the temperature-invariant enthalpy ∆H°(T0) is a primary source of the interaction energy essential for any
7292 J. Phys. Chem., Vol. 100, No. 17, 1996 reaction to proceed in an interacting system. The magnitude of ∆H°(T0) is an indicator of the type of macromolecular interaction taking place under experimental conditions and the basic energy level of that interaction. References and Notes (1) Benzinger, T. H. Nature (London) 1971, 229, 100-103. (2) Giauque, W. F. J. Am. Chem. Soc. 1930, 52, 4808-4815. (3) Giauque, W. F. J. Am. Chem. Soc. 1930, 52, 4816-4831. (4) Giauque, W. F. J. Am. Chem. Soc. 1941, 63, 1897-1901. (5) Meads, P. F.; Forsythe, W. R.; Giauque, W. F. J. Am. Chem. Soc. 1941, 63, 1902-1905. (6) Gibbs, J. W. Am. J. Sci. (3 series) 16, 441-458, In Trans. Conn. Acad. Sci. 1878, 3, 228. (7) Lewis, G. N.; Randall, M. Thermodynamics, 2nd ed.; (revised by Pitzer, K. S., Brewer, L.) McGraw-Hill: New York, 1961; pp 164-182, Appendix 665-668. (8) Planck, M. (Ogg, A., translator) Vorlesungen-Uber Thermodynamic, 7th ed.; as Treatise on Thermodynamics, 3rd ed. (pp 179-214, System of any number of independent constituents; pp 272-289, Nernst’s theorem); Longmans, Green and Co.: London, 1929. (9) Chun, P. W. Int. J. Quantum Chem.: Quantum Biol. Symp. 1988, 15, 247-258. K (Kelvin), one degree on the absolute temperature scale; 1 cal ) 4.184 J. h ) Planck’s constant, 6.598 × 10-27 erg s; k ) Boltzmann’s constant, 1.3808 × 10-16 erg/(molecule deg); N0 ) Avogadro’s number, 6.024 × 1023; νm, the maximum frequency of vibration of atoms in a solid as in Planck’s theory; N0k ) R ) 1.9872 cal mole-1 K-1. (10) Chun, P. W. J. Phys. Chem. 1994, 98, 6851-6861. The practical use of Benzinger’s method requires “the experimental determination of the all-important zero-point enthalpy and free entropy terms” as pointed out by Rhodes.58 As noted by Rhodes, Benzinger was pessimistic about prospects for practical use of his method, due to the difficulty of obtaining the required temperature-invariant quantities. This difficulty is addressed in the present series of papers.9-13 (11) Chun, P. W.; Oeswein, J. Biophys. J. 1982, 37, 398a. (12) Chun, P. W. Biophys. J. 1994, 66 (2), W-POS 230. (13) Chun, P. W. J. Biol. Chem. 1995, 270, 13925-13931. (14) Jarabak, J.; Seeds, A. E., Jr.; Talalay, P. Biochemistry 1966, 5, 1269-1278. (15) Ruwart, M. J.; Suelter, C. H. J. Biol. Chem. 1971, 246, 59905993. (16) Jarabak, J. Proc. Natl. Acad. Sci. U.S.A. 1972, 69, 533-534. (17) Bradbury, S. L.; Jakoby, W. B. Proc. Natl. Acad. Sci. U.S.A. 1972, 69, 2373-2376. (18) Myers, J. S.; Jakoby, W. B. Biochem. Biophys. Res. Commun. 1973, 51, 631-636. (19) Hoch, H. J. Biol. Chem. 1973, 248, 2992-3003. (20) Green, N. M.; Valentine, R. C.; Wrigley, N. G.; Alimad, F.; Jacobson, B.; Wood, H. G. J. Biol. Chem. 1972, 247, 6284-6298. (21) Behnke, O. Nature (London) 1975, 257, 709-710. (22) Shifrin, S.; Parrott, C. L. Arch. Biochem. Biophys. 1975, 166, 426432. (23) Lee, J. C.; Timasheff, S. N. Biochemistry 1975, 14, 5183-5187. (24) Lee, J. C.; Timasheff, S. N. Biochemistry 1977, 16, 1754-1764. (25) Gekko, K.; Timasheff, S. N. Biochemistry 1981, 20, 4667-4676. (26) Gekko, K.; Timasheff, S. N. Biochemistry 1981, 20, 4677-4686. (27) Timasheff, S. N. Ann. ReV. Biophys. Biomol. Struct. 1993, 22, 6797. (28) Aune, K. C.; Timasheff, S. N. Biochemistry 1971, 10, 1609-1617. (29) Aune, K. C.; Goldsmith, L. C.; Timasheff, S. N. Biochemistry 1971, 10, 1617-1622. (30) Moelwyn-Hughes, E. A. Physical Chemistry; Pergamon Press: New York, London, Paris, 1957; pp 264-279. (31) Barr, A. J.; Goodnight, J. H.; Sall, J. P.; Helwig, J. T. SAS GLM 27-GLM 131; Statistical Analysis System: University of Florida, 1979. (32) Chun, P. W. Manual for Computer-aided Analysis of Biochemical Processes, with Florida 1-2-4 spreadsheet; University of Florida: Gainesville, 1991.
Chun (33) Nichol, J. C. J. Biol. Chem. 1968, 243, 4065-4069. (34) Hancock, D. K.; Williams, J. W. Biochemistry 1968, 8, 25982603. (35) Osborne, J.; Steiner, R. F. Arch. Biochem. Biophys. 1972, 152, 849855. (36) Tung, M. S.; Steiner, R. F. Eur. J. Biochem. 1974, 44, 49-58. (37) Wang, D.; Bode, W.; Huber, R. J. Mol. Biol. 1985, 185, 595-624. (38) Gilbert, G. A. Discuss. Faraday Soc. 1955, 20, 24-32, Discussion 68-72. (39) Massey, V.; Harrington, W. F.; Hartley, B. S. Discuss. Faraday Soc. 1955, 20, 24-32. (40) Neurath, H.; Dreyer, W. J. Discuss. Faraday Soc. 1955, 20, 3243. (41) Rao, M. S.; Kegeles, G. J. Am. Chem. Soc. 1958, 80, 5724-5728. (42) Tinoco, I., Jr. Arch. Biochem. Biophys. 1957, 68, 367-372. (43) Crestfield, A. M.; Stein, W. H.; Moore, S. Arch. Biochem. Biophys. 1962, 1 (Suppl.), 217-222. (44) Brandts, J. F. J. Am. Chem. Soc. 1964, 86, 4291-4299. (45) Brandts, J. F. J. Am. Chem. Soc. 1965, 87, 2759-2760. We wish to point out a discrepancy in the literature concerning the Gibbs free energy profile for ribonuclease A conformational thermal transition, as determined using the two-state transition model. The data were originally published in 1965 by Brandts (this reference); a changed version appeared in a paper of Brandts and Hunt in 1967.46 The data in question appeared as Figure 2 in each case. The original data were republished exactly in the 1967 paper, except that the sign of the data points on the ordinate was changed so that what had appeared as a sequence of negative data now appeared to be positive. There was no explanation or comment in the second paper concerning this change. The correct understanding of the Brandts data is of considerable current importance, since these data have frequently been used as a test case for models of biochemical thermodynamics. In particular, we have examined the use of either the original Brandts data or the changed version of Brandts and Hunt in connection with studies of free energy changes in macromolecular interaction as interpreted using the PlanckBenzinger thermal work function.9-13 We have found that Brandts’ original 1965 results are entirely reasonable and are in agreement with a minimum in the free energy change in the system at the point of equilibrium, which is as it should be. On the other hand, use of the data as presented in the 1967 paper, which is the same except for sign change, would give a maximum: this is clearly an impossibility. (46) Brandts, J. F.; Hunt, L. J. Am. Chem. Soc. 1967, 89, 4826-4838. (47) Gekko, K.; Timasheff, S. N. Biochemistry 1981, 20, 4677-4686. (48) Lee, J.; Timasheff, S. N. J. Biol. Chem. 1981, 256, 7193-7201. (49) Garel, J.-R.; Baldwin, R. L. Proc. Natl. Acad. Sci. U.S.A. 1973, 709, 3347-3351. (50) Hagerman, P. J.; Baldwin, R. L. Biochemistry 1976, 15, 14621473. (51) Nall, B. T.; Garel, J.-R.; Baldwin, R. L. J. Mol. Biol. 1978, 118, 317-330. (52) Lumry, R.; Biltonen, R.; Brandts, J. F. Biopolymers 1966, 8, 917944. (53) Bull, H. B.; Breese, K. Arch. Biochem. Biophys. 1968, 128, 488496. (54) Kuntz, I. D. J. Am. Chem. Soc. 1971, 93, 514-518. (55) Reisler, E.; Haik, Y.; Eisenberg, H. Biochemistry 1977, 16, 197203. (56) Inoue, H.; Timasheff, S. N. Biopolymers 1973, 11, 737-743. (57) Kupke, D. W. In Physical Principles and Techniques of Protein Chemistry; Leach, S. J., Ed.; Academic Press: New York, 1973; Part C, pp 1-75. (58) Rhodes, W. J. Phys. Chem. 1991, 95, 10246-10251. (59) Rossini, F. D.; Wagnman, D. D. Circular of the National Bureau of Standards 500, Selected Values of Chemical Thermodynamic Properties; U.S. Government Printing Office: Washington, DC, 1952. (60) Chase, M. W., Jr.; Davies, C. A.; Downey, J. K., Jr.; Frurip, D. J.; McDonald, R. A.; Syverud, A. N. JANAF Thermodynamic Tables, 3rd ed.; American Chemical Society and the American Institute of Physics for the National Bureau of Standards: Washington, DC, 1985; Vol. 14, Parts I, II.
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