Planck−Benzinger Thermal Work Function: Thermodynamic Approach

Planck-Benzinger thermal work function in biological systems. Paul W. Chun. International Journal of Quantum Chemistry 2004 100 (10.1002/qua.v100:6), ...
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J. Phys. Chem. B 1997, 101, 7835-7843

7835

Planck-Benzinger Thermal Work Function: Thermodynamic Approach to Site-Specific S-Protein and S-Peptides Interactions in the Ribonuclease S′ System Paul W. Chun* Department of Biochemistry and Molecular Biology, Box 100245, College of Medicine, UniVersity of Florida, GainesVille, Florida 32610-0245 ReceiVed: January 28, 1997; In Final Form: July 2, 1997X

The Planck-Benzinger thermal work function ∆W(T) represents the strictly thermal components of any intraor intermolecular bonding term in a system, that is, energy other than the inherent difference of the 0 K portion of the interaction energy. The latter, the temperature-invariant enthalpy, is the only energy term at absolute zero Kelvin. The magnitude of ∆H(T0) or ∆H°(T0), the temperature-invariant enthalpy, is determined by the type of macromolecular interaction taking place under experimental conditions, and thus, this thermodynamic function should be particularly applicable to studies on the ribonuclease S-protein interaction with S-peptides with various substitutions at methionine-13. Values for ∆H(T0) for the S-protein-S-peptides interaction with such substitution at Met-13 were determined to be 25.67 ( 0.63 (Met-13 f Ala), 85.69 ( 0.35 (Met-13 f Phe), and 22.32 ( 0.19 kcal mol-1 (Met-13 f R-amino-N-butyric acid). A possible model for the fragment complementation reaction of S-peptide with S-protein if phenylalanine were to be substituted for Met-13 is proposed. The model incorporates feasible strong, site-specific Phe-Phe interaction, giving rise to the large temperature-invariant enthalpy. This thermodynamic approach should be an essential component of all future studies involving the site-directed, mutagenic approach to the examination of structurefunction problems in proteins.

I. Introduction One of the central problems in molecular biology is the formation of the native structure of protein from newly synthesized polypeptide chains. An understanding of protein foldingshow proteins achieve complex native forms from the disordered denatured statesis essential to understanding the factors that encode and stabilize particular structural features of proteins and aid in the production of their three-dimensional structures from the amino acid sequence. Under normal physiological conditions, many native protein molecules will adapt the three-dimensional structure spontaneously, but a general mechanism to explain the folding transition remains obscure. The principle of spontaneity was formulated on the basis of renaturation experiments in which the chain refolded spontaneously after the ensuing removal of the denaturant agent.1-3 The behavior of synthetic ribonuclease demonstrated that completed, and not only nascent, chains fold spontaneously under certain experimental conditions.2,3 The fully reduced, random coil polypeptide of ribonuclease can be reoxidized in air to produce the native enzyme, with full enzymatic activity. Consequently, the folding process can be regarded as a transition of the system from a state of high energy to a native conformation state of lower Gibbs free energy.1 The interaction of S-protein with S-peptide to form enzymatically active ribonuclease S′ has been extensively investigated by Richards and his colleagues.4-7 This first fragment complementing system of ribonuclease A and a second of staphylo* To whom correspondence should be addressed. Phone: (352) 3923356. Fax: (352) 392-2953. E-mail: [email protected] or http: //www.med.ufl.edu/biochem/pchun. X Abstract published in AdVance ACS Abstracts, September 1, 1997.

S1089-5647(97)00336-2 CCC: $14.00

coccal nuclease8-10 have served as excellent test subjects for the principles underlying the structure and dynamics of protein refolding. The hydrophobic residues in S-peptide thought to be particularly important for binding are Phe-8 and Met-13. In previous studies,11-14 peptides have been synthesized in which the methionine at position 13 has been replaced by seven other hydrophobic amino acids to determine the interaction among buried hydrophobic groups. This paper offers a new thermodynamic approach to sitespecific S-protein-S-peptide interaction based on the determination of the temperature-invariant enthalpy using the PlanckBenzinger thermal work function.16-20 Several cases of ribonuclease S-protein (residues 21-124) interaction with S-peptides (residues 1-13) with various substitutions at methionine 13 (Met-13) are examined.

Ala-13-S-peptide + S-protein h Ala-13-RNase S′ Met-13 f Ala Phe13-S-peptide + S-protein h Phe-13-RNase S′ Met-13 f Phe R-amino-N-butyric acid-13-S-peptide + S-protein h R-NBA-RNase S′ Met-13 f R-NBA Norleucine-13-S-peptide + S-protein h Norleucine-13-RNase S′ Met-13 f Norleucine The results for these fragment complementation systems are compared with those of the ribonuclease S′ systems.15,16

20S-peptide + S-protein h 20S-RNase S′ Met-13-S-peptide + S-protein h 13S-RNase S′ © 1997 American Chemical Society

20-S[I] 13-S[II]

7836 J. Phys. Chem. B, Vol. 101, No. 39, 1997 Knowledge of the nature and magnitude of the energetics of these interactions is needed in order to clarify their respective role in protein stabilization. II. Thermodynamic Approach to Site-Specific S-Protein-S-Peptide Interaction It has been well established in pure and applied chemistry of simple molecules that reaction energies at room temperature can be understood in terms of two contributions, one related to energy differences at 0 K and the other associated with integrals of heat capacity data over temperature. The necessity of a comparable separation of interaction energy terms for biochemical reactions, however, has not been obvious to most workers in structural and molecular biology. 1. Innate Thermodynamic Quantities. In many areas of chemistry, biochemistry, and structural and molecular biology, the temperature dependence of the point of equilibrium is of major significance. In general, the change in standard Gibbs free energy is related to the equilibrium constant Keq by the relation ∆G° ) -RTlnKeq. This important thermodynamic condition will change with temperature simply because of the temperature dependence of ∆G° (as well as ∆G).21,22 The remaining thermodynamic quantities (∆H°(T), T∆S°(T), and ∆C°p(T)) are evaluated using the Helmholtz-Kelvin expression.21,23 We note, for example, that at T ) 0 K, TS ) 0. Accordingly, because A ) U - TS and G ) H - TS23,24 it is found that at absolute 0 K, H°0 ) U°0 ) A°0 ) G°0. The residual values of all of these quantities (noting that entropy is excluded) are the same at absolute zero of the temperature scale and may be said to describe the innate thermodynamic stability of the system. For a chemical reaction, the difference in these quantities between reactants and products, that is, ∆H°0 ) ∆G°0 ) ∆A°0 ) ∆U°0 (but ∆S°0 ) 0), represent the differences in the innate thermodynamic stability between reactants and products or the change in stability between reactants and products as the reaction occurs. 2. Measuring Enthalpy Values. Enthalpies of reaction are frequently measured at or near room temperature (298 K) for a variety of theoretical and practical reasons, for instance, the relationship between ∆H°reaction and the temperature coefficient of the equilibrium constant, dlnK/d(1/T) ) -∆H°(T)/R,

∆H°298 ) ∆H°(T0) +

∫0298∆C°p dT

where this last term represents the thermal agitation energy (heat capacity integrals). Scientists dealing mainly with small molecules and wishing to derive ∆H°reaction values measured near room temperature in terms of inherent bond energy (or ∆H(T0)) may frequently ignore the difference between this and ∆H°298. The numerical error is not large. Cottrell24 has calculated that the effect on chemical bond energy calculations is only 1-2 kcal mol-1 out of a total of 80-100 kcal mol-1. In typical biological systems involving macromolecules, however, the conceptual argument is not so easily simplified. In macromolecular systems, the value of the temperatureinvariant enthalpy and heat capacity integrals are often of similar size. In research to date, however, the contribution of the temperature-invariant enthalpy has not been assessed. 3. The Giauque Function and the Planck-Benzinger Thermal Work Function. One form of the free energy function, the Giauque function,25-29 (G°T - H°0)/T ) -ψ°/T, where -ψ° ) (G°T - H°0) (see Appendix), has been extensively used in chemistry and physics. An equivalent formulation has

Chun recently found application in the biochemical literature as the Planck-Benzinger thermal work function,16-20 where the application to a given situation is quite different (Appendix). Here the Planck-Benzinger thermal work function, ∆W°(T) ) ∆H°(T0) - ∆G°(T),30 represents the strictly thermal components of any intra- or intermolecular bonding term, that is, energy other than the inherent difference of the 0 K portion of the interaction energy. The latter is the only energy term at absolute 0 K. Thus ∆W°(T) expresses completely the thermal energy difference of the process involved. Application of the thermal work function permits the separation of 0 K energy differences and energy differences associated with heat capacity integrals for a fuller understanding of reaction energies. The magnitude of ∆H°(T0) (see footnote 1 in Appendix), the temperature-invariant enthalpy, is determined by the type of macromolecular interaction taking place under experimental conditions, and thus, this thermal work function should be particularly applicable to studies on the ribonuclease S-protein interaction with S-peptides with various substitutions at methionine-13. III. Methods and Procedures (See Footnote 2 in Appendix) 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. The enthalpy, entropy, and heat capacity terms are evaluated as partial derivatives of the Gibbs free energy function defined by Helmholtz-Kelvin’s expression,22,23 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 ) ∆Cp(T), ∂∆S(T)/∂T ) ∆Cp(T)/T It has been shown in our laboratory that the third-order polynomial function provides a good fit in the temperature ranges accessible in biochemical systems;16-20 in fact, it is shown to be correct in the low-temperature limit. The rationale for selecting the third-order (T 3 model) polynomial functions for ∆G°(T) ) R + βT 2 + γT 3 and ∆H(T) ) R + βT 3eγT are found in the fundamentals of relevant quantum theory.23 At low temperature, the specific heat of a simple solid becomes Cv ) (12π4/5)(N0K)(kT/hνm)3. With a proper substitution of θ ) hνm/k and R ) N0k, Cv ) (12π4R/5)(T/θ)3. Clearly, the energy and specific heat are universal functions of (kT/hνm) or (T/θ).22,23 Cv is the specific heat at constant volume, N0 is Avogadro’s number, k is the Boltzmann constant, R ) 1.9872 cal mol-1 K-1, and νm is the maximum frequency of vibration of an atom in a solid state as in Planck’s theory.23 We have successfully applied the linear (T 3) model for ∆G°(T) ) R + βT 2 + γT 3 to numerous cases of protein folding, protein self-association, and protein-ligand or protein-DNA interaction.16-20 To use this model successfully, however, it is essential to have very accurate experimental data for the Gibbs free energy change, which shows the direction of the chemical change toward the minimum Gibbs function. As long as this is the case, this equation can be applied in the standard or nonstandard state. In cases of protein unfolding or DNA unwinding, however, the nonlinear (T 3) model for ∆H(T) ) R + βT 3eγT must be applied. The approach described here is consistent with the Occam razor principle, which states that the simplest possible description should be used in explaining a physical phenomenon and

Thermodynamics of the S-Protein-S-Peptides Interaction

J. Phys. Chem. B, Vol. 101, No. 39, 1997 7837 TABLE 1: Evaluation of ∆H°(T0) and Expansion Coefficients of the Planck-Benzinger Thermal Work Function in the Ribonuclease S′ Complementation Reaction, Based on Data Reported by Hearn et al.15 a reaction

R [∆H°(T0)] (kcal mol-1)

β (kcal mol-1 K-2)

γ (kcal mol-1 K-3)

20-S[I] 13-S[II]

34.05 ( 1.30 19.49 ( 1.08

-1.8563 × 10-3 -1.2034 × 10-3

4.5447 × 10-6 3.0989 × 10-6

a Compiled using the general linear model procedure of a statistical analysis of an IMSL subroutine adapted for use in an IBM or Macintosh PC. Each data point between 0 and 360 K was evaluated with extrapolation of F-statistics.31,32 20-S[I]: R2 ) 0.9995; SD ) 0.01342; PR>F ) 0.0001. 13-S[II]: R2 ) 0.990; SD ) 0.02436; PR>F ) 0.0002.

Figure 1. Thermodynamic plots of the standard Gibbs free energy change of the fragment complementation reaction of a ribonuclease S′ system as a function of temperature in the temperature range 273313 K at pH 7.0 in 0.3 M NaCl based on data reported by Hearn et al.(15) The experimental data for equilibrium and calorimetric measurements were evaluated with the general linear model procedure of statistical analysis of IMSL subroutine. The solid line represents fitted data. F ) 0.001, thus the goodness of fit of the experimental data was 99.9% or better in each case. 13-S[II] represents the formation of 13SRNase S′, and 20-S[I] the formation of 20S-RNase S′.

that the model should be made fancier only if the results require it. It is clear that a temperature-dependent model simpler than the third-order Gibbs polynomial model (T 3 model) is inadequate at low temperature and has been found to be unacceptable at room temperature; therefore it would be reasonable to apply a more complex model in the intervening temperature region only if the facts demand a more complex fit. Indeed the data do not require a different function; the model as described is very successful. 1. Determination of the Gibbs Free Energy Change as a Function of Temperature. The binding constants, KB (M-1), as a function of temperature were evaluated from the calorimetric titration of the S-protein of RNase S with various substitutions at Met-13-S-peptide, as reported by Naghibi et al.14 (Table 3). We converted these binding constants into the Gibbs free energy change as a function of temperature. The Gibbs free energy data for the fragment complementation reactions of S-peptide with S-protein and of Met(O2)-13-Speptide with S-protein were also extracted from Figure 4 and 5, respectively, of Hearn et al.15 2. Computational Procedure for Ribonuclease S′ System. In this treatment, the Gibbs free energy data, as shown in Figure 1 in the standard state, were fitted to a three-term linear polynomial function in the 273-320 K temperature range, the range in which experiments have been conducted. 2

∆G°(T) ) R + βT + γT

3

(1)

Once evaluated as shown in Table 1, the coefficients R, β, and γ were fitted to other thermodynamic parameters. ∆H°(T), ∆C°p(T), T∆S°(T), and ∆W°(T) are defined as follows,

∆H°(T) ) R - βT 2 - 2γT 3 ∆C°p(T) ) -2βT - 6γT

2

(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. Values of these

thermodynamic parameters were regenerated from the fitted coefficients of R, β, and γ using the International Mathematical Subroutine Library (IMSL) program in which each equation was iteratively executed 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 for the computer-aided analysis of biochemical processes, and each data point was evaluated with extrapolation of F-statistics in an IBM personal computer,31,32 as shown in Figures 2 and 3A,B. A built-in restriction 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.23 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. 3. Computational Procedure for the Interaction of SProtein with S-Peptides with Various Substitutions at Methionine-13. In this treatment, both the enthalpy and Gibbs free energy data in the nonstandard state, shown in Figure 4AD, were fitted to the third-term, linear polynomial function given in eq 1 in the 273-298 K temperature range, the range in which experiments were conducted. Once evaluated as shown in Table 2, the coefficients R, β, and γ were fitted to the other thermodynamic polynomials given in eqs 2-5. Each data point between 0 and 400 K was evaluated with extrapolation of F-statistics. F ) 0.001, thus the goodness of fit of the experimental data was 99.8% or better in each case (Figure 5A-C). The values for these thermodynamic quantities were further extrapolated as a function of temperature down to 0 K. IV. Results 1. Gibbs Free Energy Change as a Function of Temperature. Plots of the Gibbs free energy change as a function of temperature of ribonuclease S′ and S-protein-S-peptides interaction shown in Figures 1 and 4A-C are typical of biological reactions in that they show a Gibbs free energy change minimum in the system at the point of equilibrium, which is as it should be (Gibbs free energy minimum at 〈Ts〉, the stable temperature at which T∆S°(T) ) 0). This is not the case, however, for the S-protein-S-peptide reaction in which norleucine has been substituted for Met-13, shown in Figure 4D. As shown in Table 2, the fitted coefficients of R, β, and γ were found to be -33.52 ( 0.78, 6.997 × 10-4 and 1.403 × 10-6, respectively. No values for 〈Th〉, 〈Ts〉, or 〈Tm〉 could be determined. In this case, it is inferred that no fragment complementation reaction takes place, that is, the S-peptide does not realign with the ribonu-

7838 J. Phys. Chem. B, Vol. 101, No. 39, 1997

Chun

TABLE 2: Evaluation of ∆H(T0) and Expansion Coefficients of the Planck-Benzinger Thermal Work Function in the S-Protein-S′-Peptides Interaction at Met-13 as a Function of Temperature in the Range 278-313 K, Based on Data Reported by Naghibi et al.14 a substitution

R [∆H(T0)] (kcal mol-1)

β (kcal mol-1 K-2)

γ (kcal mol-1 K-3)

Met-13 f Ala f Phe dMet-13 f RNBA eMet13 f norleucine

25.05 ( 0.36 85.68 ( 1.43 22.32 ( 0.74 -33.52 ( 0.78

-1.358 × 10-3 -3.587 × 10-3 -1.269 × 10-3 6.997 × 10-4

3.384 × 10-6 8.545 × 10-6 3.111 × 10-6 -1.403 × 10-6

b

cMet-13

a Compiled using the general linear model procedure of statistical analysis of an IMSL subroutine for use in an IBM or Macintosh PC from which coefficients R, β, and γ for the S-protein-S′-peptides interaction were derived based on eqs 6-10. Each data point between 0 and 400 K was evaluated with extrapolation of F-statistics.31,32 b R2 ) 0.9998; SD ) 4.122 × 10-4; PR>F)0.001. c R2 ) 0.9934; SD ) 0.0143; PR>F ) 0.01. d R2 ) 0.9994; SD ) 5.498 × 10-4; PR>F ) 0.001. e R2 ) 0.9999; SD ) 1.6319 × 10-5; PR>F ) 0.001.

Figure 2. Thermodynamic plot of the Planck-Benzinger thermal work function: The fragment complementation reaction of ribonuclease S′ system as a function of temperature, based on data reported by Hearn et al.15 in the temperature range 273-313 K. Each data point between 0 and 360 K was evaluated with extrapolation of F-statistics. The values of ∆W°(T) and ∆G°(T) exhibit a positive maximum and negative minimum, respectively, at 〈Ts〉. Values for the temperature-invariant enthalpy at 〈Th〉, 〈Ts〉, and 〈Tm〉 are compared with the value obtained at 0 K.

clease S′ system to form the native structure, although it may bind elsewhere as part of a transfer reaction. 2. Evaluation of ∆H°(T0) or ∆H(T0) at 〈Th〉, 〈Ts〉, 〈Tm〉, and 0 K. Plots of the Gibbs free energy change, ∆G°(T), and the Planck-Benzinger thermal work function, ∆W°(T), as a function of temperature for the fragment complementation reactions of ribonuclease S′ systems 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 Kelvin. A plot of the Gibbs polynomial function, ∆G°(T) ) R + βT 2 + γT 3, as a function of temperature exhibits an initial value of 0 for ∆G°(T) at 〈Th〉, a minimum value for ∆G°(T) at 〈Ts〉, and the ∆G°(T) value again reaches 0 at 〈Tm〉. Here 〈Ts〉 is the stable temperature at which T∆S°(T) ) 0; 〈Tm〉 is the melting temperature; and 〈Th〉 is the harmonious temperature at which ∆G°(T) is 0, ∆C°p(T) approaches 0 and ∆H°(T) and T∆S°(T) are equal (i.e., these curves intersect) (Figure 3A,B). Values of the temperature-invariant enthalpy at 〈Th〉, 〈Ts〉 and 〈Tm〉 are compared with values obtained at 0 K as shown in Figures 2 and 3A,B (Figure 5A-C in the nonstandard state). 1. ∆H°(T0) ) ∆W°(Th), ∆G°(T) ) 0 at 〈Th〉 2. ∆H°(T0) ) ∆W°(Ts)max + ∆G°(Ts)min at 〈Ts〉 3. ∆H°(T0) ) ∆W°(Tm), ∆G°(T) ) 0 at 〈Tm〉 and one can define ∆H°(T) ) ∆H°(T0) + ∫0T∆C°p(T) dT 4. ∆H°(T0) at 0 K The values of ∆W°(T) and ∆G°(T) exhibit a positive maximum and negative minimum, respectively, at 〈Ts〉; therefore, the temperature-invariant enthalpy ∆H°(T0) ) ∆W°(Ts)max + ∆G°(Ts)min at 〈Ts〉. The temperature-invariant enthalpy at the

Figure 3. Thermodynamic plot of the fragment complementation reaction of 20-S[I] RNase S′ (A) and of 13-S[II] RNase S′ (B) as a function of temperature at pH 7.0 in 0.3 M NaCl. Each data point between 0 and 360 K was evaluated with extrapolation of F-statistics. 〈Th〉: (A) 180, (B) 160 K. T∆S°(Th): (A) 40.77, (B) 24.89 kcal mol-1. ∆H°(Th): (A) 41.19, (B) 25.59 kcal mol-1. 〈Tm〉: (A) 345, (B) 345 K. T∆S°(Tm): (A) -117.97, (B) -88.95 kcal mol-1. ∆H°(Tm): (A) -118.25, (B) -88.62 kcal mol-1. ∫0TC°p(T) dT: (A) -118.25, (B) -88.62 kcal mol-1.

melting temperature is, by Kirchhoff’s definition, ∆H°(T) ) ∆H°(T0) + ∫0T∆C°p(T) dT. Where ∆H°(Tm) and T∆S°(Tm) are of the same magnitude, ∆W°(T) ) ∆H°(T0) and ∆G°(T) approaches 0 (see Figure 5A,B). 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 (Figure 3A,B and 5A-C). Between 〈Th〉 and 〈Ts〉, ∆H°(T0) is compensated by T∆S°(T), the unavailable energy of the system, while between 〈Ts〉 and 〈Tm〉, it is the heat integrals (or thermal agitation energy) which is compensated by T∆S°(T). In some instances, the values of ∆H°(T0) and the heat integrals are of nearly equal magnitude, as in Figure 5B. 3. Thermodynamic Compensation in Biological Systems. For biological systems, ∆G°(T) ) 0 is not a preferred situation. To live, any biological system must do work (transpiration, digestion, reproduction, locomotion, etc.). When ∆G°(T) ) 0

Thermodynamics of the S-Protein-S-Peptides Interaction

J. Phys. Chem. B, Vol. 101, No. 39, 1997 7839

A

B

C

D

Figure 4. Thermodynamic plot of the nonstandard Gibbs free energy change of the fragment complementation reaction of the S-protein of RNase S′ with various substitutions at methionine-13 as a function of temperature, based on data reported by Naghibi et al.14 and by Varadarajan et al.13 in the temperature range 288-303 K in 50 mM sodium acetate with 100 mM NaCl, pH 6.0. The experimental data for differential scanning calorimetry were evaluated with the general linear model procedure of statistical analysis of IMSL subroutine. The solid line represents fitted data. F ) 0.001, thus the goodness of fit of the experimental data was 99.9% or better in each case. Gibbs free energy change as a function of temperature for the S-protein-S-peptide interaction: (A) Met-13 f Ala, (B) Met-13 f Phe, (C) Met-13 f R-amino-N-butyric acid, (D) Met-13 f Norleucine. These plots represent only a small portion of the thermodynamic curves shown in Figure 5A-C.

for a biological (or any energy) system, such work is impossible, as shown in Figures 2, 3A,B, and 5A-C.

∆H°(T) ) + T∆S°(T) ) +

∆H°(T) ) f

T∆S°(T) ) -

The possibility of the existence of life processes is not a clear and urgent demand of the physical universe. In fact, life exists only over a limited temperature range when the balance of energy and entropy demands are favorable. There is a lower cutoff point, 〈Th〉, when entropy is favorable but energy is unfavorable, and also an upper cutoff above which energy is favorable but entropy is unfavorable. Only between these two limits is the net chemical driving force (indicated by ∆G°(T)) favorable for such biological processes such as protein folding, subunit-subunit interaction, or protein self-assembly. Within this temperature range there will be a point with minimum (negative) free energy change, 〈Ts〉, at which the maximum work can be done. In protein unfolding or DNA unwinding, the process obviously differs over the full temperature range. There is a single cutoff point, 〈Tm〉, at which energy is unfavorable but entropy is favorable and ∆G ) 0.17,20

∆H(T) ) + T∆S(T) ) +

∆H(T) ) + f

T∆S(T) ) +

4. Fragment Complementation Reaction of the Ribonuclease S′ System. On the basis of Figure 3A,B, values for ∆H°(T0) at 0 K, 〈Th〉, 〈Ts〉 and 〈Tm〉 for the fragment comple-

mentation reactions of 20-S[I] RNase S′ and 13-S[II] RNase S′ in conformational transition at neutral pH were determined to be 34.05 ( 1.30, 34.05 ( 1.30, 34.05 ( 0.54, 34.02 ( 0.36 and 19.49 ( 1.10, 19.49 ( 1.00, 19.49 ( 0.42, 19.01 ( 0.13 kcal mol-1, respectively, as shown in Table 3. It is apparent that the values for each of the two reactions deviate by less than 0.02%. Within the limits of statistical error, it is also evident that the temperature-invariant enthalpy for 20-S[I] is also nearly equivalent to ribonuclease A in 20 or 30% glycerol,19 being 37 and 33 kcal mol-1, respectively. This is assumed to be the dimeric form of ribonuclease A as well as the formation of 20-S[I]. The ∆H°(T0) value for 13-S[II] is about 19 kcal mol-1, and one can assume it is in monomeric form. This would be consistent with findings reported by Crestfield et al.35 Evidence for the presence of several different aggregates of ribonuclease A of Mr range of 14 000 (monomer), 29 000 (dimer), and greater than 51 000 (higher aggregations) was obtained by gel filtration on a column of Sephadex G-75 in 0.2 M sodium phosphate buffer at pH 6.47. In a previous examination of self-associating protein systems16-18 a point, 〈Ts〉, has been designated on the temperature scale where ∆W°(T) and ∆G°(T) are at a positive maximum and a negative minimum, respectively. In such systems, 〈Ts〉 falls at about 313 K.18,19 In contrast, 〈Ts〉 in these fragment complementation reactions occurs at about 265 K in 13-S[II] and 272 K in 20-S[I], a drop of some 40°. At 〈Tm〉 of 345 K, T∆S°(Tm) and ∆H°(Tm) values are equivalent to the heat integral, ∫0T∆C°p(T) dT, and were determined to be -118 kcal mol-1 in 20-S[I] and -88 kcal mol-1 in 13-S[II] RNase S′. Observed values for 〈Ts〉 and 〈Tm〉 differ from previously reported

7840 J. Phys. Chem. B, Vol. 101, No. 39, 1997

Chun substitutions at methionine-13 as shown in Table 4. As seen in the table, substitution of Phe at Met-13 results in a decrease in 〈Tm〉 of 20 K, from 345 to 325 K, thus increasing the stability of the Phe-S-peptide-S-protein interaction. It is apparent that T∆S(Th) = ∆H(Th) and T∆S(Tm) = ∆H(Tm) for S-protein-Speptide (Met-13 f Phe), indicative of balanced thermodynamic compensation among ∆H(T0) and the heat integral and T∆S(T) as seen in Figure 5B. In S-protein-S-peptides (Met-13 f Ala and Met-13 f RNBA) (Figure 5A,C), the thermodynamic quantity ∆H(T0) and heat integral ∫0T∆Cp(T) dT are not identical. In this case, the magnitude of the heat capacity integral is much larger than that of ∆H(T0). Similar results were observed in the fragment complementation reaction of the ribonuclease S′ system described earlier. In Figure 3A,B and 5A-C, both ∆H(T) and T∆S(T) are positive from 0 K up to 〈Ts〉. These molecules require an energy input to preconfigure the individual protein chains (or separate parts of the same chain) so they are in the correct configuration to interact. Making the “preconfigured” chain will require breaking the secondary or tertiary structure, requiring energy and chain flexibility. As the chains do interact, ∆H(T) and T∆S(T) become negative with the release of energy and loss of individual freedom of motion. If ∆Cp(T) is negative, then the reactants have more modes of motion and freedom, leading to higher entropy as is seen in interacting protein systems. If the heat capacity change, ∆Cp(T), between product and reactants is positive, it means the product has more modes of motion and freedom, thus increasing entropy. In the case of protein unfolding, as we have previously observed,17 ∆H(T) is always positive, because a bonded structure is being disrupted, and T∆S(T) is always positive because extra freedom is gained as individual proteins (or portions of the same chain) are disrupted. V. Discussion

Figure 5. Thermodynamic plot of the fragment complementation reaction of S-protein of RNase with various substitutions at Met-13 of S-peptide as a function of temperature in 50 mM sodium acetate with 100 mM NaCl, pH 6.0. Each data point between 0 and 400 K was evaluated with extrapolation of F-statistics. Substitutions: (A) Met-13 f Ala, (B) Met-13 f Phe, and (C) Met-13 f RNBA. T∆S(Th): (A) 185, (B) 225, (C) 180 kcal mol-1. ∆H(Th): (A) 25.67, (B) 85.69, (C) 22.32 kcal mol-1. 〈Tm〉: (A) 340, (B) 325, (C) 350 K. T∆S(Tm): (A) -84.01, (B) -106.99, (C) -82.15 kcal mol-1. ∆H(Tm): (A) -85.09, (B) -105.36, (C) -81.19 kcal mol-1. ∫0T∆Cp(T) dT: (A) -85.00, (B) -105.83, (C) -82.03 kcal mol-1.

experimental values, as shown in Table 3. Thermodynamic compensation occurs between 〈Th〉 and 〈Tm〉, ranging from 160 to 345 K. 5. S-Protein Interaction with S-Peptides with Various Substitutions at Methionine-13. As shown in Table 4, ∆H(T0) values at 〈Th〉 for the S-protein-S-peptides interaction with various substitutions at methionine-13 were determined to be 25.67 ( 0.63 (Met-13 f Ala), 85.69 ( 0.35 (Met-13 f Phe), and 22.32 ( 0.19 (Met-13 f RNBA) kcal mol-1. The value of each of the three ∆H(T0) values of the Met-13 f Phe complementation reaction is 3-4 times greater than those of Met-13 f Ala or Met-13 f RNBA. This suggests that with substitution of phenylalanine-13 strong site-specific interaction prevails, resulting in dimer or higher aggregates. The harmonious temperature, 〈Th〉, and melting temperature, 〈Tm〉, of the S-protein-S-peptide interaction varied with various

As to the question of the heat capacity integral and whether it is a continuous function, it should be noted that no phase change occurs in the systems studied over the accessible temperature range. In the method used here, values of the various thermodynamic properties are mathematically derived, based on thermodynamic principles, from the available data. The enthalpy, entropy, and heat capacity terms are calculated as partial derivatives of the Gibbs free energy defined by Helmholtz-Kelvin’s expression.22,23 The computational procedure requires extrapolation down to 0 K. Since a phase change of water to ice is not implicit in the data, it is necessary to take the viewpoint that the reference state of water at 0 K in the present case is a supercooled liquidsthat is, the glassy state of water. Our evaluation of the temperature-invariant enthalpy for hydrogen-bonded water in equilibrium with nonhydrogen bonded water molecules37 is based on Helmholtz free energy data reported by Nemethy and Scheraga.38 The entropy of the system appears to remain independent of temperature, suggesting that there is no significant temperature-dependent difference in the degree of orientation between unbonded and hydrogen-bonded water molecules in equilibrium in this system.37 A similar conclusion could be reached based on the dielectric relaxation of water as a function of temperature as reported by Collie, Hasted, and Ritson39 and by Kell and McLaurin.40 It has been established that the two components of the heat of reaction are ∆H(T0) and the heat capacity integrals, but in research reported to date, values for the heat of reaction have considered the heat capacity integrals, while the contribution of the temperature-invariant enthalpy, ∆H(T0), has not been

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J. Phys. Chem. B, Vol. 101, No. 39, 1997 7841

TABLE 3: Comparison of ∆H°(T0) at 〈Th〉, 〈Ts〉, 〈Tm〉, and 0 K for the S-Protein-S-Peptide Interactiona reaction

∆H°(T0) at 〈Th〉 (kcal mol-1)

∆H°(T0) at 〈Ts〉 (kcal mol-1)

∆H°(T0) at 〈Tm〉 (kcal mol-1)

∆H°(T0) at 0 K (kcal mol-1)

〈Th〉 (K)

〈Ts〉 (K)

〈Tm〉 (K)

20-S[I] 13-S[II]

34.05 ( 1.30 19.49 ( 1.00

34.05 ( 0.54 19.49 ( 0.42

34.02 ( 0.36 19.01 ( 0.13

34.05 ( 1.30 19.49 ( 1.10

180 160

273 265

345 345

a Compiled using the general linear model procedure of statistical analysis of the IMSL subroutine. F ) 0.001, thus the goodness of fit of the experimental data was 99.8% or better in each use. Observed values for 〈Tm〉 are consistent with experimental values.

TABLE 4: Comparison of ∆H(T0) at 〈Th〉, 〈Ts〉, 〈Tm〉, and 0 K for the S-Protein-S-Peptides Interaction with Various Substitutions of Methionine-13a substitution Met-13 f Ala Met-13 f Phe Met-13 f RNBA Met-13 f Norleucine

∆H(T0) at 〈Th〉 (kcal mol-1)

∆H(T0) at 〈Ts〉 (kcal mol-1)

∆H(T0) at 〈Tm〉 (kcal mol-1)

∆H(T0) at 0 K (kcal mol-1)

〈Th〉 (K)

〈Ts〉 (K)

〈Tm〉 (K)

25.67 ( 0.63 85.69 ( 0.35 22.32 ( 0.19

24.46 ( 0.39 85.39 ( 0.13 22.33 ( 2.32

25.21 ( 0.57 85.39 ( 0.38 22.01 ( 0.57

25.05 ( 0.57 85.55 ( 0.38 23.06 ( 0.57 -33.52 ( 0.13

185 225 186

270 280 270

340 325 350

a Compiled using the general linear procedure of statistical analysis of the IMSL subroutine. F ) 0.001, thus the goodness of fit of the experimental data was 99.9% or better in each case.

assessed. Our thermodynamic studies of a number of interacting protein systems16-20 have shown that in some cases these two quantities can be close to equivalent. Ignoring the contribution of the temperature-invariant enthalpy, therefore, can lead to an inaccurate assessment of the heat of reaction in any macromolecular system. In many texts, the room temperature enthalpy change is written in terms of the corresponding value at 0 K, as follows:

∆H°298 ) ∆H°(T0) +

∫0298∆Cp dT

Scientists who deal mainly with small molecules frequently ignore the difference between ∆H°298 and ∆H°(T0) (∆H°0 in some texts); in the case of small molecules, the difference is not large. In systems involving macromolecules, we have shown in several recent publications that in typical biochemical reactions, for instance, protein self-association, the heat capacity integral may be as large as 50% of the experimentally measured value for ∆H°298,18,19 as shown in Figure 5B. These considerations are particularly relevant to current theoretical work involving computer modeling of molecular energies and heats of reaction. The key point is that energy minimization calculations are inherently based on a reference condition of zero vibrational energysthis is, in essence, the energy of the molecular assembly at 0 K. In other words, when the modeling calculations are used to estimate a ∆H°reaction, the computation automatically produces a quantity similar to ∆H°(T0) but with all atomic vibrators at the bottom of their respective potential wells. Most of the modern molecular mechanics programs can make the implied correction involving a sum of (1/2)hν energy terms. At this point, the computations would have furnished an appropriate estimate of ∆H°(T0). As we have explained, the directly measured ∆H°298 is not useful in making any comparison with this computed quantity. It is absolutely essential to apply the best available information about heat capacity integrals, either to correct the experimentally measured ∆H°T value from 298 K (or thereabouts) down to 0 K, or alternatively, to correct the theoretical quantity from 0 K up to a temperature range near 298 K. A theoretical extrapolation of the computed ∆H°(T0) up to experimental temperatures could be attempted using the methods of statistical thermodynamics, but the computation would be difficult and prone to serious error due to the large number of very low-energy backbone vibrations of the macromolecules. When data for protein-peptide interactions in the ribonuclease S system reported by Hearn et al.15 and Varadarajan et al.13 are evaluated to give the usual thermodynamic parameters

∆H and ∆G°, little difference is observed. However, our analysis shows substantial and well-defined differences in the temperature-invariant enthalpies and other parameters unique to this approach. The thermodynamics of the ribonuclease fragment complementation reaction at neutral pH has been previously examined,18 based on data from Hearn et al.,15 and the results from these fragment complementation systems have been compared with the thermal transition behavior of bovine pancreatic ribonuclease A as a function of temperature, based on data reported by Brandts.36 In these studies, it was found that ribonuclease A undergoes self-association to dimer or higher aggregates if the pH is lowered from 3.15 to 1.15; for these associations, ∆H°(T0) values range from 58 to 60 kcal mol-1.16-18 Such differences in the magnitude of the temperature-invariant enthalpy can be attributed to site-specific changes in the solvent ordering in the immediate domain of the protein, rather than a transition from the native to denatured state. The magnitude of ∆H°(T0) values for 20-S[I] (34 kcal mol-1) and 13-S[II] (19 kcal mol-1) is an effective measure of the ease with which these molecules can be refolded to resemble the native protein structure. In the case of 13-S[II], a dramatic lowering of the S-peptide-S-protein binding constant has been reported upon folding although the resulting complexes have nearly normal catalytic activity.4,41,42 In the treatment of ribonuclease A with cyanogen bromide, chain cleavage occurs at Met-13, yielding C-peptide which, when added to S-protein at a molar ratio of 600:1, gave 50 and 80% recovery of maximum enzymatic activity.43 It has been demonstrated that when denatured ribonuclease A is combined with S-protein, the system forms a dimer of ribonuclease A-S-protein complex.41 The substrate specificity was altered, but the pH optima remained the same with RNA but differed with CpA as a substrate.41 Comparison of the temperature-invariant enthalpy required for the complementation process of the ribonuclease S′ system and the S-protein-S-peptides interaction suggests that the pattern of the refolding process and site-specific interaction differs in each of these systems. The substitution or deletion of a single amino acid within the catalytically-active site of ribonuclease S′, at Met-13 of S-peptide, will alter the enzymatic activity, and this change will be reflected by a change in the temperature-invariant enthalpy. Such a single amino acid substitution is reflected by a change in the temperature-invariant enthalpy, as is apparent in Table 4. If norleucine is substituted for Met-13, for example, no fragment complementation can take place since the Gibbs free

7842 J. Phys. Chem. B, Vol. 101, No. 39, 1997

Chun and Phe-120 of the C-terminal end, forming a network of interacting aromatic side chains. It is assumed that the entire S-peptide would assume a certain conformation, stabilizing the active site of His-119, His-12, and Lys-41. Natural selection, however, has not favored a substitution of phenylalanine for methionine. VI. Conclusion

Figure 6. (A, top) Line model of RNase A, drawn using Pro-3D software developed in our laboratory and available on the Internet at http://www.med.ufl.edu/biochem/pchun. Brookhaven data bank X-ray crystallographic data file PDB9RAT.ENT at http://www.pdb.pnl.gov was down-loaded to an IBM-PC. (B, bottom) Enlarged view of the active site of RNase A, His-119, His-12, and Lys-41, also showing the relative position of Phe-8, Phe-120, Met-13, and Phe-46. If Met-13 were replaced by phenylalanine, note the close proximity of Phe-46, as well as Phe-8 and Phe-120, aligned with a β-carbon at the centroid position.

energy change must be at a minimum at the point of equilibrium and this is clearly not the case here. Although norleucine-13S-peptide may bind to the S-protein, it does not realign with the ribonuclease S′ system to form a biologically active molecule. Substitution of R-amino-N-butyric acid (RNBA) gives a temperature-invariant enthalpy of 22 kcal mol-1, which is consistent with a monomeric form of 13S-RNase S′. When alanine is substituted, an increase in ∆H°(T0) to 26 kcal mol-1 reflects an increase in site-specific interaction. When phenylalanine is substituted for Met-13, the magnitude of ∆H°(T0) increases to 86 kcal mol-1, suggesting a more dominant role for this amino acid in site-specific interaction. Burley and Petsko,46 in analyzing neighboring aromatic groups in four biphenyl peptides or peptide analogs and 34 proteins, reported that 60% of aromatic side chains in proteins are involved in aromatic pairs, 80% of which form networks of three or more interacting aromatic side chains. In phenyl ring centroids, the dihydral angles approached 90°. In our examination of the interaction of Phe-8-Phe-120, we found that the phenyl group at position 120 was aligned perpendicular to the phenyl group at position 8 with an β-carbon at the centroid position. In one hypothetical model for refolding upon complementation, shown in Figure 6, Phe-13 will come in contact with Phe-46 of the β-sheet of region II37,38 of S-protein (Phe-13-Phe-46 interaction). A second possibility is that Phe13 may come in contact with existing phenyl groups at Phe-8

The temperature-invariant enthalpy, ∆H°(T0), has been evaluated for two ribonuclease S′ systems, 20-S[I]-RNase S′ and 13S[II]-RNase S′, at neutral pH and S-protein-S-peptides interaction at pH 6.0, and was found to be a useful measure of the energy level of the molecule in the fragment complementation reaction. Application of the Planck-Benzinger thermal work function to evaluate the temperature-invariant enthalpy should be an essential future component of all studies involving the sitedirected, mutagenic approach to the examination of structurefunction problems in proteins. One example is the model presented for the refolding of S-peptide with S-protein if phenylalanine were to be substituted for Met-13. The model incorporates feasible strong, site-specific phenylalanine-phenylalanine interaction, giving rise to the large temperature-invariant enthalpy. The magnitude of ∆H°(T0), a fundamental energy quantity, is determined by (i) the intrinsic thermodynamic stability and integrity of the macromolecule, (ii) the thermodynamic stability of genetically-engineered mutant proteins, (iii) the specific macromolecular folding and unfolding transition pathway, (iv) the type of macromolecular assembly process, and (v) the sitespecific macromolecular interaction enhanced by solvent ordering. 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 their enlightening discussions and suggestions. This work was supported by a faculty development award, Division of Sponsored Research, University of Florida, and in part by NSF Grant BMD 83-12101. Appendix: The Giauque Function and the Planck-Benzinger Thermal Work Function It should be noted that the Gibbs free energy for an ideal diatomic molecule in the standard state is 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 the chemical equilibrium obtainable from the partition function, f.23 The Gibbs free energy function was defined as (G°T E°0)/T, and subsequently defined by Giauque25-29 as (G°T H°0)/T. Letting (G°0 - H°0)/T ) -Ψ°/T, then G°T - H°0 ) -Ψ° ) -{T ∫0T(C°p/T) dT - ∫0TC°p dT} when expressed as the heat capacity term. Values for -Ψ° for pure solids are tabulated in the National Bureau of Standards’ Circular 500 33 and JANAF thermodynamic tables.34 For an ideal gas or pure solid, this function would be equivalent to the Planck-Benzinger thermal work function, W°(T) ) H°0 - G°(T).30 The Giauque function, G°T - H°0 ) -Ψ°, has so far been sterile in terms of its application to biological systems. By contrast, the equivalent (although independently derived) Planck-Benzinger work function has been quite fruitful. In 1971, T. H. Benzinger30 proposed a thermal work function to take into account both Boltzmann statistical energy effects and the energies of quantum mechanical bonds. While the former are usually not altered significantly in micromolecular

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reactions, it was Benzinger’s conjecture that the large-scale and long-range changes of confirmation 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 function could have been used and would have been appropriatesit simply was not. Rather, Benzinger pursued a separate path, although one which can be related to the work of Giauque and other chemists. Planck defined Nernst’s heat theorem as ψ ) ∫0T(Cp/T) dT H/T,45 which may also be defined as

-∆G°/T )

∫0T(∆C°p/T) dT - ∆H°/T

(6)

Substituting the integrated Kirchhoff equation,22,23 ∆H° ) ∆H°(T0) + ∫0T∆C°p dT into eq 6 yields

-∆G°/T )

∫0T(∆C°p/T) dT - ∆H°(T0)/T - 1/T ∫0T∆C°p dT

(7)

Rearrangement of this expression gives

∆G° - ∆H°(T0) ) -{T

∫0T(∆C°p/T) dT - ∫0T∆C°p dT}

(8)

Letting ∆W° ) {T ∫0T(∆C°p/T) dT - ∫0T∆C°p dT},

∆W° ) ∆H°(T0) - ∆G°

(9)

and then ∆W° represents the strictly thermal components of any intra- or intermolecular bonding term in a system, that is, energy other than the inherent difference of the 0 K portion of the interaction energy. Thus, ∆W° expresses completely the thermal energy difference of the process involved. From this expression, it is possible to determine the temperature-invariant enthalpy, ∆H°(T0) ) ∆W°〈Ts〉max + ∆G°〈Ts〉min at 〈Ts〉, the stable temperature where ∆G° is at a minimum and ∆W° at a maximum. This relationship has been designated as the Planck-Benzinger thermal work function. Footnote to the Planck-Benzinger Thermal Work Function 1. Temperature-invariant enthalpy, ∆H°(T0) (or in some texts, ∆H°0) is an equivalent terminology for this quantity in the inherent 0 K enthalpy. It represents the chemical bond energy difference corrected to 0 K. 2. The practical use of Benzinger’s method requires “the experimental determination of the all-important zero-point enthalpy and free entropy terms” pointed out by Rhodes.44 As noted by Rhodes, Benzinger was pessimistic about the prospects for practical use of his method, due to the difficulty of obtaining the required (temperature-invariant) quantities. Our solutions to this difficulty are addressed in the present series of papers.16-20 References and Notes (1) Anfinsen, C. B. Science 1973, 181, 223-230. (2) Anfinsen, C. B.; Huber, E.; Sela, M.; White, F. M. Proc. Natl. Acad. Sci. U.S.A. 1961, 47, 1309-1313. (3) Epstein, C. J. Goldberg, R. F.; Anfinsen, C. B. Cold Spring Harbor Symp. Quantum Biol. 1963, 27, 439-449. (4) Richards, F. M. Proc. Natl. Acad. Sci. U.S.A. 1958, 44, 162-166.

(5) Richards, F. M.; Vithayathil, R. J. J. Biol. Chem. 1959, 234, 14591465. (6) Wyckoff, H. W.; Tsenoglou, D.; Hanson, A. W.; Knox, J. R.; Lee, B.; Richards, F. M. J. Biol. Chem. 1970, 245, 305-328. (7) Wyckoff, H. W.; Karl, D.; Hardman, N. M.; Allewell, N. M.; Inagami, T.; Johnson, L. N.; Richards, F. M. J. Biol. Chem. 1970, 242, 3984-3988. (8) Taniuchi, H.; Anfinsen, C. B. J. Biol. Chem. 1971, 246, 22912301. (9) Taniuchi, H.; Anfinsen, C. B. J. Biol. Chem. 1968, 243, 47784786. (10) Taniuchi, H.; Parker, D. S.; Bohnert, J. L. J. Biol. Chem. 1977, 252, 125-140. (11) Connelly, P. R.; Varadarajan, R.; Sturtevant, J. M.; Richards, F. M. Biochemistry 1990, 29, 6108-6114. (12) Varadarajan, R.; Richards, F. M.; Connelly, P. R. Curr. Sci. 1990, 59, 819-824. (13) Varadarajan, R.; Connelly, P. R.; Sturtevant, J. M.; Richards, F. M. Biochemistry 1992, 31 1421-1426. (14) Naghibi, H.; Tamura, A.; Sturtevant, J. M. Proc. Natl. Acad. Sci. U.S.A. 1995, 92, 5592-5599. (15) Hearn, R.; Richards, F. M.; Sturtevant, J. M.; Watt, G. D. Biochemistry 1971, 10, 806-817. (16) Chun, P. W. Int. J. Quantum Chem.: Quantum Biol. Symp. 1988, 15, 247-258. (17) Chun, P. W J. Phys. Chem. 1994, 98, 6851-6861. (18) Chun, P. W. J. Biol. Chem. 1995, 270, 13925-13931. (19) Chun, P. W. J. Phys. Chem. 1996, 100, 7283-7292. (20) Chun, P. W. Biophys. J. 1996, 70 (2), M-POS 421. (21) Gibbs, J. W. Am J. Sci. 1878, 16, 441-458 (via Trans. Conn. Acad. Sci. 3). (22) Lewis, G. N.; Randall, M. In Thermodynamics; Pitzer, K. S.; Brewer, L.; Eds.; McGraw-Hill: New York, 1961; pp 164-182; appendix, pp 665-668. (23) Moelwyn-Hughes, E. A. Physical Chemistry; Pergamon Press: New York, 1957; pp 90-103, 264-279, 560-563. (24) Cottrell, T. L. In The Strength of Chemical Bonds; Acad. Press, Inc.: London, 1958; Chapter 3, pp 21-46, Chapter 4, pp 47-70. (25) Giauque, W. F. J. Am. Chem. Soc. 1930, 52, 4808-4815. (26) Giauque, W. F. J. Am. Chem. Soc. 1930, 52, 4816-4831. (27) Giauque, W. F.; Blue, R. W. J. Am. Chem. Soc. 1936, 58, 831837. (28) Giauque, W. F.; Kemp, J. D. J. Chem. Phys. 1938, 6, 40-52. (29) Giauque, W. F.; Meads, P. F. J. Am. Chem. Soc. 1941, 63, 18971901. (30) Benzinger, T. H. Nature 1971, 29, 100-103. (31) Chun, P. W. Manual for Computer-Aided Analysis of Biochemical Processes with Florida 1-2-4; University of Florida (copyright reserved), 1991. (32) Barr, D. J.; Goodnight, J. H.; Sall, J. P.; Helwig, J. T. SAS GLM 27 and GLM 131, Statistical Analysis System, University of Florida, NERDC CIRCA, 1985. (33) Rossini, F. D.; Wagnman, D. D. Circular of the National Bureau of Standards 500, Related Values of Chemical Thermodynamic Properties; US Government Printing Office: Washington, DC, 1952. (34) Chase, M. W., Jr.; Davies, C. A.; Downey, J. K., Jr.; Frurip, D. J.; McDonald, R. A.; Syuverud, A. N. JANAF Thermodynamic Tables, 3rd ed.; American Chemical Society and the American Institute of Physics for the National Bureau of Standards, 1985, Vol. 14, Part I, II. (35) Crestfield, A. M.; Stein, W. H.; Moore, S. Arch. Biochem. Biophys. 1962, 1, (Suppl.) 217-222. (36) Brandts, J. F. J. Am. Chem. Soc. 1965, 87, 2759-2760. (37) Chun, P. W. Thermodynamic Studies on Hydrogen Bond Energy of Liquid Water. Poster 283, Biophysical Chemistry, 212th National ACS Meeting, Orlando, Florida, 1996. (38) Nemethy, G.; Scheraga, H. A. J. Chem. Phys. 1962, 6, 3382-3400. (39) Collie, C. H.; Hasted, J. B.; Ritson, D. M. Proc. Phys. Soc. 1948, 60, 145-160. (40) Kell, G. S.; McLaurin, G. E. J. Chem. Phys. 1969, 51, 4345-4352. (41) Richards, F. M.; Wyckoff, H. W. In The Enzymes; Boyer, P. D., Ed.; Academic Press: New York, 1971; Vol. 4, pp 647-806. (42) Richards, F. M. In Structure and ActiVity of Enzymes; Goodwin, T. W., Harris, J. I., Hartley, B. S., Eds.; Academic Press: London, pp 5-12. (43) Parks, J. M.; Baranick, M. B.; Wold, F. J. Am. Chem. Soc. 1963, 85, 3519-3521. (44) Rhodes, W. J. Phys. Chem. 1991, 95, 10246-19251. (45) Planck, M. Vorlesungen-Uber Thermodynamics, 7th ed.; (Treatise on Thermodynamics, 3rd ed.; Ogg, A., Trans.; Longmans, Green and Co.: London, 1927; pp 179-214, 272-289). (46) Burley, S. K.; Petsko, G. A. Science 1985, 229, 23-28.