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NOTE BREVI

On the temperature-invariant enthalpy change and thermodynamics

of protein’s transformations

G. P. JOHARI(1) and G. SALVETTI(2)

(1_{) Department of Materials Science and Engineering, McMaster University} Hamilton, Ontario, L8S 4L7, Canada

(2_{) Istituto di Fisica Atomica e Molecolare del CNR - Via del Giardino 7, 56127, Pisa, Italy}

(ricevuto il 31 Gennaio 1997; approvato il 19 Marzo 1997)

Summary. — To gain insight into the reasons for a large enthalpy of

protein-unfolding and of protein-association, the currently used method for deducing a temperature-invariant enthalpy by extrapolation of the limited free energy data by a cubic polynomial equation has been critically examined. This is necessary in the light of the now well-known observations that proteins vitrify gradually on cooling and thus behave like a glass. Because of this kinetic freezing, a protein retains both a configurational enthalpy and entropy at 0 K. By using the accepted thermodynamic functions, we show here that the method used in such calculations needs to be revised to include a term TtD S0, i.e. the temperature at which the equilibrium constant is

unity (or the Gibbs free energy change is zero), multiplied by the difference between the entropy of the products and reactants at 0 K. A large enthalpy of protein-transformations is anticipated generally, because a vast number of configurational and conformational degrees of freedom are available to a hydrated protein system containing buffering agents. The parametrized third-order polynomial equation used for deducing the Gibbs free energy change from data available in the 270–350 K range, as done presently, may be valid only when the contributions to the heat capacity, enthalpy and entropy are entirely vibrational, as in a completely ordered crystal. It should not be used for proteins and biopolymers. PACS 65.50 – Thermodynamic properties and entropy.

PACS 87.10 – General, theoretical and mathematical biophysics (including logic of biosystems, quantum biology, and relevant aspects of thermodynamics, information theory, cybernetics, and bionics).

It is generally known that the enthalpy of a chemical or physical transformation, DHT, of a condensed phase is relatively large. Its magnitude increases rapidly when

the number of ways by which energy can be distributed in a molecular system is increased. This is particularly remarkable in proteins whose D HT is relatively

large [1-3]. Benzinger [4] was concerned that these D HT values are too high in view of

the fact that the number of bonds in the protein, that break or form at the transformation, is relatively small. To resolve his concern in 1971, he suggested that it 753

is more appropriate to consider a temperature-invariant enthalpy, DH0, identical to that which is used in the Gibbs-Helmholtz function. He concluded that DH0 may be deduced by constructing plots of the change in the Gibbs free energy, DGT, against

temperature and determining its value at the temperature where DGT becomes zero.

This suggestions was put to use in 1994, when Chun [5] calculated D H0for 11 different protein-unfolding and protein-association transitions, by using a value of DGT, which

was extrapolated to 0 K by a parameterized, third-order polynomial equation because the experimental values of DGT are known only at high temperatures, usually above

250 K.

Because of the importance an understanding of the energetics of biological processes has, it is necesssary that such calculations be reliable. Here we show that this manner of calculation seriously underestimates the true value of D H0for proteins and biopolymers and that a recalculation of this value is required by further experiments.

In his classical treatment in the 1920s, Planck [6] had connected a physical quantity

Cp, the heat capacity, with a chemical quantity K, the equilibrium constant of a chemical

reaction, by 2RT ln KT4 DGT4 D H01





where R is the gas constant (8.3146 J/mol K), DGT is the change in the Gibbs free

energy, DCpis the difference between the heat capacities of the products and reactants

for a chemical reaction, or between the two phases involved in the phase transformation, and T is the temperature. Thus,

2RT ln KT4 DGT4 D H02 T DWT, (2) where DWT4

u





v

is referred to as Planck function, with units of J/mol K. (This unit was apparently chosen so that on division by the Boltzmann constant k, a unit less number is obtained.) Benzinger[4] effectively suggested that T DWT be written as DWT, which has the same

units as DGT, and D H0. Hence,

2RT ln KT4 DGT4 DH02 DWT.

(4)

Equation (4) implies that a plot of DGT against T will yield DWTfD H0 at that

temperature where DGT4 0 , or K 4 1 . The same plot will yield DWT4 0 , and DGT4

DH0, when DGT reaches its maximum value at 0 K, i.e. DG04 D H0 at 0 K. Hence,

according to eq. (4), the magnitude of DWT(4TDWT) and DGT are related by a

geometrical inversion, and that procedure is exact. However, it is acceptable only when both the reactants and the products involved in the transformation are ordered crystalline phases, completely devoid of such configurational degrees of freedom that may lead on cooling to a finite residual, or frozen-in, entropy and enthalpy at 0 K. In such cases, WTis exactly equal to the negative of the Giauque function, FT(4GT2 H0), whose values are tabulated in the National Bureau of Standards Circular 500. This anology with the Giauque function was overlooked in ref. [4]. (See also p. 7291 in ref. [5].) Thus, strictly speaking, WT4 TWT4 2FT. It should be pointed out that both

Planck and Giauque had explicitly used WTand FT for completely ordered crystals, i.e.

the ones with only a single configuration, or which obey the third law of thermo-dynamics.

A remarkably large number of studies since 1970 [7-23] have unambiguously demonstrated that a considerable degree of molecular freedom is present in proteins and biopolymers. Molecular segmental and other diffusional motions, localized or otherwise, which involve these degrees of freedom, become slower on cooling and ultimately become kinetically frozen at a temperature which is characteristic of that particular degree of molecular freedom (see refs. [7-15], for a review of the subject). The same degrees of molecular freedom become gradually unfrozen on heating the protein from a cryogenic temperature. Hence, hydrated proteins, although crystalline, share the molecular kinetic features of a glass, just as orientationally disordered crystals do. Therefore, on cooling towards 0 K, a protein, just as a glass-forming liquid and an orientationally disordered crystal, contains a kinetically frozen-in configurational entropy and enthalpy, S0 and H0, at 0 K. An appropriate analysis requires that these be included in eq. (1). Therefore,

2RT ln KT4 DGT4 D H01



y



z

where DH0in eq. (5) now includes two contributions: i) the difference between the bond enthalpies of the reactants and products and, ii) the difference between the frozen-in configurational enthalpy of the two at 0 K. D S0is the difference between the S0of the two states involved. By combining eqs. (1), (4) and (5),

DGT4 D H02 DWT2 T D S0, (6)

D H0_{4 DG}T1 DWT1 T D S0. (7)

Here DWT4 T DWT4 2DFT. (We retain the use of DWT here for comparison with the

analysis in refs. [4, 5], although its use, when the equivalent Planck or Giauque functions are well known, seems unnecessary, and complicating to the discussion.) According to eq. (7), when T is such that DGT4 0 or K 4 1 ,

D H0_{4 DW}T1 T D S0 (8)

or

D H0_{4 T(DW}T1 D S0); or D H04 2DFT1 T D S0. (9)

The thermodynamically appropriate value of D H0 for a protein determined from eqs. (7), (8) or (9) is therefore: DWT1 T D S0J/mol, i.e. T DS0J/mol more than the D H0 value determined from eq. (3), and done in refs. [4, 5]. This difference may be substantial. For example, a reasonable value for D S0_{of 8 J/mol K and T 450 K (where} DGT4 0 ), gives 2.8 kJ/mol for the T D S0 contribution, which is comparable to D H0 values for some proteins in ref. [5].

To illustrate this calculation, we have summarized the values of the difference between the various thermodynamic functions for cyclohexanol’s crystal I and crystal II, as calculated from the Cp data given in refs. [24-26]. Crystal I is an

Fig. 1. – The various thermodynamic functions for cyclohexanol used for calculating the temperature invariant enthalpy: a) the difference DCpvs. T between the heat capacities of the two

phases (crystal I and crystal II) from data in refs. [24-26]. b) The shape of sT

0DCpdT and

sT

0DCpd ln T plots against T calculated from data in a). c) The shape of 2DGTand DWTplots

calculated from data in b) and equations: DWT4 TsT0DCpd ln T 2sT0DCpdT ; DGT4 DWT2 DH0,

as used in refs. [4, 5]. The enthalpy change at the phase transformation is 8827 J/mol and the entropy change is 33.25 J/mol K [24-26].The correct value of DH0 is 4570 J/mol [25] and not

3285 J/mol as calculated from the methods used in refs. [4, 5]. The value of D S₀is 4.84 J/mol K and

proteins. It transforms to an ordered crystal II at 265.51K [24-26]. Crystal I also supercools below 265 K and, during the supercooling, molecular motions in it become progressively more slow, and ultimately, like that in a protein, its configurational state becomes kinetically frozen. The values of the difference between the respective thermodynamics quantities and related functions for crystal I and crystal II are shown in fig. 1. They clearly show that DH0value of 3285 J/mol calculated from eq. (4) by using the method in refs. [4, 5] is substantially less than the measured value of 4570 J/mol [25]. The difference is equal to TtD S0, where Tt4 265.5 K and D S04

4.84 J/mol K.

It should be stressed that the thermodynamics of proteins is expected to admit the same equations as the thermodynamics of other disordered materials. Hence, the difference between the shape of the DCp,

s

s

temperature for cyclohexanol and the shape of the corresponding plots for a protein, which would be noted from the data in fig. 1, is not meaningful for our purpose here. Instead this difference is expected in view of the fact that kinetic freezing occurs over a narrow temperature range near 148 K for cyclohexanol [24-26], and over an indiscernably broad temperature range in proteins [13-17, 19, 21, 23].

Benzinger’s suggestion for use of the term DWT (or equivalently, T DW , or 2DFT)

had stemmed from his concern that

s

unfolding is unreasonably high. But in view of the now well-known conclusions that a very large number and a vast variety of molecular, both configurational and conformational, motions occur in the hydrated state of a protein [7-23], a resolution of Benzinger’s concern lies in recognizing that these configurational motions contribute a substantial amount of configurational enthalpy and entropy to the overall thermo-dynamics of hydrated proteins. The configurational contributions resulting from the ability of water molecules, organic and inorganic additives and ions of the buffering agents to diffuse locally and/or over a long-range add to the configurational contributions arising already from segmental motions of a protein molecule itself.

It has been known for at least three decades that large decreases in such configurational and vibrational contributions to thermodynamic functions occur, for both chemical reactions and phase transformation, when at least one of the condensed phases is a liquid, and this knowledge has provided a basis for configurational thermodynamics of liquids and glasses. It is also well known that the configurational entropy is in principle computable from the equation, Sconf_{4 k ln V , where k is the} Boltzmann constant and V is the number of (non-degenerate) accessible configurations. The configurational enthalpy can be calculated once the configurational entropy is known. In this view, the use of a third-order polynomial equation for estimating the DGT values for proteins by extrapolation over the range from 0 K to

250 K [5] and the justification of this third-order polynomial equation (p. 7284 in ref. [5]) in terms of the Einstein-Debye theory that the heat capacity is a linear function of T3seems inappropriate, because the Einstein-Debye theory is only for the vibrational contributions to the heat capacity of solids.

* * *

We are grateful to Dr. C. BAUERand G. CERCIGNANIof Dipartimento di Fisiologia e Biochimica, Università di Pisa, Pisa, for bringing to our attention the work reported in ref. [5] of this paper. This study was supported by a grant from the Natural Sciences and Engineering Research Council of Canada.

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