13050 J. zyxwvutsrqpo Phys. Chem. 1995, 99, zyxwvu 13050 Reliability of the van’t Hoff Plots Raffaele Ragone*.’ and Giovanni Colonna Dipartimento di Biochimica e Biofisica, Seconda Universith di Napoli, via Costantinopoli 16, 80138 Naples, Italy Luigi Ambrosone DISTAAM, Universith del Molise, via Tiberio zyxwvutsr 21/A, 86100 Campobasso, Italy Received: March 31, 1995 We write to comment on the recent paper by Weber,’ which raises the issue of whether enthalpy values calculated by means of the van’t Hoff plot are actually correct. Weber denies the general validity of the van’t Hoff enthalpy in those systems, in which the bond energy is of the same order as the thermal energy. As protein chemists, we feel quite confused, since the use of that mathematical procedure is widespread among people that usually deal with systems involving the formation and/or the disruption of a large number of internal low-energy bonds. For example, the ratio between the van’t Hoff and the “true” calorimetric denaturation enthalpy is considered a useful criterion for evaluating the extent to which the protein unfolding departs from the ideal two-state mechanism (infinite cooperat- ivity), as first suggested by Privalov in the middle 1 9 7 0 ~ . ~ Weber himself, in a preceding paper on the entropy-driven association of protein subunit^,^ develops his reasoning by the van’t Hoff enthalpy. It seems therefore useful to examine in detail the source of Weber’s present criticism. First of all, we have some conceptual remarks. Weber’s statement’ that the differential dG zyxwvutsrq = -d(TS) + p dV expresses the change in the Gibbs free energy in terms of the change in total heat content, d(TS), of the system of reagents and surroundings and the external work performed by the system at constant pressure, p dV, is incorrect. The Gibbs free energy, G, is a thermodynamic potential of the system, defined as G= E + PV- TSr H - TS EA + PV (1) This definition comes directly from the second law of thermo- dynamics. G is a criterion of spontaneity for any process under the constraints of constant pressure and temperature, without any assumption about the other thermodynamic variables (E, S, V). Moreover, the differential of (1) (dG = dH - T dS) regards only transformations between equilibrium states at constant pressure and temperature. Far from equilibrium the equality takes the form of the Clausius’ inequality and the < sign must be applied. Weber’s equation (5) (AG = AH - TAS) is a direct derivation of (1) applied to chemical reactions. At this stage Weber’ goes further, dividing by T, AGIT = AHIT - AS and subsequently claiming that ‘‘8 and only 8 AH and AS are independent of temperature, then” (2) d(AGIT)ld( 11T) = AH (3) Since, in the more general case, both AH and AS are temperature dependent, in addition to AH (the van’t Hoff enthalpy) we should consider the term (4) which, in Weber’s opinion, hampers the evaluation of AH by differentiation of (2), except in the limiting case in which T(r) T(T) = zyxwvut (UT) dAHld( UT) - dAS/d( llT) ’ Phone: f39-8 1-5665869. Fax: +39-81-5665869/5665863. E-mail: COLONNA@AREANA. AREA.NA.CNR.IT 0022-365419512099- 1 3050$09.00/0 << AH. Weber also notes that the derivation of (4) “is independent of any thermodynamic consideration and is solely a matter of application of the rules of the differential calculus”. Everybody, going further into the differentiation, obtains T(T) = -T dAHldT + zyx P dAS1dT (5) The right-hand side of (5) is identically 0, given that both dAHl dT and T dASldT are equal to ACp at constant pressure (at every temperature). This is indeed implicit in the general derivation of the van’t Hoff equation, as reported in physical chemistry textbooks. On this point, in their textbook of 1958 Edsall and Wyman4 state that the knowledge of the temperature dependence of AH is required for integration purposes, whenever we want to evaluate the change of the equilibrium constant with temperature. If AGIT = -R In Keq is plotted against UT, the resulting curve is a straight line when AH (the slope) is temperature independent. If the curve is not linear, the tangent to it at any given value of UT gives AH for the corresponding value of T. Usually, AH is found constant over a limited temperature range, but this neither implies that AH is temper- ature independent nor restricts the significance of (3). Thus, a matter of calculus cannot be used to disprove the general validity of the van’t Hoff equation. Interestingly, when applied to a weak aromatic complex held together by 10 identical bonds with AG = -1 kcal mol-’, Weber’s model predicts an enthalpy of - 1.7 kcal mol-’, which is very close to the van’t Hoff enthalpy (-2 kcal mol-’). Since Weber also states that “the largest errors in the enthalpy obtained by means of van’t Hoff plots ought to occur when AH approaches 0”’ we can conclude that in this unfavorable case the van’t Hoff enthalpy evaluation seems to be reliable. Nevertheless, Weber’s conclusion is that disagreement between his model and van’t Hoff enthalpy “will not be nearly as evident as in the case of protein oligomer association, and aromatic association cannot provide as stringent a test of the general validity of the van’t Hoff plot as in the case with entropy driven associations like those of protein subunits”.’ In our opinion, the point at issue is to explain why Weber’s model fails dealing with such complex a system as protein association. As a matter of fact, the current knowledge on the role of weak interactions in “nanosystems” focuses on the impossibility of evaluating their energetics by the additive schemes adopted for “macrosystems”. When the local concen- tration of weak bonds is very high, as in the case of biological macromolecules, it is unavoidable to be engaged with correlation among interactions. In other words, from a statistical mechan- ical point of view, $, and only $, the Hamiltonian of the system (the protein oligomer, in the present case) can be expressed as a sum of components that operate on different coordinates, then the total free energy can be expressed as a sum of independent contributions. When the Hamiltonian components are coupled, the entropy of the system is represented by all first and higher order correlations, whose further separation is allowed to the extent to which the degrees of freedom of the system are ~ncorrelated.~ Thus, we shall continue to feel confident of the van’t Hoff enthalpy, at least until the hidden thermodynamics of “nanosystems” remains hidden. References and Notes (1) Weber, G. J. Phys. Chem. 1995, 99, 1052. (2) Privalov, P. L.: Khechinashvili, N. N. J. Mol. Bid. 1974, 86, 665. (3) Weber, G. J. Phys. Chem. 1993, 97, 7108. (4) Edsall, J. T.; Wyman, J. Biophysical Chemistry: Academic Press: (5) Mark, A. E.: van Gunsteren. W. F. J. Mol. Bid. 1994. 240, 167. New York. 1958: Vol. I, Chapter 4. JP950947 1 . . - 0 1995 American Chemical Society