Enthalpy van’t Hoff

Biochemists often refer to enthalpies obtained in this manner as van t Hoff enthalpies and attach the subscript i/H to the AH to distinguish it from an enthalpy obtained directly from calorimetric measurements. In practice, one need not actually obtain the K values to extract the enthalpy change. Rather, there is a relationship between da/d T and d nK/dT. For the equilibrium of equation (16.15) (at constant pressure so that the partial derivative can be replaced by the total derivative),... 

The expression for dlnK/dT developed in equation (16.17) can be substituted into equation (16.16) to obtain the enthalpy. The van t Hoff enthalpy is often... 

Optical Methods and the van t Hoff Enthalpy The optical method relies upon the change in the UV absorption of both proteins and nucleic acids as the transformation from one molecular state to the other takes place. For example, the absorbance A at 260 nm of a solution containing a nucleic acid duplex is found to increase significantly from some baseline value as the duplex melts until it reaches a second baseline where the nucleic acid is completely melted. 

The Calorimetrically Obtained van t Hoff Enthalpy In a manner analogous to that used to obtain the van t Hoff enthalpy from the fractional change in the optical absorbance, one can use the temperature dependence of the fractional enthalpy as a function of temperature to determine an effective enthalpy. We will adopt the notation to represent the total enthalpy associated with the denaturation transition. It can be obtained from an integration of the excess heat capacity, corrected for the baselines, as discussed before ... 

If the process follows the two-state model, then the calorimetric enthalpy, A //, the calorimetrically obtained van t Hoff enthalpy, AHeff, and, where optical measurements have been made, the optical van t Hoff enthalpy, AHuh, should agree. Figure 16.9 shows the ratio, AH /AHvi obtained from denaturation studies of five globular proteins under various conditions, plotted against the corresponding denaturation temperature. This plot shows that the ratio is very close to unity. Similar results for numerous other small globular proteins have been obtained and most show a similar relationship. It is clear, however, that the ratio is closer to 1.05 than to 1. Freire and Biltonen11 have shown from a consideration of the partition functions associated with the native and the denatured state that this relative excess is... 

Calorimetric (DSC) measurements yield thermodynamic properties of duplex melting in these oligonucleotides independent of any assumptions concerning the model of melting, such as a cooperative all-or-none process versus a noncooperative, multiple-stage melting process. Comparison of calorimetric enthalpies with van t Hoff enthalpies obtained either from the manipulation of heat capacity curves outlined in equations (16.19) to (16.22), or from optical or NMR measurements [equations (16.14) to (16.17)] allows conclusions to be drawn concerning the size of the cooperative unit. If the two... 

Let us consider a general melting process. If one defines a as the concentration of monomer strands in the duplex and cjot as the total monomer strand concentration, then one can derive relationships between the van t Hoff enthalpy and the temperature derivative of a at Tm to show that there is a concentration dependence of Tm for the melting of some duplexes and not others. 

From the temperature derivative of K and the corresponding derivative of a, one can get a relationship for the van t Hoff enthalpy at Tm as a function of da/dT. These relationships are summarized in Table 16.2. For the specific case of a reaction like (16.26) involving two different types of strands, the table gives only the case where ca = cb-... 

Note that the self-complementary and nonself-complementary processes of the same molecularity (n) give identical expressions for the van t Hoff enthalpy of the melting process, but not for the equilibrium constant. The factors in the expressions for K arise from the indistinguishability of 2A monomers in a process like A2 = 2A, while the A and B monomers in an AB = A + B process are distinguishable. 

Table 16.2 A summary of expressions for the equilibrium constant and van t Hoff enthalpies for duplex meltings of various... 

This van t Hoff enthalpy was obtained by an analysis based on the shape of the DSC curve. See the reference for details. (Data from L. A. Marky, K. S. Blumenfeld, S. Kozlowski, and K. J. Breslauer, Biopolymers, 22, 1247-1257 (1983). 

Abbreviations calor = AH obtained from integration of the integrated DSC curve vH - a van t Hoff enthalpy obtained from UV absorbance or NMR chemical shift measurements unless otherwise noted NR indicates that results were not reported. 

Since a comparison of various van t Hoff enthalpies (calculated by analysis of da/dT at Tm, the slope of /Tm vs In cjot. and the shape of the calorimetric curve)0 with the calorimetric enthalpy shows that A// h A//cai, it was concluded that the melting transition in this duplex is cooperative and well represented by a two-state model. 

The d(CGCGAATTCGCG)2 Duplex. When studied at comparable salt (NaCl) and duplex concentrations, the duplex of d(CGCGAATTCGCG) was found to have about the same Tm by DSC measurements (344 K) and by the temperature-dependent changes in the NMR chemical shifts of nonexchangeable GC and AT protons (345 K). At 0.01 M NaCl, the calorimetric enthalpy was 376 kJmol-1 of duplex, while the van t Hoff enthalpy evaluated from equation (16.28) was found to be 393 kJ-mol-1 of duplex. At a 10-fold higher salt concentration, the calorimetric enthalpy was 427 kJ-mol-1 of duplex, while the van t Hoff enthalpy was 310 kJ-mol-1 of duplex. From the ratio of the van t Hoff enthalpy to the calorimetric enthalpy at the higher salt concentration, it was estimated that 9 out of 12 base-pairs melt together in the cooperative unit, while all melt simultaneously at the low salt concentrations. 

The sharpness or cooperativity of the gel-to-liquid-crystalline phase transition can also be evaluated from the DSC trace. The sharpness of the phase transition is often expressed as the temperature width at half-height, AT1/2, or as the temperature difference between the onset or lower boundary of the phase transition, Ts, and the completion or upper boundary, Ti, or AT = Ti — Tg. The AT1/2 values may range from <0.1°C for very pure synthetic phospholipids to as much as 10-15°C for biological membranes. From the Tm and AT1/2 values determined for a particular phase transition, the van t Hoff enthalpy, AHvh, can be approximately determined from the relationship ... 

The temperature dependence of In can be used to define a van t Hoff enthalpy, that has... 

As in the case of the van t Hoff enthalpy per residue the effective m values are statistical quantities equal to the difference between the average m value for the conformations in which the residue is not folded and the average m value for the conformations in which that residue is folded. At low... 

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