Free energy relationships, linear validity

A.R. Fersht, Linear Free Energy Relationships are Valid, Protein Engineering, 1987,1, 442. 

The necessity of the statistical approach has to be stressed once more. Any statement in this topic has a definitely statistical character and is valid only with a certain probability and in certain range of validity, limited as to the structural conditions and as to the temperature region. In fact, all chemical conceptions can break dovra when the temperature is changed too much. The isokinetic relationship, when significantly proved, can help in defining the term reaction series it can be considered a necessary but not sufficient condition of a common reaction mechanism and in any case is a necessary presumption for any linear free energy relationship. Hence, it does not at all detract from kinetic measurements at different temperatures on the contrary, it gives them still more importance. 

If equation 5 is valid, if a linear relationship exists between AH and the calculated AE(t) parameters, and if a linear free energy relationship exists between AH and EA , we might expect that the following linear relationship might hold for the decomposition of reactant Y to produce free radicals R(Y) ... 

The success of equations such as (7), (10) and (11) depends on the validity of linear free energy relationships and of the applicability to these of the variation of the solvent as a perturbation. 

Hammett, after illustrating the existence of linear relationships among the data for a variety of side-chain reactions, defined the (7-constants to characterize the behavior of substituent groups. In the application of the Hammett equation the cr-parameters are assumed to be constant. The assessment of the validity of this same assumption for substituents in aromatic substitution reactions is the major problem which must be considered prior to the adoption of a simple two-parameter linear free-energy relationship for these reactions. Preliminary evaluations of linear relationships were undertaken through somewhat modified procedures as discussed in Section IV. Now, however, with many quantitative data available it is no longer necessary to rely on the less direct Selectivity Relationship. Rather, the more straightforward conventional Hammett approach is applicable. This procedure requires the adoption of the a1 -constants derived from the study of substituted phenyldimethylcarbinyl chlorides and the assumption of constancy of the values. This assumption is shown to be fully justified in subsequent tests of the relationship. 

The procedure adopted to portray the scope and utility of a linear free-energy relationship for aromatic substitution involves first a determination of the p-values for the reactions. These parameters are evaluated by plotting the values of log (k/ka) for a series of substituted benzenes against the values based on the solvolysis studies (Section IV, B). The resultant slope of the line is p, the reaction constant. The procedure is then reversed to assess the reliability and validity of the Extended Selectivity Treatment. In this approach the log ( K/ H) observations for a single substituent are plotted against p for a variety of reactions. This method assays the linear or non-linear response of each substituent to variations in the selectivity of the reagents and conditions. Unfortunately, insufficient data are available to allow the assignment of p for many reactions. It is more practical in these cases to adopt the Selectivity Factor S as a substitute for p and revert to the more empirical Selectivity Treatment for an examination of the behavior of the substituents. 

The catalyzed isomerization of the malonate complex was found to proceed by a different mechanism from that of the oxalate complex. The rate constants and the activation parameters are given in Table 7.24. No linear free energy relationship was found to be valid in the malonate system. 

Finally, it is interesting to note that the free energy relationships elicited in this work might have quite general implications for other enzyme reactions. In fact, the validity of such relationships in enzymes and solutions can be examined by computer simulation methods as has been illustrated in several preliminary studies from this laboratory [9,12b]. It appears that polar sites in enzymes obey to some extent the linear response approximation (the system polarisation is proportional to the applied local field) and therefore follow linear free energy relations. 

Another simple approach assumes temperature-dependent AH and AS and a nonlinear dependence of log k on T (123, 124, 130). When this dependence is assumed in a particular form, a linear relation between AH and AS can arise for a given temperature interval. This condition is met, for example, when ACp = aT" (124, 213). Further theoretical derivatives of general validity have also been attempted besides the early work (20, 29-32), particularly the treatment of Riietschi (96) in the framework of statistical mechanics and of Thorn (125) in thermodynamics are to be mentioned. All of the too general derivations in their utmost consequences predict isokinetic behavior for any reaction series, and this prediction is clearly at variance with the facts. Only Riietschi s theory makes allowance for nonisokinetic behavior (96), and Thorn first attempted to define the reaction series in terms of monotonicity of AS and AH (125, 209). It follows further from pure thermodynamics that a qualitative compensation effect (not exactly a linear dependence) is to be expected either for constant volume or for constant pressure parameters in all cases, when the free energy changes only slightly (214). The reaction series would thus be defined by small differences in reactivity. However, any more definite prediction, whether the isokinetic relationship will hold or not, seems not to be feasible at present. 

Nevertheless the general conclusions discussed here, as well as the overall experimental design for their validation, still follow the same unifying trends. For example, linear extrathermodynamic expressions can be proposed between the free energy change of a polypeptide or protein molecule involved in such hydrophobic interactions and particular molecular property parameters %j. This relationship takes the form of... 

The quadratic form of the free energy implies a linear relationship between force and the end-to-end vector, that is valid for small extensions ... 

Since the difference is in the free energy of activation, AAGt, for two concurrent reactions is AGtAB - AG aC- And, since there is a linear relationship with kA/kB, the selectivity is proportional to AAGt. This is a very simplified approach to selectivity explanation and it must be noted that many assumptions must be fulfilled for its validity. The fundamental assumption for this conclusion is that the reaction under consideration obeys a rate-equilibrium relationship. For example, the principle cannot be applied for reactions that are diffusion controlled. It is also doubtful that this principle can be applied for reactions that involve very reactive species such as carbenes, radicals, and carbonium ions [1]. 

The free-energy dependence of many PT rates follows the Bronsted relationship described above, but the success of this free-energy linear relationship is unexpected. In principle, it should not be valid over a large range of acidity constants, because the Brdnsted coefficient for acid catalysis should vary dramatically from = 0 to =1 when the reaction becomes endothermic, as shown in Figure 13.15, where acid-base catalysis is not observed. This was assigned to the small size of the barrier for the forward PT, but an explanation for the size of the barriers based on molecular stfucture of the reactants is still lacking. 

---