Extrathermodynamic derivation

It thus seems appropriate to use partition coefficients between aqueous and organic phases to represent molecular properties related to the hydrophobic interactions between small molecules and the biophase. This choice is well established at the level of a second order approximation in the extrathermodynamic derivation (see section A. 3). Numerous examples (137, 138) illustrate the utility of the partition coefficient in correlations of biological activity with chemical structure. 

Theoretically, the problem has been attacked by various approaches and on different levels. Simple derivations are connected with the theory of extrathermodynamic relationships and consider a single and simple mechanism of interaction to be a sufficient condition (2, 120). Alternative simple derivations depend on a plurality of mechanisms (4, 121, 122) or a complex mechanism of so called cooperative processes (113), or a particular form of temperature dependence (123). Fundamental studies in the framework of statistical mechanics have been done by Riietschi (96), Ritchie and Sager (124), and Thorn (125). Theories of more limited range of application have been advanced for heterogeneous catalysis (4, 5, 46-48, 122) and for solution enthalpies and entropies (126). However, most theories are concerned with reactions in the condensed phase (6, 127) and assume the controlling factors to be solvent effects (13, 21, 56, 109, 116, 128-130), hydrogen bonding (131), steric (13, 116, 132) and electrostatic (37, 133) effects, and the tunnel effect (4,... 

It follows that for a special value of one parameter, the observed value of y is independent of the second parameter. This happens at Ii= a2/ai2 or I2 = -ai/ai2 any of these values determines y= a -aia2/ai2, the so called isoparametrical point. The argument can evidently be extended to more than two independently variable parameters. Experimental evidence is scarce. In the field of extrathermodynamic relationships, i.e., when j and 2 are kinds of a constants, eq. (84) was derived by Miller (237) and the isoparametrical point was called the isokinetic point (170). Most of the available examples originate from this area (9), but it is difficult to attribute to the isoparametrical point a definite value and even to obtain a significant proof that a is different from zero (9, 170). It can happen—probably still more frequently than with the isokinetic temperature—that it is merely a product of extrapolation without any immediate physical meaning. 

The second extrathermodynamic method that we discuss here differs from Hansch analysis by the fact that it does not involve experimentally derived substitution constants (such as o, log P, MR, etc.). The method was originally developed by Free and Wilson [29] and has been simplified by Fujita and Ban [30]. The subject has been extensively reviewed by Martin [7] and by Kubinyi [8]. The method is also called the de novo approach, as it is derived from first principles rather than from empirical observations. The underlying idea of Free-Wilson analysis is that a particular substituent group at a specific substitution site on the molecule contributes a fixed amount to the biological activity (log 1/C). This can be formulated in the form of the linear relationship ... 

We conclude this section by a few general remarks about extrathermodynamic approaches. These quantitative methods involve empirical approaches that cannot be derived strictly from thermodynamic theory. They are widely used to predict and/ or to evaluate partition constants and/or partition coefficients (see Box 3.2 for nomenclature) of organic compounds. There are many situations in which some of the data required to assess the partitioning behavior of a compound in the environment are not available, and, therefore, have to be estimated. For example, we may need to know the water solubility of a given compound, its partition coefficient between natural organic matter and water, or its adsorption constant from air to a natural surface. In all these, and in many more cases, we have to find means to predict these unknown entities from one or several known quantities. 

The two extrathermodynamic assumptions used in Table 9 to derive solvent activity coefficients of anions, lead to different values of y cicHsi-)+ The assumption (i) that caesium cation is similarly solvated in methanol and in DMF, suggests that the large rate difference between reaction (27) in methanol and in DMF is as much due to differences in transition state solvation as to differences in solvation of chloride ion. This is the situation shown qualitatively in Fig. 1. On the other hand, the somewhat smaller rate difference between reaction (27) in formamide and in DMF is due entirely to differences in solvation of chloride ion, if the caesium assumption is applied to formamide and to DMF. 

We derive the extrathermodynamic relationships as a basis for the definition and understanding of the parameters for QSAR. This derivation is similar to that of Grunwald and Leffler Cl). 

The theoretical derivation of extrathermodynamical relationships is based on the arbitrary division of a molecule into regions, usually two the basic structure R, and the substituent X (1, 2). An additivity postulate then enables one to express the partial free energy of the substance (or other related property) as a sum of independent contributions, each representing the arbitrarily defined parts, and an interaction term between the two parts (equation 1). 

In the early 1950s Taft (10) outlined a sound quantitative basis for the estimation of steric effects and for separating them from polar and resonance effects. This derivation follows the standard extrathermodynamic approach and is therefore empirical. The definition of steric substituent constants is closely related to polar substituent constants, for they are obtained from the same reference system (10). The polar substituent constants, however, have been shown to arise from electronic effects, and they strongly correlate with the inductive substituent constants (10, 43, 61). 

The connectivity indices x and xv (102, 103), quantitatively characterize the size of the molecule and the degree of branching. In that sense, and also because they are derived from a theoretical procedure (104), they lack the usual extrathermodynamic meaning. The strong correlation of X with molar refractivity, MR,... 

The analysis of the parameters representing the effect of the medium in terms of extrathermodynamical relations has shown (see above, section A.2) that the hydrophobic substituent constant it depends on the molecules from which it is derived. For example, hydrophobic constants derived from the octanol/water partition coefficients of aromatic molecules differ from those obtained from aliphatic molecules (112). Collander s equation (111) provides the empirical basis for the evaluation of logP values for the same molecule in different solvents. However, solvents with markedly different solvation properties (e.g., hydrogen bonding ability) do not conform to Collander s equation (see below, section C.3). 

Several attempts have been made (217, 218, 219, 220) to establish the relationship between the extrathermodynamic approach of Hansch (114) and the mathematical approaches (198, 212, 213, 215, 216). The numerical equivalence and the theoretical interrelation between the two approaches derives from the assumption that the contributions of the individual substituents to the biological activity is represented as a weighted sum of several physicochemical properties (e.g., the extrathermodynamic parameters in the Hansch equation) as expressed in equation 103... 

We discussed the extrathermodynamic relationships that provide a common basis for most of the parameters used in QSAR. We have presented a critical analysis of the empirical framework for the derivation of these parameters and a rationale for choosing them. The analysis of the physicochemical basis of the parameters and of the methods shows the caution needed in interpreting molecular mechanisms from QSAR correlations. 

The Hansch-Fujita approach is also called the linear free-energy relationship (LFER) or extrathermodynamic approach, since most of the descriptors are derived from rate or equilibrium constants. 

In the context of the reinvestigation of Hansch models for the substituted N,N-dimethyl-a-bromophenethylamines, Unger and Hansch [28] formulated recommendations for the proper derivation of extrathermodynamic equations (supplementary comments are given in parentheses) ... 

Thermodynamic Transition-State Theory and Extrathermodynamic Correlations for the Liquid-Phase Kinetics of Ethanol Derived Ethers... 

An analogous approach to the solvation quantities derived for non-electrolyte systems [41,181], based on Kusalik and Patey s version of the Kirkwood-Buff fluctuation theory of mixtures [210], was developed recently [42] to make explicit contact not only between the solvation structure of individual ions and its corresponding macroscopic properties, but also between the individual ion s and the salt s properties without invoking any extrathermodynamic assumption [173,212]. 

Again, similar expressions can be derived to account for the effect of organic solvent concentration on the log k[ values of proteins or peptides in RPC. The solvophobic theory and the preferential interaction theory thus have a common linkage through the slopes of these log versus log[l/3] plots. From these dependences, and the relationship of log A to the free energy change, AGj. a unified theoretical approach is also provided to the thermodynamic and extrathermodynamic relationships of the interaction of proteins or peptides with hydrocarbonaceous ligands in both RPC and HlC. 

The corresponding three-dimensional grid plots for the S-values [derived according to Eq. (5)] of these two polypeptides as the temperature and j/ values were systematically varied are shown in Fig. 26a-d. In each case, the S values for polypeptides 1 and 2 were derived by regression analysis methods from the gradient of the experimental plots of log k, vemus i/> at the specified j/ and T values with the regression eoefficients >0.9985. In turn, the S value of a peptide or protein in the presence of an RPC sorbent can be related [16,20,211,212] to extrathermodynamic parameters, such as the accessible molecular surface area, A/l, ui, through the expression... 

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