Parabolic Hansch model

A non-linear Hansch model has been applied to the bactericidal concentrations (O of 17 doubly substituted phenols (Table 37.2) which have been reported by Klarmann et al. [20]. By means of multiple linear regression we obtain the parabolic Hansch model of eq. (37.9) ... 

The parabolic Hansch model is a good approximation of observed nonlinear structure-activity relationships. However, whereas the left and right sides of a parabola are always nonlinear, many nonlinear lipophilicity relationships show linear left and right sides, as also observed for the function describing the rate constants of... 

Figure 12 Comparison of the parabolic Hansch model (left curve) and Franke s protein binding model (right curve). Log P, is the lipophilicity limit, where steric hindrance or other unfavorable interactions cause a change of the linear lipophilicity-activity relationship to a parabola (reproduced from Figure 9 of ref. [175] with permission from Birkhauser Verlag AG, Basel, Switzerland).

Frequently, the relationship between biological activity and log P is curved and shows a maximum [ 18]. In that case, quadratic and non-linear Hansch models have been proposed [19]. The parabolic model is defined as ... 

The optimum value of log P for a given system is log Pq and it is highly influenced by the number of hydrophobic barriers a drug encounters in its walk to its site of action. Hansch and Clayton formulated the following parabolic model to elucidate the narcotic action of alcohols on tadpoles (189). 

In order to also take into account some nonlinear contributions of the properties [Hansch and Clayton, 1973 Kubinyi, 1993b], a Hansch parabolic model is defined as ... 

This equation shows the relationship between the Hansch and the Free-Wilson equations. It has been suggested (217, 220) that the Bocek-Kopecky interaction model holds for the parabolic dependence of biological activity on the partition coefficient. 

Besides the nonlinear models and, specifically, the parabolic model, other models were proposed for nonlinear dependence of the biological response on hydrophobic interactions. Among them, the most important are the Hansch bilinear models [Kubinyi, 1977,1979] such as... 

Hansch-Fujita hydrophobic substituent constants Hansch linear model —> Hansch analysis Hansch nonlinear model —> Hansch analysis Hansch parabolic model —> Hansch analysis Harary-Balaban index —> distance matrix... 

The ester derivatives showed a similar relationship, the optimum log P values being 6.3 -6.7, which was higher than that of the ether derivatives. The relationship among the ILS (150 and 160%), the minimum lethal dose (MLD) and the hydrophilic coefficient of the ether series of RA-V were analyzed according to both the Hansch-Fujita and the bilinear models of Kubinyi. When the parabolic model obtained from the Hansch-Fujita equation was applied to the ILS and MLD, no significant results were obtained. 

Hansch formulated a parabolic model (eq. 7, chapter 1.1) [15, 17—19] for the mathematical description of nonlinear relationships. He was aware that the sides of a parabola are always more or less curved, while in most cases at least the left side of the structure-activity relationship (i.e. the lipophilicity dependence of the more hydrophilic analogs) is strictly linear equations including a third-order lipophilicity term did not produce much improvement [19]. A computer simulation of the transport of drugs in a biological system, using hypothetical rate constants,... 

The discipline of quantitative structure-activity relationships (QSAR), as we define it nowadays, was initiated by the pioneering work of Corwin Hansch on growthregulating phenoxyacetic acids. In 1962—1964 he laid the foundations of QSAR by three important contributions the combination of several physicochemical parameters in one regression equation, the definition of the lipophilicity parameter jt, and the formulation of the parabolic model for nonlinear lipophilicity-activity relationships. 

In (eco)toxicological QSAR studies, the molecular descriptor of choice is the n-octanol/water partition coefficient (log P), generally used in a simple regression equation. However, sometimes a simple linear regression model is inadequate to model properly the dependence of biological activity (BA) on logF. For example, fish exposed to very hydrophobic chemicals for a limited test duration have insufficient time to achieve a pseudo-steady state partitioning equilibrium between the toxicant concentration in aqueous circumambient phase and the hydrophobic site of action within the fish. Hansch initiated the use of a parabolic model in log P (equation 1) to overcome... 

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