Hansch parabolic equation

Hundreds of equations later, the failure of linear equations in cases with extended hydrophobicity ranges led to the development of the Hansch parabolic equation (20) ... 

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. 

In these equations, Dmax is the larger of the summed values of STERIMOL parameters, Bj, for the opposite pair 68). It expresses the maximum total width of substituents. The coefficients of the ct° terms in Eqs. 37 to 39 were virtually equal to that in Eq. 40. This means that the a° terms essentially represent the hydrolytic reactivity of an ester itself and are virtually independent of cyclodextrin catalysis. The catalytic effect of cyclodextrin is only involved in the Dmax term. Interestingly, the coefficient of Draax was negative in Eq. 37 and positive in Eq. 38. This fact indicates that bulky substituents at the meta position are favorable, while those at the para position unfavorable, for the rate acceleration in the (S-cyclodextrin catalysis. Similar results have been obtained for a-cyclodextrin catalysis, but not for (S-cyclodextrin catalysis, by Silipo and Hansch described above. Equation 39 suggests the existence of an optimum diameter for the proper fit of m-substituents in the cavity of a-cyclodextrin. The optimum Dmax value was estimated from Eq. 39 as 4.4 A, which is approximately equivalent to the diameter of the a-cyclodextrin cavity. The situation is shown in Fig. 8. A similar parabolic relationship would be obtained for (5-cyclodextrin catalysis, too, if the correlation analysis involved phenyl acetates with such bulky substituents that they cannot be included within the (5-cyclodextrin cavity. 

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. 

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. 

Recalculating the published data on bio-depressants, Hansch and his colleagues found that the following regression equation made a good fit (the squared term ensured a parabolic relationship) ... 

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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