Hansch parabolic

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) ... 

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 ... 

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... 

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. 

Hansch C, Clayton JM. Lipophilic character and biological activity of drugs. II. The parabolic case. J Pharm Soc 1973 62 1-21. 

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 ... 

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 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). 

Hansch, C. and Clayton, J.M. (1973). Lipophilic Character and Biological Activity of Drugs II The Parabolic Case. J.Pharm.Sci, 62,1-21. 

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. 

The measurement of partitioning is only practical if the compound shows some solubility insoluble compounds are difficult to characterize and often prove less valuable anyway. A relationship between log P and the observed biology is frequently found in a series where structural modifications do not significantly affect the pK values. The classical work of Hansch showed that these relationships were often parabolic hence, the relationship often leads to an optimum value for the log P for a desired activity or selective distribution. Relationships of this type include ... 

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... 

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... 

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,... 

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).

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) ... 

This hypothesis still forms the basis of current thinking, but it is now recognized that the correlation of paragraph (c) is not linear but parabolic (Hansch 1968, Hansch, 1971, pp. 297,300). This follows from the fact that any substance that is so lipophilic that it has virtually no solubility in water cannot have depressant properties, because it will accumulate in the first oily site of loss which it encounters and be unable to leave it. 

In the four decades of discussion that followed its introduction, Ferguson s principle has not been significantly enlarged, or even further clarified. Its daring assumption that partial pressures can be substituted for thermodynamic activities has not been falsified on the other hand, it did not predict parabolic biological response to increasing partition coefficients (Hansch, 1971). At the present time, the incorporation of partition coefficients into a quantitative... 

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