Parabolic lipophilicity-activity relationship

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. 

An alternative to the parabolic model is the bilinear model (equation 6) which was derived from computer simulations, using experimental rate constants of drug transport in simple in vitro systems. " In most cases it describes nonlinear lipophilicity-activity relationships more accurately than the parabolic model. 

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... 

Quantitative Structure Activity Relationships. With the large number of compounds that has been synthesized and tested, it is surprising that several quantitative structure activity relationship (QSAR) analyses have not been attempted as a means of correlating those structural attributes and physicochemical parameters that significantly affect potency. At least one detailed QSAR study has been reported.(20.21) Initially the pertinent physicochemical parameters were obtained for each position where substituents were varied. Then the study was extended to multiple substituted analogues. The parameters evaluated included the original STERIMOL parameters L(l) for N-substituents at position 1 L(8) and B4(8) at position 8 steric influence at position 6, Es(6) hydrophobicity at position 7, x(7) and lipophilicity of the whole molecule, log P. Parabolic... 

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,... 

However, the parabolic model is still valuable for structure-activity analyses. It is the simpler model, easier to calculate, and most often a sufficient approximation of the true structure-activity relationship. The calculation of bilinear equations is relatively time-consuming, as compared to the parabolic model strange results may be obtained in ill-conditioned data sets. On the other hand, in many cases the, bilinear model gives a better description of the data, especially if additional physicochemical parameters are included in the regression equation. The lipophilicity optimum of symmetrical curves is precisely described by both, the parabolic model (optimum log P = — b/2a) and the bilinear model (optimum log P = — log P). In the case of unsymmetrical curves the site of the lipophilicity optimum is described much better by the bilinear model (optimum log P = log a — log p — log (b — a) eq. 93) than by the parabolic model. 

C in equation (2) is a molar concentration which produces a certain biological effect, P is the -octanol/water partition coefficient (see Octanol/Water Partition Coefficients), and a is the electronic Hammett constant (equation 1). The definition of a parabolic model and the combination of different physicochemical properties in one model allowed for the first time the description of structure-activity relationships which could not be correlated with a single linear term. As an alternative to log P values, a lipophilicity parameter n can be used which is defined in an analogous manner as the electronic a parameter (1 equation 3). " ... 

A parabolic relationship between membrane activity and lipophil-icity of nonionic surfactants is clearly established in series of surfactants in which either the hydrocarbon chain length or ethylene oxide chain length is varied. Activity at low and high concentrations should be considered separately. 

Lipophilicity parameters, whether included as it- values or log P, are often included twice once to the first power and again to the second power. This accommodates the common parabolic relationship between biological activity and lipophilicity (log P) (Figure 12.3). The curve will have a maximum at a lipophilicity value that balances the molecule s need to be both sufficiently polar to dissolve in the cytosol and nonpolar enough to enter and cross a cell membrane. 

Equation 42 is included because it has a negative slope—i.e., this set of congeners was selected so that all members had a superoptimal lipophilic character. Evidence now indicates 22, 25-27) that the general relationship one should expect between log 1/C and log P is parabolic. The apex of the parabola has been termed log P0. Molecules having this value of log P have ideal lipophilic character for the system under consideration. Since all of the above equations are linear in log P (the addition of a term in (log P)2 does not improve the correlation), greater activity could have been obtained in each example by designing molecules with better log P values. In all examples except Equation 42 this means increasing the lipophilic character. Equation 42 calls for less lipophilic molecules. 

Equation 6 (Table XIV) indicates that lipophilicity and molar reffactivity of the substituents at position 6 are important determinants of activity against S. aureus. There is a parabolic relationship seen with these same descriptors for the substituents at position 7. Comparison of equation 6 for S. aureus with equation 7 (Table XV) for Ps. aeruginosa indicates a different QSAR. An ethyl substituent at position 1, minimiun width (Bl) of the substituent at position 6, and the appearance of a piperazinyl ring in position 7 all appear in equation 7. The parabolic relationship of lipophilicity and MR seen in equation 6 for substituents at position 7 is found also in equation 7 indicating that there are optimiun lipophilicity and molar refraction ranges for the 7-substituent. At the same time, it must be noted that many of the values for the parameters listed in Tables IV and V show clustering which can bias the results. 

The same parabolic relationship for a-p electronic interactions is seen with E. coli, but it may be biased for the reasons already discussed. In contrast, the LFER model for Ps. aeruginosa (Table XXII, Eq. 4) indicates that only lipophilicity and molar refractivity of the substituents at position 7 are important determinants of activity. In the initial analysis for this subset using the Ps. aeruginosa test system, A34 (norfloxacin) was an outlier. Because it is the only compound showing such high activity, it was deleted. The latter s calculated activity was 1.96 versus the observed 1.64. 

Since Verloop had identified the 4-position of the aniline portion to be important for insecticidal activity, we investigated a series of 3-methoxy-4-(4 -substituted-phenyl)phenyl benzo rureas (5). The results of this study are reported in Figure 5. The regression equation indicates that resonance (R) and the lipophilic properties (pi) of the 4 - substituent are important for insecticidal activity. The positive coefficent for the R term indicates that an electron withdrawing substituent has a favorable effect on the insecticidal activity. A positive coefficent was also found for the pi term, indicating that as lipophicity is increased the insecticial activity will increase. However, the pi term indicates that a parabolic relationship exists between lipophilicity and insecticidal activity the optimxjun value for pi is 0.8. 

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