Linear 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).

Nonlinear relationships between biological activities and lipophilicity are very common. While biological activity values most often linearly increase with increasing lipophilicity [182], such an increase is no longer obtained if a certain range of lipophilicity is surpassed biological activities remain constant or decrease more or less rapidly with further increase of lipophilicity [7, 19]. Many reviews deal with nonlinear lipophilicity-activity relationships [19, 175, 178, 345, 433]. 

Franke developed another empirical model to bridge the gap between so many linear relationships and a nonlinear model (Figure 12). He considered binding of ligands at a hydrophobic protein surface of limited size as being responsible for nonlinear lipophilicity-activity relationships and formulated two equations, one for the linear left side (eq. 82) and the other one for the right side, the nonlinear part (eq. 83 log P = critical log P value, where the linear relationship changes to a nonlinear one) [435]. 

Symmetrical curves with linear ascending and descending sides, having their optimum at log P = 0, result from eq. O (Figure 13). No practical application of eq. 90 was possible because most often nonlinear lipophilicity-activity relationships are unsymmetrical and their optimum log P values are different from zero. 

Eqs. 93 and 94 may be considered as extensions of eqs. 90—92. In contrast to these equations, the bilinear model is generally applicable to the quantitative description of a wide variety of nonlinear lipophilicity-activity relationships. In addition to the parameters that are calculated by linear regression analysis, it contains a nonlinear parameter p, which must be estimated by a stepwise iteration procedure [440, 441]. It should be noted that, due to this nonlinear term, the confidence intervals of a, b, and c refer to the linear regression using the best estimate of the nonlinear term. The additional parameter P is considered in the calculation of the standard deviation s and the F value via the number of degrees of freedom (compare chapter 5.1). The term a in eq. 93 is the slope of the left linear part of the lipophilicity-activity relationship, the value (a — b) corresponds to the negative slope on the right side. 

The above Hansch equations are also generally referred to as linear free energy relationships (LFER) as they are derived from the free energy concept of the drug-receptor complex. They also assume that biological activity is linearly related to the electronic and lipophilic contributions of the various substituents on the parent molecule. 

Hansch equation A Hansch equation is a linear free-energy relationship that correlates biological activity (log 1/C) to molecular and substituent parameters. The parameters describe properties such as sterics, lipophilicity, and electronic effects, and the coefficients on the parameters determine the relative importance of each parameter. 

The oldest publication on structure-activity relationships (SARs) known to us from the literature is a paper by Cros from 1861. He compared the toxic effect of alcohols in various species after different routes of administration. He found an increase in toxic effect with decreasing water solubility, that is increasing lipophilicity. Cros was also the first to detect a maximum in activity followed by a decrease, i.e. a non-linear relationship, as a function of solubility of alcohols in water. 

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 acute toxicity of the compounds increases with the extent of organic substitution at the tin atom and reaches a maximum with tributyltin (TBT). The reason for this is the strong increase in their lipophilic properties that causes greater accumulation in membranes, especially in the mitochondria, and results in a breakdown of the organism s energy supply through the formation of ion channels. But quantitative structure-activity relationships (QSARs) are not deduced solely from linear or logarithmic... 

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 simplest form of a Free Wilson analysis is presented in eq. 192 [22], which describes the antibacterial activities of phenol and isomeric chlorophenols (51, R = H, Cl one to five chlorine atoms) vs. Staphylococcus aureus at least the linearity of the structure-activity relationship can be derived from eq. 192 on the other hand, although most probably lipophilicity is responsible for the variance in the biological activities, no Hansch equation can be derived, because each other physicochemical property of the chlorine atom will give identical results. 

The linear models used most often represent only a restricted section of structure-activity relationships. There is no reason to assume, for example, that the enhancing effect of the increase in lipophilicity on a compound s toxicity to fish will perpetuate for chemicals representing the extremes of the lipophilicity scale (log 6 or 0). Rather, non-linearity in these relationships is to be expected because of the establishment of a highly complex set of equilibria between the aqueous and non-polar phases at different rates from the release of the chemical into the system until the interaction with the target site. The linearly increasing section of the QSAR may progress into a plateau... 

Sustaining the transport model, further non-linear representations of the observed structure-activity relationships were derived. According to the McFarland model (Seydel and Schaper, 1979 Kubinyi 1993), the probability of a drug reaching the receptor after passing several membranes depends on its lipophilicity in a symmetrical manner with linearly increasing and decreasing sides of the curve ... 

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). " ... 

C. Hansch and W.J. Dunn III, Linear relationships between lipophilic character and biological activity of drugs. J. Pharmaceut. Sci., 61 (1972) 1-19. 

Lipophilicity in particular, as reflected in partition coefficients between aqueous and non-aqueous media most commonly water (or aqueous buffer) and Z-octanol,has received much attention [105,141,152,153,176,199,232,233]. Logic )W for the octanol-water system has been shown to be approximately additive and constitutive, and hence, schemes for its a priori calculation from molecular structure have been devised using either substituent tt values or substructural fragment constants [289, 299]. The approximate nature of any partition coefficient has been frequently emphasized and, indeed, some of the structural features that cause unreliability have been identified and accommodated. Other complications such as steric effects, conformational effects, and substitution at the active positions of hetero-aromatic rings have been observed but cannot as yet be accounted for completely and systematically. Theoretical statistical and topological methods to approach some of these problems have been reported [116-119,175,289,300]. The observations of linear relationships among partition coefficients between water and various organic solvents have been extended and qualified to include other dose-response relationships [120-122,160,161,299-302]. 

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