Lipophilicity-activity relationships

Free-Wilson analysis can be used for a first inspection of biological activity data [30-32]. The values of the group contributions indicate which physicochemical properties might be responsible for the variations in biological activity values and whether nonlinear lipophilicity-activity relationships are involved. Free-Wilson contributions can be derived from Hansch equations (e.g., by Eq. (18) from Eq. (14), or by Eq. (19) from Eq. (15)) [30] ... 

The bilinear model, in combination with other physicochemical properties (Eq. (30)) [24,62], was the first mathematical expression to describe nonlinear lipophilicity-activity relationships, precisely and in a flexible manner. Besides pharmacokinetic properties, such as absorption, distribution, elimination, and permeation of the... 

Last but not least, many lipophilicity-activity relationships, using n-octanol/water partition coefficients or lipophilicity parameters derived therefrom, prove the relevance of this system e.g. [18, 19, 182]). 

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]. 

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).

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 use of lipophilicity similarity matrices for the quantitative description of nonlinear lipophilicity-activity relationships is discussed in chapter 9.4. 

Other nonlinear relationships are known in addition to nonlinear lipophilicity-activity relationships. Most common are nonlinear dependences on molar refractivity e.g. resulting from a limited binding site at the receptor for examples see chapter 7.1), but also other types of nonlinear relationships, e.g. with steric parameters, are frequently obtained. Even electronic parameters (eq. 48 chapter 3.5) or molecular weight terms (eq. 56 chapter 3.7) have been used in nonlinear equations. 

The most, comprehensive review of quantitative structure-pharmacokinetics relationships [452] tabulates about 100 equations, including absorption, distribution, protein binding, elimination, and metabolism of drugs. Since many of these equations and those included in other reviews e.g. [472, 761]) have been derived before appropriate mathematical models for nonlinear lipophilicity-activity relationships (chapter 4.4) and for the correct consideration of the dissociation and ionization of acids and bases (chapter 4.5, especially eqs. 107—110) were available, some of the older results should be recalculated by using the theoretical models (chapters 4.4 and 4.5) instead of the empirical ones. 

The use of Free Wilson-type indicator variables in Hansch analysis has been discussed in chapters 3.8, 4.3, and 7 [21, 390, 391, 393]. Nonadditivities in Free Wilson analyses due to nonlinear lipophilicity-activity relationships have been discussed in chapter 4.3 [22, 390—392, 394]. 

The CoMFA methodology was also used to describe nonlinear lipophilicity-activity relationships, c.g. the inhibitory activities of quaternary alkylbenzyl-dimethylammonium compounds vs. Clostridium welchii (eqs. 206—208) [1025], other antibacterial and hemolytic activities [1026, 1027], and toxic activities of alkanes in mice (eqs. 209-211) [1026] the results of classical QSAR studies (eqs. 206, 207, 209, and 210) [23, 440] were compared with the corresponding CoMFA results (eqs. 208 and 211) [1025-1027] only homologous series of compounds were investigated. 

In addition to the similarity indices described above, other similarity indices may be defined and used in QSAR studies. A simple lipophilidty similarity index aij = — log Pi — log PjI (log Pi, logPj = logarithms of the partition coefficients of molecules i and j) can be applied to describe nonlinear lipophilicity-activity relationships of any type by the corresponding lipophilidty similarity matrices [1013]. For different data sets excellent results were obtained (Table 31), not only in homologous series (as in CoMFA studies [1025 — 1027]) but also in heterogeneous sets of compounds, where 3D QSAR approaches must fail. A selection procedure based on genetic algorithms was developed for fast and efficient variable elimination in the PLS analyses [1013]. Also in these examples the similarity matrices produced improved Tpress values in fewer components after elimination of variables which did not contribute to prediction (Table 31). 

While such lipophilidty similarity matrices do not consider the 3D structures of the molecules, they seem to be appropriate for the incorporation of nonlinear lipophilicity-activity relationships in 3D QSAR analyses, e.g. in CoMFA studies. At least from a theoretical point of view lipophiUcity similarity matrices should be preferred when the nonlinear lipophilicity-activity relationship does not result from binding but from transport and distribution of the drugs in the biological system, which most often is the case. 

Table 31. Nonlinear lipophilicity-activity relationships. Comparison of the bilinear model (r values are given instead of r values) with results from PLS analyses, using lipophilicity similarity matrices a,j= -llogPi-logPjl [1013]...

The first lipophilicity-activity relationship was published by Charles Richet in 1893, exactly 100 years ago. From his quantitative investigations of the toxicities of ethanol, diethyl ether, urethane, paraldehyde, amyl alcohol, acetophenone, and essence of absinthe ( ) he concluded plus Us sont solubles, mains Us sont toxiques (the more they are soluble, the less toxic they are). One year later Emil Fischer derived the lock and key model of ligand-enzyme interactions from his results on the stereospecificity of the enzymatic cleavage of anomeric glycosides. 

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. 

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