Biological activity values

Prediction of the in vitro activity for the racemic pair of compounds leads to a less clear interpretation of the data. Is only one enantiomer active If so, should the biological activity value be doubled, assuming only half of the assayed material is active If the enantiomers are unequally active, how should the data be treated This final model was used to predict the activities of each enantiomeric pair, providing some insight into the solution to these questions. 

Log 1/C = Ea + i (ai are the group contribution of the individual substituents X to the biological activity values and p, is the calculated biological activity of a reference compound, most often the unsubstituted analogue). 

In Equation 1.6, u is defined as the calculated biological activity value of the unsubstituted parent compound of a particular series. rep-... 

Where G.. is the activity contribution of a substituent in position i, and y represents the overall average of biological activity values. For example, Xjj 1 if the substituent is in position otherwise it is equal to zero. The most convenient way to construct equation 101 is by generating a matrix in which each row represents a specific compound. The columns are divided into as many groups as there are substituted positions on the molecule and each column within a group represents a certain substituent. For a compound in which two sites, a and b, have substituents Xla, X2a,...,Xna, and Ylb,Y2b, >Ymb> the following matrix can be constructed ... 

It is clear that for an unsymmetrical data matrix that contains more variables (the field descriptors at each point of the grid for each probe used for calculation) than observables (the biological activity values), classical correlation analysis as multilinear regression analysis would fail. All 3D QSAR methods benefit from the development of PLS analysis, a statistical technique that aims to find the multidimensional direction in the X space that explains the maximum multidimensional variance direction in the F space. PLS is related to principal component analysis (PCA)." ° However, instead of finding the hyperplanes of maximum variance, it finds a linear model describing some predicted variables in terms of other observable variables and therefore can be used directly for prediction. Complexity reduction and data... 

In the following decades, various a scales were derived for different systems and several attempts were made to derive such relationships also for biological activities of organic compounds. Bruice et al. [10] formulated group contributions to biological activity values in a series of thyroid hormone analogs, which may be considered as a first Free-Wilson-type analysis. Zahradnik and Chvapil [11] and Zahradnik [12,13] tried to apply the concept of the Hammett equation also to biological data (Eq. (5)) ... 

In 1962, Hansen [14] derived a first Hammett-type relationship between the toxicities of substituted benzoic acids and the electronic a constants of their substituents. However, later, it turned out that this was a chance correlation that only resulted from a close interrelationship between the Hammett a parameter and the lipophilicity constant % (Sec. 4 Eqs. (42) and (43)). In the same year, for the very first time, a nonlinear multiparameter equation (Eq. 6) [15] was used to describe biological activity values ... 

The proper application of Hansch and Free-Wilson analyses and the associated problems can best be illustrated with a well-investigated example. The antiadrenergic activities of meta-, para-, and met<2,/><2ra-disubstituted A/A-dimethyl-a-bromophene-thylamines have been investigated by Hansch and Lien [27], Unger and Hansch [28], Cammarata [29], and Kubinyi and Kehrhahn [30]. Table 1 presents the substituents, experimentally observed activity values, the parameter values for n,a+ and Efeta, as well as biological activity values calculated from Eqs. (14) and (16), respectively. To perform a Hansch analysis, the biological data are taken as Y values and an... 

Free Wilson analysis [31,32] is much easier to apply. Biological activity values are correlated with indicator variables, which, for each position of substitution and every substituent, indicate the presence (value 1) or absence (value 0) of the corresponding substituent (Table 2). If there is more than one substituent in a certain position or if symmetrical positions (e.g., meta,metd-disubstituted compounds) are condensed into one variable, numbers of two or higher are used instead of one. Regression analysis leads to Eq. (17) [30-32] ... 

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] ... 

Sometimes variable filters are applied before the real variable selection is performed (e.g., variables that have no or nearly no variance or variables that are highly intercorrelated with another variable both procedures are fine). On the other hand, the elimination of variables that, taken alone, show no correlation with the biological activity values is a procedure that should not be applied. There is a certain chance that this variable might be able to explain the data set in combination with another variable. A better preselection procedure is the selection from the best of all possible models with three different X variables thousands of such models can be calculated within seconds, using Eq. (25) (rY Yi ... vm)—multiple correlation coefficient rYx vector of rYxt correlation coefficients Rvv matrix of rxi,xj correlation coefficients) [49]. If necessary, highly intercorrelated variables can be eliminated afterward ... 

Many published structure-activity relationships do not meet generally accepted standards in scientific research and statistics. Most often hypotheses are not justified by the experimental data and, even worse, in some cases the results only reflect the patience of the authors to investigate many different variables to describe the biological activity values of a small number of compounds, until a certain combination of these variables gives a delusively good result. 

Richet [2] discovered that the toxicity of organic compounds inversely follows their water solubility. Such a relationship corresponds to eq. 2, where A<[) are the differences in biological activity values, caused by corresponding changes in the chemical and especially the physicochemical properties, AC. 

Tj in this biological Hammett equation stands for the activity value of the i" member of a series, Te, is the biological activity value of the ethyl compound of the same series, P is a substituent constant (corresponding to the electronic a parameter... 

While the success of QSAR analyses may be taken as sufficient evidence for the additivity of group contributions to biological activity values, the following question arises are these group contributions more or less constant from one system to the other or do they depend on the choice of the compounds and/or the biological system ... 

Also the affinity constants of the phosphonamidate analogs (1, Table 1) confirm the additivity concept of group contributions to biological activity values. The different residues R are separated from the bridge atom X by two carbon atoms all affinity values of the three series with X = —NH—, —O —, and — CH2— are closely correlated [137]. 

Correlation of biological activity values with the first few components (which can be interpreted in terms of lipophilicity, bulk, and electronic properties) gives a clear picture of the inherent properties which might be responsible for the variation in the activity values. 

Indicator variables have also been used to account for other struetural features, e.g. for intramolecular hydrogen bonds, hydrogen bond donor and acceptor properties, ortho effects, cis/trans isomerism, different parent skeletons, different test models, etc. [22, 390]. Some precautions are necessary indicator variables should not describe a single compound (in this case the corresponding group contribution includes the experimental error of this one biological activity value) and the use of indicator variables should be justified from a theoretical point of view otherwise, continuous variables will be mixed with meaningless dummy variables, just to fit the data. 

Indicator variables are especially useful in the early phases of a QSAR analysis and for large, complex data sets. Different subsets can be combined with their help, until the real dependence of biological activity values on some physicochemical parameters can be derived from a more extensive structural variation. 

The version described by Fujita and Ban (eq. 8, chapter 1.1) [20, 390, 391] is a straightforward application of the additivity concept of group contributions to biological activity values. As nowadays only this modification is used, no details of the original formulation of the Free Wilson model and its complicated symmetry equations are discussed here. 

Every substituent which only once occurs in the data set, leads to a single-point determination the corresponding group contribution contains the whole experimental error of this one biological activity value. 

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]. 

The Franke model is a definite improvement as compared to the parabolic model. In many practical cases it gives a much better fit, e.g. eqs. 85 and 86 for the spasmolytic activities of mandelic acid esters (Table 14 RA = relative biological activity values, based on cyclandelate (15), RA = 100%) [345]. 

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